3 The Monte Carlo Method, an Introduction
Detlev Reiter
Institute I`ur Lncrgrctorschung 7 Plastnaphysrk, iʼorschungs/cntrumOilerch Grnhl I. $2425
iuhch. Gcrrntiny
'l'his chapter presents the basic principles of stochastic algorithms, usually called
Monte (`aril methods. After some historical notes, the generation of random num
bers is discussed. `l'hen, as a tits non-trivial example, the concept is applied to the
ex Aleutian to integrals. More im lox ed porcelains will he discussed in the two sushi—
qrrent chapters oI`this part.
3.1 What is a Monte Carlo Calculation?
Len an early lecture note (around 1960, hut see also lll; one of the pioneers lo` the
Monte (`aril technique M.H. Kalos, quotes the two "del’initions": ti) ALAsr resort
when doing numerical integration, and (ii) a w ay oI`wasteI`ugly using computer time.
Today common assumptions characterize it as a numerical method inxolxing ran—
dom numbers in a significant way.
Len a certain sense any large computer calculation has random aspects, due to
roundofferrors. Also deterministic molecular dynamics (MD) calculations, in which
the interaction of a large number of moving particles is followed by integrating
Newton`s equation can have random results, due to randomly chosen initial condi
tins and/or due to the large number of particles. 'l'hese are usually excluded from
that definition oI`Monte Carlo techniques. The im olxement oI`randomness in Monte
(`aril methods is rendered more precisely to mean deliberate its of random num—
hers in a calculation which has the structure of a stochastic process.
Two major areas lo` application are in statistical mechanics (many particle sys—
teems) and in linear kinetic (particle) transport theory. The first type oI`calculations
are einhodied in a eery specific sampling technique, and are nor discussed here.
`l'he second, for example traffic flow, hanse, genetics, but in particular neutrinos,
radiation transport, cosmic rays, neutral and charged particle transport in plasmas,
etc., rely on a study of randy interesting stochastic processes by imitating the ran
dom processes directly on the computer. Although intuition, and the resulting high
transparency of the procedure, is an important ingredient in this type of stochastic
analysis, a sound mathematical hasps also exists. This allows rigorous mathematical
proofs to he ix en that certain methods actually provide solutions to certain generic
mathematical equations, e.g., to Fredholm integral lacerations oI`second kind in case
of transport problems. l)instinct IRO Morse numerical schemes, in rheas stochastic
methods error estimates are provided hy the method Israeli` rather than requiring
6 4 D. Re n te r
anlditiuml considerations. Some matlucmmical background and basic statistical re
sulks are also needed to analyze results of Monte (`aril sinmnlutions, for estimation
of errors and for obtaining more economical uppmuclucs beyond simple sinmnlutiun
o f mu m.
Them exam a x us mxmunt nI`intmducmry Iitumuxm, I]·0m a basic wxvlmok Icx cl
up m mmmgraphs focussing on wry <pccia1izud applicaticms lc, I`arc:<mx1plc, Il,
2, 3. 4lp, probably now hundreds 0I` wulnmscd lecture umm and uuccmntcdjmxrnal
mxiclc s. This pox·c<cnrintx*0d1|cr0x*ychapr certainly d¤1plicatc<1xm<t,iI`n0r all 0 {thm
nmncml. Our aims are no inumluce me ncrminolugy. and no convey mho message uhm
Moms Aural Metlwds do have u solid basis in measure theory (with the theory of
probability as special case mlucreof). Strict nmnhennumicul proofs of convergence of
nh: method no he emma solution emu bum also. and disnincm from num numerical
concepts, implementation can be be strongly guided by intuition and retain an high
tmnspamncy cx cu in wry complex dumtimxs.
We will, after mms <hm·r hi<tm·ica| remarks below, sum wirh introducing the
cnuccprs nI`mud0m exams, 0|`crr0rc<ti1xmtc< and unbiased procedures for comma
tin. Pmcrical implcumxmriom of Ixinum Carlo techniques rely on our ability to
draw random number {mm any probability law wc wi<h. Only A {caw, um hzmic
farces and cnuccpm in hi< regard will hc rcpuamd hum iu Scar. 5.2. lu Sect. 3.3
the central limit theorem and variance reduction techniques will be dc11wns1ra1ed ur
work using the generic example for Monte (`Urdu memlwds: Imegmtiun by stochastic
sampling. 'l'l1crelzuion ufmhis very general but inunimively clear and transparent up
plicumion no nhc nmhenmmicully and snmimcully more involved uwmspmn pmblenns
(Mom: ('aril particle sinnulzuion) will be frequently used as guidance here and will
hc di<cu<<du in more derail in Chap. 5.
3.1.1 Historical Notes
lximc Carlo cnuccpm {all imo the branch n{u:<pcrimcura| umrhcnmricx lu m·dinm·y
mathematics ccmclmicm s are deduced {mm posuxlams 4[)caducei<m). Len c:<pcrimuma|
mathematics conclusions are inferred I]·<>m 0h<urxmi0n< (Induction;.Axiomc Carlo
mcth0d< cnnxprisc [hat branch of uxpurinxcnml marhcmarics, which is concerned
with experiments on random events (mainly random numbers). Mum(`ur1u meth
odds can be of probabilistic or deterministic type.
Usually the Gm reference to the Mum: ('Urdu Method is the famous needle
experiment of(`oomph de Buffoon (1733). a 1 wrench biologist (17()7—I788). 1 ig. 3.1.
Buffon pointed out that if u needle of length L is tossed on u plane with parallel
lines A dhtmxcc D apart (D > I/).i1h¤<pr<>\m\>i|i1y p 2]//(rrD)t0{aIIs1chetaha[
i< crowns one of the lines. Later, also Laplace mggcsmd [his pmccdum m dcmmuinc
rr hy counting the mnmhur of caucus n in `\' rcpcritimxs of the c:<pcrimcur. Then
n
2 L
2 L
n
—:—:~n¤—-—. `\' rrD D `\'
Hi< hismx·ica1u<c nI`I\’Imuc Carlo has all.kcvIOCauu·u< 0{rhc mcrhmlz
(3.I)
ʼi ThEOMmv Carlo Method, an Immducuon 65
` 4A` X
Fig, 3.1. Bu1`1k>uʻs ucudlcsz What is the pmhuxbility p, that a vmxlle (length L ), which hulls
randomly ou a sheet, crosses one of the hues (distance U)? (Left: @Ompy1·ig,ht IQOKZOO3:
The Regents uf the Uniwexsity of California)
( 'vnvergenoes About ./\' : 100 OOO trials are needed bit only two digits air the
cmnmxa, Convergence is slow, but ibmlpromf,
Transparent)? The method is imuitivcly1mde1·s\a1.xdab1e, even without any math
ematical reasoning,
Error eszimales, optimization: Error estimates and optimal choice mf L, D are
provided by theory 0f probability. ( Bi.n01.nia1 distribution. statistical variance as
2"d central moment em).
I\/Immured use I\/[mute ('aril wchuiques, lu the age mf digital computers, was initiated
by the pioneering work 0f John mV Nee and Stanislaw U lam in \hem10m1c1ear
weapon develmpnxeut, They are also credited Mt having coined the phrase Moony
(`urlrz.
Many 1m»n<gump|1s onNolanc (Alo Nlclluods sum with an introduction lnMaca—
sure theory mL in auricular to elementary probability theory. Although we will
introduce and us theca proper nmlhcnuelicul mcahulzuy loo, we will, with rcspccl to
purely matlwmatical uxpccls, refer to those and largely rely upon the imuiliw mcm—
ing. Wh refrain purlicularto the la»¤iu1m>n<>gmp|1 by Hammcmlcy and Handsuomh
[2]. This book provides a short and very readable overview 0fM0me ('arm, Remark
ably, the theoretical fumndatinnus today remain rather similar t0 those from 1964.
when this b00k was [first published, Just the applications are Ihr namer smpldsticated
today. The illustrative examples 011 I\/[mute (`arm iutegratinnu and some 0f the ad
Vance techniques in this present i.utr0duc\i0u will be based upon this next' 4
‘ A pdlimu ofathm book, which i» out of prImomma long, cam hc duvmloudud from the
Emmett, u.g.. http;//v.vw..uimnc.de/html/tux1h¤¤ks.h1¤n1.
6 6 D. Re n te r
3.1.2 The Basic Principle
The principle i< to find lcmumtcp mean \a|uc<, Lo. cxpcctatinvx ax|uu<, I of ma
<yams cnnxpovxcvm. [I` a dcrcrnxiuiqic pmlxlum i< m hc mlxcd. one tim has m im cm
A qochasric <yswm such [hat a mean x alum = cxpucmrion x Alice coincides with [hu
domed solution I nfrhc dcturmiuisric pmhlcm.
In any case: 1 is a single numerical quantity ufimcrcst (nom an entire functional
dependence), and one might always think of 1 as some dehnite integral.
‘1hc simple imuinivc inncrprcmions are given below,bu1in abs1mu111u1he11m1icul
menus this stochastic model is given by the probability space {1'2.or.p. X). 1'2 is a set
0I`c1!mcnl2u‘y (random) events .:. the or-Held is u set of subsets of J2 to which the
measurable function p assigns u value (theca probability) from the imervul [0, I]. such
that the K0|m0gm·0{I`axi<m1< for A pmhahiliry are I`u|til|cd. X i< a random xariahlc
on 12, assigning A 4u<ua1|y real; uumhcr mr xccm; m each randnnx mom, c.g.:
Yaw) A R, such [hat I IT(X),1h c expected oxalic of X.
The expectation x la /`Z(X) and x aria az are defined a< the Hr<t mmcm
and second ccmml mmcm, uupcctix clay, ad, ulna<< otherwise sm., we assume
mlm they bun me
1;(X) ;: /¤1px.
MCP); /dap(,><»n(.><3)2. (1.2;
Nm mm mx) iZ,,(X). erg (X) a;(X)_ a.¤._ memammau Or X of mum
depend upon the probability measure p.
A smchasric appmximatinu m I is than obtained by producing an independent
wqucncc 0I` random cx cum w,.i I ..... \ ` according to probability law p and
cx alluring
1;‘(2<A~> : JA : Exp;). cmu
'lc estinmxwr lie isjusm the arithmetic mean of many (N) outcomes of the random
experiment.
Even wimlwum any of his absence 111uthe1nuticu1backgmundil is emotively clem
(sec examples below) that IA will converge to 1;{X), hence to 1 by cunsmnctiun.
as the number of samples N is increased. lluwcvcr the laws of large numbers and
the ccmml limber rhcorcms nI`pm\hi|ity theory um only Mx ide scum} nmrhcmarical
proofs [hat hi< Monte Carlo procedure is emu mnhiascd) but lm them it com args:
lie A I for .\` A oc, albeit slowly with 1// \//N;. lu particular the ccmml limit roc—
mom 0I` probability rhcoryg a<<cr.< them [hu probability dies·i\>uti<m 0I` ly, for large
enough `\', c<mxurgc< to A Gau<<au disuihminn, with mean oxalic I /`Z(X) and
2 Sue any mxtlxmk on Monte Furl. or Pmhahuhty Thumb.
`S 'I'hc Monte (`urI0 Method. uu Immduclmu 67
variance aʼ(1A—) : ug (X)/N. llcncc nhs typicallersulms from suuisnical emu- analy
sis under Glassine dislrib\lli0n laws apply, mg., also the resulting Conhdcntc I!\‘!Is.
Il is, lhércforé, Common practice in Monte (`urlo applications lo quote results as
I¤1\·in(Iy)
O.-
1¤1vi2»¤(i,\~).
cm;
which lmvc Confidanté levels of about 66% and 95%, réspéclivcly.
Of Coursé, in applications the variance UQ is \ls\1uIIy c\ʻcn more diI`GCS\lh
lo Compute lhzm the mean value E[X). Il is hl€r€I`0r€ replacEdby lhé empirical
—> 1 I » 2 ·“` Z [XWM 7 b(-XM]
N }
) x” D, 7 7 x Momsu V1; <> ZM
and (mc has also, under [hen: zxssumptiom mxdc. {nr large maple <izc Y
gap/A) ¤ : K.
