Contents 6 Classical Interacting Systems 1 6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.2 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2.2 Ising model in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2.3 H=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6.2.4 Chain with free ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6.2.5 Ising model in two dimensions : Peierls’ argument . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2.6 Two dimensions or one? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.2.7 High temperature expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Nonideal Classical Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.3.1 The configuration integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.3.2 One-dimensional Tonks gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.3.3 Mayer cluster expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.3.4 Cookbook recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.3.5 Lowest order expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.3.6 Hard sphere gas in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3.7 Weakly attractive tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.3.8 Spherical potential well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3.9 Hard spheres with a hard wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Lee-Yang Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4.1 Analytic properties of the partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4.2 Electrostatic analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.3 6.4 i CONTENTS ii 6.4.3 6.5 6.6 6.7 6.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Liquid State Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.5.1 The many-particle distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6.5.2 Averages over the distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.5.3 Virial equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.5.4 Correlations and scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.5.5 Correlation and response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.5.6 BBGKY hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.5.7 Ornstein-Zernike theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.5.8 Percus-Yevick equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.5.9 Ornstein-Zernike approximation at long wavelengths . . . . . . . . . . . . . . . . . . . . . . 42 Coulomb Systems : Plasmas and the Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.6.1 Electrostatic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.6.2 Debye-Hückel theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.6.3 The electron gas : Thomas-Fermi screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.7.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.7.2 Polymers as random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.7.3 Flory theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.7.4 Polymers and solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Appendix : Potts Model in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.8.2 Transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 6 Classical Interacting Systems 6.1 References – M. Kardar, Statistical Physics of Particles (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts. – L. E. Reichl, A Modern Course in Statistical Physics (2nd edition, Wiley, 1998) A comprehensive graduate level text with an emphasis on nonequilibrium phenomena. – M. Plischke and B. Bergersen, Equilibrium Statistical Physics (3rd edition, World Scientific, 2006) An excellent graduate level text. Less insightful than Kardar but still a good modern treatment of the subject. Good discussion of mean field theory. – E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics (part I, 3rd edition, Pergamon, 1980) This is volume 5 in the famous Landau and Lifshitz Course of Theoretical Physics . Though dated, it still contains a wealth of information and physical insight. – J.-P Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, 1990) An advanced, detailed discussion of liquid state physics. 1 CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 2 6.2 Ising Model 6.2.1 Definition The simplest model of an interacting system consists of a lattice L of sites, each of which contains a spin σi which may be either up (σi = +1) or down (σi = −1). The Hamiltonian is X X σi . (6.1) σi σj − µ0 H Ĥ = −J i hiji When J > 0, the preferred (i.e. lowest energy) configuration of neighboring spins is that they are aligned, i.e. σi σj = +1. The interaction is then called ferromagnetic. When J < 0 the preference is for anti-alignment, i.e. σi σj = −1, which is antiferromagnetic. This model is not exactly solvable in general. In one dimension, the solution is quite straightforward. In two dimensions, Onsager’s solution of the model (with H = 0) is among the most celebrated results in statistical physics. In higher dimensions the system has been studied by numerical simulations (the Monte Carlo method) and by field theoretic calculations (renormalization group), but no exact solutions exist. 6.2.2 Ising model in one dimension Consider a one-dimensional ring of N sites. The ordinary canonical partition function is then Zring = Tr e−β Ĥ = N X Y eβJσn σn+1 eβµ0 Hσn (6.2) {σn } n=1 = Tr RN , where σN +1 ≡ σ1 owing to periodic (ring) boundary conditions, and where R is a 2 × 2 transfer matrix, ′ ′ Rσσ′ = eβJσσ eβµ0 H(σ+σ )/2 βJ βµ H e e 0 e−βJ = e−βJ eβJ e−βµ0 H (6.3) = eβJ cosh(βµ0 H) + eβJ sinh(βµ0 H) τ z + e−βJ τ x , where τ α are the Pauli matrices. Since the trace of a matrix is invariant under a similarity transformation, we have N Z(T, H, N ) = λN + + λ− , where λ± (T, H) = eβJ cosh(βµ0 H) ± q e2βJ sinh2 (βµ0 H) + e−2βJ (6.4) (6.5) are the eigenvalues of R. In the thermodynamic limit, N → ∞, and the λN + term dominates exponentially. We therefore have F (T, H, N ) = −N kB T ln λ+ (T, H) . (6.6) From the free energy, we can compute the magnetization, ∂F N µ0 sinh(βµ0 H) M =− = q ∂H T,N sinh2 (βµ0 H) + e−4βJ (6.7) 6.2. ISING MODEL 3 and the zero field isothermal susceptibility, χ(T ) = 1 ∂M µ20 2J/kB T e . = N ∂H H=0 kB T (6.8) Note that in the noninteracting limit J → 0 we recover the familiar result for a free spin. The effect of the interactions at low temperature is to vastly increase the susceptibility. Rather than a set of independent single spins, the system effectively behaves as if it were composed of large blocks of spins, where the block size ξ is the correlation length, to be derived below. The physical properties of the system are often elucidated by evaluation of various correlation functions. In this case, we define Tr σ1 Rσ1 σ2 · · · Rσn σn+1 σn+1 Rσn+1 σn+2 · · · RσN σ1 C(n) ≡ σ1 σn+1 = Tr RN (6.9) Tr Σ Rn Σ RN −n = , Tr RN where 0 < n < N , and where Σ= 1 0 0 . −1 (6.10) To compute this ratio, we decompose R in terms of its eigenvectors, writing Then where R = λ+ |+ih+| + λ− |−ih−| . (6.11) N −n n −n 2 N 2 Σ+− Σ−+ λN λ− + λn+ λN + Σ++ + λ− Σ−− + λ+ − , C(n) = N N λ+ + λ− (6.12) Σµµ′ = h µ | Σ | µ′ i . (6.13) 6.2.3 H = 0 Consider the case H = 0, where R = eβJ + e−βJ τ x , where τ x is the Pauli matrix. Then | ± i = √12 | ↑i ± | ↓i , i.e. the eigenvectors of R are and Σ++ = Σ−− = 0, while Σ± = Σ−+ 1 1 ψ± = √ , ±1 2 = 1. The corresponding eigenvalues are λ+ = 2 cosh(βJ) , λ− = 2 sinh(βJ) . (6.14) (6.15) (6.16) The correlation function is then found to be |n| N −|n| λN −|n| λ|n| − + λ+ λ− C(n) ≡ σ1 σn+1 = + N λN + + λ− = tanh|n| (βJ) + tanhN −|n| (βJ) 1 + tanhN (βJ) ≈ tanh|n| (βJ) (N → ∞) . (6.17) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 4 This result is also valid for n < 0, provided |n| ≤ N . We see that we may write C(n) = e−|n|/ξ(T ) , (6.18) where the correlation length is ξ(T ) = Note that ξ(T ) grows as T → 0 as ξ ≈ 1 2 1 . ln ctnh(J/kB T ) (6.19) e2J/kB T . 6.2.4 Chain with free ends When the chain has free ends, there are (N −1) links, and the partition function is Zchain = X σ,σ′ = RN −1 Xn σ,σ′ σσ′ o −1 −1 λN ψ+ (σ) ψ+ (σ ′ ) + λN ψ− (σ) ψ− (σ ′ ) , + − (6.20) where ψ± (σ) = h σ | ± i. When H = 0, we make use of eqn. 6.15 to obtain RN −1 = 1 2 N −1 1 1 1 1 2 cosh βJ + 1 1 2 −1 N −1 −1 2 sinh βJ , 1 (6.21) and therefore Zchain = 2N coshN −1 (βJ) . (6.22) There’s a nifty trick to obtaining the partition function for the Ising chain which amounts to a change of variables. We define νn ≡ σn σn+1 (n = 1 , . . . , N − 1) . (6.23) Thus, ν1 = σ1 σ2 , ν2 = σ2 σ3 , etc. Note that each νj takes the values ±1. The Hamiltonian for the chain is Hchain = −J N −1 X n=1 σn σn+1 = −J N −1 X νn . (6.24) n=1 The state of the system is defined by the N Ising variables {σ1 , ν1 , . . . , νN −1 }. Note that σ1 doesn’t appear in the Hamiltonian. Thus, the interacting model is recast as N −1 noninteracting Ising spins, and the partition function is Zchain = Tr e−βHchain X XX eβJν1 eβJν2 · · · eβJνN −1 ··· = σ1 = νN −1 ν1 X X σ1 ν eβJν !N −1 = 2N coshN −1 (βJ) . (6.25) 6.2. ISING MODEL 5 6.2.5 Ising model in two dimensions : Peierls’ argument We have just seen how in one dimension, the Ising model never achieves long-ranged spin order. That is, the spin-spin correlation function decays asymptotically as an exponential function of the distance with a correlation length ξ(T ) which is finite for all > 0. Only for T = 0 does the correlation length diverge. At T = 0, there are two ground states, | ↑↑↑↑ · · · ↑ i and | ↓↓↓↓ · · · ↓ i. To choose between these ground states, we can specify a boundary condition at the ends of our one-dimensional chain, where we demand that the spins are up. Equivalently, we can apply a magnetic field H of order 1/N , which vanishes in the thermodynamic limit, but which at zero temperature will select the ‘all up’ ground state. At finite temperature, there is always a finite probability for any consecutive pair of sites (n, n+1) to be in a high energy state, i.e. either | ↑↓ i or | ↓↑ i. Such a configuration is called a domain wall, and in one-dimensional systems domain walls live on individual links. Relative to the configurations | ↑↑ i and | ↓↓ i, a domain wall costs energy 2J. For a system with M = xN domain walls, the free energy is N F = 2M J − kB T ln M ( ) (6.26) h i = N · 2Jx + kB T x ln x + (1 − x) ln(1 − x) , Minimizing the free energy with respect to x, one finds x = 1 e2J/kB T + 1 , so the equilibrium concentration of domain walls is finite, meaning there can be no long-ranged spin order. In one dimension, entropy wins and there is always a thermodynamically large number of domain walls in equilibrium. And since the correlation length for T > 0 is finite, any boundary conditions imposed at spatial infinity will have no thermodynamic consequences since they will only be ‘felt’ over a finite range. As we shall discuss in the following chapter, this consideration is true for any system with sufficiently shortranged interactions and a discrete global symmetry. Another example is the q-state Potts model, X X δσ ,1 . (6.27) δσ ,σ − h H = −J i i j i hiji Here, the spin variables σi take values in the set {1, 2, . . . , q} on each site. The equivalent of an external magnetic field in the Ising case is a field h which prefers a particular value of σ (σ = 1 in the above Hamiltonian). See the appendix in §6.8 for a transfer matrix solution of the one-dimensional Potts model. What about higher dimensions? A nifty argument due to R. Peierls shows that there will be a finite temperature phase transition for the Ising model on the square lattice1 . Consider the Ising model, in zero magnetic field, on a Nx × Ny square lattice, with Nx,y → ∞ in the thermodynamic limit. Along the perimeter of the system we impose the boundary condition σi = +1. Any configuration of the spins may then be represented uniquely in the following manner. Start with a configuration in which all spins are up. Next, draw a set of closed loops on the lattice. By definition, the loops cannot share any links along their boundaries, i.e. each link on the lattice is associated with at most one such loop. Now flip all the spins inside each loop from up to down. Identify each such loop configuration with a label Γ . The partition function is X Z = Tr e−β Ĥ = (6.28) e−2βJLΓ , Γ where LΓ is the total perimeter of the loop configuration Γ . The domain walls are now loops, rather than individual links, but as in the one-dimensional case, each link of each domain wall contributes an energy +2J relative to the ground state. Now we wish to compute the average magnetization of the central site (assume Nx,y are both odd, so there is a unique central site). This is given by the difference P+ (0) − P− (0), where Pµ (0) = δσ0 , µ is the probability that 1 Here we modify slightly the discussion in chapter 5 of the book by L. Peliti. CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 6 ++++++++ + −− − + + + + + + + −− + + + + + −−− + + + + +−+ + + + + + + + +−+ + + + + + +−− + + ++++++++ + + + + −− + + + + −− − − + + + + −− − + + + + + −− + + + + ++++++++ + −− − + + − + ++++++++ +++++++ + −− + + + + + −− + + + + +++++++ +++++++ + +−+ + + + + + −−− + + + + −−− + + + + −−+ + + + + + −−− + + + + + −−+ +++++++ + + +−+ + + + + +−+ + + +++++++ +++++++++++++++ + −− − + + + + + −− + + + + + + + −− + + + + −− + + + + + + −−− + + + + + + + + + + + +−+ + + + + + + + + + + + + + + +−+ + + + +−+ + + + + + + + − − + −− + − − − + + + + + + + + + − + + −−− + + + + + + −− + −− + − − + + + + + −− − − + + − + + − − − + + + −− − + + + − + + + − − + + + −− + + + + −− − + + + + + + + + + + + + + + +−+ + + + −− − + + − + + + + − + + + +++++++++++++++ Figure 6.1: Clusters and boundaries for the square lattice Ising model. Left panel: a configuration Γ where the central spin is up. Right panel: a configuration Cγ ◦ Γ where the interior spins of a new loop γ containing the central spin have been flipped. the central spin has spin polarization µ. If P+ (0) > P− (0), then the magnetization per site m = P+ (0) − P− (0) is finite in the thermodynamic limit, and the system is ordered. Clearly P+ (0) = 1 X −2βJLΓ e , Z (6.29) Γ ∈Σ+ where the restriction on the sum indicates that only those configurations where the central spin is up (σ0 = +1) are to be included. (see fig. 6.1a). Similarly, P− (0) = 1 X −2βJL e Γ , e Z (6.30) e∈Σ Γ − where only configurations in which σ0 = −1 are included in the sum. Here we have defined n o Σ± = Γ σ0 = ± . (6.31) I.e. Σ+ (Σ− ) is the set of configurations Γ in which the central spin is always up (down). Consider now the construction in fig. 6.1b. Any loop configuration Γe ∈ Σ− may be associated with a unique loop configuration Γ ∈ Σ+ by reversing all the spins within the loop of Γe which contains the origin. Note that the map from Γe to Γ is many-to-one. That is, we can write Γe = Cγ ◦ Γ , where Cγ overturns the spins within the loop γ, with the conditions that (i) γ contains the origin, and (ii) none of the links in the perimeter of γ coincide with any of the links from the constituent loops of Γ . Let us denote this set of loops as ΥΓ : n o ΥΓ = γ : 0 ∈ int(γ) and γ ∩ Γ = ∅ . (6.32) Then X 1 X −2βJLΓ −2βJLγ . 1− e m = P+ (0) − P− (0) = e Z Γ ∈Σ+ γ∈ΥΓ (6.33) 6.2. ISING MODEL 7 P If we can prove that γ∈Υ e−2βJLγ < 1, then we will have established that m > 0. Let us ask: how many loops Γ γ are there in ΥΓ with perimeter L? We cannot answer this question exactly, but we can derive a rigorous upper bound for this number, which, following Peliti, we call g(L). We claim that 2 · 3L · g(L) < 3L 2 L L L = ·3 . 4 24 (6.34) To establish this bound, consider any site on such a loop γ. Initially we have 4 possible directions to proceed to the next site, but thereafter there are only 3 possibilities for each subsequent step, since the loop cannot run into itself. This gives 4 · 3L−1 possibilities. But we are clearly overcounting, since any point on the loop could have been chosen as the initial point, and moreover we could have started by proceeding either clockwise or counterclockwise. So we are justified in dividing this by 2L. We are still overcounting, because we have not accounted for the constraint that γ is a closed loop, nor that γ ∩ Γ = ∅. We won’t bother trying to improve our estimate to account for these constraints. However, we are clearly undercounting due to the fact that a given loop can be translated in space so long as the origin remains within it. To account for this, we multiply by the area of a square of side length L/4, which is the maximum area that can be enclosed by a loop of perimeter L. We therefore arrive at eqn. 6.34. Finally, we note that the smallest possible value of L is L = 4, corresponding to a square enclosing the central site alone. Therefore X e−2βJLγ < γ∈ΥΓ ∞ 2k 1 X x4 (2 − x2 ) ≡r, = k · 3 e−2βJ 12 12 (1 − x2 )2 (6.35) k=2 where x = 3 e−2βJ . Note that we have accounted for the fact that the perimeter L of each loop γ must be an even integer. The sum is smaller than unity provided x < x0 = 0.869756 . . ., hence the system is ordered provided 2 kB T < = 1.61531 . J ln(3/x0 ) (6.36) The exact result is kB Tc = 2J/ sinh−1 (1) = 2.26918 . . . The Peierls argument has been generalized to higher dimensional lattices as well2 . With a little more work we can derive a bound for the magnetization. We have shown that P− (0) = 1 X −2βJLΓ X −2βJLγ 1 X −2βJLΓ = r P+ (0) . e e <r· e Z Z Γ ∈Σ+ γ∈ΥΓ (6.37) Γ ∈Σ+ Thus, 1 = P+ (0) + P− (0) < (1 + r) P+ (0) and therefore m = P+ (0) − P− (0) > (1 − r) P+ (0) > 1−r , 1+r (6.38) (6.39) where r(T ) is given in eqn. 6.35. 6.2.6 Two dimensions or one? We showed that the one-dimensional Ising model has no finite temperature phase transition, and is disordered at any finite temperature T , but in two dimensions on the square lattice there is a finite critical temperature Tc below which there is long-ranged order. Consider now the construction depicted in fig. 6.2, where the sites of 2 See. e.g. J. L. Lebowitz and A. E. Mazel, J. Stat. Phys. 90, 1051 (1998). CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 8 -32 -33 -34 -35 -31 18 19 20 21 -30 17 -8 -9 -10 -29 16 -7 2 3 4 -28 15 -6 1 0 -1 -27 14 -5 -4 -3 -26 13 12 11 -25 -24 -23 -22 22 23 24 25 -11 -12 -13 26 5 -14 27 6 -15 28 -2 7 -16 29 10 9 8 -17 30 -21 -20 -19 -18 31 35 34 33 32 Figure 6.2: A two-dimensional square lattice mapped onto a one-dimensional chain. a two-dimensional square lattice are mapped onto those of a linear chain3 . Clearly we can elicit a one-to-one mapping between the sites of a two-dimensional square lattice and those of a one-dimensional chain. That is, the two-dimensional square lattice Ising model may be written as a one-dimensional Ising model, i.e. square lattice Ĥ = −J X hiji linear chain σi σj = − X Jnn′ σn σn′ . (6.40) n,n′ How can this be consistent with the results we have just proven? The fly in the ointment here is that the interaction along the chain Jn,n′ is long-ranged. This is apparent from inspecting the site labels in fig. 6.2. Note that site n = 15 is linked to sites n′ = 14 and n′ = 16, but also to sites n′ = −6 and n′ = −28. With each turn of the concentric spirals in the figure, the range of the interaction increases. To complicate matters further, the interactions are no longer translationally invariant, i.e. Jnn′ 6= J(n−n′ ). But it is the long-ranged nature of the interactions on our contrived one-dimensional chain which spoils our previous energyentropy argument, because now the domain walls themselves interact via a long-ranged potential. Consider for example the linear chain with Jn,n′ = J |n − n′ |−α , where α > 0. Let us compute the energy of a domain wall configuration where σn = +1 if n > 0 and σn = −1 if n ≤ 0. The domain wall energy is then ∆= ∞ X ∞ X 2J . |m + n|α m=0 n=1 (6.41) Here we have written one of the sums in terms of m = −n′ . For asymptotically large m and n, we can write R = (m, n) and we obtain an integral over the upper right quadrant of the plane: Zπ/2 Z∞ dR R dφ 1 0 2J = 2−α/2 Rα (cos φ + sin φ)α Zπ/4 dφ cosα φ −π/4 Z∞ dR . Rα−1 (6.42) 1 The φ integral is convergent, but the R integral diverges for α ≤ 2. For a finite system, the upper bound on the R integral becomes the system size L. For α > 2 the domain wall energy is finite in the thermodynamic limit 3A corresponding mapping can be found between a cubic lattice and the linear chain as well. 6.2. ISING MODEL 9 L → ∞. In this case, entropy again wins. I.e. the entropy associated with a single domain wall is kB ln L, and therefore F = E − kB T is always lowered by having a finite density of domain walls. For α < 2, the energy of a single domain wall scales as L2−α . It was first proven by F. J. Dyson in 1969 that this model has a finite temperature phase transition provided 1 < α < 2. There is no transition for α < 1 or α > 2. The case α = 2 is special, and is discussed as a special case in the beautiful renormalization group analysis by J. M. Kosterlitz in Phys. Rev. Lett. 37, 1577 (1976). 6.2.7 High temperature expansion Consider once again the ferromagnetic Ising model in zero field (H = 0), but on an arbitrary lattice. The partition function is ) ( P Y NL βJ hiji σi σj 1 + x σi σj , (6.43) = cosh βJ Tr Z = Tr e hiji where x = tanh βJ and NL is the number of links. For regular lattices, NL = 21 zN , where N is the number of lattice sites and z is the lattice coordination number, i.e. the number of nearest neighbors for each site. We have used ( n o ′ e+βJ if σσ ′ = +1 eβJσσ = cosh βJ · 1 + σσ ′ tanh βJ = (6.44) e−βJ if σσ ′ = −1 . We expand eqn. 6.43 in powers of x, resulting in a sum of 2NL terms, each of which can be represented graphically in terms of so-called lattice animals. A lattice animal is a distinct (including reflections and rotations) arrangement of adjacent plaquettes on a lattice. In order that the trace not vanish, only such configurations and their compositions are permitted. This is because each σi for every given site i must occur an even number of times in order for a given term in the sum not to vanish. For all such terms, the trace is 2N . Let Γ represent a collection of lattice animals, and gΓ the multiplicity of Γ . Then L N X (6.45) Z = 2N cosh βJ L gΓ tanh βJ Γ , Γ where LΓ is the total number of sites in the diagram Γ , and gΓ is the multiplicity of Γ . Since x vanishes as T → ∞, this procedure is known as the high temperature expansion (HTE). For the square lattice, he enumeration of all lattice animals with up to order eight is given in fig. 6.3. For the diagram represented as a single elementary plaquette, there are N possible locations for the lower left vertex. For the 2 × 1 plaquette animal, one has g = 2N , because there are two inequivalent orientations as well as N translations. For two disjoint elementary squares, one has g = 21 N (N − 5), which arises from subtracting 5N ‘illegal’ configurations involving double lines (remember each link in the partition sum appears only once!), shown in the figure, and finally dividing by two because the individual squares are identical. Note that N (N − 5) is always even for any integer value of N . Thus, to lowest interesting order on the square lattice, o 2N n 1 + N x4 + 2N x6 + 7 − 52 N x8 + 12 N 2 x8 + O(x10 ) . Z = 2N cosh βJ (6.46) The free energy is therefore i h F = −kB T ln 2 + N kB T ln(1 − x2 ) − N kB T x4 + 2 x6 + 29 x8 + O(x10 ) n o 8 10 = N kB T ln 2 − N kB T x2 + 32 x4 + 37 x6 + 19 4 x + O(x ) , (6.47) again with x = tanh βJ. Note that we’ve substituted cosh2 βJ = 1/(1−x2 ) to write the final result as a power series in x. Notice that the O(N 2 ) factor in Z has cancelled upon taking the logarithm, so the free energy is properly extensive. CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 10 Figure 6.3: HTE diagrams on the square lattice and their multiplicities. Note that the high temperature expansion for the one-dimensional Ising chain yields Zchain (T, N ) = 2N coshN −1 βJ Zring (T, N ) = 2N coshN βJ , , (6.48) in agreement with the transfer matrix calculations. In higher dimensions, where there is a finite temperature phase transition, one typically computes the specific heat c(T ) and tries to extract its singular behavior in the vicinity of Tc , where c(T ) ∼ A (T − Tc )−α . Since x(T ) = tanh(J/kB T ) is analytic in T , we have c(x) ∼ A′ (x − xc )−α , where xc = x(Tc ). One assumes xc is the singularity closest to the origin and corresponds to the radius of convergence of the high temperature expansion. If we write −α x c(x) = an x ∼ A 1 − , xc n=0 ∞ X n ′′ (6.49) then according to the binomial theorem we should expect 1−α 1 an 1− . = an−1 xc n Thus, by plotting an /an−1 versus 1/n, one extracts 1/xc as the intercept, and (α − 1)/xc as the slope. (6.50) 6.3. NONIDEAL CLASSICAL GASES 11 Figure 6.4: HTE diagrams for the numerator Ykl of the correlation function Ckl . The blue path connecting sites k and l is the string. The remaining red paths are all closed loops. High temperature expansion for correlation functions Can we also derive a high temperature expansion for the spin-spin correlation function Ckl = hσk σl i ? Yes we can. We have i h P Tr σk σl eβJ hiji σi σj Y h i ≡ kl . (6.51) Ckl = P Z Tr eβJ hiji σi σj Recall our analysis of the partition function Z. We concluded that in order for the trace not to vanish, the spin variable σi on each site i must occur an even number of times in the expansion of the product. Similar considerations hold for Ykl , except now due to the presence of σk and σl , those variables now must occur an odd number of times when expanding the product. It is clear that the only nonvanishing diagrams will be those in which there is a finite string connecting sites k and l, in addition to the usual closed HTE loops. See fig. 6.4 for an instructive sketch. One then expands both Ykl as well as Z in powers of x = tanh βJ, taking the ratio to obtain the correlator Ckl . At high temperatures (x → 0), both numerator and denominator are dominated by the configurations Γ with the shortest possible total perimeter. For Z, this means the trivial path Γ = {∅}, while for Ykl this means finding the shortest length path from k to l. (If there is no straight line path from k to l, there will in general be several such minimizing paths.) Note, however, that the presence of the string between sites k and l complicates the analysis of gΓ for the closed loops, since none of the links of Γ can intersect the string. It is worth stressing that this does not mean that the string and the closed loops cannot intersect at isolated sites, but only that they share no common links; see once again fig. 6.4. 