In this chapter we establish the celebrated Jordan decomposition theorem which allows us to reduce a linear mapping over ℂ into a canonical form in terms of its eigenspectrum. As a preparation we first recall some facts regarding factorization of polynomials. Then we show how to reduce a linear mapping over a set of its invariant subspaces determined by a prime factorization of the characteristic polynomial of the mapping. Next we reduce a linear mapping over its generalized eigenspaces. Finally we prove the Jordan decomposition theorem by understanding how a mapping behaves itself over each of its generalized eigenspaces.

Some useful facts about polynomials

Let P be the vector space of all polynomials with coefficients in a given field F and in the variable t. Various technical computations and concepts involving elements in P may be simplified considerably with the notion ‘ideal’ as we now describe.

Definition 7.1 A non-empty subset I ⊂ P is called an ideal of P if it satisfies the following two conditions.

1. f + g ∈ I for any f, g ∈ I.

2. fg ∈ I for any f ∈ P and g ∈ I.

Since F may naturally be viewed as a subset of P, we see that af ∈ I for any a ∈ F and f ∈ I. Hence an ideal is also a subspace.

Let g1, …, gk ∈ P. Construct the subset of P given by

{f1g1 + ··· + fkgk | f1, …, fk ∈ P}.

It is readily checked that the subset defined in (7.1.1) is an ideal of P. We may say that this deal is generated from g1, …, gk and use the notation I(g1, …, gk) to denote it.

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. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).

Introduction

Modern Monte Carlo methods have their roots in the 1940s when Fermi, Ulam, von Neumann, Metropolis and others began considering the use of random numbers to examine different problems in physics from a stochastic perspective (Cooper, 1989); this set of biographical articles about S. Ulam provides fascinating insight into the early development of the Monte Carlo method, even before the advent of the modern computer). Very simple Monte Carlo methods were devised to provide a means to estimate answers to analytically intractable problems. Much of this work is unpublished and a view of the origins of Monte Carlo methods can best be obtained through examination of published correspondence and historical narratives. Although many of the topics which will be covered in this book deal with more complex Monte Carlo methods which are tailored explicitly for use in statistical physics, many of the early, simple techniques retain their importance because of the dramatic increase in accessible computing power which has taken place during the last two decades. In the remainder of this chapter we shall consider the application of simple Monte Carlo methods to a broad spectrum of interesting problems.

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. Since analytical treatments also usually require mathematical approximations whose accuracy is difficult to assess or control, one does not know whether discrepancies between theory and experiment should be attributed to shortcomings of the model, the approximations, or both. The goal of this text is to provide a basic understanding of the methods and philosophy of computer simulations research with an emphasis on problems in statistical thermodynamics as applied to condensed matter physics or materials science. There exist many other simulational problems in physics (e.g. simulating the spectral intensity reaching a detector in a scattering experiment) which are more straightforward and which will only occasionally be mentioned.

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. 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.

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 crystallized proteins. One limitation is thus that not all proteins can be crystallized, 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. Thus, realistic simulations that include water explicitly and take into account polarization effects are inordinately difficult. There have also been many attempts to handle this task by means of molecular dynamics simulations, but the necessity of performing very long runs of very large systems makes it extremely difficult (if not impossible) to reach equilibrium. We are thus in the quite early stages in the study of biological molecules via Monte Carlo methods, but, as we shall see, results are quite promising for the future. The field is developing rapidly, and in some ways we are passing through the same kind of maturation period as Monte Carlo enthusiasts in physics did 30–40 years ago in which simulations were, at first, not taken too seriously by experimentalists. This will surely change.

Within this book we have attempted to elucidate the essential features of Monte Carlo simulations and their application to problems in statistical physics. We have attempted to give the reader practical advice as well as to present theor-etically based background for the methodology of the simulations as well as the tools of analysis. New Monte Carlo methods will be devised and will be used with more powerful computers, but we believe that the advice given to the reader in Section 4.8 will remain valid.

In general terms we can expect that progress in Monte Carlo studies in the future will take place along two different routes. First, there will be a continued advancement towards ultra high resolution studies of relatively simple models in which critical temperatures and exponents, phase boundaries, etc., will be examined with increasing precision and accuracy. As a consequence, high numerical resolution as well as the physical interpretation of simulational results may well provide hints to the theorist who is interested in analytic investigation. On the other hand, we expect that there will be a tendency to increase the examination of much more complicated models which provide a better approximation to physical materials. As the general area of materials science blossoms, we anticipate that Monte Carlo methods will be used to probe the often complex behavior of real materials. This is a challenge indeed, since there are usually phenomena which are occurring at different length and time scales. As a result, it will not be surprising if multiscale methods are developed and Monte Carlo methods will be used within multiple regions of length and time scales. We encourage the reader to think of new problems which are amenable to Monte Carlo simulation but which have not yet been approached with this method.

Commentary

In the preceding chapters we described the application of Monte Carlo methods in numerous areas that can be clearly identified as belonging to physics. Although the exposition was far from complete, it should have sufficed to give the reader an appreciation of the broad impact that Monte Carlo studies has already had in statistical physics. A more recent occurrence is the application of these methods in non-traditional areas of physics related research. More explicitly, we mean subject areas that are not normally considered to be physics at all but which make use of physics principles at their core. In some cases physicists have entered these arenas by introducing quite simplified models that represent a ‘physicist's view’ of a particular problem. Often such descriptions are oversimplified, but the hope is that some essential insight can be gained as is the case in many traditional physics studies. (A provocative perspective of the role of statistical physics outside physics has been presented by Stauffer (2004).) In other cases, however, Monte Carlo methods are being applied by non-physicists (or ‘recent physicists’) to problems that, at best, have a tenuous relationship to physics. This chapter is to serve as a brief glimpse of applications of Monte Carlo methods ‘outside’ physics. The number of such studies will surely grow rapidly; and even now, we wish to emphasize that we will make no attempt to be complete in our treatment.

In recent years the simulation of relatively realistic models of proteins has become a ‘self-sufficient’ enterprise. For this reason, such studies will be found in a separate, new chapter (Chapter 14), and in this chapter we will only consider models that are primarily of interest to statistical physicists.

NVT ensemble 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; for the consideration of hard rods, spherocylinders, etc., the reader is referred elsewhere (Frenkel and Smit, 1996).