(3.6)
lcé, for large enough N, inilhé Gallssiun based error cslimulés (3.4) U Can safely
be replaced by JA , al lousel for large sample size N Z 100. In the opposite Case
N f 100 S¥\1d!nl`s l-dislribluion should be émploycd in error analysis instead.
3.2 Random Number Generation
The Monte (`aril method rocs[< nu our ability to pmducc rzxndnm numbers drawn
|]ʻ0m any pzmicnxlar prohzxhility di<trihuti<m mknifeit ¤rbislS. |]ʻ0m [hc pmhahility
delusory {unction (pdf) f(.r), wish F(.r) dt f(L).
lhcmnplcs arc wetting 0{ a sur|`acc hy rain infirm disuihutivm, radioactixn:
decay 4P0iss0u distrihminm, Or [hc distribution 0{ xclncilics 0I` nmlcculcs in 21 gm
((}a\lessen dislrib\lli0n). In Geneml, lhé probability law must be known, éilhér from
theory, Sxpérimcnl or plmlsibilily.
A theorem from mcusllrc theory slums:
Theorem I. liar`}! fpmhahilfxfv) M11»·a,mn/· ;1 run by r]1»·rm11]m,wrI in: u wrdg1/berIxu1n
p pl. p, > pg. pd > p_;;1(, qfrlxnwpam.
(I) IRAʼs1e¤ has u rmuirmx rlix/rfhmlfrm (w1]1bmbubi[il_v r]r»·r1,x[»fv
(Ii) Parr lvm hux :1 r]i,xrʻn/·/1/ r]ir/xfhm/[rm.
(ii) Parr rhino/·e¤ [A :1 par/112[r1g[rʻu[ rmuriburirm.
68 I). Renter
The third part is required for me ambiance mathematical case only (general measure
able spaces, 0-algebras. ). but il does not occur in practical Monte (`aril apple
cations. This means, for any distribution law arising in an application We can obtain
random numbers iu [wo scrap<: Fir<taram10nxdccisi0vx(ha<cd cm [hc two remaining
weighting facmxi P1·P2) whcthcrrhc c0miuu0u< or the discrete diqrilxutimx i< to hc
sampled, and second than gcncmriug A random number {mm [hc chosen di<u·i\>uti<m
M or pd. We will <h0w clmv that for huh casey c0miuu0u< and discrete dies‘ihu—
[inns, gcucml pmccdurc< for random n¤11n\>crgcncrari0nCPIsr, atclassr iu principle.
We refer to the standard reference on the pmduction ufnonuniform random numbers
[5]. This book deals with the myriad number of Ways to lrunsfoun the uniform run
dom numbers imo anything else one might want. Also the Hrs! section (pp. 1493)
of [6] is 21 very comprclucnsive immduclion to random number generation.
3.2.1 Uni l brm Random Numbers
Uniform random numbers are the basis for generation of rundown numbers with all
macro dhtrilxutimx laws. A madman xariahlc is um`m*mlv disuihumd on an imcx al
la, h], ifrhc di<u·ihuri0ndousery f i<
fg!)
gym,
NR
(am
]
with XW y, I iI`:r i< iu theca iumxw al \0, H and () clwwhcrc.
The cal<<icaI method m gcucmtc uniI`0m1 ramlcmx num\>cr< on |O,l] is by so
called ICU congrucmial ramlcmx number gcncramry which arc defined hy [hc
{HH
\n
E,,
}
M
mor]
new
(5.8)
Hem r1 i< A magic multiplicand, new il` 0I`tcu chow to hc [hc Is·gc<timccr rcprcr
scnmhlc on me mchauc pm 23% cm, anu n <hm.1u hc prime m m. Proofs rm.
particular choices of (large) pamnxcmrs r1 and new that [hc gcncmmr achicxcs theca
largcq p0<<ilc period 0I` new — 1 diIcr.cm random uumhcr< are quite cumlxcrscmxc.
Optimal pammctcr ch0icc< arc typically found experimentally, we again I6]. The
Glint periodicity limits precision only in very large calculations, ag. on modem
massively parallel computing systems. A rather subtle issue is also irrzleperrzlemw of
an entire sequence 0I`mndu1n numbers (l0c.cil.).
3.2.2 Non-uni form Random Numbers
As stated above We only need to consider discrete distributions and continuous
alimibxxmions.
Finding random um\>cr< with A gig diwrcrc disuihurion i< rectal: Lc A
discrete dhtrilxutimx, with lv elementary 0utc0mc< labelled by natural numbers
{0,1,2 ...., Iv},\>cgixcnhy
1 The Monte Furl IVIu1h0d` uu Immductmu 69
l’(X:[) :p,;O.
Zn, 1 .
mi) mx g i) Ep,
mm
with Fʼ(X ij [hc pmhahiliry 0IOUwm i. F i< the (cmxmlmixc; disuihmion. Lu 5
hc a uniform random number nu [O, 1], ruche theca random xariahlc X with X i if
FQ — 1] < { § FQ) is distributed according to F.
3.2.2.1 Inversion Method
'l'l1cim‘ersion method provides random samples z from u disuibxuion F by convert
ing uniform random numbers §. '1 his is simply alone by setting
;: min(.r\F(:r) 2 E) ~ F. (3.10)
[I` F i< <u·ict|y m0n0mn0u<, than z F *(E). For example, il` is [hcidsu·i—
button density function (pdf) no be genemmcd, hen msn Gnu nhc chumming function
1·'{1·) : fx ri! [fz], pick u uniform random number { on[(),1]zmdscl§ :
and Hnully invert this to Und random number z, which is then distributed according
no [Qu).
The same transfurnxwmion rules as for any density function apply also for a pdf.
Hence [hc gcucml qmtcgy is: Try m transform A gixcu pdf ff.) m another disbar
Huron f, <ouch [hat theca imcrw of [hc new cnunulmixc diurilamion F is cxplicirly
known. Then apply [hc method of micro<i<m and u·an<{m·m hack. Figure 5.2 llu<—
tmtc<thcmcthm1 of im Carson for [hc vmrnml (Gaussian; diswihmion
,
1
”
.¤
V.,
m q
I .r d$(:r) dt m(t) T 1 4 cr. T . (3.11) . 2 V 2
Unfortunately, the Gaussian error function curfew] and hence cannot be inverted
in closed form. We will show how to generate Gaussian random numbers. even
without numerical inversion. further below.
lmm this procedure follows directly the natural and best forename for storing (also
umlrhdiumxsional; mhulatcd dam for random sampling in Mum Carlo appliqué
dom: Form theca iuxcrsc cmnulatixc disuihmion function F"(.r) (Lu.: [hc quintile
fuucriom and smirk hi< for .r uniformly spaced in IO, 1 |. Theca tack {mm a uniI`m·m
diswihmion 0u|O,l|am1 hud F'1(€) by imcrpolaricm iNathimholc.
70
I).
Rein:
¤¤<¤>
[
<¤>
tOB0 5 5 E
/
j
··[
<¤¤
72
IWO2O·}%XJ
Hg. 3.2. (2) Imprison of ('achy (zluslxml line) ad nmmnl distnibumnu (will line),
( lv) Cu n xu la u xu d mr n b u tn o n fu g u e o f mmm] d mr n h u mm ( il I) , ( c) [mu m e mu
lutixc distribution of murmur distribution G" Uniform muddy numbers guy ,§2 unwiseu)
um com cried Lo mndmu numb cm sy , :2 from a num. dmrihutigm (urdiuutu)
3.2.2.2 Rejection
Amulet general mclmd for generating 11011-u1.\j.f0rm random numbers is the 1*0
ejection method (Lv. Ncuma1.m, 1947), This method Las always applicable, although
it may sometimes be rather ixncihcicnl. F0t distributions with Unite support, Le.,
_/(us) # O Only mu a Unite dome 1\1 (say, IU [11, b]), [ind the maximum cr Of
f(1·), sample za rz•m\<>1npz•ir(E|,E2) with 5 unil`<>rm on }\I am E2 unil`<>rm on |O,c|.
II` fy 3 f(£y ), accept E. . Otherwise rcjccl this pair and pick an new pain Repeal [his
procedure until a pair is ucccplcd. Clearly, theca cflicicncy <>I` this mulched (ugh. macro
surd ax an crag number <>I` accepted random pain lo number lo` pain produced)
may he quite poor, in particular iflhc distribution f(n*) has sharp maxima.
A more gcncral, and mnnclinncm more cfiicicnl rejection mulched, working own
011 iuhuitz sampling dummies M , results if One [ids a second distribulimu g(.::) and a
numerical constant 1: such that f § 4: - g(.::). Again [ind a pair (21, z;) 0fm1.\d0m
numbers, howcvcr with zl Mt sampled lmiibtmly mu A1 but 11*0111 distribulimu g(z)
instead, 2; is uniform mu the interval [O, 4:]. The randmm variable 21 is accepted if
22 § _/(xl)/gfzl), ()Lhc1ʼwisc a new pair (21, z;) ls generated. Sec ('hap. 5 Ibr au
imp<>rlanla1¤plicali<>n in particle simulation.
3.2.2.3 Examples
3.2.2,5,1 I r nw r sivii
I.mp0rtamcxa111p1cs iu which the inversion method can be applied are, eg., the expo
lentil distribulimu (Of the mean 11*0:: flight length 0f radiation iu matter), the cosine
distribution 0I` polar emissary angles against surface uotmals, the surfaces crams
bug Maxwellian flux disrribulimu [ (wl) cx m_fM¤u¤ (wlmonoal velocity compm
ncnl¤<>l`gz•x1m>|cculc¤ with Maxwcllizm vclocily distribution (fM_,,,,). Wk cxplicilly
illmlralc the inversion mulched here Ihr theca (`duchy distribution: The Cauchy dicer
lrihulion, sc. |·ig. 3.Z(u), in physical applications also culled Lorentz distribution, is
am cxumplc <>I`a distribution I`nuclei<>n that ham no moments. ll arise; often in rudizv
lion transfer, ugh., ax |inc.—¤Impcs <>I` nalurullyr or Stark hmmlcncd lines or in other
tesmuauce phenomena
1 The Monte Furl Mu1h0d‘ uu Immductmu 7I
"
1
f < > 0 nm 4 F by - by + (2
Hem<thcmcdiau(1iuc <hi{r;, and v is [hc halfwidth at hall` nmrdumm (HWHM;.
Gcncrariug mmlm number with A (`achy di<u·i\>uti<m is usually done byImow
mm. Fir<1 tmusfnrm m Z. srandardimd (`achy, hy S (T 7 /J)/¢. The cmnulmixc
disuihnxrion is [hun gh cu a<
I
'
I
I
1
r r
—
/ 1
F`~ .r — <rln — > —aʻcmn » . .15) my Mnzw ( ¤
'1l1creforc the random number z : b + 1*- Lau{vow{§ — 1/2)}. with § a uniformly
disuibxuenl mndomnumber on [0. 1], has a (`duchy (b,c)dies·ibu1iun.
.%.2.2.}.2 The Bm Muller /Vlerh01l_/br Guussizm Rur11imr1Nu1r1l>ers
Because the Gaussian error function cannot be inverted in closed form, the following
cnmhinaticm of u·aus{mnorm0n, rcjcctimx and im cr<i<m method is typically applied:
Nm cnc, hm two independent vmrnmlly dimilautud random uumhcr< 4;,. zg) are
produced hy Hm u·au<I`m·ming randcmx x m·ia\>1c< Z; , Z; I}·0m cartu<iau to polar c0—
m·diuatc< I?. <D. The angle <Pi1h cu uniform in [O. 2rr]. Only ¤:0s(<D) and sin(d$) are
needed, and a rcjccti<m marched (comparing a unix circle and A mrrcmndiug <quark)
can bcx1sedfor1l1em.'1h c variable H has, due to the Jacobin of the 1rumI`orma1ion1.
u Gaussian flux distribution (see above) rather than a Gaussian itself, and this can be
directly generated by the Melted of Inversion. 'l'ransfor1ning buck Z1 : H · uos[<P)
and Z; : H ·isu[<P) provides u pair of independent Gaussian random numbers.
3.3 Integration by Monte Carlo
lumgmticm by Nlmuc Carlo i< A smchzmtic marched for the dcmmxuiniqic pmhlcm nl`
{Ending an integral, which in sxxfikicntly complex high dinncnsional situations can
be competitive or even superior to numerical methods.