6.3 Nonideal Classical Gases Let’s switch gears now and return to the study of continuous classical systems described by a Hamiltonian Ĥ {xi }, {pi } . In the next chapter, we will see how the critical properties of classical fluids can in fact be modeled by an appropriate lattice gas Ising model, and we’ll derive methods for describing the liquid-gas phase transition in such a model. CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 12 6.3.1 The configuration integral Consider the ordinary canonical partition function for a nonideal system of identical point particles interacting via a central two-body potential u(r). We work in the ordinary canonical ensemble. The N -particle partition function is Z Y N ddpi ddxi −Ĥ/kB T e hd i=1 N dZ Y 1 X λ−N d T d xi exp − u |xi − xj | . = N! kB T i<j i=1 Z(T, V, N ) = 1 N! (6.52) Here, we have assumed a many body Hamiltonian of the form Ĥ = N X X p2i u |xi − xj | , + 2m i=1 i<j in which massive nonrelativistic particles interact via a two-body central potential. As before, λT = is the thermal wavelength. We can now write d Z(T, V, N ) = λ−N QN (T, V ) , T where the configuration integral QN (T, V ) is given by Z Z Y 1 e−βu(rij ) . ddx1 · · · ddxN QN (T, V ) = N! i<j (6.53) p 2π~2 /mkB T (6.54) (6.55) There are no general methods for evaluating the configurational integral exactly. 6.3.2 One-dimensional Tonks gas The Tonks gas is a one-dimensional generalization of the hard sphere gas. Consider a one-dimensional gas of indistinguishable particles of mass m interacting via the potential ( ∞ if |x − x′ | < a ′ u(x − x ) = (6.56) 0 if |x − x′ | ≥ a . Let the gas be placed in a finite volume L. The hard sphere nature of the particles means that no particle can get within a distance 12 a of the ends at x = 0 and x = L. That is, there is a one-body potential v(x) acting as well, where 1 ∞ if x < 2 a v(x) = 0 if 21 a ≤ x ≤ L − 21 a (6.57) ∞ if x > L − 12 a . The partition function of the 1D Tonks gas is given by λ−N Z(T, L, N ) = T N! ZL ZL dx1 · · · dxN χ(x1 , . . . , xN ) , 0 0 (6.58) 6.3. NONIDEAL CLASSICAL GASES 13 where χ = e−U/kB T is zero if any two ‘rods’ (of length a) overlap, or if any rod overlaps with either boundary at x = 0 and x = L, and χ = 1 otherwise. Note that χ does not depend on temperature. Without loss of generality, we can integrate over the subspace where x1 < x2 < · · · < xN and then multiply the result by N ! . Clearly xj must lie to the right of xj−1 + a and to the left of Yj ≡ L − (N − j)a − 12 a. Thus, Z(T, L, N ) = λ−N T ZY2 ZYN ZY1 dx1 dx2 · · · dxN a/2 x1 +a xN −1 +a YN −1 ZY2 Z ZY1 −N dx1 dx2 · · · dxN −1 YN −1 − xN −1 = λT a/2 = λ−N T xN −2 +a YN −2 ZY2 Z ZY1 dx1 dx2 · · · dxN −2 a/2 λ−N T x1 +a 1 2 YN −2 − xN −2 xN −3 +a 2 = ··· λ−N T (L − N a)N . N! N! p factor comes from integrating over the momenta; recall λT = 2π~2 /mkB T . = The λN T x1 +a X1 − 21 a N The free energy is F = −kB T ln Z = −N kB T = ( − ln λT + 1 + ln L −a N ) , (6.59) (6.60) where we have used Stirling’s rule to write ln N ! ≈ N ln N − N . The pressure is p=− ∂F nkB T k T = = LB , ∂L 1 − na N −a (6.61) where n = N/L is the one-dimensional density. Note that the pressure diverges as n approaches 1/a. The usual one-dimensional ideal gas law, pL = N kB T , is replaced by pLeff = N kB T , where Leff = L − N a is the ‘free’ volume obtained by subtracting the total ‘excluded volume’ N a from the original volume L. 6.3.3 Mayer cluster expansion Let us return to the general problem of computing the configuration integral. Consider the function e−βuij , where uij ≡ u(|xi −xj |). We assume that at very short distances there is a strong repulsion between particles, i.e. uij → ∞ as rij = |xi − xj | → 0, and that uij → 0 as rij → ∞. Thus, e−βuij vanishes as rij → 0 and approaches unity as rij → ∞. For our purposes, it will prove useful to define the function f (r) = e−βu(r) − 1 , (6.62) called the Mayer function after Josef Mayer. We may now write Z Z Y 1 d QN (T, V ) = 1 + fij . d x1 · · · ddxN N! i<j (6.63) A typical potential we might consider is the semi-phenomenological Lennard-Jones potential, σ 12 σ 6 . − u(r) = 4 ǫ r r (6.64) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 14 Figure 6.5: Bottom panel: Lennard-Jones potential u(r) = 4ǫ x−12 − x−6 , with x = r/σ and ǫ = 1. Note the weak attractive tail and the strong repulsive core. Top panel: Mayer function f (r, T ) = e−u(r)/kB T − 1 for kB T = 0.8 ǫ (blue), kB T = 1.5 ǫ (green), and kB T = 5 ǫ (red). This accounts for a long-distance attraction due to mutually induced electric dipole fluctuations, and a strong short-ranged repulsion, phenomenologically modelled with a r−12 potential, which mimics a hard core due to overlap of the atomic electron distributions. Setting u′ (r) = 0 we obtain r∗ = 21/6 σ ≈ 1.12246 σ at the minimum, where u(r∗ ) = −ǫ. In contrast to the Boltzmann weight e−βu(r) , the Mayer function f (r) vanishes as r → ∞, behaving as f (r) ∼ −βu(r). The Mayer function also depends on temperature. Sketches of u(r) and f (r) for the Lennard-Jones model are shown in fig. 6.5. The Lennard-Jones potential4 is realistic for certain simple fluids, but it leads to a configuration integral which is in general impossible to evaluate. Indeed, even a potential as simple as that of the hard sphere gas is intractable in more than one space dimension. We can however make progress by deriving a series expansion for the equation of state in powers of the particle density. This is known as the virial expansion. As was the case when we investigated noninteracting quantum statistics, it is convenient to work in the grand canonical ensemble and to derive series expansions for the density n(T, z) and the pressure p(T, z) in terms of the fugacity z, then solve for z(T, n) to obtain p(T, n). These expansions in terms of fugacity have a nifty diagrammatic interpretation, due to Mayer. We begin by expanding the product in eqn. 6.63 as X X Y fij + fij fkl + . . . . 1 + fij = 1 + i<j i<j (6.65) i<j , k<l (ij)6=(kl) As there are 12 N (N − 1) possible pairings, there are 2N (N −1)/2 terms in the expansion of the above product. Each such term may be represented by a graph, as shown in fig. 6.7. For each such term, we draw a connection between dots representing different particles i and j if the factor fij appears in the term under consideration. The contribution for any given graph may be written as a product over contributions from each of its disconnected 4 Disambiguation footnote: Take care not to confuse Philipp Lenard (Hungarian-German, cathode ray tubes, Nazi), Alfred-Marie Liénard (French, Liénard-Wiechert potentials, not a Nazi), John Lennard-Jones (British, molecular structure, definitely not a Nazi), and Lynyrd Skynyrd (American, ”Free Bird”, possibly killed by Nazis in 1977 plane crash). I thank my colleague Oleg Shpyrko for setting me straight on this. 6.3. NONIDEAL CLASSICAL GASES 15 Figure 6.6: Left: John Lennard-Jones. Center: Catherine Zeta-Jones. Right: James Earl Jones. Figure 6.7: Diagrammatic interpretation of a term involving a product of eight Mayer functions. component clusters. For example, in the case of the term in fig. 6.7, the contribution to the configurational integral would be Z 1 ∆Q = ddx1 ddx4 ddx7 ddx9 f1,4 f4,7 f4,9 f7,9 N! Z Z Z (6.66) × ddx2 ddx5 ddx6 f2,5 f2,6 × ddx3 ddx10 f3,10 × ddx8 ddx11 f8,11 . We will refer to a given product of Mayer functions which arises from this expansion as a term. The particular labels we assign to each vertex of a given graph don’t affect the overall value of the graph. Now a given unlabeled graph consists of a certain number of connected subgraphs. For a system with N particles, we may then write X N= mγ n γ , (6.67) γ where γ ranges over all possible connected subgraphs, and mγ = number of connected subgraphs of type γ in the unlabeled graph nγ = number of vertices in the connected subgraph γ . Note that the single vertex • counts as a connected subgraph, with n• = 1. We now ask: how many ways are there of assigning the N labels to the N vertices of a given unlabeled graph? One might first thing the answer is simply N !, however this is too big, because different assignments of the labels to the vertices may not result in a distinct graph. To see this, consider the examples in fig. 6.8. In the first example, an unlabeled graph with four CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 16 vertices consists of two identical connected subgraphs. Given any assignment of labels to the vertices, then, Q we can simply exchange the two subgraphs and get the same term. So we should divide N ! by the product γ mγ !. But even this is not enough, because within each connected subgraph γ there may be permutations which leave the integrand unchanged, as shown in the second and third examples in fig. 6.8. We define the symmetry factor sγ as the number of permutations of the labels which leaves a given connected subgraphs γ invariant. Examples of symmetry factors are shown in fig. 6.9. Consider, for example, the third subgraph in the top row. Clearly one can rotate the figure about its horizontal symmetry axis to obtain a new labeling which represents the same term. This twofold axis is the only symmetry the diagram possesses, hence sγ = 2. For the first diagram in the second row, one can rotate either of the triangles about the horizontal symmetry axis. One can also rotate the figure in the plane by 180◦ so as to exchange the two triangles. Thus, there are 2 × 2 × 2 = 8 symmetry operations which result in the same term, and sγ = 8. Finally, the last subgraph in the second row consists of five vertices each of which is connected to the other four. Therefore any permutation of the labels results in the same term, and sγ = 5! = 120. Q Q m In addition to dividing by the product γ mγ !, we must then also divide by γ sγ γ . We can now write the partition function as Z Y λ−N d X N! Z= T Q mγ · N! m γ ! sγ γ {mγ } d d d x1 · · · d xnγ γ Y fij i<j !mγ · δN , P m γ nγ , (6.68) where Pthe last product is over all links in the subgraph γ. The final Kronecker delta enforces the constraint N = γ mγ nγ . We next define the cluster integral bγ as bγ (T ) ≡ Z 1 1 · sγ V ddx1 · · · ddxnγ γ Y fij . (6.69) i<j Q Since fij = f |xi − xj | , the product γi<j fij is invariant under simultaneous translation of all the coordinate vectors by any constant vector, and hence the integral over the nγ position variables contains exactly one factor of the volume, which cancels with the prefactor in the above definition of bγ . Thus, each cluster integral is intensive, scaling as V 0 .5 If we compute the grand partition function, then the fixed N constraint is relaxed, and we can do the sums: P mγ nγ Y 1 X m βµ −d −βΩ e λ Ξ =e = V bγ γ T mγ ! γ {mγ } m 1 βµ −d mγ nγ e λT V bγ γ m ! γ γ mγ =0 X nγ . eβµ λ−d b = exp V γ T = ∞ Y X (6.70) γ Thus, Ω(T, V, µ) = −V kB T and we can write p = kB T n= X γ 5 We X X eβµ λ−d T γ zλ−d T γ nγ zλ−d T nγ nγ assume that the long-ranged behavior of f (r) ≈ −βu(r) is integrable. nγ bγ (T ) , (6.71) bγ (T ) (6.72) bγ (T ) , (6.73) 6.3. NONIDEAL CLASSICAL GASES 17 Figure 6.8: Different assignations of labels to vertices may not result in a distinct term in the expansion of the configuration integral. Figure 6.9: The symmetry factor sγ for a connected subgraph γ is the number of permutations of its indices which Q leaves the term (ij)∈γ fij invariant. where z = exp(βµ) is the fugacity, and where b• ≡ 1. As we did in the case of ideal quantum gas statistical mechanics, we can systematically invert the relation n = n(z, T ) to obtain z = z(n, T ), and then insert this into the equation for p(z, T ) to obtain the equation of state p = p(n, T ). This yields the virial expansion of the equation of state, n o p = nkB T 1 + B2 (T ) n + B3 (T ) n2 + . . . . (6.74) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 18 6.3.4 Cookbook recipe Just follow these simple steps! • The pressure and number density are written as an expansion over unlabeled connected clusters γ, viz. βp = X zλ−d T γ n= X nγ nγ zλ−d T γ bγ nγ bγ . • For each term in each of these sums, draw the unlabeled connected cluster γ. • Assign labels 1 , 2 , . . . , nγ to the vertices, where nγ is the total number of vertices in the cluster γ. It doesn’t matter how you assign the labels. Q • Write down the product γi<j fij . The factor fij appears in the product if there is a link in your (now labeled) cluster between sites i and j. • The symmetry factor sγ is the number of elements of the symmetric group Snγ which leave the product Qγ i<j fij invariant. The identity permutation always leaves the product invariant, so sγ ≥ 1. • The cluster integral is 1 1 bγ (T ) ≡ · sγ V Z ddx1 · · · ddxnγ γ Y fij . i<j Due to translation invariance, bγ (T ) ∝ V 0 . One can therefore set xnγ ≡ 0, eliminate the volume factor from the denominator, and perform the integral over the remaining nγ −1 coordinates. • This procedure generates expansions for p(T, z) and n(T, z) in powers of the fugacity z = eβµ . To obtain something useful like p(T, n), we invert the equation n = n(T, z) to find z = z(T, n), and then substitute into the equation p = p(T, z) to obtain p = p T, z(T, n) = p(T, n). The result is the virial expansion, n o p = nkB T 1 + B2 (T ) n + B3 (T ) n2 + . . . . 