Moshe Gitterman

A non-markovian model for calcium kinetics in the body

We present a new generalized compartmental model for calcium kinetics as measured by tracer concentration in blood plasma. The parameter measuring incorporation of calcium in bone discriminates between different levels of physical development in female teenagers and between teenagers and adults.

“Escape” of a periodically driven particle from a metastable state in a noisy system

We consider the statistical properties of the escape time of a particle initially sitting in a potential well subject to a combination of white noise and a periodic forcing term. As one finds in the case of the much-studied bistable potential, different kinds of resonant effects can occur, as measured by the survival probability and the average residence time. When this time is considered as a function of the noise strength, then we show that for small amplitudes of the forcing term there are no resonant effects, while for large amplitudes such effects can appear. We also show that a resonant phenomenon is possible in terms of the amplitude of a periodic forcing term.

Motion in a Periodic Potential Driven by Rectangular Pulses

Exact solutions are found for the averaged velocity of an overdamped pendulum driven by a series of square pulses. The formalism developed can be applied to either periodic or random switching as well as cases intermediate between the two. The phenomenon known as phase locking can be shown to exist when periodic switching is used.

A Comment on Early-Time Solutions of the Smoluchowski Equation

We present a simple derivation of classes of early-time solutions of the Smoluchowski equation in the presence of boundaries, simplifying and generalizing an analysis by van Kampen.

A transition in a noisy linear system driven by a periodic signal

We consider a one-dimension linear random walk between two trapping points in which the transition probabilities vary periodically in time. An earlier analysis of this system showed that the mean time to trapping of a particle in this system exhibits a minimum when considered as a function of frequency. In this note we show that this parameter makes a transition in behavior from a monotonic decrease with increasing amplitude of the periodic term to a monotonic increase with this parameter depending on the frequency. A physical argument is suggested to explain this behavior. Confirmation of this crossover can also be derived from a diffusion model.

The behavior of a periodically-forced nonlinear system subject to additive noise

We continue the study of a nonlinear first-order dynamical system first considered by Chen. This model is characterized by a multiplicative periodic forcing term and additive dichotomous noise in place of the white noise of Chens analysis. Two parameters are used to characterize the qualitative properties of such a system, the mean first-passage time to the ends of the interval and the Fourier spectrum generated by the solution of the equation. We show that the mean first-passage time is monotonic in the amplitude of the periodic force and exhibits a resonant dependence on its frequency. In addition the substitution of dichotomous for white noise leads to a systematic change in the ability to smooth out the peaks in the Fourier spectrum of the solution.

Two-state system in an oscillating field

We consider two states connected by two channels, one of which is activated but the other has no potential barrier. The height of the barrier is oscillating together with an external ac-field. We find that an external field shifts the average populations of the states and tends to equalize them. The steady-state flux circulates along the close circuit formed by these channels.

On the entropy of a class of constrained random walks

We define and calculate the entropy of some random walks which have two endpoints fixed, and for which displacements are allowed to take all possible values. An example is given in which the entropy can either be increased or decreased by imposing a constraint. It is also shown, by example, that when the constrained entropy approaches its unconstrained value, the rate of approach is asymptotically O((ln n)/n).

Effects of an oscillating field on a diffusion process in the presence of a trap

AbstractConsider a diffusion process on an infinite line terminated by a trap and modulated by a periodic field. When the frequency is equal to zero the mean time to trapping will be finite or infinite, depending on the sign of the field. We ask whether this behavior can be changed by an oscillatory field, and show that it cannot for pure Brownian motion. We suggest that transition can appear when the signal propagation velocity is finite as for the telegraphers equation. We further suggest that the asymptotic time dependence of the survival probability is proportional tot−1/2 just as in the case of ordinary diffusion. The same conclusion is shown to hold for a system whose dynamics is governed by the equation