Leafs consider the source rate of particles (likewise, of momentum. heal. ct..)
in u nnucmscopic system (c.g.. u fluid flow). in which these particles innicmscopic
objects) are ruled by u kinetic, i.e. nnicmscopit (Buhzman1n)!q\1u1i¤¤n. Examples are
chemical sources (particle. momentum. energy) in plasma chemistry. or radiative
heat murk in emu nI`mdimi<m transfer rhcorv.
Such mmm then mad
/) /ir i
1 ; dx-y(.¤ )j(`¤ ;; ajyg-3. @.14;
Hem f i< the one particle diswihmion ldumiryp fuucrion _f(r,v. i, t) or where
[hc sm. .r nfrhc clcx am phmwspacc may, mg., hc chamcmrizcd hy a position we
my 1*, A x clncity xucmr v, theca [imc t, Lu. cumin< x m·ia\>lc<, and further A discrete
72 I), Renter
chemical species index [, u1soforc>`a111p1c for internal quail stares. gh) is again
some weighing function dcncrmincd by me pmicular moment of imcrcsn. In man
emuticul terms one would refer to this as Lebcsgue-Smieltjes Integral of mcasurabcl
I`uncri0n g(:r) with Rupert m (pmhabiliry) measure defined hy di<u·i\>uti<mdousery
f <-¤)
We will dews lntcgrmion by Nalco Carlo ming [hc example {mm [2]: lm the
iumgmticm dcmmi V hc [hu unit iumrxal |O, 1 |, [hc uniform di<u·i\>uti<m on
[0,1] (Lu.: 1 on |O,1|, and f(:r) () elsewhere) and g(:r) (¤—xp(:r) —
1]/fe — 1).('1eurly.
e" — 1
I Karl ().418(),.. . (3.15; 0 — 1
We will now integrate this same function by Monte ('aril. Our HST method dues not
require any theory, but instead, inspired by Buffoon`snced1c mperimcm, we will just
use pairs {lf; of independent uniform random numbers and compare the known
urea (the unit square [0.1] X [0.1]) with the unknown urea 1, which is the urea
underneath function gfx), in [0.1]. Le., we count u hi! if the point diced by the
pair ofrandmn numbers is under the curve gfx), and an miss omlucrwisc.
A< can clearly be sec on Fig. 5.3 the ratio 0I`hit<mmm1 uumhcrofsamplcs com
xcrgm m [hc emu oxalis of the imcml, a< c:<pcctcd_ and also the <ti<tica1 error.
indicated a< empirical smndard dcxiaticm sy. 45.5; scale with 1/v/.\` a< cxpucmd.
OI` cmxrsc such A Monte Carlo imugraticm murhml i< pmcurly {<m|i<h. By hi<
method we ham, in principle, replaced the <ing|c imcml mer fnxncrion g hy a dmv
Bel integral over the area between abscissa and function gfx]. '1 he convectional
text-book mewled (crude Mums ('urlo) can be obtained from this one by the obsess
ovation that once the Gm random number {1 of the pai: is known, we do not have
to rely upon {Z no decide about counting zero or one. Given {L then an one will be
counted with probability Paz p : g{§1]. Hence instead we can use than (conditional)
cxpucmd oxalic p nfrhc binomial disuihmion h(l,p) directly. This i<, admittedly A
quite obscure uxplauaticm for wmucthiug really trivial. Bur it is also the underlying
idea behind A powerful x aria reducing Iximc Carlo technique known under di{—
{umm Nam< in diI`{cram umm of applicmiouz (hnditicmal cxpccrarion c<timer (in
ncurmn <hi|ding). |4|, ax cmgiugtr2uxs{<u·1xmri0n4u*an<I`crrhc0rymainlyn Russian
Iitcmuxrc), |7|, or energy parriticming murhml in radimixc hum {mus fur [Xl.
'his method is opposite to mndunnizzuiunz We have replaced a sampled result
(zero or ounce) by its expectation value. In our particular example we have carried our
one of the two immigrations analytically, conditional on the outcome {1. '1 he second
mundane number is nun needed an all in his pmicular civil case bum mlm nm me
xelcvampuin1.VVha1 is impugn also in general terms is lm one (generally: some)
of [hc two (generally: 1xmuy;imcgral< has been done analytically, and only [hc rcr
meringue mms by random sampling. The general rule i<: Alway< u·y to do as many
imugraticms analytically or numerically and resort to Mum Carlo only for the run.
Len paretic|cu·auspm·1theory this concept will lead to p0wurI`u| hybrid methodsacnm—
hinging iufcwrnmion gained analytically mr numerically) and nochasrically, bridging
74 I). Railcar
Inlegralmn of |( x) with CRUDErM¤nte Car|¤—Meth¤d
2*-1
M=Q
0.70
0.65
0.60
¤ 0.55 .9
E 0.50
% 0.45
a 0.40
SD 35 ʻ
E 0.30
|’* 0.25
0.20
0.15
0.10
number cl samples, Iogamhmnc scaling
integral
apprcxnmatncn
O-4159233
Fig. 3.4. 1.uva1umjng Integral of (cap(uʼ) i 1)/(u i 1) on [0,1]meldod: crude Monte Carlo
statistical urhisé is reduced $0181y bymodifyL\g (sm0OIhjJ1g) the !sEi.ma[O1’ g(u:), in
inlxprmauce sampling the lludétlyhng Iandrhm \a1'iabI! (Ot m1.\d0m p1’0C&ss) is
allcrcd lo zm<>lI1cr<>nc, in order lo ucI1i&wc varizmcc rcnhnclion. A compcmaling
wcighl correction [actor in introduced in lc cslimalor lo nminluin lc mmc mean
value T : ]T(g(X))
7 ,. . . ' .f(*)#. . ~,;*. . 1 1 » g(J.) - f(.L)d.:. g(.1.)7_/ (J.)d.:. g(J.)_/ (.1.)d.:. (. .16) f (T)
V
V
Y
Hence we have g(.1:)f / The name mf this nléthrhd, importance
mnwpling originates from lc spacial lcchniquca 0l`lc used lo End optimal hissing
achcnmm (Lu.: <>I`lIu: rum\<>n1pr<>cuss, in parliculurin grumbler lechery. A mom
general, hul also somewhat hnprccimc lcrminology would rcfcr lo his concept an
n<>n—anz•lgamonlc Carlo, us comparedlo lhc analog I\’|<>nlc Carlschcmc. In lhc
Iallcr lc underlying pmhuhilily di¤lriIVali<>n law isdirectyluckn [mmhlcappliquév
lion, whcrcm in lc I`0rmcr onc umm a diI`1k;rcnldi¤lrih1|li<>n, molivalcd by pmclical,
cc<>n<>n1icul0r<>lI1cr rcmons, and mlalislicul wcighls lo compcmaln; him.
As seen Hmm (3,16), the value mf I is independent Of hOw the integrantl is
decsmxpmsed imo a p1’0dUCt mf it p1’Obabi.H\y density and a ésprhusé IImC\i0I1, but
me vamps, ¤§ (y) and U; @3Emmamy can be (mmm.
Le->\’s lake, again, 0111* example, I0 illustrate the Conccpli In Order [0 reduce the
variance UE Of 5 with respect I0 pmbidnility law we shrhuld try I0 make Z as
c<>nslzu1lasp<>»ihlc on [O,] |. The 'lhylor cxpumion ol` our purlicular lunclei<>n gfx)
indicate; umm me mm) §(m) ; gu)/T should hc mommownnHzm g(m) itscll`.
Hence we \1'y f(.::) cx us, Le., f(.::) Z 2.:: S0 [hal is umrmalizéd I0 One rm [0,1].
Our impmrlmxcc sampling pmcedllré [0 evaluate 1 110w pmcecds as f0laws:
Draw randmm numbers §IWWm ByHlé method Ofinvétsimu, this issuedmué by
setting § V/E, with § a1u jfr>rm m1.\d0m number 011 [0,1]. Then, againIBCtm the
aritlmiélic a\c.m.g: Of many (N) random Variables Figure 3,5shows the result
O1` such an ixnlegration, again vs, [\` , (`liary the Crhuvétgéucé is (ij I0 the C01T©C[
value, (ii) still only Rex 1/\qN), but (iii) the error bars my am; much malice lhzm in
both preciously discussed Nlonlc (Tz•r|<>in1cgmti<>n methods.
Again, il ccdm lo hc pointed out lhul lc cmcicncy 0I` lc procedure; is nci—
lucre dclcrmxncd by lc varizmcx, not by N per (`PU—limc, hul only by lhc Hgurc
0I` Muriel: xuriuncc pur (`PU lima;. And hence. importance sampling. more generally.
n<>n—anul<>g sampling, can go both way; in Monte: Carlo. Il; pcrlbrnnancx; ham lo hc
assessed Ou a case by case basis,
As a géucral Observation, Oneshambled 110[! that lu mm-analog 1VIOm! Carlo
schemes the 01*1*01* assessment simply based uprhu the empirical Variance, and €H0t
bars Obtained fm m the central Limit \h!01‘eI11, canheIéss Idiablé than iu analog sim
ulalirms, Allhmugh the variance may be decreased by a Clever impmrlmxcc sampling
Integration cl f(><) wnh Importance Samp|e—Mcme Car1¤—Meth¤d
v<¤>=
0.70
0.65
0.60
5 0.55
E 0.50
*:5 0.45
E; 0.40
0.35
6 E 0.30
¤>< 0.25
0.20
0.15
0.10
number of samples, logarithmic scaling
mtgs.
apprcxnmatncn
0.4180233
Fig. 3.5. Same integral as in big. 3.-1, method: importance sampling Monte Carlo
76
D, Renter
method, the variance of the variance may increase. thus invulidzuing convectional
error bar estimates. sec [9].
As in the case of conditional expectation Monte (`urlo We can design an cmrcnne
case 0{impmtancc sampling with zum srarisrical corm Africa only one maple: Let u<
scar ff.) gfx),/[_ Chuck: I cams. Monte Carlo integration pmcccds hy
wnmpling {mm [his di<u·ihuri0nff.r) which, inMs.c ofourpmricem*c:<mx1plc can
hc done by [hc rqicctinu technique. Then, indcpcndcm 0I`[hc implying, I is <cm*cd.
Unfortunately we needed the knowledge of the Gal result 1 already to design this
perfect zero variance scheme.
3.3.0.5 5f Monte Carlo
Finally we use our simple integral to ilhlslrale the concept of the 6/ lv[¤n1c(`m·1¤>
method, which is widely used in kinetic particle uzmsporl sinmllulions. Starting point
is the idea to split the unknown purunnctcr imo u large known nearby quantity and
<mal| unknown pcrturhaticm. Len particle <Imo|ari0n< hi< can also hc theca <ing|c parti—
clc dktriluutimx I`uncut<m ff.) sox ing some kinetic cumin or moments nfrhis pd.`.
lu near equilibrium situariom wc haw
[Qu) : [€W(.¤) + ¤5_f(w) c3.17>
with, for example, the Muxwelliun equilibrium distribution _fWm and a small pcmlr
balun 6f.Ilcuntl1cn be udvanlugcotxs to solvé. by lV[on1c(`ur1u sampling, only for
{if rather [han for theca {ull disuihmion.
S0 lc us consider our integral again, and write. accordingly, lv Lv 4 6] with
In [hc known pan
`
`
I
I
,
2
I
1 + lu q I ) lu T » rw 7) <3.1s» U &_14 ¤<> e_14 2 3€_1
and 6] [hc roc<t. Clearly.
»! 1 2H / ` 0
— I — rr — rr /2 0] dm . (3.19;
0 — I
Figure 3.6 <haws theca roc<u|t 0{roc csrimarc for I, with In known and {XI cx alumcd
by crude Nlnurc Carlo. Clearly by eliminating a large, known, cmurihmion m I theca
rclariw errors of [hc csrimarcs for any gig sample <i/c `\' are greatly reduced a<
ccmxparcdm prcxi0u<mc[hm1s.
