Information Landscape and Flux, Mutual Information Rate Decomposition and Entropy Production

We explore the dynamics of information systems. We show that the driving force for information dynamics is determined by both the information landscape and information flux which determines the equilibrium time reversible and the nonequilibrium time-irreversible behaviours of the system respectively. We further demonstrate that the mutual information rate between the two subsystems can be decomposed into the time-reversible and time-irreversible parts respectively, analogous to the information landscape-flux decomposition for dynamics. Finally, we uncover the intimate relation between the nonequilibrium thermodynamics in terms of the entropy production rates and the time-irreversible part of the mutual information rate. We demonstrate the above features by the dynamics of a bivariate Markov chain.

Renewal characterization of Markov modulated Poisson processes

A Markov Modulated Poisson Process (MMPP) M(t) defined on a Markov chain J(t) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov chain J(t) is in state i. M(t) is called strongly renewal (SR) if M(t) is a renewal process for an arbitrary initial probability vector of J(t) with full support on P={i:λi>0}. M(t) is called weakly renewal (WR) if there exists an initial probability vector of J(t) such that the resulting MMPP is a renewal process. The purpose of this paper is to develop general characterization theorems for the class SR and some sufficiency theorems for the class WR in terms of the first passage times of the bivariate Markov chain [J(t),M(t)]. Relevance to the lumpability of J(t) is also studied.

The coalescent in two partially isolated diffusion populations

SummaryThe n0 coalescent of Kingman (1982a, b) describes the family relationships among a sample of n0 individuals drawn from a panmictic species. It is a stochastic process resulting from n0 − 1 independent random events (coalescences) at each of which n (2 ≤ n ≤ n0) ancestral lineages of a sample are descended from n − 1 distinct ancestors for the first time. Here a similar genealogical process is studied for a species consisting of two populations with migration between them. The main interest is with the probability density of the time length between two successive coalescences and the spatial distribution of n − 1 ancestral lineages over two populations when n to n − 1 coalescence takes place. These are formulated based on a non-linear birth and death process with killing, and are used to derive several explicit formulae in selectively neutral population genetics models. To confirm and supplement the analytical results, a simulation method is proposed based on the underlying bivariate Markov chain. This method provides a general way for solving the present problem even when an analytical approach appears very difficult. It becomes clear that the effects of the present population structure are most conspicuous on 2 to 1 coalescence, with lesser extents on n to n − 1 (3 ≤ n) coalescence.This implies that in a more general model of population structure, the number of populations and the way in which a sample is drawn are important factors which determine the n0 coalescent.

A correlated random walk for the transport and sedimentation of particles

This paper considers a bivariate random walk modelon a rectangular lattice for a particle injected into a fluid flowing in a tank. The numbers of jumps of the particle in thexandydirections in this particular model are correlated. It is shown that when the random walk forms a bivariate Markov chain in continuous time, it is possible to obtain the state probabilitiespxy(t) through their Laplace transforms. Two exit rules are considered and results for both of them derived.

Correlated random walks with stay

A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability distribution. In a correlated random walk (CRW) the movement follows a Markov chain and induces correlation in the state of the walk at various epochs. Then, the walk can be modelled as a bivariate Markov chain with the location of the particle and the direction of movement as the two variables. In such random walks, normally, the particle is not allowed to stay at one location from one step to the next. In this paper we derive explicit results for the following characteristics of the CRW when it is allowed to stay at the same location, directly from its transition probability matrix: (i) equilibrium solution and the fast passage probabilities for the CRW restricted on one side, and (ii) equilibrium solution and first passage characteristics for the CRW restricted on bath sides (i.e., with finite state space).

A correlated random walk for the transport and sedimentation of particles

This paper considers a bivariate random walk model on a rectangular lattice for a particle injected into a fluid flowing in a tank. The numbers of jumps of the particle in the x and y directions in this particular model are correlated. It is shown that when the random walk forms a bivariate Markov chain in continuous time, it is possible to obtain the state probabilities pxy(t) through their Laplace transforms. Two exit rules are considered and results for both of them derived.