Magnetisation Probabilities And Metastability In The Ising-Books Pdf

Magnetisation probabilities and metastability in the Ising
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238 L S Schulman, where the sum over configurations CL includes only those having magnetisation m p. the inverse temperature will throughout this paper be taken larger than pc 0 44 so. there is a first order transition For h f 0 the constrained free energy merely picks up a. factor exp phVm since the factor exp ph a is the same for all having the same m. The partition function as a function of h is,Z h exp PF mk phVmk exp P 3. where mk is the magnetisation when k spins are down i e. mk V 2k V 4,Equation 3 also defines the total free energy F h. In equilibrium the probability of finding the magnetisation with the value m with an. external field h is,Ph m exp pF m p h m V j Z h 5, The apparently trivial h dependence in the numerator is deceptive and for meta. stability in some sense false the issue being what states should enter the sum in 2. For the present we avoid this murky issue and use the licence suggested by 5 to. concentrate on Po the zero field distribution,Po will be calculated by three methods.
ij Heuristically arguing in terms of droplets large and small to identify the largest. contributions in the sum 2, ii Doing the sum in 2 exactly for mk 0 s k s 6 and 0 s V k s 6 This sum for. k s 3 is in fact the only work we have been able to find on P m in the literature Siegert. 1955 An exact calculation of a quantity related to P but not equivalent to it is to be. found in Gaunt and Baker 1970 j, iii A Monte Carlo technique applicable for all m which looks at virtual changes in. m and computes P m from the transition probability and the principle of detailed. A stochastic evolution can be assigned to the Ising model and the associated master. equation can with rather strong assumptions be projected to give a Fokker Planck. equation for the diffusion of the quantity m The potential in which m diffuses is. essentially log P m and the Fokker Planck equation turns out to yield the Arrhenius. formula for the lifetime of the metastable state, Throughout the paper and in particular towards the end we point out the pitfalls of. trying to understand metastability in terms of probability peaks Notwithstanding the. derivation of the quite reasonable Arrhenius formula our overall conclusion is that this. is not the best approach to metastability,2 Heuristic calculation. Suppose T 1 p is well below T and h 0 For the infinite system there are two. values of spontaneous magnetisation and for the finite system we expect POto have. maxima near fm the values of the spontaneous magnetisation corresponding to some. Magnetisation probabilities and metastability in the Ising model 239. Consider Po m for m m E E being a fixed small positive number A configura. tion contributing to this probability represents a fluctuation away from the most likely. configurations The most likely configurations at a given T have some distribution of. spin down clusters and a configuration with m m E will have fewer and smaller. clusters If the probability of such a fluctuation occurring in some large volume is p then. the probability of its occuring in twice the volume isp Hence log Po ms E Po m. is proportional to V In terms of free energy this says approximately that F m E. gois proportional to V goitself is also proportional to V. Now consider Po m for m m E A magnetisation below m can be obtained by a. homogeneous fluctuation throughout the volume and configurations of this sort give. contributions in the sum 2 which decrease with increasing V exactly like those in. Po m for m m However for E not too small there is a cheaper fluctuation in free. energy currency to reduce m Most of the volume is left at the equilibrium value of m. Le m while a small part is at m m a condition in which its free energy is just as. low as the m state The only free energy cost comes from the interface If a volume. V1 is in the m phase we have,m V m E V m V VI m VI 6.
The length of the interface is 4 Etimes a geometrical factor g Solving for VI from 6. and letting IT be the surface tension free energy length. F m so J 2 m J vJm m, For m well away from m and near zero the droplet stretches from one end of the. system to the other and the length of the interface is just 2N independent of m For. such m we expect,F m 90 2Nu 8, What stands out in equations 7 and 8 is that F m gobehaves as JVrather than. V Consequently for large enough V two phase contributions to the free energy are. more important than ho mogeneous fluctuations The depth of the minima at m for. h 0 is therefore O JV Turning on a magnetic field introduces a contribution. hmV O V For large enough V the magnetic field triumphs and the local minimum. shrinks to insignificance This is an important limitation on the idea that a metastable. state is a local peak in the probability distribution or local minimum of free energy. Note that this limitation would persist had we taken the surface contribution to be V. with any U 1 not just f as suggested by some droplet models One can also see at this. stage why metastability can be more easily defined for long range forces since the. definition of an interface essential to the derivation of the dvfactor can only be made. with forces that decrease sufficiently rapidly with distance. 3 Exact calculation of Po mk k s 6, The sum to be evaluated is given in equation 2 Note that we are actually calculating. F m not P For k 0 1 2 3 the probabilities have been calculated by Siegert 1955. and Yang With increasing k the numbers and kinds of configurations increase rapidly. What must be determined for each k is the number of reversed bonds With the help of. a computer we have evaluated these numbers and our results are presented in table 1. 240 L S Schulman,Table 1 Values of expi PF k for k N Jv. on the N x N Ising lattice Q, exp 4 T Let exp PF k VQ H k Then we further define.