Hi< method is mlm relaxed to theca <0 called correlation sampling ruchuiquc. in
which one would evahmc both 1 and lg. by Monte (`urlo techniques. but using the
same random numbers. Both EMS's are men positively corrclumcd and me Swiss
Lilac precision of the Monte (`urlo cstinmtc for the difference 61 can be subslamiully
better than in indcpcndcm cstinmtcs of! and lg. or ufl alone.
3 'l`h¢: Munir CarloMclmd, an Immducuon 77
Integration nl f(x) with L¢M¤n\e C:u|¤—Ms\h¤d
vow = 5
D.70
0.65
D.E0
§ 0.55
E D.50
0.45
% D.40
3 0.es
2 D.30
|ʼ< 0.25
D.20
0.15
D.10
number cf samples, Icganthrruc scaling
integral
appruxnmaucn
Q-418*233
Fig. 3.6. Sams: innegml as m1·ig.3.-1, method: 6] Monte Carlo
3.4 Suers
The paraphrase Of this introduction was I0 shrew that tandrhm numbers can be generated
11*01*11 any given pmbidsility density distribution, and that IVIOmc (`a1ʼ10 I\/I€\h0ds can
hc rcgurdcd ax sloclmmlic (rather lhzm numerical) pmccdurcs For integration. Nlonlc
Carlo consists <>I`im cling za rumlom game; such lhul lc cxpcclcd value; lo` za proper
rum\<>n1 xuriuhlc is exactly equal lo lc paramclcr which is lo hc computed. Au;rag—
ing over rcpculcd imlcpcmlcnl Monte Carlo samples from Hal gam com urges (in
lc proper mcuxurc lhcorclicul acme) lo the desired solution.
The additional Cmmplicalirhu arising iu many particle physics applicalirhus and
iu trausikir theory is due [0 One fact Only: Distinct Hum! the material lu this present
chapter the amp]jug distributive is s0m!\i.m!s uml kumwn explicitly, Instead
it will be given Only iiuplicitly as s01Uti0n 01* ii, usually, vc q C0mp]icatz cqualirhu
(mag.: the Bmltzxuaxm equalirm, the Fokkcr-Planck équalirhu,ct.c,). Wk will see that
this extra c0mp].icati011 can be dealt with by sampling 11*0111 Curian stochastic p1ʻ0
mum (generating purliclc lrzjccloricm), rulhcr lhzm mzunpling from a ix can pd.`, sc.
(`hup. 5. Bul lc Mt: hslinmlion <>I`mull—dimcnsional integrals, the unbiased nulurc
lo` lc mulched, pr0<>l`<>I`c<>nvcrgcn¢x;cr.r<>r hum, xuriuncc reductionmulledm,racemem
csscnlially lc same um in this prcscnl inlmduclion.
7x 1> . Rum
References
I I. Kulm. PA, Whxt10ck.Mnn/c (`1u·lnMe=II1mI.r4 Wl. I.· Ravine(\N11cyrIutcr~cmucuphr
hcutmuw. John Welch und Som. New York. WSG) 6%. 64
.|.M, I |mummer~lcy` |).<T, II:md>c<m1h. Marry Furl NCI//md; (Fhapnnun und I lull. I.<mr
don & New York. 1964) 64. 65. 72
R.Y, Rubenslem. m \Wl¢{v Series in Pmlmbiliry <111<lMu//rvrmxfiral S/mix/iu; (JohnWaley
and Sons. New York. 1981) 64
.I, Spumier. |.. Gclhurd. Marry (`mho Prinriplcv nm! Ncurmn Tmnspmr Pmlzlmrs (Adr
Dawn Wuwlcy Puh11cut10vx<T0xnpany·. WG9) 64. 72
|.. Dm my. lv'nnrI/n[;‘m~m Rxmdnm Vnrin/e= Gmm·urinn (SpnugcrrVur1ug. Bcrlm Hur
clelberg New York. 1986) 68
D.E. Knurl. m mV:im1mw·ir<1l Algurirlrnu. Vol. Z (¤\cld1son WesleyReadmg. 1998) 68
G. M1khml0v. Up/imi;u/ion qf\wi,g/1mIMunrc (`Mia M0//wzlx (Sprmger Verlag. Bream
Iimdclhcrg New York. I992) 72
A. Wang. |V|.I*. Mod. 1. Quant, Spccxmwc, R, A 104. 288 (2007) 72
K. N0uck` Ann. nucl. Iincrgy l8(6). $09 (1991) 76
. K&p1tOl& 4
’ Zaklady metody Monte Carlo
4.1 Monte Carlo integrate
Jak Hilo icier, >smyslc1x1 ]>oEiLaEo\ʻ§ch simulaci je generovx-xt koufiguravc systému
muumuu Gistc n tycoconfiguree pakpoiit ke stanoveni riiznyclx terrnodyuaxuickfncln Gi
struktumich velau, coi obvykle pfedstavuje vypoéet néjaké stfedni hoduoty. I\iIet0da·1 MC
geuerujc koufiguracc proveé s ohledem na efektivni vvpoéet stiednich hodnot. Nézcv m0—
marly pak poochi zoho, ic mmrhizoidl oddeterministicé MD pouiivé generétor Il21h0(]l\S'(Jh
risel.
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
szatiszickymi vlaszncszmi. Podrobnéjéi informace najde ézenéf v dadazku 12.1
Jako nivodni piffled memory MC si ukase vypoéer éisla rr, viz algorixmus 4 1 Budeme sxfiler néhodné
body do jednotkového Etverce a szanovime.Jessei bodpadl do ézvnkmhu se sziedem v jsdnom vrcholu irverce
Za pieapaklnan rovncmérného mznaiena aren
poise bad ve Civnkruhu rr
poet viceh bean 4
(4 U °
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
nam, kvnmni, pennona by mmcaissonabyu kcrelované. Jen vzaalenasn mnam baan oa puéétku saniaaNamenii
mei Jean, zapoétsmeJennieku do poécu iispéénightcapkusill Tenw elementérni pokesmusime opakVaz mno
hank. abychom ams na nejpnesnéjgi vysleaek;fitc>m pam, ischubbya je nepFim0demurnélannné aanmenane
polite pokes
Princip melody Monte Carlo ve statisticképhysice vysvétlimc na simulaci v kauonickém
NVT sahuaro. Simulate v jinych suuborech bumble uvedeny v samostatnécapitolz
Uvaiujmc vjpoéetstipendiwhodunityfunke X v kanouickém souboru, rov. (2.10). a
piepiéme .i1\E!gl'?u do diskrétnich ]>ron1§n11v<‘h nap pomoci liclxobéiuikového prexvirllu).
abychom ho mull vyéislit na puéitaéiz
.\ʻ(n·"·<k>) (·np{-aL¤(n="·<*>)]
<.\ʼ> = E
V Z ·*x1>l—HU(r" "kʼ>l
nm
CAPITAL 4. Z.~{KL.-\DY XIETODY MONTE CARLO
INTEGER n zvelkmlgj pat bud!]
INTEGER i
xwrsczn nu point bum; we ahem.
REAL x.y svufmlniec lewd vu Etwmi
”‘~···e$
.··JJgg__,._ _f, _;:;,."`
REAL rend() /1mkm m-uuejici mi/nmluéislu u xmmvnlu (0,1) %*-I 1-. ·:ʼ *J "I I! ·ʼC-·¥**=i*wi"*";·€?-ʻ
. a Legumefi-__;;
. ."'
nu ;= O
mn 1 ;= 1 ru n uu
#..;;.3.}?;·ʼf"zf1·}1·ʼQ; ~
X := man
y ;= rend()
IF x·x+y·y < 1 T1-mw nu ;: nu + 1
`·*¤ *r·°f"¢t‘
»s;;Q·\ =ʻ·
,,_;aL;.-.-:;,;
h ·¥_,'·_{_q;$;ʼi$.=g·jj=.u{L:$j
" 2x.'··.;x v ·. ·".*
PRINT "p1=", 4·•nu/11
Chyhu (;»hmu?; ml/md smmmluzmilmz:IElky) uypuimm; z rif = ((f — (j))’/(n — 1)). kd]; / : 1
.4 pmmlépm1¤»Im»mi vr/4 é nu/n u _/ = 0 S pmmlépmlubmmi 1 — ar/4 é 1 » nu/n.
PRINT ·c.h.yb¤=··, 4·sqr¤((1-nu/n>~(nu/n)/<n·1>)
Algorithms 441: VjpcéetESLu nmeteduManee Carlo.
de rN*(*) zooisymbolicy k-tfhaml vdiskr$£ui111 $011}:01*11 bodfi I kclybychom mxdélnli
iutegraiui ixxtnrval kaidéprotonné pnuZiv uu 10 ilku". rl0s\:u1<·1n<* ii pfi N : 100 vsruy
slnéelkfopuc konfigvxmci. n = l(l;m". V praxxitvdy n<·u1F12<·1n<· 11\ʻz\2 0vs1t pfi \·§ʻ]>uPt11
(X) vouchy konHguracc, alepoufv jm·_ji<·l1 116_jzxk011 vylxrmmxn porl1111m2i11u. Otzixkou pak
jv. jzxk 11<·j<rf`0ktiv1xéjillum p<>d1111mii1u1 \‘yI>mt.
Nc-jh5in6j§i l]l(!£O(]()\l jc vyl>imt Lyin ko11figx11·a<:<· zu-ln milnmlxni. <·<>M Jo Piipurl xueinué
n1atv11mri<·l<c* m0to<ly Monte Carlo vfpuéux muhomesul>11i·cl1 im <·gr¢ilf’ cubcboli mxz xmivui
ruetunly XIm1t<· Carlo) [32],
I Pm adhered mzegrélufunke ju-, ..4,4. 4-,,) pFesbolasm {2 v D—r0zmémém prostoru plan
f(¤¤¤,-»-,wn)<\<r» dsrn ~ %Zf(x·§Aʼ,.. ,.z-W), (-1.3)
’
)
de (.#,...,x§)Zanei
kay néhodny bod v oblaszi Qjeeii D-objem je |$2|. V évodnim piikladu vypuifcu
éisla wjameintegralifunki f(x,y) = {1 : :r* +y” <1, 0 : xi + y2 > 1) pFes (2 : [0,1] >< [0,1}
V pfipadé v_fʻpuGt11stFc<h1fh0<I1mrysystéx1111 uumlxax Uaistiv rmx (1.2). véuk tz1t01~\r1rla1
zccla selhzivzi z jednoho prostciho <l1'1vmlu; nél1mlu§· v§·l>G1· 11<—<lDlAMxlil mozi km1fip_1mx—
ccmi, ketch<*1nu_jivclk011 pravdéporlnlmost vskyu1 ax Ludii pucl>tat11r‘ pFx>pivu4)i k lm<lm»r5
(X}. an mail pravdépoclubujuxi Eipieo n<:n102115?111i. N1·j_j<·<ln0<lu§v_j1 sv mm skutw-Guu>t
<\r~111011st1`\1j<· mxpip:·1<lu systému N tulxfrh kuuli. Gerwrovnt uzil10<IuF kn11fig11m<·<e mlmto
$\'SH1lll\l%ll2\lll(ʻl1ZiIiih0<]Il€ vmlit pnlohy Nkm1livol>_j<:1m1 V (nziI1u<ln?· vylximt k011fig111xu:i
rʻ\vʻUʻ) ex yopitat vidy souiin l`11uk(·X SB<>ltz11m1m<>v3”111 f`axl:m1·m11 <·x]¤(—:iI]).
Pro libuvolxxnnx kuxnfiguraci viak svulvak<>|1 I11ʻz».vd6p0<l0b11usli (pro vysoké lmstuly téunii
as jismtoxx) riojrlv k mum, 20 nlcspofx clvékoalac svbudu p1`0\i11z1t, atassely BnIlx11m1u11Wv
fzxkmr buzlc 11111.1. Pravdépodobuost ziskéui muiné k0r1figu1ʻa<:0_utéméi uUlanx vfrux (-L2)
uebudc iʼ1bc¢: dczfiuovén (clLeninnnully nulou).
