N- and V-Representability Problems in Classical Statistical Mechanics
Classical equilibrium statistical mechanics presents a class of unsolved N representability problems analogous to those in the quantum mechanical regime discussed earlier in this chapter. In this case, N refers to the number of particles (atoms or molecu les) present, rather than the number of electrons. The most straightforward version of this classical problem concerns a single-species monatomic system (i.e., spherically symmetric identical particles) and involves the pair correlation function g(r). T his nonnegative function of interparticle distance r is defined by the occurrence probability of particle pairs at r, relative to random expectation. Consequently, deviations of g(r) greater than 1 indicate that interparticle interactions have biased the distance distribution to a greater-than-1 random expectation, while deviations less than 1 indicate the opposite.
For many cases of interest, the interparticle potential energy function V can be regarded as a sum of terms arising from each pair of particles present:
The pair potentials v(r) typically are taken to satisfy the following criteria:
(a) v(r) -> +
as r -> 0;
(b) v(r) is bounded, and is piecewise continuous and differentiable for r > 0;
(c) |v(r)| < C/r
(C > 0, n > 3), for r > R > 0.
Under these circumstances, g(r) plays a special role in the thermodynamic properties of the N-particle system (Hansen and McDonald, 1976). This fundamental quantity appears in closed-form expressions giving the pressure and mean energy at the prevailing temperature and density. Furthermore, it appears in expressions for the X-ray and neutron diffraction patterns for the substance; consequently, these diffraction measurements constitute an experimental means for measuring g(r) for real substances. It sh ould be added that g(r) is also one of the traditional results reported from computer simulations of N-body systems (Ciccotti et al., 1987).
The experimentally, or computationally, adjustable parameters are temperature; particle number density; container size, shape, and boundary conditions; and number N of particles. For most cases of interest, one focuses on the infinite-system lim it, where the container size and N diverge, while temperature, number density, and container shape are held constant. The central problem then concerns the mapping between the pair of functions v(r) and g(r), where the latter is interpreted as the infini te-system limit function.
Historically, the fundamental theory of classical systems (particularly in the liquid state) concentrated heavily on prediction of g(r) for a given v(r), that is, the mapping from v to g. This has generated several well-known approximate integral equat ion predictive theories for g(r), including those conventionally identified in the theoretical chemistry literature by the names Kirkwood (1935), Bogoliubov-Born-Green-Yvon (Born and Green, 1949), Percus and Yevick (1958), and hypernetted chain (van Leeuw en et al., 1959) integral equations, each of which has spawned successor refinements. However, in all cases the respective approximations invoked have, strictly speaking, been uncontrolled. Consequently, the local structure and thermodynamic property pr edictions based on these various integral equations have had only modest success in describing the dense liquid state, they have failed to predict the so-called nonclassical singular behavior at the liquid-vapor critical point (Widom, 1965), and they have been largely useless for the study of freezing and melting transitions. Perhaps as a result of these shortcomings, the recent trend in classical statistical mechanics has been to rely heavily on direct computer simulation of condensed-phase phenomena. B ecause these simulations often require massive computational resources, a case can be made that revival of analytic predictive theory for g(r) would be favorable from the point of view of the "productivity issue" in theoretical and computational chemistry .
In some respects, the inverse mapping of g to v is even more subtle, intriguing, and mathematically challenging. At the outset, one encounters the obvious matter of defining the space of functions g(r) that in fact can be generated by a pairwise additi ve potential energy function V. A few necessary conditions are straightforward; as already remarked, g(r) cannot be negative. It is generally accepted (but not rigorously demonstrated) that g must approach unity as r diverges if the temperature is positi ve, even though the system itself may be in a spatially periodic crystalline state. In addition, the Fourier transform of g(r) - 1,
is also subject to necessary conditions stemming from the nature of the linear equilibrium response of the system to weak external perturbations: for all k > 0 one must have (Percus, 1964)
These generic conditions can be supplemented by others that are necessary if v(r) has an infinitely repelling hard core, that is,
This hard-core property prevents neighbors from clustering too densely around any given particle, and from the geometry of hard-sphere close packings it is possible to bound the integral of r
g(r) over finite int
ervals of r.
A primary challenge concerns formulation of sufficient conditions on g(r), given that V possesses the pairwise-additive form displayed above. At present we have no rational criterion for deciding whether a given g(r), however "reasonable" it may appear to be by conventional physical standards, corresponds to the thermal-equilibrium short-range order for any pairwise additive V. It is not even clear at present how to construct a counterexample, namely, a g(r) meeting the necessary conditions abo ve that cannot map to a v(r) of the class described. In any case, formulation of sufficient conditions would likely improve prospects for more satisfactory integral equation (or other analytical) predictive techniques for g(r).
Several directions of generalization exist for this classical V(r) representability problem; these include the following matters:
1. Properties of triplet and higher-order correlation functions g for occurrence probabilities of particle n-tuples;
2. Properties of correlation functions for particles (molecules) with internal degrees of freedom (rotation, vibration, conformational flexibility);
3. Effects of specific nonadditive potentials, which would be the case when including three-particle contributions in V; and,
4. Multicomponent (several species, or mixture) systems, in particular the important case of electrostatically charged particles (ions) with their long- ranged Coulo mbic interactions.
References
Born, M., and H.S. Green, 1949, A General Kinetic Theory of Liquids, Cambridge, London.
Ciccotti, G., D. Frankel, and I.R. Kirkwood, 1987, eds., Simulation of Liquids and Solids, North-Holland, Amsterdam.
Hansen, J.P., and I.R. McDonald, 1976, Theory of Simple Liquids, Section 2.6, Academic Press, New York.
Kirkwood, J.G., 1935, Statistical mechanics of fluid mixtures, J. Chem. Phys. 3:300-313.
Percus, J.K., 1964, in The Equilibrium Theory of Classical Fluids, H.L. Frisch and J.L. Lebowitz, eds., W.A. Benjamin, New York, pp. II-33 to II-170 (see particularly II-41).
Percus, J.K., and G.J. Yevick, 1958, Analysis of classical statistical mechanics by means of collective coordinates, Phys. Rev. 110:1-13.
van Leeuwen, J.M.J., J. Groeneveld, and J. De Boer, 1959, New method for the calculation of the pair correlation function, Physica 25:792-808.
Widom, B., 1965, Equation of state in the neighborhood of the critical point, J. Chem. Phys. 43:3898-3905.
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