Ido Dayan

Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis

Abstract We analyse the diffusion of photons in a two-layer turbid medium, paying specific attention to parameters suggested by the use of lasers for diagnostic purposes in a medical setting. The data produced by such experiments consists of an intensity profile measured on the surface of the medium, which is measured either as a function of time at a fixed distance from the input laser beam, or else as a function of distance keeping the measurement time fixed. In both cases we demonstrate that the presence of a second layer can be detected provided that physical properties of the two layers differ by a sufficient amount, and also show up in calculated values of such parameters as the mean travel time of a photon to the surface of the medium, and the distribution of the maximum depth reached by an arbitrary photon.

Real space renormalization group approach to the random field Ising model

The authors discuss results from two types of real space renormalization group (RSRG) calculations applied to the random field Ising model in three dimensions. Starting from a lattice of size L, the RSRG is used to reduce the lattice to a size L=2, on which the trace is done exactly. In this way, thermodynamic properties, such as the magnetization and susceptibility, can be determined approximately. They find that, for a given size, the susceptibility increases as the temperature, T, is reduced down to the transition temperature, Tc, and becomes essentially independent of temperature below Tc. Both in the vicinity of Tc and at lower temperatures, there are large sample-to-sample fluctuations in the susceptibility which grow with increasing system size. They interpret these results in terms of the droplet theory of the transition.

Some recent variations on the expected number of distinct sites visited by an n-step random walk

Asymptotic forms for the expected number of distinct sites visited by an n-step random walk, being calculable for many random walks, have been used in a number of analyses of physical models. We describe three recent extensions of the problem, the first replacing the single random walker by N→∞ random walkers, the second to the study of a random walk in the presence of a trapping site, and the third to a random walk in the presence of a trapping hyperplane.

A generalized radiation boundary condition

The radiation boundary condition was originally proposed by Collins and Kimball as a means of avoiding the unphysical prediction of the Smoluchowski model for reaction rates that the calculated rate in three dimensions k(t) has the property k(0)=∞. A microscopic model that can be used to derive the boundary conditions uses the tacit assumption that an encounter between two molecules A and B gives rise to a reaction with a probability α<1. We consider a non‐Markovian model in which the probability that exactly n encounters between A and B are required to produce a reaction is equal to θn. We show that when the expected number of such encounters is finite, one gets the usual radiation boundary condition. When the expected number is infinite, one finds a boundary condition that is nonlocal in time. The extension of our analysis to higher dimensions as well as to the Smoluchowski equation is readily generated.

Percolation in multi-layered structures

The authors study a model for layered porous media. The model is a d-dimensional lattice percolation system with different concentrations given to different layers. They find for the 2D site percolation system, a critical line on which the percolation clusters are anisotropic and characterized by two different correlation length exponents. Their results suggest that due to the introduction of long-range correlations one obtains a new universality class different from isotropic percolation and from directed percolation.

Kinetic behavior of diffusion systems in the presence of non‐Markovian boundaries

Previously, the authors have calculated a number of properties of diffusion in the presence of partially reacting boundaries, in which the probability that a particle impinging on the boundary will react is not a constant. This is reflected in the appearance of a boundary condition nonlocal in time. Here a number of consequences for kinetic behavior of such a nonlocal boundary condition are derived. The circumstances under which results derived from the Smoluchowski model or its generalization are asymptotically valid are discussed, as well as conditions that lead to deviations from the predictions of such a theory.

Number of distinct sites visited by a random walker in the presence of a trap

The authors study the number of distinct sites visited by a random walker in d=1 after t steps, S(t), in the presence of a trap. They calculate the distribution q(S, t) of S(t) in the limit of large t. They find an unusual crossover in the probability density at S approximately=Sx identical to Dt. For S >Sx, q(S, t) approximately St-3/2 exp(-S2/4Dt). Fro this crossover it follows that the mean number of distinct sites visited is (S(t)) approximately In(t).

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).