Robert Zwanzig

On the identity of three generalized master equations

Abstract Three apparently different quantum mechanical master equations, derived by Prigogine and Resibois, by Montroll, and independently by Nakajima and Zwanzig, are shown to be identical. The derivation by Zwanzig, based on projection operator and Liouville operator techniques, is repeated in greater detail than in previous articles. The results of Prigogine and Resibois, and of Montroll, are found by making changes in notation.

High‐Frequency Elastic Moduli of Simple Fluids

This article presents a calculation of the infinite‐frequency elastic moduli of monatomic fluids. When the intermolecular potential has the standard Lennard‐Jones form, the elastic moduli are shown to be related to the pressure and internal energy of the fluid. Numerical values of the elastic moduli at various densities and temperatures are presented in tabular and graphical form. An identity is derived relating the shear and bulk moduli of any isotropic material in which particles interact by means of two‐body central forces; this is a generalization of the familiar Cauchy identity occurring in the theory of elasticity of solids. The approach is based on analysis of the high‐frequency limit of exact time‐correlation function expressions for shear and bulk viscosity.

First‐Order Phase Transition in a Gas of Long Thin Rods

Onsagers prediction of a first‐order phase transition in a gas of long thin rods is verified by extensive calculations on a simple model. The molecules are rectangular parallelepipeds, of length l and square cross section d×d. The long molecular axes can point in only three mutually perpendicular directions. Virial coefficients, up to the seventh, are calculated exactly as functions of orientation, in the limit l→ ∞, d→0, and l2d=constant. The transition predicted by Onsagers theory, based on a second virial approximation to the potential of mean force in the space of relative orientations, is observed also when all virial coefficients up to the seventh are included. The virial expansion appears to converge well for the isotropic phase; but the properties of the anisotropic phase depend sensitively on the order of approximation.

Dielectric Friction on a Moving Ion

A recent derivation by R. H. Boyd of the frictional force on a slowly moving ion due to dielectric loss in the surrounding medium is corrected and generalized.

Theory of Vibrational Relaxation in Liquids

A new formulation of the theory of vibrational relaxation, based on Zeners semiclassical approximation, is presented here. The relaxation rate is shown to be proportional to the spectral density of the force exerted on the oscillator by its environment. The isolated binary collision theory is derived, but only with the condition that the collision frequency is much smaller than the oscillator frequency. This requirement is not satisfied in a liquid; we conclude that Litovitzs application of the isolated binary collision theory to liquids is not justified. A possible relation between vibrational relaxation and the self‐diffusion coefficient in a liquid is discussed.

Elementary Derivation of Time‐Correlation Formulas for Transport Coefficients

This article presents an elementary derivation of time‐correlation formulas for transport coefficients. The derivation, based on classical statistical mechanics, is modeled on one by Kubo, Yokota, and Nakajima. In agreement with theirs, it makes use of the assumption of regression of fluctuations. It differs, however, in its avoidance of Markoffian assumptions. The decisive point of the derivation is the introduction of a parameter ξ to measure the rate of the transport process. In the limit of a very slow process, i.e., in the limit ξ→0 and time t→∞ with ξt held constant, the results coincide with those obtained by Markoffian assumptions. The relation to other work, and in particular to Kirkwoods friction constant theory, is touched on briefly.

Frequency‐Dependent Transport Coefficients in Fluid Mechanics

Exact time‐correlation formulas for frequency‐dependent shear viscosity, bulk viscosity, and thermal conductivity are presented. These formulas are evaluated for a fluid composed of molecules with internal degrees of freedom, in the limit of very weak coupling of internal and translational motions. In this limit, the shear viscosity is unaffected by internal modes. The bulk viscosity consists of two parts, one due to translational motions, and the other due to relaxation of the energy contained in internal modes. The thermal conductivity consists of two parts, one due to translational motions, and the other due to diffusion of energy in internal modes (the Eucken correction). The resulting excess sound absorption is in complete agreement with the predictions of the thermodynamic theory of relaxation in fluids.

Dielectric Relaxation in a High‐Temperature Dipole Lattice

The effect of rotational Brownian motion on the dielectric susceptibility of a rigid cubic lattice of permanent point dipoles is calculated. All dipolar interactions are taken into account by means of a high‐temperature perturbation expansion. The most significant result is that dipolar interactions give rise to additional new relaxation times increasing dielectric loss at high frequencies.

Two Assumptions in the Theory of Attractive Forces between Long Saturated Chains

A recent calculation by Salem of the attractive forces between two long parallel hydrocarbon chains was based on two assumptions: (1) pairwise additivity of dispersion forces between individual groups on each chain, and (2) isotropy of the polarizability of a group on the chain. The validity of these assumptions is examined here by means of a Drude model calculation of the interaction between two parallel linear lattices of dispersion oscillators. We describe an exact calculation on this model, and also a calculation based on applying Salems assumptions to the same model. We conclude that the assumptions are not valid for the Drude model, because of strong induced‐dipole interactions between neighbors along the lattice. Salems procedure leads to essentially correct numerical results, however, due to use of an experimentally determined bond polarizability at a crucial stage of the calculation. This allows for the effects of interactions between neighbors. Our results are applicable to real hydrocarbon ch...

Anomalous Specific Heat and Viscosity of Binary van der Waals Mixtures

This article investigates the behavior of several properties of a model binary van der Waals mixture near the solution critical point. The behavior of the heat capacity is determined from an exact analysis of the partition function. For the shear and bulk viscosity, the time dependence of the currents appearing in the time correlation function formulas is determined from linearized hydrodynamic equations. For the heat capacity and the shear viscosity the results of this model are identical to those originally obtained by Fixman by a different procedure. Comparison is also made with the results obtained by Zwanzig and Mountain for these quantities in a one‐component van der Waals system near the liquid—vapor critical point. One finds, in qualitative agreement with available experiments, that the shear viscosity of a binary mixture diverges at the critical point.