6.3.5 Lowest order expansion We have Z Z 1 d b− (T ) = d x1 ddx2 f |x1 − x2 | 2V Z = 12 ddr f (r) (6.75) and b∧ (T ) = = 1 2V Z 1 2 Z Z Z ddx1 ddx2 ddx3 f |x1 − x2 | f |x1 − x3 | Z 2 d d r ddr′ f (r) f (r′ ) = 2 b− (6.76) 6.3. NONIDEAL CLASSICAL GASES and We may now write 19 Z Z Z 1 d d b△ (T ) = d x1 d x2 ddx3 f |x1 − x2 | f |x1 − x3 | f |x2 − x3 | 6V Z Z = 61 ddr ddr′ f (r) f (r′ ) f |r − r ′ | . n o 3 −d 2 p = kB T zλ−d b− (T ) + zλ−d · b∧ + b△ + O(z 4 ) T + zλT T 3 −d 2 n = zλ−d b− (T ) + 3 zλ−d · b∧ + b△ + O(z 4 ) T + 2 zλT T We invert by writing 2 3 zλ−d T = n + α2 n + α3 n + . . . (6.77) (6.78) (6.79) (6.80) and substituting into the equation for n(z, T ), yielding Thus, n = (n + α2 n2 + α3 n3 ) + 2(n + α2 n2 )2 b− + 3n3 b∧ + b△ + O(n4 ) . (6.81) 0 = (α2 + 2b− ) n2 + (α3 + 4α2 b− + 3b∧ + 3b△ ) n3 + . . . . (6.82) α2 = −2b− α3 = −4α2 b− − 3b∧ − 3b△ (6.83) We therefore conclude = 8b2− − 6b2− − 3b△ = 2b2− (6.84) − 3b△ . We now insert eqn. 6.80 with the determined values of α2,3 into the equation for p(z, T ), obtaining p = n − 2b− n2 + (2b2− − 3b△ ) n3 + (n − 2b− n2 )2 b− + n3 (2b2− + b△ ) + O(n4 ) kB T = n − b− n2 − 2b△ n3 + O(n4 ) . (6.85) Thus, B2 (T ) = −b− (T ) , B3 (T ) = −2b△ (T ) . (6.86) 6.3.6 Hard sphere gas in three dimensions The hard sphere potential is given by u(r) = ( ∞ if r ≤ a 0 if r > a . (6.87) Here a is the diameter of the spheres. The corresponding Mayer function is then temperature independent, and given by ( −1 if r ≤ a f (r) = (6.88) 0 if r > a . We can change variables b− (T ) = Z 1 2 d3r f (r) = − 32 πa3 . (6.89) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 20 Figure 6.10: The overlap of hard sphere Mayer functions. The shaded volume is V. The calculation of b△ is more challenging. We have Z Z b△ = 16 d3ρ d3r f (ρ) f (r) f |r − ρ| . (6.90) We must first compute the volume of overlap for spheres of radius a (recall a is the diameter of the constituent hard sphere particles) centered at 0 and at ρ: Z V = d3r f (r) f |r − ρ| =2 Za dz π(a2 − z 2 ) = 4π 3 3 a − πa2 ρ + π 12 ρ3 . (6.91) ρ/2 We then integrate over region |ρ| < a, to obtain b△ = − 16 Za n 3 2 · 4π dρ ρ2 · 4π 3 a − πa ρ + 0 π 12 ρ3 o (6.92) 2 6 = − 5π 36 a . Thus, n p = nkB T 1 + 2π 3 3 a n + 5π 2 6 2 18 a n o + O(n3 ) . (6.93) 6.3.7 Weakly attractive tail Suppose u(r) = ( ∞ −u0 (r) if r ≤ a if r > a . (6.94) Then the corresponding Mayer function is f (r) = ( −1 if r ≤ a βu0 (r) e − 1 if r > a . (6.95) 6.3. NONIDEAL CLASSICAL GASES Thus, b− (T ) = 1 2 21 Z 3 d r f (r) = 3 − 2π 3 a Z∞ h i + 2π dr r2 eβu0 (r) − 1 . (6.96) a Thus, the second virial coefficient is B2 (T ) = −b− (T ) ≈ 2π 3 3 a 2π − kB T Z∞ dr r2 u0 (r) , (6.97) a where we have assumed kB T ≪ u0 (r). We see that the second virial coefficient changes sign at some temperature T0 , from a negative low temperature value to a positive high temperature value. 6.3.8 Spherical potential well Consider an attractive spherical well potential with an infinitely repulsive core, Then the corresponding Mayer function is Writing s ≡ R/a, we have ∞ if r ≤ a u(r) = −ǫ if a < r < R 0 if r > R . if r ≤ a −1 βǫ f (r) = e − 1 if a < r < R 0 if r > R . Z B2 (T ) = −b− (T ) = − 12 d3r f (r) 4π 3 3 1 βǫ 4π 3 (−1) · 3 a + e − 1 · 3 a (s − 1) =− 2 3 βǫ 2π 3 = 3 a 1 − (s − 1) e − 1 . (6.98) (6.99) (6.100) To find the temperature T0 where B2 (T ) changes sign, we set B2 (T0 ) = 0 and obtain kB T0 = ǫ 3 s . ln 3 s −1 (6.101) Recall in our study of the thermodynamics of the Joule-Thompson effect in §1.10.6 that the throttling process is isenthalpic. The temperature change, when a gas is pushed (or escapes) through a porous plug from a high pressure region to a low pressure one is Zp2 ∂T ∆T = dp , (6.102) ∂p H p1 CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 22 Figure 6.11: An attractive spherical well with a repulsive core u(r) and its associated Mayer function f (r). where ∂T ∂p H " # 1 ∂V T = −V . Cp ∂T p (6.103) Appealing to the virial expansion, and working to lowest order in corrections to the ideal gas law, we have p= and we compute ∂V ∂T p N N2 kB T + 2 kB T B2 (T ) + . . . V V (6.104) by seting 0 = dp = − N kB N2 2N 2 N kB T dV + k T B (T ) dV + d kB T B2 (T ) + . . . . dT − B 2 V2 V V3 V2 Dividing by dT , we find T The temperature where ∂T ∂p set T ∗ B2′ (T ∗ ) = B2 (T ∗ ), i.e. H ∂V ∂T " # ∂B2 −V =N T − B2 . ∂T p (6.105) (6.106) changes sign is called the inversion temperature T ∗ . To find the inversion point, we If we approximate B2 (T ) ≈ A − B T, d ln B2 =1. d ln T T ∗ (6.107) then the inversion temperature follows simply: B B =A− ∗ ∗ T T =⇒ T∗ = 2B . A (6.108) 6.3.9 Hard spheres with a hard wall Consider a hard sphere gas in three dimensions in the presence of a hard wall at z = 0. The gas is confined to the region z > 0. The total potential energy is now X X u(xi − xj ) , (6.109) v(xi ) + W (x1 , . . . , xN ) = i<j i where v(r) = v(z) = ( ∞ if 0 if z ≤ 21 a z > 21 a , (6.110) 6.3. NONIDEAL CLASSICAL GASES 23 Figure 6.12: In the presence of a hard wall, the Mayer sphere is cut off on the side closest to the wall. The resulting density n(z) vanishes for z < 12 a since the center of each sphere must be at least one radius ( 21 a) away from the wall. Between z = 21 a and z = 32 a there is a density enhancement. If the calculation were carried out to higher order, n(z) would exhibit damped spatial oscillations with wavelength λ ∼ a. and u(r) is given in eqn. 6.87. The grand potential is written as a series in the total particle number N , and is given by Z Z Z ′ ′ Ξ = e−βΩ = 1 + ξ d3r e−βv(z) + 12 ξ 2 d3r d3r′ e−βv(z) e−βv(z ) e−βu(r−r ) + . . . , (6.111) where ξ = z λT−3 , with z = eµ/kB T the fugacity. Taking the logarithm, and invoking the Taylor series ln(1 + δ) = δ − 12 δ 2 + 31 δ 3 − . . ., we obtain Z Z Z h i ′ −βΩ = ξ d3r + 21 ξ 2 d3r d3r′ e−βu(r−r ) − 1 + . . . (6.112) z> a 2 a ′ z> a 2 z >2 R The volume is V = d3r. Dividing by V , we have, in the thermodynamic limit, z>0 βΩ 1 = βp = ξ + 21 ξ 2 − V V Z 3 dr Z h i ′ d3r′ e−βu(r−r ) − 1 + . . . a ′ z> a 2 z >2 (6.113) = ξ − 32 πa3 ξ 2 + O(ξ 3 ) . The number density is n=ξ ∂ (βp) = ξ − 34 πa3 ξ 2 + O(ξ 3 ) , ∂ξ (6.114) and inverting to obtain ξ(n) and then substituting into the pressure equation, we obtain the lowest order virial expansion for the equation of state, n o p = kB T n + 32 πa3 n2 + . . . . (6.115) As expected, the presence of the wall does not affect a bulk property such as the equation of state. Next, let us compute the number density n(z), given by X δ(r − ri ) . n(z) = i (6.116) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 24 Due to translational invariance in the (x, y) plane, we know that the density must be a function of z alone. The presence of the wall at z = 0 breaks translational symmetry in the z direction. The number density is N X δ(r − ri ) Tr eβ(µN̂ −Ĥ) n(z) = Tr eβ(µN̂−Ĥ) = Ξ −1 ( i=1 Z ′ ′ ξ e−βv(z) + ξ 2 e−βv(z) d3r′ e−βv(z ) e−βu(r−r ) + . . . ) (6.117) Z h i ′ ′ = ξ e−βv(z) + ξ 2 e−βv(z) d3r′ e−βv(z ) e−βu(r−r ) − 1 + . . . . ′ Note that the term in square brackets in the last line is the Mayer function f (r − r ′ ) = e−βu(r−r ) − 1. Consider the function if z < 12 a or z ′ < 12 a 0 −βv(z) −βv(z ′ ) ′ e e f (r − r ) = 0 if |r − r ′ | > a −1 if z > 21 a and z ′ > 12 a and |r − r ′ | < a . (6.118) Now consider the integral of the above function with respect to r ′ . Clearly the result depends on the value of z. If z > 32 a, then there is no excluded region in r ′ and the integral is (−1) times the full Mayer sphere volume, i.e. larger than 12 a, the integral − 43 πa3 . If z < 12 a the integral vanishes due to the e−βv(z) factor. For z infinitesimally 2 a 3a 3 is (−1) times half the Mayer sphere volume, i.e. − 3 πa . For z ∈ 2 , 2 the integral interpolates between − 23 πa3 and − 34 πa3 . Explicitly, one finds by elementary integration, Z h0 ′ 3 ′ −βv(z) −βv(z ) ′ −1 − 23 dr e e f (r − r ) = − 4 πa3 z a − 3 1 2 + 1 z 2 a − i 1 3 2 if z < 21 a · 23 πa3 if 21 a < z < 32 a if z > 3 2a (6.119) . After substituting ξ = n + 34 πa3 n2 + O(n3 ) to relate ξ to the bulk density n = n∞ , we obtain the desired result: 0 h n(z) = n + 1 − n 3 z 2 a − 1 2 + 1 z 2 a − i 1 3 2 if z < 12 a · 23 πa3 n2 if 21 a < z < 32 a if z > 3 2a (6.120) . A sketch is provided in the right hand panel of fig. 6.12. Note that the density n(z) vanishes identically for z < 12 due to the exclusion of the hard spheres by the wall. For z between 12 a and 32 a, there is a density enhancement, the origin of which has a simple physical interpretation. Since the wall excludes particles from the region z < 21 , there is an empty slab of thickness 21 z coating the interior of the wall. There are then no particles in this region to exclude neighbors to their right, hence the density builds up just on the other side of this slab. The effect vanishes to the order of the calculation past z = 23 a, where n(z) = n returns to its bulk value. Had we calculated to higher order, we’d have found damped oscillations with spatial period λ ∼ a. 6.4. LEE-YANG THEORY 25 6.4 Lee-Yang Theory 6.4.1 Analytic properties of the partition function How can statistical mechanics describe phase transitions? This question was addressed in some beautiful mathematical analysis by Lee and Yang6 . Consider the grand partition function Ξ, ∞ X z N QN (T, V ) λT−dN , (6.121) Z dd x1 · · · dd xN e−U(x1 , ... , xN )/kB T (6.122) Ξ(T, V, z) = N =0 where QN (T, V ) = 1 N! Z is the contribution to the N -particle partition function from the potential energy U (assuming no momentumdependent potentials). For two-body central potentials, we have U (x1 , . . . , xN ) = X i<j v |xi − xj | . (6.123) Suppose further that these classical particles have hard cores. Then for any finite volume, there must be some maximum number NV such that QN (T, V ) vanishes for N > NV . This is because if N > NV at least two spheres must overlap, in which case the potential energy is infinite. The theoretical maximum packing density for hard π = 0.74048. If the spheres spheres is achieved for a hexagonal close packed (HCP) lattice7 , for which fHCP = 3√ 2 √ 3 have radius r0 , then NV = V /4 2r0 is the maximum particle number. Thus, if V itself is finite, then Ξ(T, V, z) is a finite degree polynomial in z, and may be factorized as Ξ(T, V, z) = NV X N z QN (T, V N =0 ) λ−dN T NV Y z 1− = , zk (6.124) k=1 where zk (T, V ) is one of the NV zeros of the grand partition function. Note that the O(z 0 ) term is fixed to be unity. Note also that since the configuration integrals QN (T, V ) are all positive, Ξ(z) is an increasing function along the positive real z axis. In addition, since the coefficients of z N in the polynomial Ξ(z) are all real, then Ξ(z) = 0 implies Ξ(z) = Ξ(z̄) = 0, so the zeros of Ξ(z) are either real and negative or else come in complex conjugate pairs. For finite NV , the situation is roughly as depicted in the left panel of fig. 6.13, with a set of NV zeros arranged in complex conjugate pairs (or negative real values). The zeros aren’t necessarily distributed along a circle as shown in the figure, though. They could be anywhere, so long as they are symmetrically distributed about the Re(z) axis, and no zeros occur for z real and nonnegative. Lee and Yang proved the existence of the limits p 1 = lim ln Ξ(T, V, z) V →∞ V kB T ∂ 1 n = lim z ln Ξ(T, V, z) , V →∞ ∂z V 6 See (6.126) C. N. Yang and R. D. Lee, Phys. Rev. 87, 404 (1952) and ibid, p. 410 e.g. http://en.wikipedia.org/wiki/Close-packing . For randomly close-packed hard spheres, one finds, from numerical simulations, = 0.644. 7 See fRCP (6.125) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 26 Figure 6.13: In the thermodynamic limit, the grand partition function can develop a singularity at positive real fugacity z. The set of discrete zeros fuses into a branch cut. and notably the result n=z ∂ ∂z p kB T , (6.127) ∂ . In which amounts to the commutativity of the thermodynamic limit V → ∞ with the differential operator z ∂z particular, p(T, z) is a smooth function of z in regions free of roots. If the roots do coalesce and pinch the positive real axis, then then density n can be discontinuous, as in a first order phase transition, or a higher derivative ∂ j p/∂nj can be discontinuous or divergent, as in a second order phase transition. 