Space-charge-limited 2D electron flow between two flat electrodes in a strong magnetic field

An approximate analytic solution is constructed for the 2D space-charge-limited emission by a cathode surrounded by nonemitting conducting ledges of width Lambda. An essentially exact solution (via conformal mapping) of the electrostatic problem in vacuum is matched to the solution of a linearized problem in the space charge region whose boundaries are sharp due to the presence of a strong magnetic field. The current density growth in a narrow interval near the edges of the cathode depends strongly on Lambda. We obtain an empirical formula for the total current as a function of Lambda which extends to more general cathode geometries.     

Large Deviations for a Stochastic Model of Heat Flow

We investigate a one-dimensional chain of 2N harmonic oscillators in which neighboring sites have their energies redistributed randomly. The sites −N and N are in contact with thermal reservoirs at different temperature τ₋ and τ₊. Kipnis et al. (J. Statist. Phys., 27:65–74 (1982).) proved that this model satisfies Fourier’s law and that in the hydrodynamical scaling limit, when N → ∞, the stationary state has a linear energy density profile , u ∈[−1,1]. We derive the large deviation function S(θ(u)) for the probability of finding, in the stationary state, a profile θ(u) different from . The function S(θ) has striking similarities to, but also large differences from, the corresponding one of the symmetric exclusion process. Like the latter it is nonlocal and satisfies a variational equation. Unlike the latter it is not convex and the Gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state. We also briefly discuss more general models and find the features common in these two and other models whose S(θ) is known.     

The velocity autocorrelation function of a finite model system

We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particlesN and the length of the boxL approach infinity andN/L , the velocity autocorrelation function(t) is given simply by c² exp(–2ct@#@). For a finite system, the function _N(t) is periodic with period 2L/c. We also show that for more general velocity distribution functions (particles can have velocities ±c_i,i = 1,...), _N(t) is an almost periodic function oft. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keept fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our _N (t) will be very close to(t) as long ast is small compared to the effective size of the system, which is 2L/c for the first model.Supported in part by the AFOSR under Contract No. F44620-71-C-0013.     

Stochastic processes originating in deterministic microscopic dynamics

We investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics, i.e., ifQ(t) is the displacement of the test particle we letQ _A(t) =Q(At)/√A and consider the distribution of the trajectory Q_A(t) in the limit A→∞. The randomness of the motion is due entirely to the randomness of the initial state of the fluid, test particle, or both, and the process is generally non-Markovian. Nevertheless, it can be proven in some cases and we expect it to be true in many more that Q_A (t) looks like Brownian motion in the limit A→∞. Some results for simple model systems are presented.     

On the exponential decay of correlation functions

We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.     

Circle theorem for hard-core binary lattice gases

We study the distribution of zeros in a symmetric, two-component Widom-Rowlinson lattice system (any number of dimensions). We show that for sufficiently large mean activity the system partition function cannot vanish if the magnitude of the ratio of the two (complex) activities is different from one.Supported in part by National Science Foundation Grants CHE76-11253 (to LKR) and Phy 78-15920 (to JLL), and by the Petroleum Research Foundation.     

Inequalities for higher order Ising spins and for continuum fluids

The recently derived Fortuin, Kasteleyn and Ginibre (FKG) inequalities for lattice gasses are investigated for higher order Ising spin systems and multi-component lattice gasses. Conditions are given for the validity of the FKG inequalities for higher order spin systems with Hamiltonians of the form used recently as models for various physical systems, e.g.He ³–He ⁴ mixtures. We also investigate various inequalities for binary lattice gases and show how these can be carried over to continuum systems.Supported in part by U.S.A.F.O.S.R. # F 44620-71-C-0013.N.S.F. Graduate Trainee.