IY<·S<~r1i,ktkré sv nnhizi k mlstmx1611i mhom pmblémnn. jr mm: piiv§ p0Ftu stfvdni
h<><lu<>t.y m·I1 1rl<·um xxvaiovat libovolné k011Hg1u·:1r·r—. :xl<· pi:·<l11<>sm5 ty. I<tvr¢ʻ ]>z><lst;xtuD
pfispivznji k hodnoné intvgrélu (zu1gli<tl<y im]mrmnr·r< surnplvm/), (tizlu>11 [mk jo
1. junk Low 1·0alizm·ar ex
2. <last\m·1ur~-li sprzivuf 0<lhzxd
Pol<u<i sv Tyler ]>problemu (1), jo SZ\I\\<7liG_illlé ubti%n6 vytvuiit. 1n<>Zn<>u (<losLat,vP1x6 px·av<lG—
|>0<lu|>m>11) l<m1liy.g111·a<·i ]:0u!1Xʻ1u \ʻl<lA¢lei11 i111h\sti<· do lmixdnElnu p1·nst<>ru (viz v§§v11\·¤·<Ir·11§
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
11161 bit Lexkovf pmblém z této koufiguravc vytvoiit jiuou <l<>smt,r~G11c‘ p1·m*<l&pml<>bn011
k011Hguraci. IntuitivuéLizctady uavrhuout uésledujici schénm, ktcré pm_]<:rln0d11cl10st vy
S\'6(,liIll(T opt. nacystmu tuhjch kouli. V mmm pffpmlé _0 ntii B0ltzmam1€1v fukmr
bu<l' nula ucbo jcdna a tidy kaidé kouhgurace, vcketché uc<10<:hézi k pfckryvu Zziclufrh
koala, rim stnjnou pravdépodobuost vjskytu, zatimco koufigumco speek1Eveu1 sc nikxly
uvvyskytnm1. Piedpoklédcjmc Lady, ic so uém podafilo Iléjilkylll zpuisobcm vytvoiifumi
11011 krmfigur:u·i. Zménimc»li nyni polohu jedné (uebo i vice) kouli mk,icempét 11<·<loj<lc·
k pF<&kry\*11. (IUSIHIIPIIIO daléi moinou k0uHgu1ʻa<ti. ax tak u11°12c·1m·puken5nvz1t A vytvniit
pnsloupnost k<mi·igumci.kitc>1·é bduuaéi x·y\>ramu1 puslmxpuusti v kmmuivkém smxlmru.
Jak uvi<limrʻ clzilu. lzc vfév uvcdexxf priurip mzriiiit u 11:1jh, \·Im<l11<>11 vy|>1·:111<m pm
>alox1p110st k<mH;.g1x1‘z1<·i k vYpoGtu stivduf hoduoly i vv >]<>2itFjSi1u pii;>zx<I. Kaly B¤»Itxn1eu1»
uh v {axkmr nalvfvé \ʻi<·<· lnociuot A1102Jan dvou. Rigmézui sv tvnro ]>ml>l¤*111 fvéi pm11m·i
Z\I:11·k0v<>\·§*<·h i<·t6zu°1. mi mxkoxncr xznjisti i sprzivxnosl vfslvmlku (lmcl 2).
4.2 Boltzmannovské vzorkovéni
4.2.1 Markovovy ietézce
Nil\l`kO\ʼlIi\icezec je pcsloupnost nzilxodufch vcliéin (urlzilosti, cvfi) Sw, lc = 1, . . . ,00,
ktcré so vybirnji zgisté (pro jednoduchost koncéné) mnoiicy stm/G (pm nés: k01xfig11m<zi)
(.4,}, 1 : 1, . 4 , , AI. Viskyc scathe nuni pima rmzzivislj, lc urlzilost, kccrou pozorujemc
v .,6us0“ k + 1zaikaisi rm mmjabké udélsotbyln poznrnvzirmVFnsv k. K011k1·ét11?>, jostliiv
so v Pacts lie vyskytmz addlest ,-1, s prnvnlép<><h»I>u<:stinfl`) (tj. nyil1c><h1A v¢·liGiu:1S(k) xmhfvai
}10<l11nly .-i, s prawlépoclolmnstiHawy pak v Paso I: + l sn-Malianst A, \·ysl<yrm· s ]>1zn·<l6—
pu<l<>h1msri. pm ktormn Platt
_(A·¢n»
_ ,_ _(k; , ,.- Z,11, ,
vlm za1]><;h10 v<·kto1·0v5
(.1,.1)
wf =¤'*>-w.
(4,;;
kcal xuatimzv W jc LTV. vnnticn parer:}marl. jcjii privacy HQ.,] 5 U'(.—l, A .4]ranji fyzxkélui
@*211:1111 p1ʻm¤rlép0d0b110st,i pfczhcrlu zv shavu A,doestavu] Fjsotmly 11c·zzip0r11é) ax jui
musi splfxovat, normovaci podminku
Z UʼQ,+] = 1 pm vécchna i.
(4.6)
KA~\PITOL.·\ -1. Z.—Ux'L.~\DY .\Il5TODY .\[().\`TE ('ARLO
Tam pcdmiuku z11a111vr1:i. Ze x k<mHg111‘:1<·0 .—\, vzuiluw (s ]>1’:1\·zI6]m:]r1|>11<>>ti_jr·rI1m) _j1·<h1a
x kuuHgurz1c·i .-\_,, j = 1, . . , AI.
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delmuvaz i Markovuv v-new ve Kerrimmeme spojity EasMi.swanfm napiéeme tedy x,(1). Smaleavesak pied
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konfiguraci v Case t — dl, Rm. (4.6) pkpiéeme do tvaru vr,(t + dn) - n,(z) = Z;. x,(1)I·\’M, — :,0) >< 1,
de za 1 dosadime z (4.6), Po vydéleni dx dostaneme uv.Hidi1-maini (master equation)
v
M
M Z .V · #i = Z».u>w.~,
—¤,<»>_$w,».,
n
m
n
_ un
de elementary WH, maj? nyni vyznam rychlosti zménystaveu aNepali pm né (4.6) normalizeace je zajiéténa
Durham élanm v (4.7),
Markovbv process icesediscuspecialim pfipadem obecnélm .¤zapIm.¤ru-w/mnebch milmzldu/m pmre.»1»,
coi je cystm néhodnychovelin definovanych na jnstém pravdépndabnosmim proszoru s hudnmanu v jiszém
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nazyvanou téi uajekrorie. NapF. pm vyée deiinovany Markomv Ferézec je jedna konkrémi posloupnosz konfi»
Gracie, nap?. {A7.A,4,,.-1-,,.~1,_,, . . .), jednou realized, které se vyskylne sJimu pravdépodobnosw v iexézcn
{s<*¤,.$*2¤,s<i**,.s<·•> ,... }. Teprove kdyi uvaiujeme pravdépcdobmmaimmuasmdimemluvit O pm.
CEOs'
MaK111v—li XIm·kov\°xv i0n<ʻzIIIIZOIIIQ Si ]><>lugthull mmiavk. nzxpiiklzul. \·y_)rlu» li xv smvu
.-L, jzxké jc px·zw¢l6po1lolm0sc who, Ze po k krociclx clojrlu do sum1 A]? Jak buclv zaivisct
vyskyt stave A] au k'? Tight oteizky csvétlimc ucjlépo ua pfiklzulu.
Mém vcancelii pnéitaé. Tnumi.vidy fungujc, :1I¤lmr§ijc=msc sitiRexuékcly fuugujv.
knuckley ue. Dlouhudcbym pozorovéuim jsum zjistil, in
1. l`engage—li sit! dues, jc 60% pravdépndobnosm. io buds fuugovat i zitm;
2. nefungujc-li sid duns, pak s [)l'8.V(léI)0(lOhXlOSEi 70% uebudc fuugovaxn anizitm.
Stav sitétady nahjvei <lv0u ll(J(\llOf,, .-1, ;:fluI\[§\l_i(*M an .-12 =.,u<-i'u11p_11_]<·". P1·e1v<l5pmlul>
oust v Ease k IOC pnpsnt dvuumzmémim v<·ki<>r<·n1
nm _(7r(k1 TRW)
XIexl.ir·¢· ]>h·cl1ml11_j0 v {umm piipmlf
_ 0.6 (1.4 — {1.3 0.7 '
Jvstliic Lczly véerabylsm.v sité popsén vckmormn ·1r(", pak pm rlncéuistamplatiz
rm. = Wm. ·W.
Tidy nap?. jc-li nm = (1,0), pak Wm. = (046,0.4), a je-li nt.') = (0,1), pak Wm.
(0.3, 0.7). Tact mho pckraéovm, aduh§ den mém rozloicni
Ti!) Z .,,0) , W Z ,r(1)_vV2
'T0m tv1·ze·ui ncni picsué. pr<1t<>21· ]n·uvdé]>or1<>l>xms!. n·:11izzu·0 1u:kz:mz¤?nrʼ ;n>>lmnpm>s1i Ixmlv u<·j<pil
uvula. Commie 11<·r·h|`si pi<··lx•m-i kmmrmm vyhnmuu p¤s1oup¤ms•,5 hm-m< ummlm-¢»u px;m1m¤»·l.»\»m»<¤i
4.2. BOLTS.-\NNO\'SKE` \`ZORK()\i-XXI
za Led
nu, :{(0.48,0452), je~1g1r<1>=(1,0),
(0.39,0.61), _pc·l11r(‘) = (0,1)
Pokraéuji-lidelc, zjistim, ic r0zl0Zcui 1r(") pm velkzi n nebudc ui vhbccZivsct un 1r(" n
dust LTV. Iimitni rozloicui
Jig·rr(")
: 1r é (0.42%,0.5714),
o
(4.8)
tedy pri'1111émé pravdépodulmost fungcvziin jc 43ApproverLou jc t.<um1 tak axleaverly<·ky'f
Zkusme uécojiného. Byl jsem na dov<>l<»11éa11<*vImo Adavévm sit`f11ng0vz]z1 rawlm uv. Jvrlvn
kolcga oak. ic sit; uofuuguvalu. Durham Lvrrli. iv sum. mV<lF;><><Liz>h1msri pm :1 prat gins
Lcnly ansi stejné a ]oxt.<·G11i stav prom vvzmorme vcr tvaxru
TH') : (0,5.0.a),
Jake ju te-dy p1·nv<l5p¤><luI>n<1>t. iw sit` bucle fuugovm, ni duns piijdu do précc`!
ww : M') · w Z (0.45,0.as)
a. tedy pravdépudobnust jo 45%.Bud-li nyni pokraéovat vn vypoétu pmvdépodolmostd
dél, zjistim, ie pn uékolikadducezh dostanupét rczloicni (4.8).
Ve statistické viz.n mis znjimzimeekui vcliéin. Zdc I]lG%(¥ jnko piiklzul vcliéiny (pom
rovatelué)sloet vydélckz jcstliie sirunguja, vydélém X(,,f11ngujc") = 2000 KE za den,
jostle rncfuugujan u<¤nmh11procuret Elbcdupoufc X(,,r1cf1111g11_jc") = SUOKG, Pr1'}r¤x5r1x§
vfdfvlck jc din stivdni h<><lll0[Gull
(X) = vr(,,fuugu_jc").\`(,,f11ugujc") + 1r(.,ucf•111g11_jv").\`(..114-i'm1gn1_]v") i 1143 KP.
Vatic piechodnx v xmivx _jud1m<l1u·l1ér11 piiklzulnn mai tu vlzxsmost,. is- >yst, 6xu xtmyi
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z jackalm r<>zl<wi<·11i jsuw vyéli. .·\ juketvum piiklZulusvisi s p1ivm111[111 p1·ohl<$1m·m \')'l>l`2\I\l;
p<>Slm1p110st,i. s ii hurIr·u1<· pr<><1 2ixvt kuufig11ra511i p1·ust<>1"Y CI·clu<1<lu§r\. XI6j1m· p0sl<111]>—
oust stem·fʼ1 Gilienyui ku11H;.g111ʻzui {.-\(A`>}f;I vybrzmuu z XIa1ʻk<>v<>vn F<~L<ʻz<·<· {s<·>,s~> .... }
s limimim mzloivuim
cap(·-Ebb) _coxp(—/1U])
(4 9)
E2; ¤x1>(#/wk) Q
de ssmc oznzxiili U] = Z/(AJ). Pak stfcdui hideout villainy X podl whom Fvtézco.