H k Q2k Gk,Listed below are all non zero Gk for the given k. k 4 G40 V V V 30 3231 1254 24 Gdl V2 21 V 118,G42 8V 85 Gd3 18 G d 4 I. k 5 G50 V4 50 V7 995 V 2 9310 V 35424 120,G51 V3 3 13 V2 536V 3 812 G52 5 V2 132V 926. G53 30V 400 G54 V 43 G55 8,4 Stochastic evaluation of P o m. This is a variant of the Monte Carlo technique one of whose first uses was the. evaluation of the partition function for the Ising model Fosdick 1963 We also use a. process in which two spins are flipped conserving magnetisation see Kalos eta1 1978. One can interpret these spin flips as dynamics or as a way of finding the principle. contribution to the sum 2, For given k or m and a randomly selected initial configuration we consider the.
configuration generated by flipping one up spin down and one down spin up The. selection of spin flip candidates is random If the double flip lowers the energy the. configuration is so changed If the flip raises the energy it is implemented with. probability exp AE T The system then relaxes to those configurations figuring most. prominently in 2 For some k and 7 relaxation may be slowed by the kind of. metastability considered by Kalos et a1 1978 We did not study this phenomenon. Next we consider the outcome of a virtual spin flip That is we randomly select a. single spin and evaluate A E if it were to be flipped If AE 0 we record the virtual. occurrence of a transition If AE 0 the virtual transition is recorded with probability. None of these transitions k k f 1 takes place The system remains with k spins. down and only the double spin flips actually change its configuration A record is kept. of the number of k k f 1flips which would have occurred had the single flip transition. been implemented Then allowing for the variation in the numbers of available up and. Magnetisation probabilities and metastability in the Ising model 241. down spins for different k we obtain the ratio of transition probabilities k k 1 and. starting with the k 1 states k 1 k, By the principle of detailed balance which has been built into the microscopic. stochastic dynamics transition probabilities and the equilibrium distribution are. related by,Po k W k k 1 Po k 1 W k 1 k 9, where W k j is the transition probability for going from k spins down to j spins. down All that our double stochastic process obtains and indeed all that we need are. the ratios,W k k 1 W k 1 k Po k 1 P o k 10, Having obtained this ratio for all k we normalise with the condition. 5 Results of the calculations, The forthcoming results represent a combination of the exact and stochastic methods. For k c 6 exact results were used at the same time checking that the stochastic method. probability ratios came out reasonably near to the exact values see table 2 For k 3 7. stochastic ratios were used In figure 1 is a typical graph of log Po against k at T 2 0. and N 19 The minimum value is approximately log 2N and the curve flattens. towards m 0 as expected The minimum occurs at m values just a bit larger than m. Table 2 Comparison of exact and stochastic probability ratios for k from 0 to 6 spins down. Also given are the spontaneous magnetisation m as predicted by the calculated P k and. the exact theoretical value The quantity listed is Q k log P k l P k Note that. because a logarithm is tabulated it is the smallness of the difference between numbers that is. significant rather than their ratio In our units T 2 27. T 2 0 N 19 T 2 0 N 15 T 2 0 N 13, k Exact Stochastic Exact Stochastic Exact Stochastic.
