Z. Schuss

A singular perturbation approach to non-Markovian escape rate problems

We employ singular perturbation methods to examine the generalized Langevin equation which describes the dynamics of a Brownian particle in an arbitrary potential force field, acted on by a fluctuating force describing collisions between the Brownian particle and lighter particles comprising a thermal bath. In contrast to models in which the collisions occur instantaneously, and the dynamics are modeled by a Langevin stochastic equation, we consider the situation in which the collisions do not occur instantaneously, so that the process is no longer a Markov process and the generalized Langevin equation must be employed. We compute expressions for the mean exit time of the Brownian particle from the potential well in which it is confined.

Thermal activation from the fluxoid and the voltage states of dc SQUIDs

The probability density of thermal fluctuations about different types of nonequilibrium steady states of a dc SQUID are evaluated by generalizing a technique used before for the fluctuations of a single Josephson junction. Probability densities obtained for both ‘‘running’’ and ‘‘beating’’ modes are used to calculate thermal activation rates as well as the various branches of the I‐V characteristic. The results are compared with the experiments of Voss et al. and good agreement is found.

On the performance of state-dependent single server queues

New asymptotic methods for the analysis of queueing systems are introduced and applied to state-dependent

On the lifetime of a metastable state at low noise

M/G/1

Transitions from the equilibrium state of a hysteretic Josephson junction induced by self-generated shot noise

queues. The methods are used to compute approximations to the stationary density of the queue length, the mean length of a busy period, the mean number of customers served during a busy period as well as other quantities of interest. We obtain results that are superior to those obtained from diffusion approximations in that they are uniformly valid for all values of the traffic intensity while diffusion approximations are adequate only when this quantity is close to one. When specialized to state-independent queues, our approximations are shown to agree with the asymptotic expansions of known exact results. Finally, we show that the behavior of the state-dependent systems is markedly different from that of the corresponding state-independent systems.

Frequency Fluctuations in Noisy Oscillators

Abstract The mean lifetime of a metastable state of a dynamical system driven by small noise is calculated. The vector field of the dynamical system which need not be deriveble from a potential, is assumed to have a vanishing normal component on the boundary of the domain of attraction of the metastable state.

Noise-Induced Transitions in Multi-Stable Systems

Abstract We postulate the existence of self-generated normal current shot noise due to the long-lived voltage fluctuations in a hysteretic Josephson juntion. The resulting low temperature transition rates out of the (zero voltage) superconducting state are much larger than those arising from Johnson noise alone. Excellent agreement with experiments is then achieved at all temperatures. The problematic aspects of the theory are pointed out and discussed briefly.

First Passage Times for Processes Governed by Master Equations

We study frequency and period fluctuations in a nonlinear oscillator driven by Gaussian white noise. We define the random period as the random time between two consecutive zero crossings by the random phase plane trajectory, and the random frequency as the number of such zero crossings per unit of time. These quantities are shown to be related by renewal theory. We find asymptotic expressions for the means and variances of the random period and random frequency, for small damping and small noise. The formulas are particularly useful for oscillators with high frequency.

Uniform expansion of the transition rate in Kramers' problem

The damped physical pendulum driven by constant torque serves as a model for many physical systems (e.g.,the motion of an ion in a crystal subject to a uniform electrostatic field, the point Josephson junction driven by constant current, charge density waves, etc.). For certain ranges of parameters it has both stable equilibrium states and a stable non-equilibrium state. In the presence of a random driving force of thermal or shot noise type there are transitions between the stable states of the pendulum. We calculate the steady state distribution of fluctuations about the stable states and the transition rates between them. For the point Josephson junction at very low temperatures we postulate the existence of “self-generated” shot noise and obtain transition rates which agree with the experimental results of Voss and Webb. This paper summarizes the work of Ben-Jacob, Bergman, Imry, Knessl, Matkowsky and Schuss.

Lifetime of oscillatory steady states

We calculate the activation rates of metastable states of processes governed by Master Equations, by calculating mean first passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the Master Equation. In addition we discuss the validity of various diffusion approximations to the Master Equation, showing that such approximations are not valid in general.