N=’
(4.10)
z 1 N rm. .x : - Z}, ,
" k;1
do XW : X(A(*)), bud pm I'(lSN)ll<'i 71 kOIl\'(ʼl'{.§OVi\\, k souhomvé stivclni hoclumé <lem<*
vztalwm
A
x
;
A
l
(.\‘> :Z¤,x(.4J; ;X¤,.\·,,
,
:
1
,
:
(4.11)
1
Zbfvsi ureat pm>cI111f11kv. An ktvrfrln ww = ww ·VV* I<<>1m·1·p,11_j¤· k limmanimumz)rwZr·r1i
1r. .I<·stli2<·r
CAPITAL 4. ZA·XKL.-\DY METODY MONTE CARLO
1. véechnyscabysou dosaiitclné z Iibuvcluéhoistavu v koneéum Easv s ncuulovou
pravdépodobnosti a
2. ing?Sanveniperiodicf (stav .4, jcperiodicy,jostlec cxistuje pcriurla m tnkové,
ic j(».11 7rf"ʼ : 0, pak ¤§”*ʼ“ʼ = 0 a jms 7rf"ʼ ye 0, pak wf”*ʼ“ʼ ye 0),
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nest nm cxistujn Imitate 1r = limbic.,¤, nm [33, 34]. Ruzloicui pmvclépnrlnlmcmsti 1r j<Mr.dy
r es en1n1 m v nuc c
vr · W = 1r
(4.12)
a \0t0 eoni jeJedé
I Jinymi slavey. vektor stavni 1r je vlasmim Ievym vekzorem szochastickémantics W.Lizeukaseaz, ie v§wechna
daléivaletsi éisla jsou v absolutni hodnoxé men§i nei 14
4.2.2 Urni matice pfechodu
Potiebujemc LCD zkcnstrucvat posloupnost (Markovfiv ictfrzcc) konfiguraci mk, shy sc.
pmvdépodobnust vgfskytu jeduotlivfch k0ufig11rz1<:i rcvnaln Boltzmaumové vzizc (4.9)ketchzi
Lak buds: picdscavovat limimi mzl02ur1i_jisté,satim ucz11ai1né mat.i<·<· p?n·c·honlu. Tom jr
priivé Opaén)? pmblém 1102 tcu, ktcrj sr uhvyklv véi v u~<u’1i \Ia\rl<<m>\·§’<‘}1 i<·L<*z<·f’1. tj.
k dunéMercki picechodunullzt limimi r<>zdélvni.
Pro ureaimusiccpitiedumeme colkcm Lii pm11l111i11ky
Wi.,] 3 0 pro V§(‘('hllH L,] : 1 .... , Jl (-1.13)
Z U'Q_,J :1 pm v:§0<·Iu1z\ 1= 1, , . . , N [4.14)
vr · W : 1r .
(4.15)
Poslcdui roving vyjadiuje podminku tzvl detailing rovuovaihy. Toto jc nutnzi podmiuku za—
ji§rjuicei vlastnostilimitingo (stacionérnihoMazolani pravdépudobrmsci. Umimc wi splnic,
puiadujeme-li silnéjéi podminku, tzv. pndminku mikroskopické revcrzibility. Staéi, nby
1r,U@_,, = wjlija,
(4.16)
a pudmiuka detailui rovnovéhy jcsplaynu.
I Plainest podminky (4.15) za piedpokladu palmist (4 16) sand ukéieme. aphkujeme-Vioperaor E;]
na obéstrainyrovinge (4.16) a dsaidme»Ii jedméku za Z;} H}., (viz rov. (4,14))napvav.suané
Rovuivv (4,13)—»(4.15) 11<=111·?uji xuatici pi0<·h0cl11jwlznozmxixné. Smstava (4.14) an (415)
Mr. pic<lsmvujc calked 2M rovuic pro Al >< M ruzzualxnyclx.
Vyuiijme ay who, ie ni = exp.(-/9U,)/Q je limit roxluieni. P0 dusazcni do (-1.16)
dostanemc bud` W E = E : cap[—B(U, — U;,)],
W,.] vt]
(4.17)
.4.2. 120LTz;x»1Am=ovs1y£ vzorazwxtsxl
11<·I><>N.b.a, = HQ.,] = 0. IOUl< _}r· vi<lPt. pmlui1l<» so mim zlmxil. ul<·s;><n`1 v ]m1n(·1·u |>1·.1x··
¢l5p<><lm1b11ost,(. ll<Ziplll<1`111xkc·m· Q: v u|u·<·116r11 pFi]m<i5 (_ism1—Ii 7, jim? 102 B|lu1xzn1n|u»vx
[:1:n\·ml5]><ul<>Iu1usii) tvum v(·»l<·<l<·k x11:1111r·u.i.vusim stm‘i xemulzxl _jm·u 1·m·lu¢i\‘11{ ]>1·;1wI<“·]>¤»
cI¤1}>11¤»sri za m·1uusi111<· »<· >m1·;1t 0 sm1?¢·! Fissilen,. Sumuo sv nyui |1h—svFrl5i11u·. iv 11nzui<·<·
;>i·m·<·lm<l11 <lufix1m·.:\ vnmhy
<»,.,/ ]>1`m>1:,£_/ .17}, 2 r,.
, u — pm r ¢ _; ·17r < rr,. u,,,_/:4 "’w, ‘ »'
_ (1.1b)
1— Z Wwk p1·n1:_j.
k.k¢x
kcal rap,] jr lilmvolmi #yn1.ctrirrk1i stmhastirxlcei (tj. Platt pro ui (4.13) an (4.1-j) u1:xli<·.
splfnujv podmiuku mikmskopické 1·0v<·1·1ihiliry. Gom jc muticc ruwricxné X[cL1 ·u;>olis<—m (viz
<·iL:1<·<·1 ua str. 12) A p0112iva11;i <]ucl11<·S. P0xx1z11rxc~m:_j11x0_j<·§t. 20prvrxi clvxiéxdky V (1.18)
lzv xnpsm k0111]>nktu6_ja1kc>
. , / _ I\,A,=u,,,,11111n |.— pwrfj,
W77.
I (I IJ)
I Mead (4 18) neon jedinéSymmetricé matice znémj jako Barksrcvf (néKay téi G|auber0va°] mé war
M -JjL;· wo ¤ #/
H], : ’ 1»ru‘,A
iH X pm,:,_
je via soave Kay |ep§i ne? Metropclisova.
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gismo pcdmnoiinu Ump Cellho kcnfngurainiho prosturu (YKatery piedpoklédéme diskrémi Niatice pfechodu
melody lepelné Iézné je
II',.] : <*xp(—JJ'(,))/ Z ·-xp1—.iL';) pro .~l,.¤\, 6 C,,,.,x. H 21}
·\4 EL`.»·.·.
Abychom belie kcnkrémi, zvolme si za (',,,.,.monouViech kunfigurad Iiéicich se jen hodnozou proménné nm
jednom mFiZk¤vémobdu Ir Tum prornénnou [spin) oznaiime .~,, PkAIZenpsm
Nb], » J,) >: ··x|»{— fl <.~§`1]/VZ/·x]>{— iI`(~j)].
(13;;
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spume .~y, vybirérne tedy s Bohzmanncvou véhcu dancu energxervvi konhguraci n teplotou, cci sivze pfedslavwt
ja|<0 pinion spin do levmoslatu Ei Mtepslné Iézné"0 dnné teploté T Spin tak Japcmens" na pfedchozf stav
.1 UVM, Halvasi na .—l, Nyni pfeclbéhneme ponékud vyklad a srovnéemetcdu mepelné Iézné s Mstropolisovou
meted. Meted tepelnéIdsé je vhodnou ahernahvcujostleie Nleuopuhsuva metoda sEastne neefekuvni
pm pFi|i§montho 0dmitnul;}ch kcnhguraci. Implamenlace Je v§ak néroinéjii ga zpravldla je zaloiena na tom, ic
pro Vietnam moiné okolijednohospinupeedm vypofteme a tabelujeme rczlcienf (4 21). Kviberupsmn pak
puuiijeme nap?. algorithms 12.2, viz. odd. 12.1 2 Pro cystm Spoufedv éma hudnotami spmu (lsmghv model
rufiikcvy ply) js malled tepelné Iézné tctoiné S Barkerovou metodou
J:\ A. Barker: Ans/. J. I’I4y,¤. 18. 119. (1965),
*1}. .},Glzu1hvr: I. Mu!h. Pays 4. 294 (1903).
INI (`xvvx. L. .I:u·nl¤> In ('. Il¢·|»1»i: I'dp Ifvr. Lvl! ·l2. 13Ul>(1!1T‘)).
l(.~\P1T()L.·\ 4. Z.Ylx`I..\DY .\I1TT<)1)\` XI<).\"l`]·.` ('.\H1,<)
4.2.3 Zkuéebni zména k0nHgurace
ZI>j‘\‘ai u<l]>m·5<l5t uu <>te\xl<11. co ju 11xaLiv¤· zu, ,, x·y>r11]>11_jarv 111zxli¢·i<·l1 ]>i<·<·l1u<l11 (-LIS).
pippin<I116 (»l.20)4 Tum xrmlinv 1r le\\·;i ]mml111i11f·1m11 [11·;xx·<l5p¤><lnI>1msl gm-r1<·1·¤mi11{ zkuirelrrri
k011iʻi;;111ʻm·n· A] z ko11{ip_111ʻa<·<· .-1,: v 11eisl<·el11_]icix¤1 <»<l<lil4· sv <l<>xvi111<·. luly l>11<l<· tum xl<11§<·\m{
k011Hgm‘z1<·pieatsnaxkcly u<I111it1u1ta14 Pm kl;1si<·l<¢ spunitf syst<‘111|:11<l<·m:·mit111ist<>111a1\i<¤~
f1u1l<¢·i u(rN —> r'N).
Vw st,x11<la1·:l1xi NIcL1ʻo[isuv6 xnmtoclé jo texto 111;1tir:<· sy111<·tri<·kzi, BIA1A110li rmly k<m—
fig1|rm.·i .-1,, pak l<011fig111ʻz\ci xl] 1x1usi111L· \*ygm1<ʻ1ʻ0\·:1t s p1·z1v<lG]>0<l<>l>11<>sti st<{j110u. jul<<>11
by<¢h0111 x l<<mHg111·z1c<· AJ g<211¢·1·uvnli koxmiignxmci A,. Pmlobné jr: mrxux vv sp0ji`é111 pfipmli.
kV prevue<l6pu¢l<>l;1msL jv 11z1l11·nz::1m lu1stu\.m1 p1·zw<lFlm<l<>l>11<>s1i: 11<·s1¤1{1m·_i<·u mp<>11x<·1¤<>m
11:1 tu. iv mm lnnxstutu p1·znʻ<l6pa1<lu|1u<>>ti j<· ¤l<·ii1u>m1u3 \·xl1l:~¤lw·111 kv I<nx·t<*x>.l:§ʻ|11 ]>1ʻ<>111F~r1»
u§‘1u 1"` :1111172vsvxu1611it.jc·s!li2e· T>y<·11m11 ¢·I1l6|i pivil k jiu{*1u ><»¤ni(n<|11i¢·i111 (rmpi. >.[<*1·i<·
Kim). Délvxuusi1men1itpfi u:iv1·I111 xkuxiulnxnilm |m>1111 uxi un 1>a11¤1M1 ¤·1;;¤><l1(11<>>r. rmlv zulu
svstérxnmulll (po111o<·i<·o 11n·j111<·11§ill ]mFt. l<1ul<1i) pi<·<·}¤Am~t x_]¤·¤I1u*<'·:1sr1 Im11hgn11.1<`11ilu»
]>1<>sl<>1`\1 <lu4]i116. spool·izil115 vv >]m_]it3”m·l1 u1<>1h·l¤·¤·l1 jr- m1\u<> nlzit ;>oy¤11· na m. nhy systrhnx
Suzy<Iu<> y>i·1·l<r>1ml vr·ll<r‘ ]1<>t¢·11z·i;\l<>v6 lm1·i<*1·y. uupi. u vrnithnivlnslatpfui vnl1u>sti. I’mluI>u5
11 1x<ʻl:t<·r¢<N 111Fiil<<nʻ{·m·l1 1x1<><I¤·1 i1 xufxIvanz1>mi i1 uu<·<·. lulv 11vui nm21u*¤·¤I1111 l<n11iig11m<·i
]>h·111611il m1rl111I1<>11 ]>¤mx<·]>usi11p113ʻ1uim\111¤*m11n14]¤~¤I1mtliv(ʻ<·l1 sponnfi; pak jv 1ml1uʻ 1116111\
v _dim·<h1<>111 kmk11 <·<·l<>11 slu1pi1x11 spi11 m1jc·:l11<>v14
Pm lm1nl;1·6t110st 11vr—c\<—1m- (lm ,yapir:ké pFi1:1m1_x·,
Miiikovy cystm s diskrétnim k0nHgura6uim prostorem
[[\ʻ:xi¤1_j111<· pro _j<·cl11<><l11m·!1<1sL :<yst111joluui p1ʻr11r1G11u6 mx l<a2<l6111 v1ʼc·]u>l11 nuiilcoy 1u1I1§\·zx']i
pmxzv Iii lu><her. kt<~1·6 ¤>x11uPiu1<· {1.2. V jvelnmxxn 1:1*0]:11 NIC i¤1u1ln<·1· I>url<·1ur· 111511it
]10<l1u>u1 [m\m· r1u_j<·clx1o1x1 v1c·lu>]11 xxniiilayz. Xlaiuw-li11u11F111 1<ulr1<wrn1 I. xlmsi1u<·ji x1u611i\
s p1·zx\*¤l(·pml<ylumsti 1/2 un 2 ex s m11t6 pxzn·<l(·pmI¤¤|nm><|( un $5. |u>:l¤>|»m`- 2 mm`·ui11u~
s pr2xv<l5]>¢»<luI>11<>s!{ 1/2 un 1 Gi 321 Il 1m 1 Fi 2.