6.4.2 Electrostatic analogy There is a beautiful analogy to the theory of two-dimensional electrostatics. We write NV p 1 X z = ln 1 − kB T V zk k=1 NV h i X =− φ(z − zk ) − φ(0 − zk ) , (6.128) k=1 where 1 ln(z) (6.129) V is the complex potential due to a line charge of linear density λ = V −1 located at origin. The number density is then NV ∂ X p ∂ = −z φ(z − zk ) , (6.130) n=z ∂z kB T ∂z φ(z) = − k=1 to be evaluated for physical values of z, i.e. z ∈ R+ . Since φ(z) is analytic, 1 ∂φ i ∂φ ∂φ = + =0. ∂ z̄ 2 ∂x 2 ∂y (6.131) 6.4. LEE-YANG THEORY 27 If we decompose the complex potential φ = φ1 + iφ2 into real and imaginary parts, the condition of analyticity is recast as the Cauchy-Riemann equations, ∂φ2 ∂φ1 = ∂x ∂y , ∂φ1 ∂φ =− 2 . ∂y ∂x (6.132) Thus, − ∂φ 1 ∂φ i ∂φ =− + ∂z 2 ∂x 2 ∂y i ∂φ1 ∂φ2 ∂φ2 1 ∂φ1 + + − =− 2 ∂x ∂y 2 ∂y ∂x ∂φ1 ∂φ1 =− +i ∂x ∂y = Ex − iEy , (6.133) where E = −∇φ1 is the electric field. Suppose, then, that as V → ∞ a continuous charge distribution develops, which crosses the positive real z axis at a point x ∈ R+ . Then n+ − n− = Ex (x+ ) − Ex (x− ) = 4πσ(x) , x (6.134) where σ is the linear charge density (assuming logarithmic two-dimensional potentials), or the two-dimensional charge density (if we extend the distribution along a third axis). 6.4.3 Example As an example, consider the function (1 + z)M (1 − z M ) 1−z = (1 + z)M 1 + z + z 2 + . . . + z M−1 . Ξ(z) = (6.135) The (2M − 1) degree polynomial has an M th order zero at z = −1 and (M − 1) simple zeros at z = e2πik/M , where k ∈ {1, . . . , M −1}. Since M serves as the maximum particle number NV , we may assume that V = M v0 , and the V → ∞ limit may be taken as M → ∞. We then have 1 p = lim ln Ξ(z) V →∞ V kB T 1 1 lim = ln Ξ(z) v0 M→∞ M 1 1 M = M ln(1 + z) + ln 1 − z − ln(1 − z) . lim v0 M→∞ M The limit depends on whether |z| > 1 or |z| < 1, and we obtain if ln(1 + z) p v0 = h i kB T ln(1 + z) + ln z if |z| < 1 |z| > 1 . (6.136) (6.137) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 28 Figure 6.14: Fugacity z and pv0 /kB T versus dimensionless specific volume v/v0 for the example problem discussed in the text. Thus, ∂ n=z ∂z p kB T = If we solve for z(v), where v = n−1 , we find z= v 0 v−v0 v0 −v 2v−v0 1 v0 · 1 v0 · z 1+z h if |z| < 1 z 1+z +1 if v > 2v0 if 1 2 v0 i (6.138) if |z| > 1 . (6.139) <v< 2 3 v0 . We then obtain the equation of state, v ln v−v0 p v0 = ln 2 kB T ln v(v0 −v)2 (2v−v0 ) if v > 2v0 if 2 3 v0 < v < 2v0 if 1 2 v0 < v < 23 v0 . (6.140) 6.5 Liquid State Physics 6.5.1 The many-particle distribution function The virial expansion is typically applied to low-density systems. When the density is high, i.e. when na3 ∼ 1, where a is a typical molecular or atomic length scale, the virial expansion is impractical. There are to many terms to compute, and to make progress one must use sophisticated resummation techniques to investigate the high density regime. To elucidate the physics of liquids, it is useful to consider the properties of various correlation functions. These 6.5. LIQUID STATE PHYSICS 29 objects are derived from the general N -body Boltzmann distribution, −1 1 −β ĤN (p,x) ZN · N ! e f (x1 , . . . , xN ; p1 , . . . , pN ) = −1 1 βµN −β ĤN (p,x) Ξ · N! e e OCE (6.141) GCE . We assume a Hamiltonian of the form ĤN = N X p2i + W (x1 , . . . , xN ). 2m i=1 (6.142) The quantity ddxN ddpN ddx1 ddp1 · · · (6.143) hd hd is the propability of finding N particles in the system, with particle #1 lying within d3x1 of x1 and having momentum within ddp1 of p1 , etc. If we compute averages of quantities which only depend on the positions {xj } and not on the momenta {pj }, then we may integrate out the momenta to obtain, in the OCE, f (x1 , . . . , xN ; p1 , . . . , pN ) P (x1 , . . . , xN ) = Q−1 N · 1 −βW (x1 , ... , xN ) e , N! where W is the total potential energy, X X X u(xi − xj ) + w(xi − xj , xj − xk ) + . . . , v(xi ) + W (x1 , . . . , xN ) = i<j i (6.144) (6.145) i<j<k and QN is the configuration integral, 1 QN (T, V ) = N! Z Z d x1 · · · ddxN e−βW (x1 , ... , xN ) . d (6.146) We will, for the most part, consider only two-body central potentials as contributing to W , which is to say we will only retain the middle term on the RHS. Note that P (x1 , . . . , xN ) is invariant under any permutation of the particle labels. 6.5.2 Averages over the distribution To compute an average, one integrates over the distribution: Z Z F (x1 , . . . , xN ) = ddx1 · · · ddxN P (x1 , . . . , xN ) F (x1 , . . . , xN ) . The overall N -particle probability density is normalized according to Z ddxN P (x1 , . . . , xN ) = 1 . (6.147) (6.148) The average local density is n1 (r) = X Zi δ(r − xi ) Z = N d x2 · · · ddxN P (r, x2 , . . . , xN ) . d (6.149) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 30 Note that the local density obeys the sum rule Z ddr n1 (r) = N . (6.150) In a translationally invariant system, n1 = n = N V is a constant independent of position. The boundaries of a system will in general break translational invariance, so in order to maintain the notion of a translationally invariant system of finite total volume, one must impose periodic boundary conditions. The two-particle density matrix n2 (r1 , r2 ) is defined by X n2 (r1 , r2 ) = δ(r1 − xi ) δ(r2 − xj ) i6=j Z Z = N (N − 1) ddx3 · · · ddxN P (r1 , r2 , x3 , . . . , xN ) . (6.151) As in the case of the one-particle density matrix, i.e. the local density n1 (r), the two-particle density matrix satisfies a sum rule: Z Z d d r1 ddr2 n2 (r1 , r2 ) = N (N − 1) . (6.152) Generalizing further, one defines the k-particle density matrix as X′ nk (r1 , . . . , rk ) = δ(r1 − xi1 ) · · · δ(rk − xik ) i1 ···ik Z Z N! ddxk+1 · · · ddxN P (r1 , . . . , rk , xk+1 , . . . , xN ) , = (N − k)! (6.153) where the prime on the sum indicates that all the indices i1 , . . . , ik are distinct. The corresponding sum rule is then Z Z N! d d r1 · · · ddrk nk (r1 , . . . , rk ) = . (6.154) (N − k)! The average potential energy can be expressed in terms of the distribution functions. Assuming only two-body interactions, we have X u(xi − xj ) hW i = i<j = = 1 2 1 2 Z Z Z X ddr1 ddr2 u(r1 − r2 ) δ(r1 − xi ) δ(r2 − xj ) (6.155) i6=j Z ddr1 ddr2 u(r1 − r2 ) n2 (r1 , r2 ) . As the separations rij = |ri − rj | get large, we expect the correlations to vanish, in which case X′ nk (r1 , . . . , rk ) = δ(r1 − xi1 ) · · · δ(rk − xik ) i1 ···ik −−−−−→ rij →∞ X ′ δ(r1 − xi1 ) · · · δ(rk − xik ) i1 ···ik 1 N! · n (r ) · · · n1 (rk ) = (N − k)! N k 1 1 2 k−1 1 1− ··· 1 − n1 (r1 ) · · · n1 (rk ) . = 1− N N N (6.156) 6.5. LIQUID STATE PHYSICS 31 The k-particle distribution function is defined as the ratio gk (r1 , . . . , rk ) ≡ nk (r1 , . . . , rk ) . n1 (r1 ) · · · n1 (rk ) For large separations, then, gk (r1 , . . . , rk ) −−−−−→ rij →∞ k−1 Y j=1 1− j N (6.157) . (6.158) For isotropic systems, the two-particle distribution function g2 (r1 , r2 ) depends only on the magnitude |r1 − r2 |. As a function of this scalar separation, the function is known as the radial distribution function: g(r) ≡ g2 (r) = 1 X δ(r − xi ) δ(xj ) 2 n i6=j 1 X δ(r − xi + xj ) . = 2 Vn (6.159) i6=j The radial distribution function is of great importance in the physics of liquids because • thermodynamic properties of the system can be related to g(r) • g(r) is directly measurable by scattering experiments For example, in an isotropic system the average potential energy is given by Z Z d 1 hW i = 2 d r1 ddr2 u(r1 − r2 ) n2 (r1 , r2 ) Z Z = 12 n2 ddr1 ddr2 u(r1 − r2 ) g |r1 − r2 | Z N2 = ddr u(r) g(r) . 2V (6.160) For a three-dimensional system, the average internal (i.e. potential) energy per particle is Z∞ hW i = 2πn dr r2 g(r) u(r) . N (6.161) 0 Intuitively, f (r) dr ≡ 4πr2 n g(r) dr is the average number of particles lying at a radial distance between r and r + dr from a given reference particle. The total potential energy of interaction with the reference particle is then f (r) u(r) dr. Now integrate over all r and divide by two to avoid double-counting. This recovers eqn. 6.161. In the OCE, g(r) obeys the sum rule Z hence ddr g(r) = Z V V · N (N − 1) = V − , N2 N n ddr g(r) − 1 = −1 The function h(r) ≡ g(r) − 1 is called the pair correlation function. (OCE) . (6.162) (6.163) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 32 Figure 6.15: Pair distribution functions for hard spheres of diameter a at filling fraction η = π6 a3 n = 0.49 (left) and for liquid Argon at T = 85 K (right). Molecular dynamics data for hard spheres (points) is compared with the result of the Percus-Yevick approximation (see below in §6.5.8). Reproduced (without permission) from J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, fig 5.5. Experimental data on liquid argon are from the neutron scattering work of J. L. Yarnell et al., Phys. Rev. A 7, 2130 (1973). The data (points) are compared with molecular dynamics calculations by Verlet (1967) for a Lennard-Jones fluid. In the grand canonical formulation, we have # " Z N N (N − 1) 3 n d r h(r) = V −V · V hN i2 2 2 N − N = −1 N = nkB T κT − 1 (6.164) (GCE) , where κT is the isothermal compressibility. Note that in an ideal gas we have h(r) = 0 and κT = κ0T ≡ 1/nkB T . Self-condensed systems, such as R liquids and solids far from criticality, are nearly incompressible, hence 0 < nkB T κT ≪ 1, and therefore n d3r h(r) ≈ −1. For incompressible systems, where κT = 0, this becomes an equality. As we shall see below in §6.5.4, the function h(r), or rather its Fourier transform ĥ(k), is directly measured in a scattering experiment. The question then arises as to which result applies: the OCE result from eqn. 6.163 or the GCE result from eqn. 6.164. The answer is that under almost all experimental conditions it is the GCE result which applies. The reason for this is that the scattering experiment typically illuminates only a subset of the entire system. This subsystem is in particle equilibrium with the remainder of the system, hence it is appropriate to use the grand canonical ensemble. The OCE results would only apply if the scattering experiment were to measure the entire system. 6.5. LIQUID STATE PHYSICS 33 Figure 6.16: Monte Carlo pair distribution functions for liquid water. From A. K. Soper, Chem Phys. 202, 295 (1996). 6.5.3 Virial equation of state The virial of a mechanical system is defined to be G= X i xi · Fi , (6.165) where Fi is the total force acting on particle i. If we average G over time, we obtain 1 T →∞ T hGi = lim ZT X dt xi · Fi i 0 1 = − lim T →∞ T ZT X m ẋ2i dt 0 (6.166) i = −3N kB T . Here, we have made use of d m xi · ẋi , (6.167) dt as well as ergodicity and equipartition of kinetic energy. We have also assumed three space dimensions. In a bounded system, there are two contributions to the force Fi . One contribution is from the surfaces which enclose the system. This is given by8 X (surf) = −3pV . (6.168) xi · Fi hGisurfaces = xi · Fi = m xi · ẍi = − m ẋ2i + i 8 To derive this expression, note that F (surf) (surf) is directed inward and vanishes away from the surface. Each Cartesian direction α = (x, y, z) (surf) then contributes −Fα Lα , where Lα is the corresponding linear dimension. But Fα = pAα , where Aα is the area of the corresponding face and p. is the pressure. Summing over the three possibilities for α, one obtains eqn. 6.168. CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 34 The remaining contribution is due to the interparticle forces. Thus, X p N 1 xi · ∇i W . = − kB T V 3V kB T i Invoking the definition of g(r), we have 2πn p = nkB T 1 − 3kB T Z∞ dr r3 g(r) u′ (r) . (6.169) (6.170) 0 As an alternate derivation, consider the First Law of Thermodynamics, dΩ = −S dT − p dV − N dµ , from which we derive p=− ∂Ω ∂V T,µ =− Now let V → ℓ3 V , where ℓ is a scale parameter. Then ∂Ω 1 ∂ p=− =− ∂V 3V ∂ℓ Now Ξ(T, ℓ3 V, µ) = ∞ X N =0 Thus, 1 ∂Ω(ℓ3 V ) p=− 3V ∂ℓ ∂F ∂V . (6.172) T,N Ω(T, ℓ3 V, µ) . (6.173) ℓ=1 Z Z ∞ X 1 βµN −3N d3x1 · · · d3xN e−βW (x1 , ... , xN ) e λT N! N =0 = (6.171) ℓ3 V ℓ3 V Z Z 1 βµ −3 N 3N 3 e λT ℓ d x1 · · · d3xN e−βW (ℓx1 , ... , ℓxN ) N! V (6.174) V kB T 1 ∂Ξ(ℓ3 V ) 3V Ξ ∂ℓ ℓ=1 # " Z ∞ Z X ∂W kB T 1 X 1 N 3 3 −βW (x1 , ... , xN ) zλ−3 x · d x · · · d x e 3N − β = i 1 N T 3V Ξ N! ∂xi i = N =0 V 1 D ∂W E = nkB T − . 3V ∂ℓ ℓ=1 P Finally, from W = i<j u(ℓxij ) we have D ∂W E ∂ℓ V ℓ=1 = X i<j (6.175) xij · ∇u(xij ) Z∞ 2πN 2 = dr r3 g(r) u′ (r) , V (6.176) 0 and hence p = nkB T − 2 2 3 πn Z∞ dr r3 g(r) u′ (r) . (6.177) 0 Note that the density n enters the equation of state explicitly on the RHS of the above equation, but also implicitly through the pair distribution function g(r), which has implicit dependence on both n and T . 6.5. LIQUID STATE PHYSICS 35 Figure 6.17: In a scattering experiment, a beam of particles interacts with a sample and the beam particles scatter off the sample particles. A momentum ~q and energy ~ω are transferred to the beam particle during such a collision. If ω = 0, the scattering is said to be elastic. For ω 6= 0, the scattering is inelastic. 