Computational Strategies for Mapping Equilibrium Phase Diagrams (2003)

The phenomenon of phase behavior is generic to science. In a general way it is the organization of many-body systems into forms that reflect the interplay between constraints imposed macroscopically (through the prevailing external conditions) and microscopically (through the interactions between the elementary constituents). In addition to its most familiar manifestations in condensed matter science, the phenomenon (and its attendant challenges) features in areas as diverse as gauge theories of subnuclear structure, the folding of proteins, and the self-organization of neural networks. This chapter is concerned with the generic computational strategies needed to address the problems of phase behavior. The physical context will not extend beyond the structural organization of the elementary phases (liquid, vapor, crystalline) of matter, although the strategies are much more widely applicable than this. For the most part this chapter largely restrict to systems comprising simple classical particles in defect-free structures, interacting through a prescribed potential function. In its fullest sense the “problem” of phase behavior entails questions of kinetics: how phase transformations occur. Within a computational framework, such questions are naturally addressed with the techniques of Molecular Dynamics (MD), which numerically integrate the equations of motion associated with the many-body potential and thereby attempt to replicate, authentically, the phase-transformation process itself. This chapter's attention is restricted to the task of mapping equilibrium phase boundaries. This choice leads naturally to another. Freed of the need to capture the authentic dynamics, one finds that the natural computational tool becomes Monte Carlo (MC) rather than MD. The strategic advantage of this choice is the range of ways in which MC algorithms may be engineered to move around configuration space. Indeed, understanding the distinctive features of that “space” which are implied by the occurence of the phase behavior, and engineering an algorithm (effectively a pseudo-dynamics) to match are the key themes of recent activities, which is the highlight in this work. Section II discusses the basic equipment. Section II.A formulates the problem in the complementary languages of thermodynamics and statistical mechanics. Section II.B outlines the standard portfolio of MC tools and explains why they are not equal to the challenge posed by this core problem. Section III identifies and explores what proves to be the key concept here: the notion of a path through configuration space. Most approaches to the core problem utilize a “path” which, in some sense, “connects” the two regions of interest. They can be classified on the basis of the choices made in regard to that path: (a) the route it follows through configuration space and (b) the way in which one contrives to sample the configurations along that route. There are a limited number of generically distinct path-routing options; and these are summarized in Section III.B. In Section III.C the generically distinct path-sampling strategies are identified. There are actually fewer than one might suppose. Surveyed here is a small number of path-based techniques (combinations of routing and sampling strategies) in some detail. This survey (Section IV) extends from the standard technique of long-standing, numerical integration to reference macrostates, to one only recently introduced, in which the core problem is solved by a MC leap directly between the phases of interest. There are some strategies which do not naturally fit into this framework, but which must certainly be included here (Section V). Solving the core problem at some state point is the major part of the task of determining the phase boundary, but not quite all. To complete the job, one has to track a phase boundary (once one has found a point on it) and one has to extrapolate from the limited sizes of system that simulations can handle to the thermodynamic limit of interest. Section VI reviews the complementary tools needed. For the most part focus is on relatively idealized model systems; in Section VII issues associated with departures from the ideal are considered. Finally, Section VIII offers some thoughts on how further progress can be made.