I Mzmce xx pro vie uvedeny pffklad je obrcvskd — Ji\v>< J`, kde \' je poéet vrchclfn mF{iky Pokusmc se ju
zaps zlespoh pro ;\` : 2, tedy pro rufiitiiku sl sag, a pro zkuéebni zménupsnu ~J
111213
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1|2]
1|3|
2|1
2|2
2|a
2|1
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an
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u 1/2 1/2 n
1,2 nv 1,’2
1/2
an
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u
n
313;,33
u
u
nn an
u n 0
1/2 mn u u n 0 u an
U (l l/2 1/2 ll ll ll (lvm)
11
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( 1
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Pro napkini poiizaéovéhc programu nahésti nepoifebujeme zapsat explicstné ani rrvatncn u aniW astaéi ndm
implicit vyjédieni pcmociallegorizeu,
°4h·nnmz1um6I:¤lx· i v¢l1x>1ixu* mfunill x·iu· s >inf1 mn`<*¤Imm_ nn ni. vm- i·lvr·r<i 22 mmlmiulum 4-. .x|»x . . I . I l I
|xr.uʻrl¤`·]»u¤In|mu~II piijn-xi v )I<~tr<>p<>l1sm·Dusm hyly (los: v<·Ik<ʻ.
-1.2. l3()I.TZ.\l.-\.\'.\'<J\`SKE \'Z()I?Ix'()\Z$.\'I
Spotty classically cystm
N
\` pfipncli spu_jin!l1<>syst6uu1 sv p<><l111i11l<usy111<·11·i<·u(r"` `Av r"). : n(r'—>
rN)1u·z1n511i.
pummel z1°1s\z1n<·u1v ll ki\l'[(;ZSk§'('h s<>11Pax<l11i<r Pzistir. \'<: \‘5t§i116 siu1ulm·{ so v _]<el1 10m kmku
lnfhc pouf- s juclnou Pzismici. [)1'OL<)2C, junk vimv z ])i(!(“)5Zll#('llxivnh, so novei l<<>nHp,1u·;1<:<
110su1i pill li:Zi( od p1"1\·0Leni. jiuzxk b)‘<·h0111 0pettedsmli vvlmi 11upmv<lGp<><I0bu011 kon
1ip_L1rz1<·i (nap?. pi0l<1‘yv 1u0l<·lu1), ktuui by 11<·m<>hlzx INI, pfijaxm. Nu_j_j0<l1m<lu§§i mm·tu<l<>u
jo piiitvui 11i}m<lu6l1o Gislsx ·u(,,,_,,) r<>v¤1<>1115r115 l'(>Z]<J2(‘Il<‘]ll> v iure·1·v:1lu (ir/.z]) kv l:u2<l6
*211* Ms c6 sox uric vy >mu6 (ix i<·<· 0 yohoxo r : r,. z·, . r; . k t 1 l I ( [ l /
I", = r. +1/( ,:,4;
fl, = M, + ~(r.:..1»
(4-24)
rag : r: + 1:‘,,,;_,,).
4.2.4 Realize kook Metropolisnvy melody
\'m1·<~<· (4.lE)) z11a\1u<·11:i, 20 u15uu zkcuévbui l<o11{ig¤1c· vIanngi Prvrvkx-111 1uz\(i<:<· n,,,J I>u<l<·
piijnm s ])l`<·\\ʼ(l5l)()(lUI)ll0S|.{
. pm;] : mm
7V {1. .
_.(—l.2u)
mi v pfipz\<l5 l<a\u<mi1·k<*lmmL0%¢·11i mmGi
pm, : mi1x{1.s·x]>(—.iA(`)} .
(LEG)
AF ; (/·'(»l,) — l,`(,l,) E U, - LV,.
(4.27)
Jinfmi slavey. u1zim<·—livlcmlux I. ]<<>11{g11me·i .l,. lj. SW Z xl,. pak bu<l¤·m<· mitavk1·niu1
I.·+ 1 116l:<1y um·<>u l<u11fig111’:u·i .\, .x 116k<Iy >t;u·¤¤11 .·1,: z1hyr·l1<11 xs1·11Puili xzipis ul;;_m’iI1uH
(s \·5<I<1111im mlm. 2v¤·x]>limit11ixai|¤is 1uuri<· rx, ,, A UQ ,, m·Mt·ln1_j<·111:·). |n1<lv11u~ x11zu'·it
vfvlnoxi lu111fig.;11r.u*i .-IW. xl;u$e~|>u[ l;u11fi_s;¤11‘n<·i .-l’k"`. ax \’i'S]<'<!l|0Il lmu{ip_111·n<·i po ]>mv¢·:l¤·n[
l<1·<1k11 .W"° U.
Nigh truly ii 11uW2<·111¤· S}ll`IlU\l| s1·hrʻ1u21 j<·<l1¤1l1u Nlmmlv Carlo kmku pm r1<{jb521x5j§[
si1m1lz1<·iMTVr<>p<>lisovm1 1u0L<><!m1:
1. Zvolimv Geisticwi, k((ʻl'l)ll sm- l>u:lrwhylmt, 111Fi2l<<»v§ bod ax pn<l0Im5. Czisaycv jv xnmimi
[)l‘i)(']l2i%(‘I` systr·111:xti<:l<y v c*yl<I 11(jsm1—li sr<j11<}m rlurchu) uvbo vyhirzxt 11:ih0zlnF4
2. \'yam><ʻl.r·1m· xku§<·111 k011iig111*z1r·i .~lʼk"` mk. iv z1u511i111 m· 11zipl1o<luF px>l<»l1u (<n·iv11lm:i.
spin) x·yb1·u1nGistau viz mls}. 4.2.3.
ss. \ʻylmm·m(· AU : Lʻ(.»1**·*~) — L¤(.4<*>) E ri**·~* - U".
1. Kx>11ii;.111zu·i pfi_j1r1<·111<· (.r\'*‘U I .-Vk"`) s |>1·;mlP]1<>¤lohrMtsppm pmllv (71.26). zu
ti111<·¤mnp;x51nu11Ina1v<lGpmlu|n1<>¤1i 1 —]1|,;,,>]1 ml11ni111<·11u· (.l‘*"’ : AMY). 'I`<1\ i¤—<‘i
p<>(·i1;u‘¤>x·¤¥lm [>m;;1.x11111 m.um·ue\. Zn x5·;;m·1n<·111_j<»111m· 1;il1ml11U (slu 41 = rxml,. vm
rm. (12.1). .lm·-li u < ppm. pak nl<u§<~h11i kn1niig111`m·i |>i’ijm<·1u<·. V <vpnF116111 ]»ii]>zuI<‘
|m]c1·;u'·\1j¢·111¤·>m·>1;m»n1 |;<nnI1;;n11·z1ri4
3. lʼ<>lu·u<ʻ11_j<~111r·lmml<·1u l.
I{.—\lʼlTOLA~\ -1. Z.—YIx'L.-\DY A\Il?'!`OD`Y MONTE C.~\RLO
.-\mv§<·<l5lzi111ctex1Louhu. xisluixxw d0stext05113p.n.évt k011{ig¤11*:u·{ ]>mi<·I>u)°c;h lc vXʻpnPh1
l`vxil:2iI11i<:l1 v<·liFin.
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4. Generujeme u : lm,. Je-Ii u < <·xp(—,iAU), pkAzku§ebni Imnhguraci pfqmems, v opainém pfipadé
pokraiiujeme se starch konfiguraci
Z2 druhé,checke—|¤ u§etHt generovéni néhodnéhcElsievénmneme sul ie pro .\I' < U (zku§ebni konligurace
je enevgemicky v§h0dnéj§{) jé [qw = l` a redy zkuiebni konfnguracjeevidy phjara PodmikA nen zni
4 1a—|¤ .\l` < ll, pak zkuéebni kcnhguracu piqmeme a pckraiuyeme bcdem 1 , v opaéném pfipadé ge
nerujeme u : num) a je-Ii u < ¤·xp(-`i_\(`), pak zkuéebni kcnfngurari pinjmeme, v opainém pfipadé
pokraiujeme se steam konfiguraci,
Piilclmly i1upl<—mm1Lnc0 M<·tmp<»Iisovy mvmyjsnu \l\'(}(]i!!l}ʼ v a1l;;<u·it1u<·<·l1 -1.2 a —l.3.
4.3 Zlomek pr1_]et1 a nastavem parameter
Dfxlviitmm ha\1ʻ2 \ktm·ist,ik<>11 NIC siuu1I;u·{j<· tm: nl<111u·k piijvli (au ¤ ¤·|>1n1<·<· mtin). <`·ili pw
l\)i‘X'
]><1I`<·{ pI·i_j»uXʻ< I1 km1fi;;111m·i
p<>(·<*t vSm~m·h p_<·1u·x*<>v;111f·m·h l<n11H;:_111ʼm { l
UN)>ml
l[\ʻz\2l1_ie·1x1<~-li siuu1lm:<· klnsi¢·kXʻ<·h >]m]iI{*r·l1 syst61u1°1. Ivmlv lvduo p<mu*r zir·_jux<* aeivisvt ua
\ʻ<·li|<<>>ti Almévlmilm posuuuli xl v 1¤w11i<·i (-1.24) Gi v jim? pn<l¤>|¤11¤* l5u<lm·-linap zl piflii
1u:1l¤*.himv x11x511u 1·11c1·pgi<· v j<·<l1u>1n lqnooku uml;}. Bultx1mu1111`i\ʻ l`a1l:(o1· 1·x]1(—,iAL`) |>lixl<§ʻ
_j<·<l1niDc1~. Lé111&?kui<lzi l<<m|ig111ʻm·<· I>u<l<— piijz1La1 ax X Iyudc hlixl:0 j1·<l11i?<·c·, zxlv t;xl<6c·i2·l;1ivim
si11111lz1<·vl111<l<·uizkaiprimvk<>11fi;;1x1·;xc·r·s<·m><lscl1z·11x;\l11li§i. I3u<l<·-1i1Iv¢·ll< 6|1ml<·xu15u:1
c·11<·1·gu~ v jc<h10 kmku takVlkzi 21 VLFt§iu<>u klzulmi. B0lt,xn1num"1v {`ul<t<>r m·xp(—HAU)
Inlizkf 11ulc sax téméf knidzi l·(UI)HQ,|\l`Z\(ʻ(ʻ budc 0<]n1it1u1lz1. (:0% j<· zy]1116 tuké 11¢·<ri`m·l<tiv11{.