6.5.4 Correlations and scattering Consider the scattering of a light or particle beam (i.e. photons or neutrons) from a liquid. We label the states of the beam particles by their wavevector k and we assume a general dispersion εk . For photons, εk = ~c|k|, while 2 2 for neutrons εk = ~2mk . We assume a single scattering process with the liquid, during which the total momentum n and energy of the liquid plus beam are conserved. We write k′ = k + q εk′ = εk + ~ω , (6.178) (6.179) where k′ is the final state of the scattered beam particle. Thus, the fluid transfers momentum ∆p = ~q and energy ~ω to the beam. Now consider the scattering process between an initial state | i, k i and a final state | j, k′ i, where these states describe both the beam and the liquid. According to Fermi’s Golden Rule, the scattering rate is 2 2π h j, k′ | V | i, k i δ(Ej − Ei + ~ω) , (6.180) ~ where V is the scattering potential and Ei is the initial internal energy of the liquid. If r is the position of the beam particle and {xl } are the positions of the liquid particles, then Γik→j k′ = V(r) = N X l=1 v(r − xl ) . (6.181) The differential scattering cross section (per unit frequency per unit solid angle) is ~ g(εk′ ) X ∂ 2σ P Γ = ′ , ∂Ω ∂ω 4π |vk | i,j i ik→j k where g(ε) = Z ddk δ(ε − εk ) (2π)d (6.182) (6.183) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 36 is the density of states for the beam particle and Pi = 1 −βEi e . Z (6.184) Consider now the matrix element N Z 1 X ′ ddrei(k−k )·r v(r − xl ) i j, k′ V i, k = j V l=1 N X 1 = v̂(q) j e−iq·xl i , V (6.185) l=1 where we have assumed that the incident and scattered beams are plane waves. We then have N X X 2 ~ g(εk+q ) |v̂(q)|2 X ∂ 2σ j = e−iq·xl i δ(Ej − Ei + ~ω) P i 2 ∂Ω ∂ω 2 |∇ ε | V i j l=1 k k g(εk+q ) N |v̂(q)|2 S(q, ω) , = 2 V 4π |∇ ε | (6.186) k k where S(q, ω) is the dynamic structure factor, S(q, ω) = N X X 2 2π~ X j e−iq·xl i δ(Ej − Ei + ~ω) Pi N i j (6.187) l=1 Note that for an arbitrary operator A, ∞ X 2 XZ j A i δ(E − E + ~ω) = 1 j Ai dt ei(Ej −Ei +~ω) t/~ i A† j j i 2π~ j j = −∞ Z∞ 1 X 2π~ j iĤt/~ dt eiωt i A† j j e A e−iĤt/~ i −∞ Z∞ 1 dt eiωt i A† (0) A(t) i . = 2π~ (6.188) −∞ Thus, 1 S(q, ω) = N Z∞ X X Pi i eiq·xl (0) e−iq·xl′ (t) i dt eiωt −∞ Z∞ 1 = N dt e −∞ i X iωt l,l′ l,l′ eiq·xl (0) e−iq·xl′ (t) (6.189) , where the angular brackets in the last line denote a thermal expectation value of a quantum mechanical operator. If we integrate over all frequencies, we obtain the equal time correlator, Z∞ 1 X iq·(xl −x ′ ) dω l S(q, ω) = e 2π N ′ l,l −∞ Z = N δq,0 + 1 + n ddr e−iq·r g(r) − 1 . S(q) = (6.190) 6.5. LIQUID STATE PHYSICS 37 Figure 6.18: Comparison of the static structure factor as determined by neutron scattering work of J. L. Yarnell et al., Phys. Rev. A 7, 2130 (1973) with molecular dynamics calculations by Verlet (1967) for a Lennard-Jones fluid. known as the static structure factor9 . Note that S(q = 0) = N , since all the phases eiq·(xi −xj ) are then unity. As q → ∞, the phases oscillate rapidly with changes in the distances |xi − xj |, and average out to zero. However, the ‘diagonal’ terms in the sum, i.e. those with i = j, always contribute a total of 1 to S(q). Therefore in the q → ∞ limit we have S(q → ∞) = 1. In general, the detectors used in a scattering experiment are sensitive to the energy of the scattered beam particles, although there is always a finite experimental resolution, both in q and ω. This means that what is measured is actually something like Z Z Smeas (q, ω) = ddq ′ dω ′ F (q − q ′ ) G(ω − ω ′ ) S(q ′ , ω ′ ) , (6.191) where F and G are essentially Gaussian functions of their argument, with width given by the experimental resolution. If one integrates over all frequencies ω, i.e. if one simply counts scattered particles as a function of q but without any discrimination of their energies, then one measures the static structure factor S(q). Elastic scattering is determined by S(q, ω = 0, i.e. no energy transfer. 6.5.5 Correlation and response Suppose an external potential v(x) is also present. Then P (x1 , . . . , xN ) = where QN [v] = 9 We may write δq ,0 = 1 V (2π)d δ(q ). 1 1 −βW (x1 , ... , xN ) −β Pi v(xi ) , · e e QN [v] N ! Z Z P 1 ddx1 · · · ddxN e−βW (x1 , ... , xN ) e−β i v(xi ) . N! (6.192) (6.193) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 38 The Helmholtz free energy is then 1 ln λ−dN Q [v] . N T β (6.194) δQN δF 1 1 =− · · . δv(r) β QN δv(r) (6.195) F =− Now consider the functional derivative Using X v(xi ) = i hence δF = δv(r) Z Z X ddr v(r) i δ(r − xi ) , Z X δ(r − xi ) ddx1 · · · ddxN P (x1 , . . . , xN ) (6.196) i = n1 (r) , (6.197) which is the local density at r. Next, consider the response function, χ(r, r ′ ) ≡ = δn1 (r) δ 2F [v] = δv(r ′ ) δv(r) δv(r ′ ) δ 2QN 1 δQN δQN 1 1 1 · 2 − · β QN δv(r) δv(r ′ ) β QN δv(r) δv(r ′ ) (6.198) = β n1 (r) n1 (r ′ ) − β n1 (r) δ(r − r ′ ) − β n2 (r, r ′ ) . In an isotropic system, χ(r, r ′ ) = χ(r − r ′ ) is a function of the coordinate separation, and −kB T χ(r − r ′ ) = −n2 + n δ(r − r ′ ) + n2 g |r − r ′ | (6.199) −kB T χ̂(q) = n + n2 ĥ(q) (6.200) κT = 1 + n ĥ(0) = −nkB T χ̂(0) , κ0T (6.201) = n h |r − r | + n δ(r − r ′ ) . 2 Taking the Fourier transform, ′ = n S(q) . We may also write i.e. κT = −χ̂(0). What does this all mean? Suppose we have an isotropic system which is subjected to a weak, spatially inhomogeneous potential v(r). We expect that the density n(r) in the presence of the inhomogeneous potential to itself be inhomogeneous. The first corrections to the v = 0 value n = n0 are linear in v, and given by Z δn(r) = ddr′ χ(r, r ′ ) v(r ′ ) Z (6.202) 2 d ′ ′ = −βn0 v(r) − βn0 d r h(r − r) v(r ) . 6.5. LIQUID STATE PHYSICS 39 Note that if v(r) > 0 it becomes energetically more costly for a particle to be at r. Accordingly, the density response is negative, and proportional to the ratio v(r)/kB T – this is the first term in the above equation. If there were no correlations between the particles, then h = 0 and this would be the entire story. However, the particles in general are correlated. Consider, for example, the case of hard spheres of diameter a, and let there be a repulsive potential at r = 0. This means that it is less likely for a particle to be centered anywhere within a distance a of the origin. But then it will be more likely to find a particle in the next ‘shell’ of radial thickness a. 6.5.6 BBGKY hierarchy The distribution functions satisfy a hierarchy of integro-differential equations known as the BBGKY hierarchy10 . In homogeneous systems, we have N! 1 gk (r1 , . . . , rk ) = (N − k)! nk Z Z d xk+1 · · · ddxN P (r1 , . . . , rk , xk+1 , . . . , xN ) , d (6.203) where P (x1 , . . . , xN ) = 1 −βW (x1 , ... , xN ) 1 · e . QN N ! (6.204) Taking the gradient with respect to r1 , we have Z Z P 1 n−k ∂ d gk (r1 , . . . , rk ) = · d xk+1 · · · ddxN e−β k<i<j u(xij ) ∂r1 QN (N − k)! P P ∂ −β i≤k<j u(ri −xj ) −β i<j≤k u(rij ) , ·e e × ∂r1 where P k<i<j (6.205) means to sum on indices i and j such that i < j and k < i, i.e. X k<i<j X i<j≤k X i≤k<j u(xij ) ≡ u(rij ) ≡ u(ri − xj ) = N −1 X N X i=k+1 j=i+1 k−1 X k X i=1 j=i+1 k N X X i=1 j=k+1 u xi − xj u ri − rj u(ri − xj ) . Now P P ∂ e−β i<j≤k u(rij ) · e−β i≤k<j u(ri −xj ) = ∂r1 ) ( P P X ∂u(r1 − rj ) X ∂u(r1 − rj ) −β i<j≤k u(rij ) −β i≤k<j u(ri −xj ) , · e ·e + β ∂r1 ∂r1 1<j≤k 10 So k<j named after Bogoliubov, Born, Green, Kirkwood, and Yvon. (6.206) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 40 hence k X ∂u(r1 − rj ) ∂ gk (r1 , . . . , rk ) = −β gk (r1 , . . . , rk ) ∂r1 ∂r1 j=2 Z ∂u(r1 − xk+1 ) − β(N − k) ddxk+1 P (r1 , . . . , rk , xk+1 , . . . , xN ) ∂r1 = −β k X ∂u(r1 − rj ) j=2 ∂r1 gk (r1 , . . . , rk ) (6.207) (6.208) Z ∂u(r1 − xk+1 ) gk+1 (r1 , . . . , rk , xk+1 ) + n ddxk+1 ∂r1 Thus, we obtain the BBGKY hierarchy: k −kB T X ∂u(r1 − rj ) ∂ gk (r1 , . . . , rk ) = gk (r1 , . . . , rk ) ∂r1 ∂r1 j=2 Z ∂u(r1 − r ′ ) gk+1 (r1 , . . . , rk , r ′ ) . + n ddr′ ∂r1 (6.209) The BBGKY hierarchy is an infinite tower of coupled integro-differential equations, relating gk to gk+1 for all k. If we approximate gk at some level k in terms of equal or lower order distributions, then we obtain a closed set of equations which in principle can be solved, at least numerically. For example, the Kirkwood approximation closes the hierarchy at order k = 2 by imposing the condition g3 (r1 , r2 , r3 ) ≡ g(r1 − r2 ) g(r1 − r3 ) g(r2 − r2 ) . This results in the single integro-differential equation Z −kB T ∇g(r) = g(r) ∇u + n ddr′ g(r) g(r ′ ) g(r − r ′ ) ∇u(r − r ′ ) . (6.210) (6.211) This is known as the Born-Green-Yvon (BGY) equation. In practice, the BGY equation, which is solved numerically, gives adequate results only at low densities. 6.5.7 Ornstein-Zernike theory The direct correlation function c(r) is defined by the equation Z h(r) = c(r) + n d3r′ h(r − r ′ ) c(r ′ ) , (6.212) where h(r) = g(r) − 1 and we assume an isotropic system. This is called the Ornstein-Zernike equation. The first term, c(r), accounts for local correlations, which are then propagated in the second term to account for longranged correlations. The OZ equation is an integral equation, but it becomes a simple algebraic one upon Fourier transforming: ĥ(q) = ĉ(q) + n ĥ(q) ĉ(q) , the solution of which is ĥ(q) = ĉ(q) . 1 − n ĉ(q) (6.213) (6.214) 6.5. LIQUID STATE PHYSICS 41 The static structure factor is then S(q) = 1 + n ĥ(q) = In the grand canonical ensemble, we can write κT = 1 . 1 − n ĉ(q) 1 1 1 + n ĥ(0) = · nkB T nkB T 1 − n ĉ(0) =⇒ (6.215) n ĉ(0) = 1 − κ0T , κT (6.216) where κ0T = 1/nkB T is the ideal gas isothermal compressibility. At this point, we have merely substituted one unknown function, h(r), for another, namely c(r). To close the system, we need to relate c(r) to h(r) again in some way. There are various approximation schemes which do just this. 6.5.8 Percus-Yevick equation In the Percus-Yevick approximation, we take c(r) = 1 − eβu(r) · g(r) . (6.217) Note that c(r) vanishes whenever the potential u(r) itself vanishes. This results in the following integro-differential equation for the pair distribution function g(r): Z ′ g(r) = e−βu(r) + n e−βu(r) d3r′ g(r − r ′ ) − 1 · 1 − eβu(r ) g(r ′ ) . (6.218) This is the Percus-Yevick equation. Remarkably, the Percus-Yevick (PY) equation can be solved analytically for the case of hard spheres, where u(r) = ∞ for r ≤ a and u(r) = 0 for r > a, where a is the hard sphere diameter. Define the function y(r) = eβu(r) g(r), in which case ( −y(r) , r ≤ a c(r) = y(r) f (r) = (6.219) 0 , r>a. Here, f (r) = e−βu(r) − 1 is the Mayer function. We remark that the definition of y(r) may cause some concern for the hard sphere system, because of the eβu(r) term, which diverges severely for r ≤ a. However, g(r) vanishes in this limit, and their product y(r) is in fact finite! The PY equation may then be written for the function y(r) as Z Z (6.220) y(r) = 1 + n d3r′ y(r ′ ) − n d3r′ y(r ′ ) y(r − r ′ ) . r ′ <a r′ <a |r −r ′ |>a This has been solved using Laplace transform methods by M. S. Wertheim, J. Math. Phys. 5, 643 (1964). The final result for c(r) is r 3 r c(r) = − λ1 + 6η λ2 + 21 η λ1 · Θ(a − r) , (6.221) a a where η = 16 πa3 n is the packing fraction and λ1 = (1 + 2η)2 (1 − η)4 , λ2 = − (1 + 21 η)2 . (1 − η)4 (6.222) This leads to the equation of state p = nkB T · 1 + η + η2 . (1 − η)3 (6.223) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 42 quantity B4 /B23 B5 /B24 B6 /B25 B7 /B26 exact 0.28695 0.1103 0.0386 0.0138 PY 0.2969 0.1211 0.0281 0.0156 HNC 0.2092 0.0493 0.0449 – Table 6.1: Comparison of exact (Monte Carlo) results to those of the Percus-Yevick (PY) and hypernetted chains approximation (HCA) for hard spheres in three dimensions. Sources: Hansen and McDonald (1990) and Reichl (1998) This gets B2 and B3 exactly right. The accuracy of the PY approximation for higher order virial coefficients is shown in table 6.1. To obtain the equation of state from eqn. 6.221, we invoke the compressibility equation, ∂n 1 nkB T κT = = . ∂p T 1 − n ĉ(0) We therefore need ĉ(0) = Z (6.224) d3r c(r) Z1 = −4πa dx x2 λ1 + 6 η λ2 x + 3 0 = −4πa3 1 3 λ1 + 3 2 η λ2 + 1 12 1 2 η λ1 x3 η λ1 . (6.225) With η = 61 πa3 n and using the definitions of λ1,2 in eqn. 6.222, one finds 1 + 4η + 4η 2 . (1 − η)4 (6.226) 6kB T ∂p 1 + 4η + 4η 2 . = πa3 ∂η (1 − η)4 (6.227) 1 − n ĉ(0) = We then have, from the compressibility equation, Integrating, we obtain p(η) up to a constant. The constant is set so that p = 0 when n = 0. The result is eqn. 6.223. Another commonly used scheme is the hypernetted chains (HNC) approximation, for which c(r) = −βu(r) + h(r) − ln 1 + h(r) . (6.228) The rationale behind the HNC and other such approximation schemes is rooted in diagrammatic approaches, which are extensions of the Mayer cluster expansion to the computation of correlation functions. For details and references to their application in the literature, see Hansen and McDonald (1990) and Reichl (1998). 