J. K. Percus

Reduction of the N‐Particle Variational Problem

A variational method is presented which is applicable to N‐particle boson or fermion systems with two‐body interactions. For these systems the energy may be expressed in terms of the two‐particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground‐state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N‐particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two‐particle and one‐particle density matrices of an N‐particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ...

Nonlinear aspects of chemotaxis

Abstract A simplified Keller-Segel model for the chemotactic movements of cellular slime mold is reconsidered. In particular, we ask for the circumstances under which the cell distribution can autonomously develop a δ-function singularity. By the use of suitable differential inequalities, we show that this cannot happen in the case of one-dimensional aggregation. For three or more dimensions, we produce time developments which do become singular, while in the important special case of two-dimensional motion, we advance arguments that the possibility of chemotactic collapse requires a threshold number of cells in the system.

Statistical Thermodynamics of Nonuniform Fluids

We have developed a general formalism for obtaining the low‐order distribution functions nq(r1, …, rq) and the thermodynamic parameters of nonuniform equilibrium systems where the nonuniformity is caused by a potential U(r). Our method consists of transforming from an initial (uniform) density n0 to the final desired density n(r) via a functional Taylor expansion. When n0 is chosen to be the density in the neighborhood of the rs we obtain nq as an expansion in the gradients of the density. The expansion parameter is essentially the ratio of the microscopic correlation length to the scale of the inhomogeneities. Our analysis is most conveniently done in the the grand ensemble formalism where the corresponding thermodynamic potential serves as the generating functional [with U(r) as the variable] for the nq. The transition from U(r) to n(r) as the relevant variable is accomplished via the direct correlation function which enters very naturally, relating the change in U at r2 due to a change in n at r1. It ...

Equilibrium state of a classical fluid of hard rods in an external field

The external field required to produce a given density pattern is obtained explicitly for a classical fluid of hard rods. All direct correlation functions are shown to be of finite range in all pairs of variables. The one-sided factors of the pair direct correlation are also found to be of finite range.

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.

Structure of a Liquid-Vapor Interface

The structure of the interface of an argonlike fluid in equilibrium with its own vapor at low temperature is studied using molecular dynamics. The longitudinal pair correlations in the interface are found to be consistent with a simply defined ensemble of local thermodynamic states. However, the transverse correlations exhibit very long-range behavior not predicted by straightforward local thermodynamics. These results strongly suggest that the interface is made up of an ensemble of configurations in each of which the transition from liquid to vapor is locally sharp, but that the transition surface fluctuates strongly in space and time.

Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension

Diffusion in a narrow two-dimensional channel of width A(x), depending on the longitudinal coordinate x, is the object of our study. We show how the 2+1 dimensional diffusion equation can be projected onto a 1+1 dimensional one, governing corresponding one-dimensional density, in a steady-state approximation. Then we demonstrate the method on a nontrivial exactly solvable case for A(x)=x and discuss projection of the initial condition.

Self-diffusion of fluids in narrow cylindrical pores

Fluids under stochastic dynamics in narrow cylindrical pores exhibit a dynamical transition from single-file diffusion (SFD) to Fickian bulk diffusion. For long time, the mean square displacement will change as the pore size increases, with a transition from SFD, ∼t1/2, to Fickian, ∼t, while the diffusion coefficient (Dxx) increases from zero. We present a theory of this important process in terms of a hopping time, τhop, leading to Dxx∝(τhop)−1/2 which is verified with simulation. While the crossover is to be expected, the simple form is a priori unanticipated and is likely to be universal.

Model for density variation at a fluid surface

A fluid model with freely propagating longitudinal density waves is modified by the imposition of an external field. A relation between the resulting density inhomogeneity and the applied potential is obtained, depending only upon the uniform fluid pair distribution function. This is solved for a container-bounded fluid. The resulting surface density profiles for classical and zero-temperature Bose hard-sphere fluids compare very well with numerical experiments.

Sphericalization of nonspherical interactions

A density and temperature‐independent spherically symmetric reference potential is constructed for an interacting classical fluid of nonspherical molecules. It annuls the first order correction to the free energy and, in special cases, the second as well. The potential is a limiting form of that used successfully for N2 by Shaw et al., and reproduces numerically the Y4 approximation of Barboy and Gelbart for homonuclear dumbbell molecules.