NUMERICAL SIMULATIONS - CANONICAL AND MICROCANONICAL

Gyan Bhanot The Institute for Advanced S~udy

Princeton, NJ 08540, USA

ABSTRACT

A brief review of numerical simulation methods is given. After de-

scribing convergence criteria for the usual (canonical ensemble) Monte-

Carlo methods, a numerical study of the microcanonical ensemble for

Ising-like systems is described.

Any field theory on the lattice [i] is defined by an action S(o)

which is a function of the dynamical variables o(i). 'i' labels sites

(or links) on a lattice as well as any internal degrees of freedom (e.g

color in the case of QCD). The lattice is usually hypercubic with pe-

riodic boundary conditions to minimize finite size effects.

If O(~) is some function of the o's, its expectation value (Green'~

function) is defined by,

i f (0) e-8S (0) <0> B = [ O do (i)

with,

Z = f e-~S(°)do (2)

where 8 = I/T is the inverse temperature.

In numerical simulations [2-5], one works on a finite lattice (usu-

ally with limitations both on computer memory and time). One generates

a sequence of configurations of the o fields in which averages like the

one in Eqn. (i) can be computed. In principle, these configurations

could be generated completely randomly. However, most of these would

have a very small Boltzman weight (i.e. e -BS(o) would be very small for

them). The main idea of numerical simulation techniques is importance

sampling. That is, one tries to generate configurations distributed as

e -~s(o) so that in the averaging (Eqn. (i)) each of these is equally

important.

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The basic problem is to find {o} distributed as,

i -SS(q)do (3) Peq (o) = Z e

Let W(g÷o') be a transition matrix that generates a configuration P(o')

from P(o). Thus

P(~') = [ W(O+O') P(O) (4) o

W has the properties

W(o+o') > 0 (5a)

and

W(o÷o') = 1 (normalization) (5b)

In Eqn. (4), P(o) is the probability distribution of any particular val-

ue of each of the o's in the configuration {a}. W(o÷o') defines a Markov

chain. We want to have P (o) be an eigenvector of this chain. This is eq

guaranteed if detailed balance is satisfied; i.e. if,

W(a÷o') = P ') W(a'+o) (6) Peq (o) eq (o

The proof that P is an eigenvector of W is trivial. We have eq

W(o÷o') P(o) = [ W(g'+o) P(o') o o

= P(o') (7)

where we have used Eqns. (6) and (5b). Notice that P has eigenvalue eq

unity. A theorem due to Frobenius and Perron [6] guarantees that this

eigenvalue is non-degenerate. This theorem states that:

Theorem: A stochastic matrix with positive entries has a non-degenerate

maximum eigenvalue whose eigenvector has only positive entries.

This proves that P is the unique distribution. eq

theorem, let us prove the following:

To motivate this

Theorem: The Markov chain defined by W never diverges away from P eq"

Let us define the norm N between two ensembles E and E' where P(o) and

P' (o) are the probability distributions of configurations {o} in these

two ensembles. Let N be given by

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N def l l E - z ' l l = [ IP(~)-P'(o)l [8)

If E' is the ensemble obtained from E by applying W

P' (~) = [ W(o'+o) P(q') (9) o t

Then

I] E'-Eeq [I = [ I P' ((~) -Peq ((~) I (def) (~

= [ I 57 w < o , - ~ o ) ( P ( o , ) - p ( o ' ) ) i O O' eq

W(O'÷O) IP(o') -P (q') I GO' eq

= Jl E-Eeq II (I0)

This proves the theorem.

What is done in numerical simulations to implement this Markov chain

is to start with some distribution of the o's on the lattice and generate

new configurations using one of two popular choices for W.

a) The Metropolis method

W(~+o') = ~ exp[~(S(~) - S(o'))] if S(~') > S(~)~

1 otherwise ~ (ii)

and

b) The Heat bath method

W(o÷o') = exp[-SS(O') ] (12)

Both of these satisfy Eqn. (6) and will result in configurations distrib-

uted as Eqn. (3). For more detail on the numerical simulation procedure,

see Refs. [2-5,7].

