Historically physics was first known as ‘natural philosophy’ and research was carried out by purely theoretical (or philosophical) investigation. True progress was obviously limited by the lack of real knowledge of whether or not a given theory really applied to nature. Eventually experimental investigation became an accepted form of research although it was always limited by the physicist's ability to prepare a sample for study or to devise techniques to probe for the desired properties. With the advent of computers it became possible to carry out simulations of models which were intractable using ‘classical’ theoretical techniques. In many cases computers have, for the first time in history, enabled physicists not only to invent new models for various aspects of nature but also to solve those same models without substantial simplification. In recent years computer power has increased quite dramatically, with access to computers becoming both easier and more common (e.g. with personal computers and workstations), and computer simulation methods have also been steadily refined. As a result computer simulations have become another way of doing physics research. They provide another perspective; in some cases simulations provide a theoretical basis for understanding experimental results, and in other instances simulations provide ‘experimental’ data with which theory may be compared. There are numerous situations in which direct comparison between analytical theory and experiment is inconclusive. For example, the theory of phase transitions in condensed matter must begin with the choice of a Hamiltonian, and it is seldom clear to what extent a particular model actually represents real material on which experiments are done.

INTRODUCTION

The combination of improved experimental capability, great advances in computer performance, and the development of new algorithms from computer science have led to quite sophisticated methods for the study of certain biomolecules, in particular of folded protein structures. One such technique, called ‘threading,’ picks out small pieces of the primary structure of a protein whose structure is unknown and examines extensive databases of known protein structures to find similar pieces of primary structure. One then guesses that this piece will have the same folded structure as that in the known structure. Since pieces do not all fit together perfectly, an effective force field is used to ‘optimize’ the resultant structure, and Monte Carlo methods have already begun to play a role in this approach. (There are substantial similarities to ‘homology modeling’ approaches to the same, or similar, problems.) Of course, the certainty that the structure is correct comes primarily from comparison with experimental structure determination of crystalized proteins. One limitation is thus that not all proteins can be crystalized, and, even if they can, there is no assurance that the structure will be the same in vivo. Threading algorithms have, in some cases, been extraordinarily successful, but since they do not make use of the interactions between atoms it would be useful to complement this approach by atomistic simulations. (For an introductory overview of protein structure prediction, see Wooley and Ye (2007).) Biological molecules are extremely large and complex; moreover, they are usually surrounded by a large number of water molecules.

INTRODUCTION

In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field!), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3NkB, as T → 0, and the entropy has unphysical behavior since S → −∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0; due to zero point motions which ‘smear out’ the particles over some region around these lattice points.

INTRODUCTION AND PERSPECTIVE

In the preceding chapters of this book we have dealt extensively with equilibrium properties of a wide variety of models and materials. We have emphasized the importance of insuring that equilibrium has been reached, and we have discussed the manner in which the system may approach the correct distribution of states, i.e. behavior before it comes to equilibrium. This latter topic has been treated from the perspective of helping us understand the difficulties of achieving equilibrium. The theory of equilibrium behavior is well developed and in many cases there is extensive, reliable experimental information available.

In this chapter, however, we shall consider models which are inherently non-equilibrium! This tends to be rather uncharted territory. For some cases theory exists, but it has not been fully tested. In other situations there is essentially no theory to rely upon. In some instances the simulation has preceded the experiment and has really led the way in the development of the field. As in the earlier chapters, for pedagogical reasons we shall concentrate on relatively simple models, but the presentation can be generalized to more complex systems.

DRIVEN DIFFUSIVE SYSTEMS (DRIVEN LATTICE GASES)

Over two decades ago a deceptively simple modification of the Ising lattice gas model was introduced (Katz et al., 1984) as part of an attempt to understand the behavior of superionic conductors.

WHAT IS A MONTE CARLO SIMULATION?

In a Monte Carlo simulation we attempt to follow the ‘time dependence’ of a model for which change, or growth, does not proceed in some rigorously predefined fashion (e.g. according to Newton's equations of motion) but rather in a stochastic manner which depends on a sequence of random numbers which is generated during the simulation. With a second, different sequence of random numbers the simulation will not give identical results but will yield values which agree with those obtained from the first sequence to within some ‘statistical error’. A very large number of different problems fall into this category: 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. 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.

INTRODUCTION

In the previous chapters of this text we have examined a wide variety of Monte Carlo methods in depth. Although these are exceedingly useful for many different problems in statistical physics, there are some circumstances in which the systems of interest are not well suited to Monte Carlo study. Indeed there are some problems which may not be treatable by stochastic methods at all, since the time-dependent properties as constrained by deterministic equations of motion are the subject of the study. The purpose of this chapter is thus to provide a very brief overview of some of the other important simulation techniques in statistical physics. Our goal is not to present a complete list of other methods or even a thorough discussion of these methods which are included but rather to offer sufficient background to enable the reader to compare some of the different approaches and better understand the strengths and limitations of Monte Carlo simulations.

MOLECULAR DYNAMICS

Integrationmethods (microcanonical ensemble)

Molecular dynamics methods are those techniques which are used to numerically integrate coupled equations of motion for a system which may be derived, e.g. in the simplest case from Lagrange's equations or Hamilton's equations. Thus, the approach chosen is to deal with many interacting atoms or molecules within the framework of classical mechanics. We begin this discussion with consideration of systems in which the number of particles N, the system volume V, and the total energy of the system E are held constant.

FLUIDS

NVTensemble and the virial theorem

The examination of the equation of state of a two-dimensional model fluid (the hard disk system) was the very first application of the importance sampling Monte Carlo method in statistical mechanics (Metropolis et al., 1953), and since then the study of both atomic and molecular fluids by Monte Carlo simulation has been a very active area of research. Remember that statistical mechanics can deal well analytically with very dilute fluids (ideal gases!), and it can also deal well with crystalline solids (making use of the harmonic approximation and perfect crystal lattice periodicity and symmetry), but the treatment of strongly correlated dense fluids (and their solid counterparts, amorphous glasses) is much more difficult. Even the description of short range order in fluids in a thermodynamic state far away from any phase transition is a non-trivial matter (unlike the lattice models discussed in the last chapter, where far away from phase transitions the molecular field approximation, or a variant thereof, is usually both good enough and easily worked out, and the real interest is generally in phase transition problems).

The discussion in this chapter will consider only symmetric particles, and for the consideration of hard rods, spherocylinders, etc., the reader is referred elsewhere (Frenkel and Smit, 1996).