6.5.9 Ornstein-Zernike approximation at long wavelengths Let’s expand the direct correlation function ĉ(q) in powers of the wavevector q, viz. ĉ(q) = ĉ(0) + c2 q 2 + c4 q 4 + . . . . (6.229) 6.5. LIQUID STATE PHYSICS 43 Here we have assumed spatial isotropy. Then 1 − n ĉ(q) = 1 = 1 − n ĉ(0) − n c2 q 2 + . . . S(q) (6.230) ≡ ξ −2 R2 + q 2 R2 + O(q 4 ) , where Z∞ R = −n c2 = 2πn dr r4 c(r) 2 (6.231) 0 and ξ −2 R∞ 1 − 4πn 0 dr r2 c(r) 1 − n ĉ(0) R∞ = . = R2 2πn 0 dr r4 c(r) (6.232) The quantity R(T ) tells us something about the effective range of the interactions, while ξ(T ) is the correlation length. As we approach a critical point, the correlation length diverges as a power law: ξ(T ) ∼ A|T − Tc |−ν . (6.233) The susceptibility is given by χ̂(q) = −nβ S(q) = − nβR−2 ξ −2 + q 2 + O(q 4 ) (6.234) In the Ornstein-Zernike approximation, one drops the O(q 4 ) terms in the denominator and retains only the long wavelength behavior. in the direct correlation function. Thus, χ̂OZ (q) = − nβR−2 . ξ −2 + q 2 (6.235) We now apply the inverse Fourier transform back to real space to obtain χOZ (r). In d = 1 dimension the result can be obtained exactly: χOZ d=1 (x) = − n kB T R2 Z∞ −∞ eiqx dq −2 2π ξ + q 2 (6.236) nξ e−|x|/ξ . =− 2kB T R2 In higher dimensions d > 1 we can obtain the result asymptotically in two limits: • Take r → ∞ with ξ fixed. Then χOZ d (r) ≃ −Cd n · ξ (3−d)/2 e−r/ξ d−3 , · · 1 + O kB T R2 r(d−1)/2 r/ξ (6.237) where the Cd are dimensionless constants. • Take ξ → ∞ with r fixed; this is the limit T → Tc at fixed r. In dimensions d > 2 we obtain Cd′ n e−r/ξ d−3 χOZ (r) ≃ − . · · 1 + O d kB T R2 rd−2 r/ξ In d = 2 dimensions we obtain χOZ d=2 (r) ≃ − 1 r C2′ n −r/ξ e · 1 + O , · ln kB T R2 ξ ln(r/ξ) where the Cd′ are dimensionless constants. (6.238) (6.239) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 44 At criticality, ξ → ∞, and clearly our results in d = 1 and d = 2 dimensions are nonsensical, as they are divergent. To correct this behavior, M. E. Fisher in 1963 suggested that the OZ correlation functions in the r ≪ ξ limit be replaced by ξη e−r/ξ χ(r) ≃ −Cd′′ n · · , (6.240) kB T R2 rd−2+η a result known as anomalous scaling. Here, η is the anomalous scaling exponent. Recall that the isothermal compressibility is given by κT = −χ̂(0). Near criticality, the integral in χ̂(0) is dominated by the r ≪ ξ part, since ξ → ∞. Thus, using Fisher’s anomalous scaling, Z κT = −χ̂(0) = − ddr χ(r) (6.241) Z −(2−η)ν e−r/ξ , ∼ A ddr d−2+η ∼ B ξ 2−η ∼ C T − Tc r where A, B, and C are temperature-dependent constants which are nonsingular at T = Tc . Thus, since κT ∝ |T − Tc |−γ , we conclude γ = (2 − η) ν , (6.242) a result known as hyperscaling. 6.6 Coulomb Systems : Plasmas and the Electron Gas 6.6.1 Electrostatic potential Coulomb systems are particularly interesting in statistical mechanics because of their long-ranged forces, which result in the phenomenon of screening. Long-ranged forces wreak havoc with the Mayer cluster expansion, since the Mayer function is no longer integrable. Thus, the virial expansion fails, and new techniques need to be applied to reveal the physics of plasmas. The potential energy of a Coulomb system is Z Z U = 21 ddr ddr′ ρ(r) u(r − r ′ ) ρ(r ′ ) , (6.243) where ρ(r) is the charge density and u(r), which has the dimensions of (energy)/(charge)2, satisfies ∇2 u(r − r ′ ) = −4π δ(r − r ′ ) . Thus, −2π |x − x′ | , d=1 u(r) = −2 ln |r − r ′ | , d = 2 |r − r ′ |−1 , d=3. For discete particles, the charge density ρ(r) is given by X qi δ(r − xi ) , ρ(r) = i (6.244) (6.245) (6.246) 6.6. COULOMB SYSTEMS : PLASMAS AND THE ELECTRON GAS 45 where qi is the charge of the ith particle. We will assume two types of charges: q = ±e, with e > 0. The electric potential is Z φ(r) = ddr′ u(r − r ′ ) ρ(r ′ ) X (6.247) qi u(r − xi ) . = i This satisfies the Poisson equation, ∇2 φ(r) = −4πρ(r) . (6.248) The total potential energy can be written as Z ddr φ(r) ρ(r) X = 21 qi φ(xi ) , U= 1 2 (6.249) (6.250) i 6.6.2 Debye-Hückel theory We now write the grand partition function: Ξ(T, V, µ+ , µ− ) = ∞ ∞ X X 1 βµ+ N+ −N+ d 1 βµ− N− −N− d e λ+ e λ− · N+ ! N− ! N+ =0 N− =0 Z Z −βU(r1 , ... , rN +N ) d + − . · d r1 · · · ddrN+ +N− e (6.251) We now adopt a mean field approach, known as Debye-Hückel theory, writing ρ(r) = ρav (r) + δρ(r) (6.252) av φ(r) = φ (r) + δφ(r) . (6.253) We then have U= 1 2 Z ddr ρav (r) + δρ(r) · φav (r) + δφ(r) ≡ U0 ignore fluctuation term }| { Z }| { z Z zZ d av av d av d 1 1 = − 2 d r ρ (r) φ (r) + d r φ (r) ρ(r)+ 2 d r δρ(r) δφ(r) . We apply the mean field approximation in each region of space, which leads to Z e φav (r) d Ω(T, V, µ+ , µ− ) = −kB T λ−d z d r exp − + + kB T Z e φav (r) d − kB T λ−d z d r exp + , − − kB T where λ± = 2π~2 m± kB T , µ± z± = exp . kB T (6.254) (6.255) (6.256) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 46 The charge density is therefore ρ(r) = δΩ e φ(r) e φ(r) −d −d − e λ z exp + , = e λ z exp − − − + + δφav (r) kB T kB T (6.257) where we have now dropped the superscript on φav (r) for convenience. At r → ∞, we assume charge neutrality and φ(∞) = 0. Thus −d λ+ z+ = n+ (∞) = λ−d (6.258) − z− = n− (∞) ≡ n∞ , where n∞ is the ionic density of either species at infinity. Therefore, e φ(r) ρ(r) = −2e n∞ sinh . kB T (6.259) We now invoke Poisson’s equation, ∇2 φ = 8πen∞ sinh(βeφ) − 4πρext , (6.260) where ρext is an externally imposed charge density. If eφ ≪ kB T , we can expand the sinh function and obtain ∇2 φ = κ2D φ − 4πρext , where κD = 8πn∞ e2 kB T 1/2 , λD = (6.261) kB T 8πn∞ e2 1/2 . (6.262) The quantity λD is known as the Debye screening length. Consider, for example, a point charge Q located at the origin. We then solve Poisson’s equation in the weak field limit, ∇2 φ = κ2D φ − 4πQ δ(r) . (6.263) Fourier transforming, we obtain −q 2 φ̂(q) = κ2D φ̂(q) − 4πQ =⇒ φ̂(q) = 4πQ . q 2 + κ2D Transforming back to real space, we obtain, in three dimensions, the Yukawa potential, Z 3 d q 4πQ eiq·r Q −κD r φ(r) = = ·e . (2π)3 q 2 + κ2D r (6.264) (6.265) This solution must break down sufficiently close to r = 0, since the assumption eφ(r) ≪ kB T is no longer valid there. However, for larger r, the Yukawa form is increasingly accurate. For another example, consider an electrolyte held between two conducting plates, one at potential φ(x = 0) = 0 and the other at potential φ(x = L) = V , where x̂ is normal to the plane of the plates. Again assuming a weak field eφ ≪ kB T , we solve ∇2 φ = κ2D φ and obtain φ(x) = A eκD x + B e−κD x . (6.266) We fix the constants A and B by invoking the boundary conditions, which results in φ(x) = V · sinh(κD x) . sinh(κD L) (6.267) Debye-Hückel theory is valid provided n∞ λ3D ≫ 1, so that the statistical assumption of many charges in a screening volume is justified. 6.6. COULOMB SYSTEMS : PLASMAS AND THE ELECTRON GAS 47 6.6.3 The electron gas : Thomas-Fermi screening Assuming kB T ≪ εF , thermal fluctuations are unimportant and we may assume T = 0. In the same spirit as the Debye-Hückel approach, we assume a slowly varying mean electrostatic potential φ(r). Locally, we can write εF = ~2 kF2 − eφ(r) . 2m (6.268) Thus, the Fermi wavevector kF is spatially varying, according to the relation 1/2 2m kF (r) = ε + eφ(r) . ~2 F (6.269) The local electron number density is 3/2 kF3 (r) eφ(r) . n(r) = = n∞ 1 + 3π 2 εF (6.270) In the presence of a uniform compensating positive background charge ρ+ = en∞ , Poisson’s equation takes the form # " 3/2 eφ(r) − 1 − 4πρext (r) . (6.271) 1+ ∇2 φ = 4πe n∞ · εF If eφ ≪ εF , we may expand in powers of the ratio, obtaining ∇2 φ = 6πn∞ e2 φ ≡ κ2TF φ − 4πρext (r) . εF (6.272) Here, κTF is the Thomas-Fermi wavevector, κTF = 6πn∞ e2 εF 1/2 . (6.273) Thomas-Fermi theory is valid provided n∞ λ3TF ≫ 1, where λTF = κ−1 TF , so that the statistical assumption of many electrons in a screening volume is justified. One important application of Thomas-Fermi screening is to the theory of metals. In a metal, the outer, valence electrons of each atom are stripped away from the positively charged ionic core and enter into itinerant, planewave-like states. These states disperse with some ε(k) function (that is periodic in the Brillouin zone, i.e. under k → k + G, where G is a reciprocal lattice vector), and at T = 0 this energy band is filled up to the Fermi level εF , as Fermi statistics dictates. (In some cases, there may be several bands at the Fermi level, as we saw in the case of yttrium.) The set of ionic cores then acts as a neutralizing positive background. In a perfect crystal, the ionic cores are distributed periodically, and the positive background is approximately uniform. A charged impurity in a metal, such as a zinc atom in a copper matrix, has a different nuclear charge and a different valency than the host. The charge of the ionic core, when valence electrons are stripped away, differs from that of the host ions, and therefore the impurity acts as a local charge impurity. For example, copper has an electronic configuration of [Ar] 3d10 4s1 . The 4s electron forms an energy band which contains the Fermi surface. Zinc has a configuration of [Ar] 3d10 4s2 , and in a Cu matrix the Zn gives up its two 4s electrons into the 4s conduction band, leaving behind a charge +2 ionic core. The Cu cores have charge +1 since each copper atom contributed only one 4s electron to the conduction band. The conduction band electrons neutralize the uniform positive background of the Cu ion cores. What is left is an extra Q = +e nuclear charge at the Zn site, and one extra 4s conduction band electron. The Q = +e impurity is, however, screened by the electrons, and at distances greater than an atomic radius the potential that a given electron sees due to the Zn core is of the Yukawa form, φ(r) = Q −κTF r ·e . r (6.274) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 48 We should take care, however, that the dispersion ε(k) for the conduction band in a metal is not necessarily of the free electron form ε(k) = ~2 k2 /2m. To linear order in the potential, however, the change in the local electronic density is δn(r) = eφ(r) g(εF ) , (6.275) where g(εF ) is the density of states at the Fermi energy. Thus, in a metal, we should write ∇2 φ = (−4π)(−e δn) = 4πe2 g(εF ) φ = κ2TF φ , where κTF = q 4πe2 g(εF ) . (6.276) (6.277) The value of g(εF ) will depend on the form of the dispersion. For ballistic bands with an effective mass m∗ , the formula in eqn. 6.272 still applies. The Thomas-Fermi atom Consider an ion formed of a nucleus of charge +Ze and an electron cloud of charge −N e. The net ionic charge is then (Z − N )e. Since we will be interested in atomic scales, we can no longer assume a weak field limit and we must retain the full nonlinear screening theory, for which ∇2 φ(r) = 4πe · 3/2 (2m)3/2 εF + eφ(r) − 4πZe δ(r) . 2 3 3π ~ (6.278) We assume an isotropic solution. It is then convenient to define εF + eφ(r) = Ze2 · χ(r/r0 ) , r (6.279) where r0 is yet to be determined. As r → 0 we expect χ → 1 since the nuclear charge is then unscreened. We then have 2 Ze 1 Ze2 ′′ 2 ∇ · χ(r/r0 ) = 2 χ (r/r0 ) , (6.280) r r0 r thus we arrive at the Thomas-Fermi equation, 1 χ′′ (t) = √ χ3/2 (t) , t with r = t r0 , provided we take ~2 r0 = 2me2 where aB = ~2 me2 3π √ 4 Z 2/3 = 0.885 Z −1/3 aB , (6.281) (6.282) = 0.529 Å is the Bohr radius. The TF equation is subject to the following boundary conditions: • At short distances, the nucleus is unscreened, i.e. χ(0) = 1 . (6.283) 6.7. POLYMERS 49 Figure 6.19: The Thomas-Fermi atom consists of a nuclear charge +Ze surrounded by N electrons distributed in a cloud. The electric potential φ(r) felt by any electron at position r is screened by the electrons within this radius, resulting in a self-consistent potential φ(r) = φ0 + (Ze2 /r) χ(r/r0 ). • For positive ions, with N < Z, there is perfect screening at the ionic boundary R = t∗ r0 , where χ(t∗ ) = 0. This requires (Z − N ) e Ze2 ′ Ze2 χ (R/r0 ) r̂ = r̂ . (6.284) E = −∇φ = − 2 χ(R/r0 ) + R R r0 R2 This requires −t∗ χ′ (t∗ ) = 1 − N . Z (6.285) For an atom, with N = Z, the asymptotic solution to the TF equation is a power law, and by inspection is found to be χ(t) ∼ C t−3 , where C is a constant. The constant follows from the TF equation, which yields 12 C = C 3/2 , hence C = 144. Thus, a neutral TF atom has a density with a power law tail, with ρ ∼ r−9/2 . TF ions with N > Z are unstable. 6.7 Polymers 6.7.1 Basic concepts Linear chain polymers are repeating structures with the chemical formula (A)x , where A is the formula unit and 5 x is the degree of polymerization. In many cases (e.g. polystyrene), x > ∼ 10 is not uncommon. For a very readable introduction to the subject, see P. G. de Gennes, Scaling Concepts in Polymer Physics. Quite often a given polymer solution will contain a distribution of x values; this is known as polydispersity. Various preparation techniques, such as chromatography, can mitigate the degree of polydispersity. Another morphological feature of polymers is branching (see fig. 6.21), in which the polymers do not form linear chains. Polymers exhibit a static flexibility which can be understood as follows. Consider a long chain hydrocarbon with a −C − C − C− backbone. The angle between successive C − C bonds is fixed at θ ≈ 68◦ , but the azimuthal angle 50 CHAPTER 6. CLASSICAL INTERACTING SYSTEMS Figure 6.20: Some examples of linear chain polymers. ϕ can take one of three possible low-energy values, as shown in the right panel of fig. 6.22. Thus, the relative probabilities of gauche and trans orientations are Prob (gauche) = 2 e−∆ε/kB T , Prob (trans) (6.286) where ∆ε is the energy difference between trans and gauche configurations. This means that the polymer chain is in fact a random coil with a persistence length ℓp = ℓ0 e∆ε/kB T , (6.287) where ℓ0 is a microscopic length scale, roughly given by the length of a formula unit, which is approximately a few Ångstroms (see fig. 6.23). Let L be the total length of the polymer when it is stretched into a straight line. If ℓp > L, the polymer is rigid. If ℓp ≪ L, the polymer is rigid on the length scale ℓp but flexible on longer scales. We have ℓp 1 ∆ε/kB T = e , (6.288) L N where we now use N (rather than x) for the degree of polymerization. In the time domain, the polymer exhibits a dynamical flexibility on scales longer than a persistence time. The persistence time τp is the time required for a trans-gauche transition. The rate for such transitions is set by the energy barrier B separating trans from gauche configurations: τp = τ0 eB/kB T , (6.289) where τ0 ∼ 10−11 s. On frequency scales ω ≪ τp−1 the polymer is dynamically flexible. If ∆ε ∼ kB T ≪ B the polymer is flexible from a static point of view, but dynamically rigid. That is, there are many gauche orientations of successive carbon bonds which reflect a quenched disorder. The polymer then forms a frozen random coil, like a twisted coat hanger. 