We now turn to a more exotic way of doing numerical simulations in

the microcanonical ensemble [8-10]. Here, instead of a transition prob-

ability matrix W(~÷~'), one uses a deterministic map ~ which generates

states invariant under its mapping. The density function of these states

is the microcanonical ensemble. In most systems, there are conservation

laws. Whenever they exist, one must restrict the map to fixed values of

the conserved quantities. Let us consider a concrete example, the 2-d

Ising model. The action is

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S = [ (~i~i+l) (13) i

One way to do numerical simulations is to use the canonical ensem-

ble Monte-Carlo methods described before. A more novel way is to gener-

ate states for which S is fixed. Thus, define the "microcanonical" par-

tition function,

ZMC = {!} ~S(g),S O (14)

The "average" of any operator O(g) is

= [ O(a) ~S(g) s0/ZMc (15) {~}

The crucial question is: How does O depend on the initial state {~}0

from which we start the evolution at fixed S = $07 A system is ergodic

iff O is independent of {~}0" This happens if a single cycle traces

through the entire space of states at constant S 0. It is clear that in

a finite system, it is impossible to avoid Poincare recurrences. The

hope is that when the volume tends to infinity, ergodicity holds. How-

ever, no proofs exist for this except for very special systems.

A possible way out would be to introduce a small amount of random-

ness in the way the space of states at constant S O is mapped out. This

randomness would have the effect of ensuring that one does not get stuck

in cycles from a bad choice of initial state. In a sense, the randomness

would allow an average over all possible initial states and this would

certainly increase the possibility of ergodic behavior.

A sample microcanonical simulation for the 2-d Ising model was done

recently [i0] using a method originally proposed by Creutz [9]. In this

method [ii], instead of generating states with S = constant, the sum of

S plus a 'demon' action S d is kept constant. This allows a local algo L

rithm to be constructed to update the lattice spins. The demons are a

set of variables that act as heat exchangers between different parts of

the lattice. The demon attempts to flip the spin locally and the spin

flip is accepted if the sum STo t = S d + S can be kept fixed with the

flipped spin. The partition function of this system is

Z = [ 6 S (16) Cc (C)+Sd(C)'STot

where C and c label lattice and demon configurations respectively. Aver-

ages of a lattice quantity 0 in this partition function are defined by

386

= Cc [ O(C) 6S(C)+Sd(C) ,STot/Z (17)

Given a demon energy and the configuration around a particular spin, the

algorithm that decides whether the spin flips is totally deterministic.

However, a certain amount of randomness can be introduced by making the

choice of sites to update randomly over the lattice.

The result of the study [10] showed that if this was done, problems

with ergodicity were not encountered. Microcanonical averages (Eqn. (17))

can then be related to canonical expectation values if I/V effects are

properly taken into account. The relation between these quantities (cor-

rect to (I/V2)) is,

= i a [a<o>/~B] <°>B* = °s T 2v a~ ~ (18)

where 8" is defined implicitly by

and

= (19) <e>B* eST

e = S/V (20)

In general, there might be several advantages to the microcanonical over

the canonical formulation for numerical simulation.

i) The deterministic nature of the evolutions allows very fast computer

codes to be written for discrete systems (such as the Ising model

and possibly also SU(2) gauge theory with the Icosahedral subgroup

as gauge group).

2) The renormalization group can probably be implemented more directly

in the microcanonical ensemble.

3) In certain situations, the canonical ensemble does not exist while

the microcanonical does [12]. For such systems, there is no choice

but to use the microcanonical formulation.

Acknowledgements

I thank the National Science Foundation for a travel grant that

made it possible for me to attend the conference in Mahabaleshwar and

the organizers of the school for their hospitality.

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References

[i] K. Wilson, Phys. Rev. DI0 (1974) 2445 [2] M. Creutz, "Quarks, Gluons and Lattices" (Cambridge Univ, Press,

1983) [3] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rep. 95 (1983) 201 [4] C. Rebbi, "Lattice Gauge Theory and Monte Carlo Simulations" (World

Scientific, Singapore, 1983) [5] K. Binder in "Phase Transitions and Critical Phenomena", C. Domb

and M. S. Green, eds. Vol. 5B (Academic Press, New York, 1976) [6] F. R. Gantmacher, "Applicat~-ons of the Theory of Matrices", (Inter-

science, N.Y. 1959). Translated by J. L. Brenner. [7] G. Bhanot, "A Review of Numerical Simulation Methods", ed. A.

Faessler, in Progress in Particle and Nuclear Physics, (Pergamon Press Ltd., 1984)

[8] D. Callaway and A. Rahman, Phys. Rev. Lett. 49 (1982) 613 [9] M. Creutz, Phys. Rev. Lett. 50 (1983) 1411

[10] G. Bhanot, M. Creutz and H. Neuberger, IAS preprint, "Microcanonical Simulation of Ising Systems", Nov. 1983, to appear in Nucl. Phys. B [FS]

[ii] See [i0] for details [12] A. Strominger, Ann. Phys. 146 (1983) 419