First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises

We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis' $q$-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge-Kutta algorithm and the first passage times are recorded. The first passage time density is determined along with the mean first passage time. Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the mean first passage time are discussed.

TRANSITION AND EVOLUTION OF INFORMATION AT THE MICROSCOPIC LEVEL

A simultaneous development of the fundamental research areas of the information theory is needed for efficient development in the information technologies. It is known that for the complicated macroscopic systems information evolution may be shaped on the basis of the principal thermodynamics laws (the second law of thermodynamics, etc). At the same time it is not known whether the fundamentals of the information theory for the macroscopic systems may be applicable to the microscopic systems. The study works out a mathematic model of the discrete phase space adapted to describing the evolution of information (entropy) of the microscopic systems. The discrete phase-space model rests on the indeterminacy principle and fundamental properties of the discrete continuous-time Markovian systems. The Kolmogorov equations represent the main mathematical tools technique. The suggested model refers to the smallest metric scale when the external macroscopic observation is possible. This scale can be viewed as a quasiclassical level. The research results are the following. The structure of the phase space of the elementary signal is revealed. It is demonstrated that the entropy of the microscopic systems increases, i.e. for the microscopic systems the second law of thermodynamics is true. There has been demonstrated transition from the microscopic model to the macroscopic one thus proving the former’s adequacy. The discrete phase-space model is promising in the aspect of further development. For example, it can be applied to the physical systems “particle – field”. The approach represented by the model will allow to study electromagnetic and gravity fields at the quasiclassical level. The above model of the discrete phase space and its application in the study of the evolution of the microscopic systems is a proprietary design of the authors.

The Limit Theorems for Function of Markov Chains in the Environment of Single Infinite Markovian Systems

This paper is intended to study the limit theorem of Markov chain function in the environment of single infinite Markovian systems. Moreover, the problem of the strong law of large numbers in the infinite environment is presented by means of constructing martingale differential sequence for the measurement under some different sufficient conditions. If the sequence of even functions gnx,n≥0 satisfies different conditions when the value ranges of x are different, we have obtained SLLN for function of Markov chain in the environment of single infinite Markovian systems. In addition, the paper studies the strong convergence of the weighted sums of function for finite state Markov Chains in single infinitely Markovian environments. Although the similar conclusions have been carried out, the difference results performed by previous scholars are that we give weaker different sufficient conditions of the strong convergence of weighted sums compared with the previous conclusions.