ELSEVIER Physica A 208 (1994) 31-34

m g l

Energy relaxation in Ising models Markus Siegert a, Dietrich Stauffer b

a Computer Center, Cologne University, D-50923 K6ln, Germany b Institute for Theoretical Physics, Cologne University, D-50923 K6ln, Germany

Received 16 April 1994

Abstract

Starting with all spins parallel, s imulations show the energy to relax at the critical point as t ~- l) /~z with z near 2.2 for 1699802 (no logari thm) and 2.1 for 28003.

Most of the recent attempts to determine the dynamical critical exponent for the Glauber-Ising model from large lattices and moderate times concentrated on the decay of the magnetization [1] with time, right at the critical point. For cluster flips [2,3] or Creutz cellular automata [4], also the energy was investigated, and we do the same here for Glauber kinetics.

Right at the critical point of the two-dimensional Ising model, the energy is expected [2,3] to relax with time t as t-Uz log(t) to its equilibrium value, where z is the Glauber kinetic exponent for single-spin dynamics [1]. Monte Carlo results from parallel computers, for systems nearly as large as those of Miinkel et al. [4], determine an effective 1/Sz = - d l n ( M ) / d l n ( t ) from magnetization M and analogously from energy E at consecutive time steps [3,4]. The results from 1024 transputers of a Parsytec Gigacluster, Fig. la, give slightly higher z than from 140 Intel Paragon i860 nodes, Fig. lb, for lattices with more than 101° sites. Similar results are obtained from longer runs and smaller lattices, like Fig. lc from 64 Intel nodes. None indicate the presence of a log(t) factor in E; Fig. lc shows explicitely the bad results if a log(t) factor is taken into account in the analysis.

For 28003 similar convergence to z = 2.10 was found on 140 nodes, Fig. 2; the magnetization and energy are then supposed [3] to relax as t-Mzv and t (a-1)/zv where a = 0.11, fl = 0.32 and u = 0.63 are the critical exponents for specific heat, spontaneous magnetization, and correlation length; a = 0, fl -- 1/8, u = 1 for two dimensions.

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32 M. Siegert, D. Stauffer / Physica A 208 (1994) 31-34

This lack o f evidence for a logarithm is surprising; but we know since decades that F~d/kBT and a ~ d - 1 / k s T approach in 2 < d < 4 dimensions two universal

constants at the critical point, whereas for the two-dimensional Ising model the quantity involving the nonanalytic part F of the free energy has a logarithmic divergence and the one involving the interface tension a does not. (Here ~ is

2 . 2 5

2 . 2

2 . 1 5

2 .1

2 . 0 5

1 9 5

1.9

169980"169980 critical Ising: M(diam.) and E(+) without logs

(a) 'M' o o ' d E ' +

o

<> o +

o o < > o o

o o

+ o

o +

t +

+ +

+

+

+

+

i i i t i

0.02 0.04 0.06 0.08 0.1 1 / t i m e

N 2.1 >=

2 0 5

144480^2 (diam.,+) and 57600^2 ( s q , , x ) : M ( d i a m , s q ) a n d E ( + , x ) without logs 2 . 2 5

(b)

2 . 2

+ [] [] o o

2 . 1 5 x % o o x + ~ + o o , O o o [ ]

o o [ ] o

++ +

+ ~ + x

+

+

2

1 . 9 5

1.9 I

0 . 0 2

o o o o o

o o o o o

o

+

+ +

+ + +

+ +

+

+

i i t i

0.04 0.06 0.08 0.1 1/time

Fig. 1. Effective kinetic exponent z from magnetization and energy. Part a for 1699802 divided among 1024 nodes, part b for 1444802 and 140 processors (and 576002 on 60 processors for 30 < t < 359), and part c for 172802 and 64 processors. The time varied up to 40 (a), 100 (b) and 8000 (c) sweeps through the lattice, and comparison with 86402 showed no significant differences.

N >=

2,4

2.2

¢.

