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).

Within the contents of 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 theoretically 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.

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 recent, provocative perspective of the role of statistical physics outside of 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’ of 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.

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.

COMPARISONS OF METHODS FOR NUMERICAL INTEGRATION OF GIVEN FUNCTIONS

Simple methods

One of the simplest and most effective uses for Monte Carlo methods is the evaluation of definite integrals which are intractable by analytic techniques. (See the book by Hammersley and Handscomb (1964) for more mathematical details.)

When the temperature is raised above absolute zero, the amplitudes of both the weak-localization, universal conductance fluctuations and the Aharonov–Bohm oscillations are reduced below the nominal value e2/h. In fact, the amplitude of nearly all quantum phase interference phenomena is likewise weakened. There is a variety of reasons for this. One reason, perhaps the simplest to understand, is that the coherence length is reduced, but this can arise as a consequence of either a reduction in the coherence time or a reduction in the diffusion coefficient. In fact, both of these effects occur. In Chapter 2, we discussed the temperature dependence of the mobility in high-mobility modulation-doped GaAs/AlGaAs heterostructures. The decay of the mobility couples to an equivalent decay in the diffusion constant, where d is the dimensionality of the system, through both a small temperature dependence of the Fermi velocity and a much larger temperature dependence of the elastic scattering rate. The temperature dependence of the phase coherence time is less well understood but generally is thought to be limited by electron–electron scattering, particularly at low temperatures. At higher temperatures, of course, phonon scattering can introduce phase breaking.

Another interaction, though, is treated by the introduction of another characteristic length, the thermal diffusion length. The source for this lies in the thermal spreading of the energy levels or, more precisely, in thermal excitation and motion on the part of the carriers.

In this chapter we discuss a variety of issues related to the phenomenon of one-dimensional conductance quantization, probably one of the most important phenomena exhibited by mesoscopic conductors. The quantization is observed in one of the simplest of structures, namely the quantum point contact (QPC) that can be straightforwardly realized by means of the split-gate technique. The QPC is essentially a nanoscale constriction, connected at either end to macroscopic reservoirs, through which electrons may travel ballistically at low temperatures. In this chapter, we discuss how the strong lateral confinement that electrons experience as they pass through the QPC quantizes their energy into a series of discrete one-dimensional subbands. Through a simple analysis, based on a noninteracting model of transport that assumes linear response, we show that the conductance associated with these subbands takes the universal value 2e2/h, independent of the subband index. This results in the observation of a universal staircase structure in the conductance of QPCs, as their gate voltage is used to change the number of occupied subbands one at a time. An important requirement for the observation of this effect is that electron transport through the QPC should be ballistic, and we will see how this typically limits its observation to low temperatures (≤ 4.2 K). The conductance quantization provides a striking demonstration of the validity of the Landauer approach to electrical conduction, and in this chapter we also extend the discussion to consider the influence of scattering and non-vanishing source–drain bias on the conductance.

The original edition of this book grew out of our somewhat disorganized attempts to teach the physics and electronics of mesoscopic devices over the past decade. Fortunately, these evolved into a more consistent approach, and the book tried to balance experiments and theory in the current, at that time, understanding of mesoscopic physics. Whenever possible, we attempted to first introduce the important experimental results in this field followed by the relevant theoretical approaches. The focus of the book was on electronic transport in nanostructure systems, and therefore by necessity we omitted many important aspects of nanostructures such as their optical properties, or details of nanostructure fabrication. Due to length considerations, many germane topics related to transport itself did not receive full coverage, or were referred to only by reference. Also, due to the enormity of the literature related to this field, we did not include an exhaustive bibliography of nanostructure transport. Rather, we tried to refer the interested reader to comprehensive review articles and book chapters when possible.

The decision to do a second edition of this book was reached only after long and hard consideration and discussion among the authors. While the first edition was very successful, the world has changed significantly since its publication. The second edition would have to be revised extensively and considerable new material added. A decision to go ahead was made only after welcoming Jon Bird to the author's team.