Irwin Oppenheim

Molecular Theory of Brownian Motion for Several Particles

A molecular derivation is presented for the coupled Langevin equations that describe the motion of heavy particles in a fluid. In contrast to the case of a single heavy particle, the friction tensors which appear depend upon the instantaneous separations between the particles. A Fokker–Planck equation describing the reduced distribution function for the heavy particles is also obtained. Calculations with these equations require evaluation of the friction tensors. The friction tensors are evaluated in two ways, by an approximate macroscopic hydrodynamic calculation and an approximate hydrodynamic fluctuation calculation. Both calculations lead to identical expressions for the friction tensors which are shown to have a long‐range character at large interparticle separations. Finally, it is shown that these long‐range effects cancel in the calculation of the diffusion constant for each particle but remain in the calculation of the relative diffusion constant.

Molecular theory of Brownian motion

A molecular theory of Brownian motion is presented which starts from the Hamiltonian equations of motion for a system consisting of one heavy particle of mass M and N light particles of mass m (bath). A projection operator which averages over the bath variables is utilized. Expansions in powers of λ=(m/M)12 are obtained for the equation of motion for the heavy particle. It is rigorously demonstrated that the Langevin equation is valid up to O(λ2) for all times. The magnitude of the momentum of the heavy particle is restricted to be of order λ-1. It is also shown using linear response theory that the friction constant appearing in the Langevin equation is identical to the friction coefficient which characterizes the drag on a macroscopic body moving with a prescribed velocity.

Rotational Relaxation in Fluids

In this paper a number of expressions for angular momentum and reorientation correlation times are obtained. The inverse of the angular momentum correlation time is related to a sum of two terms, one of which involves the shear viscosity of the medium and the other of which describes the precessional relaxation of the molecule. The derivation is based on the similarity of the time dependences of the torque and the force exerted on the molecule of interest. The reorientation correlation time is related to the angular momentum relaxation time in the diffusion and the near‐diffusion limits. The expressions obtained here have been confirmed by a number of ESR experiments.

Memory effects in the relaxation of quantum open systems

A close examination of the validity of the Markovian approximation in the context of relaxation theory is presented. In particular, we examine the question of positivity of various approximations to the reduced dynamics of an open system in interaction with a heat reservoir. It is shown that the Markovian equations of motion obtained in the weak coupling limit (Redfield equations) are a consistent approximation to the actual reduced dynamics only if supplemented by a slippage in the initial conditions. This slippage captures the effects of the non‐Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of O(N0) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is O(eN). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.

On the theory of concentrated hard-sphere suspensions

A systematic theory for the dynamics of hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. A generalized diffusion equation is derived for concentrated suspensions. The volume fraction (φ) dependence of the short- and long-time self-diffusion coefficients are thus explored from a unifying point of view. The long-range hydrodynamic interactions due to the Oseen tensor are shown to play a crucial role in both coefficients, while the short-range hydrodynamic interactions just lead to corrections. The importance of the correlation effects between particles due to the long-range hydrodynamic interactions is also stressed. The nonlocal correlation effect is an important factor, leading to the behavior of the long-time self-diffusion coefficient (DSL) as DSL ∼ (1 − φ/φ0)2 near the volume fraction of φ0 = 0.5718. The direct interactions are also found to be drastically reduced by the short-range hydrodynamic interactions.

Stochastic and Deterministic Formulation of Chemical Rate Equations

It is shown, on the basis of some examples, that the commonly used stochastic theory of gas‐phase chemical rate equations reduces to the deterministic formulation in the thermodynamic limit, N → ∞, V → ∞, N / V fixed. Since the commonly used deterministic collision theory of chemical kinetics is derivable from the Boltzmann equation with reactive scattering terms, the stochastic formulation of chemical kinetics is thus shown to be equivalent to the results of the Boltzmann equation in the thermodynamic limit. However, since the stochastic theory has not been derived from the Liouville equation for finite systems, the validity of the calculations of the deviation of the stochastic mean from the deterministric results and the fluctuations about that mean for finite N has not been established.

Nonlinear response I: General considerations

Abstract The linear response technique of Kubo is extended to treat the response of a system to quadratic order in the external field. Nonlinear equations of motion for a set of slowly varying macroscopic variables, a ( t ), are obtained. These equations express the time derivative of a ( t ) in terms of a ( t ) and a ( t ) a ( t ) and time-dependent coefficients which involve equilibrium correlation functions. The coefficient of a ( t ) is identical to that found from linear response theory. A discussion of the long-time behavior of the coefficients is presented.

Exact Conditions for the Preservation of a Canonical Distribution in Markovian Relaxation Processes

Necessary and sufficient conditions have been determined for the exact preservation of a canonical distribution characterized by a time‐dependent temperature (canonical invariance) in Markovian relaxation processes governed by a master equation. These conditions, while physically realizable, are quite restrictive so that canonical invariance is the exception rather than the rule. For processes with a continuous energy variable, canonical invariance requires that the integral master equation is exactly equivalent to a Fokker‐Planck equation with linear transition moments of a special form. For processes with a discrete energy variable, canonical invariance requires, in addition to a special form of the level degeneracy, equal spacing of the energy levels and transitions between nearest‐neighbor levels only. Physically, these conditions imply that canonical invariance is maintained only for weak interactions of a special type between the relaxing subsystem and the reservoir. It is also shown that canonical ...

Nuclear Spin Relaxation in Gases and Liquids. IV. Interpretation of Experiments in Gases

The theory of nuclear spin relaxation in gases and liquids developed by Bloom and Oppenheim is used to interpret measurements of spin lattice relaxation times for various gaseous systems over wide ranges of temperature and density. The systems discussed include: molecular H2 with different ortho and para concentrations; mixtures of H2 with He, Ne, Ar, N2, CO2, O2, CO, NO, and N2O; and normal D2. The analysis leads to a determination of the form of the anisotropic intermolecular potentials providing that the form of the isotropic part of the intermolecular potentials are known. Values of quadrupole moments for H2, O2, N2, CO, NO, N2O, CO2, and D2 are obtained which are in good agreement with previous values. Thus, nuclear spin relaxation becomes a valuable tool for obtaining anisotropic intermolecular potentials.