Monte Carlo Simulations in Statistical Physics: Percolation, Diffusion, and Applications, Slides of Statistical Physics

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Chapter 1

Topic: Introduction

Simulations in

Statistical Physics

Percolation

• A very large number of different problems fall into this category:
• For example, in percolation an empty lattice is gradually filled

with particles by placing a particle on the lattice randomly with

each 'tick of the clock'.

• Lots of questions may then be asked about the resulting 'clusters‘

which are formed of neighboring occupied sites.

• Particular attention has been paid to the determination of the

'percolation threshold', i.e. the critical concentration of occupied

sites for which an 'infinite percolating cluster' first appears.

• A percolating cluster is one which reaches from one boundary of a

(macroscopic) system to the opposite one.

• The properties of such objects are of interest in the context of

diverse physical problems such as conductivity of random mixtures,

flow through porous rocks, behavior of dilute magnets, etc.

Diffusion Limited Aggregation (DLA)

• Another example is diffusion limited aggregation (DLA) where a

particle executes a random walk in space, taking one step at each

time interval, until it encounters a 'seed' mass and sticks to it.

• The growth of this mass may then be studied as many random

walkers are turned loose.

• The 'fractal‘ properties of the resulting object are of real interest,

and while there is no accepted analytical theory of DLA to date,

computer simulation is the method of choice.

• In fact, the phenomenon of DLA was first discovered by Monte

Carlo simulation!

Applications in Statistical mechanics

• Considering problems of statistical mechanics, we may be

attempting to sample a region of phase space in order to estimate

certain properties of the model,

• although we may not be moving in phase space along the same

path which an exact solution to the time dependence of the model

would yield.

• Remember that the task of equilibrium statistical mechanics is to

calculate thermal averages of (interacting) many-particle systems:

• Monte Carlo simulations can do that, taking proper account of

statistical fluctuations and their effects in such systems.

WHAT PROBLEMS CAN WE SOLVE WITH IT?

• The range of different physical phenomena which can be explored

using Monte Carlo methods is exceedingly broad.

• Models which either naturally or through approximation can be

discretized can be considered.

• The motion of individual atoms may be examined directly; e.g. in a

binary (AB) metallic alloy where one is interested in inter-diffusion

or unmixing kinetics (if the alloy was prepared in a

thermodynamically unstable state) the random hopping of atoms

to neighboring sites can be modeled directly.

• This problem is complicated because the jump rates of the

different atoms depend on the locally differing environment.

WHAT DIFFICULTIES WILL WE ENCOUNTER?

1.3.1 Limited computer time and memory

• Because of limits on computer speed there are some problems which are inherently not suited to computer simulation, at this time.
• A simulation which requires years of cpu time on whatever machine is available is simply impractical.
• Similarly a calculation which requires memory which far exceeds that which is available can be carried out only by using very sophisticated programming techniques which slow down running speeds and greatly increase the probability of errors.
• It is therefore important that the user first consider the requirements of both memory and cpu time before embarking on a project to make sure whether or not there is a realistic possibility of obtaining the resources to simulate a problem properly.
• Of course, with the rapid advances being made by the computer industry, it may be necessary to wait only a few years for computer facilities to catch up to your needs.

WHAT DIFFICULTIES WILL WE ENCOUNTER?

1.3.2 Statistical and other errors

• Assuming that the project can be done, there are still potential

sources of error which must be considered.

• These difficulties will arise in many different situations with

different algorithms so we wish to mention them briefly at this

time without reference to any specific simulation approach.

• All computers operate with limited word length and hence limited

precision for numerical values of any variable.

• Truncation and round-off errors may in some cases lead to serious

problems.

WHAT STRATEGY SHOULD WE FOLLOW

• Most new simulations face hidden pitfalls and difficulties which may not be apparent in early phases of the work.
• It is therefore often advisable to begin with a relatively simple program and use relatively small system sizes and modest running times.
• Sometimes there are special values of parameters for which the answers are already known (either from analytic solutions or from previous, high quality simulations) and these cases can be used to test a new simulation program.
• By proceeding in this manner one is able to uncover which are the parameter ranges of interest and what unexpected difficulties are present.
• It is then possible to refine the program and then to increase running times.
• Thus both cpu time and human time can be used most effectively. It makes little sense of course to spend a month to rewrite a computer program which may result in a total saving of only a few minutes of cpu time.

HOW DO SIMULATIONS RELATE TO THEORY AND EXPERIMENT?

• In many cases theoretical treatments are available for models for which there is no perfect physical realization (at least at the present time).
• In this situation the only possible test for an approximate theoretical solution is to compare with 'data' generated from a computer simulation.
• As an example we wish to mention recent activity in growth models, such as diffusion limited aggregation for which a very large body of simulation results already exists but for which extensive experimental information is just now becoming available.
• It is not an exaggeration to say that interest in this field was created by simulations.
• Even more dramatic examples are those of reactor meltdown or large scale nuclear war: although we want to know what the results of such events would be we do not want to carry out experiments!
• There are also real physical systems which are sufficiently complex that they are not presently amenable to theoretical treatment.

RELATE TO THEORY AND EXPERIMENT?

• In simulation processes which is as complete as possible, making use of the perfect control of 'experimental' conditions in the 'computer experiment‘.
• They make use of possibility to examine every aspect of system configurations in detail.
• The desired result is then the explanations of the physical mechanisms that are responsible for the observed phenomena.
• We therefore view the relationship between theory, experiment, and simulation to be similar to those of the vertices of a triangle, as shown in Fig. 1.1.
• Here each is distinct, but it is strongly connected to the other two.