+

1.8 ~-D

1.6

1 .4

1.2

M. Siegert, D. Stauffer / Physica A 208 (1994) 31-34

1 7 2 8 0 " 1 7 2 8 0 M(+) . d E ( d i a m . ) , d E / I o g ( s q . ) , t < 8 0 0 0 r r

0 ÷

+

q~ 0

o D

o

[]

i i i I i 0.005 0.01 0.015 0.02 0.025

f / t i m e

Fig. 1 - - continued.

(c)

0.03

33

2.25

2.2

N >=

2 . f 5

==

2.1

2800"2800*2800 critical Ising: M ( d i a m . ) and E(+) 2.3 j ,

0 0

<)

0

0 ° 0

<> O

O

+ + 2.05 + • + + + + + +

+ + +

+ +

2 J I L I I

0.02 0.04 0.06 0.08 0.1 t / t i m e

Fig. 2. Same for three dimensions: 28003 on 140 processors and 40 sweeps through the lattice.

the correlation length.) Of course, empirical data never exclude the possibility of an effect smaller than the errors. For example, we can fit our energies using log(t + const) instead of log(t), provided the constant is appreciably larger than our observation times between 10 2 and 10 4. Fig. l merely shows that the data fit much better without log(t) than with log(t). (Simulations by authors of Ref. [3] are consistent with this lack of logarithms.)

34 M. Siegert, D. Stauffer / Physica A 208 (1994) 31-34

The slight discrepancy in the z values estimated from Figs. I a and 1 b might be due to parallelism effects, since the number of lines per processor in Fig. 1 a was much smaller than in Fig. lb. Already Mtinkel [5] found that the effective exponent z depends on the number of processors and lattice size, although Ito showed the asymptotic exponent z to be independent of the type of updating. Normally we go through the square lattice like a typewriter, starting in the upper left and ending in the lower right corner; an upper and lower buffer ensure periodic boundary conditions. This order of updating is already changed somewhat due to multi-spin coding, affecting the dynamics. Now, on a parallel computer, many spins or spin groups of the lattice are updated simultaneously, affecting the time-dependence even stronger [5]. Figs. lb and lc followed at least within each strip simulated by a single processor the typewriter principle and updated the buffers immediately after the corresponding line was finished. Fig. 1 a, on the other hand, updated both buffers at the same time, and Miinkel [5] in two and three dimensions started the updating in the interior of the strip. Fig. lb agrees with the "desired" [ 1 ] value of z near 2.17 whereas Fig. la agrees with the slightly higher value of Ref. [ 5]. The error bars are of the order of this difference. (Changing random number generators for the Intel computer gave no significant change, but selecting a linear lattice dimension which is not a proper multiple of the number of processors increased the effective z.)

We tried to see if the slight discrepancies [5] between Monte Carlo and series expansions in the five-dimensional critical temperature, which are also of the order of the Monte Carlo error, are due to parallelity effects. However, for lattices with 805 spins on 40 processors we saw no difference to the Monte Carlo data of Ref. [5] for smaller lattices; and varying the number of processors simulating 645 spins gave no statistically significant changes. Also in three dimensions, our z in Fig. 2 and Miinkel's z '--- 2.08 i 0.03 agree nicely.

Perhaps neither the effects of parallel computing nor of logarithmic factors [6] on the Ising dynamics are fully understood.

We thank ZPR at Cologne University and HLRZ at KFA Jiilich for computer time, H.Gould, A.Aharony, D.W.Heermann and G.A.Kohring for comments, and the German-Israeli Foundation for partial support.

References

[1] N. Ito, Physica A 196 (1993) 591 and literature cited there. [2] N. Ito and G.A. Kohring, Physica A 201 (1993) 547. [3 ] L. Colonna-Roman, A.I. Melcuk, H. Gould and W. Klein, Relaxation to equilibrium of single

cluster dynamics of the Ising model, Physica A, in press. [4] B. Kutlu and N. Aktekin, Critical slowing down in Ising model for Creutz algorithm, Physica

A, in press. [5] Ch. Miinkel, D.W. Heermann, J. Adler, M. Gofman and D. Stauffer, Physica A 193 (1993)

540. [6] R. Matz, D.L. Hunter and N. Jan, J. Stat. Phys. 74 (1994) 903.