Exisfujv p1·nt<¤_jistei optixmilui h0<[110t:1 zlomlcu pfijmxt§ʻ1·l1 ]{<>IlH|;lll'%\(. 0 kitcré sc clloulm
1121ivu5 pFn<IPA>klzirlz1l0. 20 musi bf! blfxkei 1/2. xwvéjéi \·fʻzk11111)ʻ [15] véuk nkuzuji: 21- mm
<wpti11u1m lvii vv v5t§iu5pip1<l1'iOlliw \ : l/3.
I PiesnéjiFeeno, je nutno optimalizovat nejen délku krcku pudle (4 24), alcely (var (pravdépodobnosmi
rozloieni) zkuéebniho p0sunuti"Jibi jen fyzikélni Uvaha vede k pcznatkn, ie rovuomémé rozloieni v krychh
poodle (4 24) as nebulae pro Iwcmoganni A uzotropni xekutiny Lim nej\ep§im a ie zkuéebni pcsunuxi by mélo by:
sphericysymmetricéAndto je neefeklivni mit piilné krétké posqunuti zérovefx s d|0uhymi,jakje1cmu pil pouiiti
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knack mileage p0tFeb0vat nlznédelky pro vzorkovéni rhznychastephh volncstiLze proto dopomiit nap?
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Ve specnélnkh pfipadechMalie bytcpmu:Hn€ ` jiné. Nap?. pro tekutiny tuhych téles je mnhemniiéi
[ai 0 1), cci ovamsoavesi s typnckym zpnlscbemimplemente,JinnmpickledmJouFid.é sysiémy, zvléité
blink kvitickéhobcdzzede je efektivnéjéi mit délku zkuiebniho posunuti srovnatelnou s velikcsti simulcvaného
systemu i zaenuvellumimaleho zlcmku piijatich konfiguraciFeba 1%) mei malou zménu konfigurace vjednom
ki c k
".l I(lulu|`u. Mn!. Phys. G3, 559 (1988),
La. zr,0m?1< PRIVET A NAS'1?\\`E.\’[ PAR.METROlé'
INTEG ER N pu ul umm
VECTOR r[1. 4N] ku11fiqumr1·{r,};;,
QREAL Epa:ir[1. .N, 1. .N] inlmllsu Iuimnju/z wrwyyif
REAL Ezkus[1. .N] ;unwire’ 4-m·vyu· ;An.!··lmi I.¤»u/hgmmw
RE A L U t e: . U zk u s , U
REAL u(VECTUR r1 ,1*2) funkwz 1nm:c]I1;i pviywvnu rrmyxz u(|r; — rl \tunui .s ymlalmmi r1 u r2
(vzzlzlmlrm k vkrujaugfvn purlminkrim Mx ulynrzlmm 5.1 nrlm 5 2}
REAL rend() funk vrnrcjisi nzilwrlnéisu ¢rmlcmnl1z (0.1)
INTEGER i , j
REAL d vwlxkvst Ku.§r·I1ui!w pus1muLi
[ʼv'mlrm.· 1uq:l1uYui Lmlky pzimzqjclz cucvyvi
FOR 1: =1TD N— 1 DCI
FDR j := i+1 TD N DCI nigh (1.;} ]n·m·/nizi r·.§wr·/my prim zT¢I.slz<·, 1 < J
Epa ir [ i, j] ;= ¤<r [ a ] , 1- [ 1] )
1z p¤1r[J,1] ;= 1zpa1r[¤,j]
Jmlrn reckless AIC swmlzuw (I MC km}; » lm5:l1]m utmnwwn}
Utica := 0
F ¤R i : =1T U N D D
:I.u.§z·!mi yuisunuli ufunm 1
VECTOR rusks
rzkus.x ;= rh].; + ¤·<m¤1(>·0.5>
rzkus.y := r[1].y + d*(rend()·0.5)
mms .; ; = rh ].; + d ·<md ()—0 ,s >
erg]pm?1eL starr? u um? u1m1yu:.· .·mmEuL pics n§m:Imightumy 1·1Zz1ui ml i
U zk u s : = 0
U := 0
F UR j : = 1' 1`UNDDIF j < > i THEN
Ezkusijl ;= ¤<r[j],rzk¤s>
uzkus ;= uzkus + 12zkus[j]
U ;= U + 1;pa1r [ 1,j]
lllrwugmlzwiu mL: k yr /?ullz1mmnum ku1mt1mm
IF um < exp.<-<uzkus-U>/<k»‘r>> man
knvrfiymvurr; phjnlu
r[i] := rusks
U tah := U na; + U z k us
F UR ] : = 1c c > Nod
Epair[j,i] ;= Ezkustjl
Epair[i,j] ;= Ezkustjl
ELSE km1jiy;n1·ur:u rnlrrzfmulu
Ute: := Utica + U
PRINT "energie configure =", Tot/2
p1”is;u?w:I: mIku5rI1*]10 p1i1mjs1nc zupnéclli rlvakrril, protofucot/2
Algcnritmus 4.2: Metrupolisova MC simulate N awxm'1 n vjpoéctvanillini cucrgic.
1{AP1T(JL.~\ ·I. Z_—U(L.-KDY.\Ili`T()D)' .\l().\'TE ('.\}?L() [
INTEGER L mzlxkast Harry m%icky
SPINTYPE s[1. .L,1. .L] kun/immx 5,,; SPINTYPE In {yap apunumfyu·u1ummʼ
REAL u(SP!NTYPE s1,s2) funnels m·mEliz:iinlay1·nk:?ni ¤·1unyii m:jblzi~Ti»·/1 .»m4.w:lvl
INTEGER 1, j
xrrrzcza p1us[1_ _LJ. m1¤us[1. .1.] ¤m1my.`m.m11z
REAL Uzkus, U
Iv:¢lcm.· mrlczy sauserhi
FUR 1 := 1 T0 L-1 D0 p1us[i] := i+1
p1us[L] ;= 1
FU R i : = 2 T D L D U mi n u s [i ] : = i -1
mi n u s [ 1 ] ; = L
jcricn reckless MC szvuulnvn (1 .'\ICk1·nk s kuidyjm poxm·m}
F D R1:=1T¤LD 0
F OR j := 1TOLD0
zLn§1:b1AizmE1m spinn uuMichaeln i
SPINTYPE szkus
szkus := {nuvziruilnulnzilnulnvr;1}
jimmy mui1m.vt.· szkus := s[i,j] + (nmhis_v11wrrirʻk:iznuʻus|)
pm :lmm.mmmnj Ismq1iufz:1·u1nnyvu:lsulfai szkus := -s[i ,j]
vyjpnfct surd a novéchummiersSaont pics u§c¤·lmy neybhiii sunuzly
U ;= u(s[i,j] ,s[p1.us[i] ,j]) + u(s[i.j] ,¤[minus[i] ,j])
+ u(s[i,j],s[i,p1us[j]]) + u(s[i,j],s[i,minus[j]])
Uzku s : = u (szku s, s[ p 1 u s[ i] , j] ) + u (szku s, s[ min u s[ i] , j] )
+ u(szkus,s[i,p1us[j]]) + u(szkus,s[i,minus[j]]).
.M¢·!v·a;ml1svZu test; k ye Bnltzyrnmmmna kmmtuum
ur rend() < exp.<—<uzku¤-U)/<k·T>> man
km:/!y1mu:1piffleum
s[i,j] := szkus
A` iJ
Alguritmus 4.3: Metmpclisova MC simulacra nu puriodické Etvurcové miiicc s iutcrakci
ngzjbliiisiclx suusedmi.
()h1e3x¤·k 4.1 11kz1xujm· mivi>a1<¤>t <·l1_v}>\’ mGou[tulleu (pmllv virizilwvé stuvnwi 1·uv11im·<·
[5.12)) pm s_vs1.611x G4 Lr·1u1auʻ<I-.I<:u<·s<>v§ʻ<·l1 z\r¤11m“1 \ʻ pc·1ʻi<>rliek_{ʻ<·h uk1·z1_jmʻf<·l1 p<><i1u[11l<;i<·l1
au dlcu xkusiclmiho posuunxti :1 zlomku pfi_jz1l§<·l1 kunHgu1‘n<·i. \'irlin1<~, 20 cpzimziLuisy loii
v i11tc1ʻvz\lu [0.2, 0.3].
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MD, nap?. Verletovy Prctoie bud stejné provedenMetropolisv {est, nezélsii na kvalité integrate a je moiné
pout dlouhéintegralnirkkyAtio mstoda. v které seeméni najednou polohy v§ech ééstic, se nazyvé Ixylwulni
Mints Carla? a je pro nepFi|i§elké systémy pFekvapivé efekzivni.
U 1IlHZl(0\'5'(’l1 systénnh x diskrétuimi slwupni volnosti zpmvirllu<-mzimc si1m1lzxFN{ pn
1ʻzum:t1ʻ, jimi hyclxom mohli zlomck pfijvti <>vli\ʻuiKTun vézxk xzivisi nufoxyk:ilr1i<·l1 pod
Illillkéifh. Nap?. pii uixkich mplutaicclnbumh-Bolusu;u11u'i\· i}xl<t<>1· ¤—xp(-Jil`) vvlmi mu|(· u
si11111Izn·:~|n1<lr· m·<·f}·l<tixʻ11i. X6 k<1y lzv t<·11m ]>1·ol>|<111 xmimittnap? vfiv n111i11611<m 1m·1<»¤1m1
Tsp Duzmc. A. D, l{0u11z·rh·. B. J, Pvxncllvtun zx DRm.th Ph!sirtn L1·LL1·1.¤ R 195. 216 (1987)
—1.3. ZLOAIEK PIXEIJETI A ,\`.—\5I-\\V`ENi I?-XRAAIETRU
0,020 I?
0.015
0.020 <>
W 0.015 O O
` {11110 ,— 0 4) -{ l)(llU 5 OO
0O 0
0OO
lmnax .. 4 W z _,,, A ,,;r,,.._A (mos W`;. 1 ,,4, x ,7 .».. ..4
ll U.1 11.2 U3 U.-I U.;} \l.G 0.7 U U1 U.2 ll.;} U.·| U.5 ll.6 0.7 11.8
Orbsk 4.1: Zévislost. smérodatné chyby méfeni viriélohu tlaku dl" na maxEmilni délcc
{ zkuéebnihoziskén
pusunuti
d valveo) an na zlomku piijatjchconfigurei X (vpruvo).Kidy budbyl 7
z 10MC krokni Kimk = pokusPhnomtJeduEastici) na systému 64 LJ atom;}
(a = 1, s : 1) pii T :1.2,;;:0.8.
LcpclnéLizi. zvlziit,5jostlev 111511imv v jvrlucmx k1·ul<11 <·:·l¤>u :»l:upi spi1u”1. Jini ulgmix
mus spoilé vv vytvuicui s<¤z11sum1 v§<u·I1 u1o2n§<·h :·lu1uc·ur;i1 ·ui<·l1 uv _i<·<lu<>k1ukn\ʻy“<·l1
ZIIIFII ku11Ggx11ʻzu·0.Z uich so pak vy|>rr1ʻc· jv<h1:1 xm zziklmlé >a1·m·n;h1i_j<·_jib1·I1 p1·zw<l6]w:Iw|¢—
nest. \:}'l>l'?lllé zrnéuzx _j<· pak virly piijzntzx. \‘:u‘ieu1t:1 mmimic: ;1lgm·ih1u1 hm Il(‘\’lS[)§§ll{'i'|l
;><>]<11s1`K h11<l<· [>0pszi11:1 v kzxp. 9.24
Pro s{>c·<·iz\lui x1xiiikmʻ1* >\·>I61\ʻ vxihluji i mv. klzxsmmi1l g»1·itn1vR. k<Ir· sv mGui <·c·I:i
r1c·p1ʻn\ʻi<l<·111zioblastspnnui. lvm p•»»l\1py_]rI<>¤1 v§uk jii r11i11m1ʻzi11n~<· XI<·tr<>p<>lix<>\ʻy 1x1r·tu<l)ʻ
A ¥(%<ʻ]1I0 skript.
"\’iz [15] uvlm R. HWSun<ls<en, J »S. \\'uup; zn A. M. Fcrrcubcrg v [18].
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