Probability Distribution Of Residence Times Research Articles

Probability distribution of residence times of grains in models of rice piles

We study the probability distribution of residence time of a grain at a site, and its total residence time inside a pile, in different rice pile models. The tails of these distributions are dominated by the grains that get deeply buried in the pile. We show that, for a pile of size L, the probabilities that the residence time at a site or the total residence time is greater than t, both decay as 1/t(ln t)x for L(omega) << t << exp(L(gamma)) where gamma is an exponent > or = 1, and values of x and omega in the two cases are different. In the Oslo rice pile model we find that the probability of the residence time T(i) at a site i being greater than or equal to t is a nonmonotonic function of L for a fixed t and does not obey simple scaling. For model in d dimensions, we show that the probability of minimum slope configuration in the steady state, for large L, varies as exp(-kappaL(d+2)) where kappa is a constant, and hence gamma=d+2.

Probability distribution of residence times of grains in sand-pile models

We show that the probability distribution of the residence times of sand grains in sand-pilemodels, in the scaling limit, can be expressed in terms of the probability of survival of asingle diffusing particle in a medium with absorbing boundaries and space-dependent jumprates. The scaling function for the probability distribution of residence times isnon-universal, and depends on the probability distribution according to which grains areadded at different sites. We determine this function exactly for the one-dimensionalsand-pile when grains are added randomly only at the ends. For sand-piles with grainsadded everywhere with equal probability, in any dimension, and of arbitrary shape, weprove that, in the scaling limit, the probability that the residence time is greater thant is , where is the average mass of the pile in the steady state. We also study finite size corrections tothis function.

Escape and synchronization of a Brownian particle

The escape and synchronization of a 1-\ensuremath{\mu}m Brownian particle from an optical trap is presented. The probability distribution of residence times within a given well is exponential with a cutoff at short time. Temporal modulation of the well leads to synchronization.

Escape and synchronization of a Brownian particle.

The escape and synchronization of a 1-\ensuremath{\mu}m Brownian particle from an optical trap is presented. The probability distribution of residence times within a given well is exponential with a cutoff at short time. Temporal modulation of the well leads to synchronization.

Flux ratio theorems for nonstationary membrane transport with temporary capture of tracer

It has been shown recently that the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane which has transport properties varying arbitrarily with the distance from a boundary, is independent of time from the very first appearance of the two outfluxes from the membrane. This surprising proposition has been proved for boundary conditions defining standard unidirectional fluxes, and then generalized to classes of time-dependent boundary conditions. The operational meaning of all the resulting theorems is that when any of them appear to be refuted experimentally, the presence of more than one parallel transport pathway (that is, of membrane heterogeneity transverse to the direction of transport) can be inferred and analyzed. Recent experimental data have been interpreted accordingly. However, the proofs of the theorems given so far have not taken into account the possibility of temporary capture of tracer at sites fixed in the membrane (including also entrances to microscopic culs-de-sac). The possible presence of such a process, which would not affect fluxes in the steady state, left a fundamental gap in the aforementioned inferences. It is shown here that all the theorems previously proved for the flux ratio under unsteady conditions remain valid when temporary capture of tracer is admitted, no matter how the rate of capture, and the probability distribution of residence times of tracer at capture sites, may depend on the distance from a membrane boundary. The validity of the aforementioned inferences from observed time-dependence of the flux ratio is thereby extended to a much wider class of membrane transport processes.