6.7. POLYMERS 51 Figure 6.21: Branched polymers. 6.7.2 Polymers as random walks A polymer can be modeled by a self-avoiding random walk (SAW). That is, on scales longer than ℓp , it twists about randomly in space subject to the constraint that it doesn’t overlap itself. Before we consider the mathematics of SAWs, let’s first recall some aspects of ordinary random walks which are not self-avoiding. We’ll simplify matters further by considering random walks on a hypercubic lattice of dimension d. Such a lattice has coordination number 2d, i.e. there are 2d nearest neighbor separations, δ = ±a ê1 , ±a ê2 , . . . , ±a êd , (6.290) where a is the lattice spacing. Consider now a random walk of N steps starting at the origin. After N steps the position is N X δj , (6.291) RN = j=1 where δj takes on one of 2d possible values. Now N is no longer the degree of polymerization, but more like L/ℓp, which is the number of persistence lengths in the chain. We assume each step is independent, hence δj · δj ′ = a2 δjj ′ and R2N = N a2 . The full distribution PN (R) is given by PN (R) = (2d)−N X δ1 =a d Zπ/a dk1 ··· 2π −π/a ≈ X δ R, P j δj δN Zπ/a −π/a " d #N dkd −ik·R 1 X e cos(kµ a) 2π d µ=1 Z " # 1 2 2 ddk −ik·R e exp N ln 1 − k a + ... (2π)d 2d a 2d d Z = ad ··· Ω̂ d dke −N k2 a2 /2d −ik·R e = d 2πN d/2 e−dR (6.292) 2 /2N a2 . This is a simple Gaussian, with width R2 = d · (N a2 /d) = N a2 , as we have already computed. We define √ R0 = N a as the radius of gyration. The total number of random walk configurations with end-to-end vector R is then (2d)N PN (R), so the entropy of CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 52 Figure 6.22: Left” trans and gauche orientations in carbon chains. Right: energy as a function of azimuthal angle ϕ. There are three low energy states: trans (ϕ = 0) and gauche (ϕ = ±ϕ0 ). a chain at fixed elongation is h i dk R2 S(R, N ) = kB ln (2d)N PN (R) = S(0, N ) − B 2 . 2N a If we assume that the energy of the chain is conformation independent, then E = E0 (N ) and dkB T R2 . 2N a2 In the presence of an external force Fext , the Gibbs free energy is the Legendre transform F (R, N ) = F (0, N ) + G(Fext , N ) = F (R, N ) − Fext · R , (6.293) (6.294) (6.295) and ∂G/∂R = 0 then gives the relation N a2 R(Fext , N ) = F . dkB T ext (6.296) This may be considered an equation of state for the polymer. Following de Gennes, consider a chain with charges ±e at each end, placed in an external electric field of magnitude E = 30, 000 V/cm. Let N = 104 , a = 2 Å, and d = 3. What is the elongation? From the above formula, we have eER0 R = = 0.8 . (6.297) R0 3kB T Structure factor We can also compute the structure factor (see eqn. 6.190), N N 1 D X X iq·(Rm −Rn ) E . S(q) = e N m=1 n=1 (6.298) 6.7. POLYMERS 53 Figure 6.23: The polymer chain as a random coil. For averages with respect to a Gaussian distribution, We then have h 2 i iq·(R −R ) m n . = exp − 21 q · (Rm − Rn ) e N 1 X −|m−n|q2 a2 /2 S(q) = e = N f N q 2 a2 /2 , N m,n=1 (6.299) (6.300) where f (x) = x x2 2 −x e − 1 + x = 1 − + + ... . x2 3 12 (6.301) Self-avoiding walks On a lattice of coordination number z, the number of N -step random walks starting from the origin is ΩN = z N . If we constrain our random walks to be self-avoiding, the number is reduced to γ−1 N ΩSAW y , N = CN (6.302) where C and γ are dimension-dependent constants, and we expect y < ∼ z − 1, since at the very least a SAW cannot immediately double back on itself. In fact, on the cubic lattice one has z = 6 but y = 4.68, slightly less than z − 1. One finds γd=2 ≃ 43 and γd=3 ≃ 76 . The RMS end-to-end distance of the SAW is RF = a N ν , (6.303) where a and ν are d-dependent constants,with νd=1 = 1, νd=2 ≃ 34 , and νd=3 ≃ 53 . The distribution PN (R) has a scaling form, R 1 (a ≪ R ≪ N a) . (6.304) PN (R) = d f RF RF One finds f (x) ∼ with g = (γ − 1)/ν and δ = 1/(1 − ν). ( xg x≪1 δ exp(−x ) x ≫ 1 , (6.305) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 54 6.7.3 Flory theory What is missing from the random walk free energy is the effect of steric interactions. Suppose we have a chain of radius R. Then the average monomer density within the chain is c = N/Rd . Assuming short-ranged interactions, we should then add a term to the free energy which effectively counts the number of near self-intersections of the chain. This number should be roughly N c. Thus, we write F (R, N ) = F0 + u(T ) N2 R2 + 21 dkB T . d R N a2 (6.306) The effective interaction u(T ) is positive in the case of a so-called ‘good solvent’. The free energy is minimized when 0= ∂F R dvN 2 , = − d+1 + dkB T ∂R R N a2 which yields the result RF (N ) = ua2 kB T 1 d+2 3 N d+2 . (6.307) (6.308) Thus, we obtain ν = 3/(d + 2), and νd=3 = 35 , which is extremely close to the correct value ν = 0.5880 obtained from numerical work and renormalization group calculations. For a SAW under an external force, we compute the Gibbs partition function, Z Z d Fext ·R/kB T Y (Fext , N ) = d R PN (R) e = ddx f (x) esn̂·x , (6.309) where x = R/RF and s = kB T /RF Fext and n̂ = F̂ext . One than has R(Fext ) = RF Φ(RF /ξ), where ξ = kB T /Fext 2 and R(Fext ) = Fext RF /kB T . For small values of its argument one has Φ(u) ∝ u. For large u it can be shown that R(Fext ) ∝ (Fext RF /kB T )2/3 . 6.7.4 Polymers and solvents Consider a solution of monodisperse polymers of length N in a solvent. Let φ be the dimensionless monomer concentration, so φ/N is the dimensionless polymer concentration and φs = 1 − φ is the dimensionless solvent concentration. (Dimensionless concentrations are obtained by dividing the corresponding dimensionful concentration by the overall density.) The entropy of mixing for such a system is given by eqn. 2.352. We have 1 Vk φ ln φ + (1 − φ) ln(1 − φ) , (6.310) Smix = − B · v0 N where v0 ∝ a3 is the volume per monomer. Accounting for an interaction between the monomer and the solvent, we have that the free energy of mixing is 1 v0 Fmix = φ ln φ + (1 − φ) ln(1 − φ) + χ φ(1 − φ) . V kB T N (6.311) where χ is the dimensionless polymer-solvent interaction, called the Flory parameter. This provides a mean field theory of the polymer-solvent system. The osmotic pressure Π is defined by ∂Fmix , Π=− ∂V N p (6.312) 6.7. POLYMERS 55 which is the variation of the free energy of mixing with respect to volume holding the number of polymers constant. The monomer concentration is φ = N Np v0 /V , so φ2 ∂ ∂ = − . ∂V Np N Np v0 ∂φ Np Now we have Fmix = N Np kB T and therefore 1 −1 χ ln φ + (φ − 1) ln(1 − φ) + (1 − φ) , N i kB T h −1 (N − 1) φ − ln(1 − φ) − χ φ2 . v0 Π= (6.313) (6.314) (6.315) In the limit of vanishing monomer concentration φ → 0, we recover Π= φ kB T , N v0 (6.316) which is the ideal gas law for polymers. For N −1 ≪ φ ≪ 1, we expand the logarithm and obtain 1 v0 Π = φ + 12 (1 − 2χ) φ2 + O(φ3 ) kB T N ≈ 1 (1 − 2χ) φ2 . (6.317) 2 Note that Π > 0 only if χ < 12 , which is the condition for a ’good solvent’. In fact, eqn. 6.317 is only qualitatively correct. In the limit where χ ≪ 21 , Flory showed that the individual polymer coils behave much as hard spheres of radius RF . The osmotic pressure then satisfies something analogous to a virial equation of state: 2 Π φ φ 3 +A = RF + ... kB T N v0 N v0 φ = h(φ/φ∗ ) . N v0 (6.318) 3 This is generalized to a scaling form in the second line, where h(x) is a scaling function, and φ∗ = N v0 /RF ∝ 3 −4/5 ∗ N , assuming d = 3 and ν = 5 from Flory theory. As x = φ/φ → 0, we must recover the ideal gas law, so h(x) = 1+O(x) in this limit. For x → ∞, we require that the result be independent of the degree of polymerization N . This means h(x) ∝ xp with 45 p = 1, i.e. p = 45 . The result is known as the des Cloiseaux law: v0 Π = C φ9/4 , kB T (6.319) where C is a constant. This is valid for what is known as semi-dilute solutions, where φ∗ ≪ φ ≪ 1. In the dense limit φ ∼ 1, the results do not exhibit this universality, and we must appeal to the theory of liquids as discussed in §6.5. CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 56 6.8 Appendix : Potts Model in One Dimension 6.8.1 Definition The Potts model is defined by the Hamiltonian H = −J X δσ i ,σj hiji −h X δσ i ,1 . (6.320) i Here, the spin variables σi take values in the set {1, 2, . . . , q} on each site. The equivalent of an external magnetic field in the Ising case is a field h which prefers a particular value of σ (σ = 1 in the above Hamiltonian). Once again, it is not possible to compute the partition function on general lattices, however in one dimension we may once again find Z using the transfer matrix method. 6.8.2 Transfer matrix On a ring of N sites, we have Z = Tr e−βH X βhδ βhδ βJδ βJδ e σ1 ,1 e σ1 ,σ2 · · · e σN ,1 e σN ,σ1 = {σn } = Tr RN , where the q × q transfer matrix R is given by 1 1 Rσσ′ = eβJδσσ′ e 2 βhδσ,1 e 2 βhδσ′ ,1 In matrix form, eβ(J+h) eβh/2 βh/2 e R= . .. eβh/2 e βh/2 eβ(J+h) βJ e = eβh/2 eβh/2 1 eβh/2 eβJ 1 .. . eβh/2 1 eβJ .. . 1 1 1 1 if σ if σ if σ if σ if σ = σ′ = 1 = σ ′ 6= 1 = 1 and σ ′ = 6 1 ′ 6= 1 and σ = 1 6= 1 and σ ′ 6= 1 and σ 6= σ ′ . ··· ··· ··· .. . · · · eβJ ··· 1 eβh/2 1 1 .. . 1 eβJ (6.321) (6.322) (6.323) The matrix R has q eigenvalues λj , with j = 1, . . . , q. The partition function for the Potts chain is then Z= q X λN j . (6.324) j=1 We can actually find the eigenvalues of R analytically. To this end, consider the vectors βh/2 1 e 1 0 −1/2 , ψ = q − 1 + eβh φ = . . . .. .. 0 1 (6.325) 6.8. APPENDIX : POTTS MODEL IN ONE DIMENSION 57 Then R may be written as R = eβJ − 1 I + q − 1 + eβh | ψ ih ψ | + eβJ − 1 eβh − 1 | φ ih φ | , where I is the q × q identity matrix. When h = 0, we have a simpler form, R = eβJ − 1 I + q | ψ ih ψ | . (6.326) (6.327) From this we can read off the eigenvalues: λ1 = eβJ + q − 1 λj = e βJ −1 , (6.328) j ∈ {2, . . . , q} , (6.329) since | ψ i is an eigenvector with eigenvalue λ = eβJ + q − 1, and any vector orthogonal to | ψ i has eigenvalue λ = eβJ − 1. The partition function is then Z = eβJ + q − 1 N + (q − 1) eβJ − 1 N . In the thermodynamic limit N → ∞, only the λ1 eigenvalue contributes, and we have F (T, N, h = 0) = −N kB T ln eJ/kB T + q − 1 for N → ∞ . (6.330) (6.331) When h is nonzero, the calculation becomes somewhat more tedious, but still relatively easy. The problem is that | ψ i and | φ i are not orthogonal, so we define |χi = where | φ i − | ψ ih ψ | φ i q , 2 1 − hφ|ψi x ≡ hφ|ψi = eβh q − 1 + eβh 1/2 (6.332) . Now we have h χ | ψ i = 0, with h χ | χ i = 1 and h ψ | ψ i = 1, with p | φ i = 1 − x2 | χ i + x | ψ i . and the transfer matrix is then R = eβJ − 1 I + q − 1 + eβh | ψ ih ψ | p βh 2 2 βJ 2 χ χ χ χ + e − 1 e − 1 (1 − x ) | ih | + x | ψ ih ψ | + x 1 − x | ih ψ | + | ψ ih | " # βh e | ψ ih ψ | = eβJ − 1 I + q − 1 + eβh + eβJ − 1 eβh − 1 q − 1 + eβh q−1 + eβJ − 1 eβh − 1 | χ ih χ | q − 1 + eβh βh (q − 1) eβh 1/2 βJ χ ih ψ | + | ψ ih χ | , + e −1 e −1 | q − 1 + eβh which in the two-dimensional subspace spanned by | χ i and | ψ i is of the form a c R= . c b (6.333) (6.334) (6.335) (6.336) CHAPTER 6. CLASSICAL INTERACTING SYSTEMS 58 Recall that for any 2 × 2 Hermitian matrix, M = a0 I + a · τ a0 + a3 a1 − ia2 = , a1 + ia2 a0 − a3 (6.337) the characteristic polynomial is and hence the eigenvalues are P (λ) = det λ I − M = (λ − a0 )2 − a21 − a22 − a23 , λ± = a0 ± q a21 + a22 + a23 . For the transfer matrix of eqn. 6.335, we obtain, after a little work, h i λ1,2 = eβJ − 1 + 21 q − 1 + eβh + eβJ − 1 eβh − 1 rh i2 1 ±2 q − 1 + eβh + eβJ − 1 eβh − 1 − 4(q − 1) eβJ − 1 eβh − 1 . (6.338) (6.339) (6.340) There are q − 2 other eigenvalues, however, associated with the (q − 2)-dimensional subspace orthogonal to | χ i and | ψ i. Clearly all these eigenvalues are given by λj = eβJ − 1 The partition function is then , j ∈ {3 , . . . , q} . N N Z = λN 1 + λ2 + (q − 2) λ3 , (6.341) (6.342) and in the thermodynamic limit N → ∞ the maximum eigenvalue λ1 dominates. Note that we recover the correct limit as h → 0.

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