1 1 26 1 31 0 827 0 832 0 572 0 552,2 0 919 0 949 0 517 0 582 0 289 0 268. 3 0 691 0 752 0 320 0 386 0 116 0 158,4 0 526 0 519 0 183 0 204 0 003 0 012. 5 0 400 0 373 0 085 0 040 0 075 0 013,m 0 911 0 910 0 911 0 914 0 911 0 914. T 1 8 N 19 T 1 8 N 15 T 1 8 N 13, k Exact Stochastic Exact Stochastic Exact Stochastic. 1 0 836 0 883 0 411 0 421 0 165 0 141,2 0 510 0 550 0 126 0 147 0 088 0 001.
3 0 298 0 404 0 048 0 048 0 232 0 226,4 0 145 0 062 0 162 0 149 0 317 0 348. 5 0 038 0 011 0 238 0 252 0 368 0 280, m 0 95686 0 95836 0 956 86 0 956 43 0 956 86 0 956 64. 242 L S Schulman, Figure 1 log POagainst k number of spins dowrr for T 2 0 N 19. To check the heuristic assertions of 2 we first note table 3 There we study the N. dependence of the height of tke maximum at m 0 We think the case is good for that. difference behaving as N JV rather than NZ In figure 2 is a plot of log Po 2against. m again T 2 0 N 19 By equation 7 for m m and until the curve flattens. logPo should be a straight line and indeed the fit to a straight line seems good. Moreover for various N values of gcr can be deduced from the straight line fit An. estimate of T alone can be obtained from table 3 and formula 8 Thus at T 2 0 cr is. about 0 7 For lower temperatures U is found to rise data not in table 3 and at T 1 0. approaches 2 0 which is the energy cost of a broken bond showing essentially no. entropy contribution at this low temperature, Finally we check that for Im I m log Po does indeed scale as N2 V In figure 3 is. plotted logPo logPo min N2 for various N at T 2 0 The constant is put in to. take care approximately of overall normalisation The curves are seen to be. Table 3 Probability of m 0 for systems with spontaneous magnetisation T T as a. function of system size Listed below is AF T log P log in for various N on. an N x N lattice where the maximum occurs near m 0 and the minimum at m near m. the spontaneous magnetisation,T 2 0 T 1 8,N AF AFIN AFIN2 AF AFIN AFIN.
7 12 9 1 84 0 26 15 2 2 17 0 31,9 13 9 1 55 0 17 19 9 2 21 0 25. 10 14 4 1 44 0 14 21 4 2 14 0 21,12 16 5 1 37 0 11 25 3 2 10 0 18. 14 19 7 1 4 0 10 27 6 1 97 0 14,15 19 8 1 32 0 08 28 8 1 92 0 13. 17 23 0 1 35 0 08 33 1 1 95 0 11,19 25 5 1 34 0 07 38 9 2 04 0 11. Magnetisation probabilities and metastability in the Ising model 243. Figure 2 log Po against k for T 2 0 N 19,Figure 3 log PO against m for T 2 0 various N.
reasonably parallel to one another for m m while for m C m the incorrectness of the. 1 N 2 scaling causes them to separate, There is one significant if puzzling feature of figure 2 to which we call attention. Note that the straight line fit to log Po seems to pass through 0 at m m rather than. through log N This feature is borne out by other graphs at other T and N not. reproduced here Of course it is only the extrapolation of the line that hits zero as the. form 7 does not hold so close to m We remark that this property does not arise. because of any confusion of k versus m dependence as the graphs in question have the. same normalisation for all values of k i e equation 11 It may seem strange to worry. about log N terms but in the next section this will be seen to be crucial for the recovery. of the Arrhenius formula,244 L S Schulman, We summarise our results for Po expressed in terms of the function F m. F m NZr m p logN 95 12, Equation 1 2 defines r as a function independent of N but depending on T and m We. take r m 0,b For m E m m l S g im E to be specified below. F m gdV J 2 m Jm m go 13,c For 0 S m S m l S g,F m 2Nu go 14.
For m 0 F m F m An overall constant may be added to all terms to allow for. or dm normalisation,For h 0 defining Fh m in the obvious way. Fh m F m hmN 9h 90 15, There is a small but important range m E m m within which we have not. given F m nor have we given a precise estimate of E Within this range the single large. droplet competes with the volume distribution of smaller droplets as major contribu. Magnetisation probabilities and metastability in the Ising model 239 Consider Po m for m m E E being a fixed small positive number A configura tion contributing to this probability represents a fluctuation away from the most likely configurations The most likely configurations at a given T have some distribution of

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