Far-From-Equilibrium Time Evolution between Two Gamma Distributions

Abstract

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem and show that as the strength of the stochastic noise increases, the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise, a transition occurs whereby the PDF never reaches a stationary state, but instead, forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analysed in terms of entropy and information length, the total number of statistically-distinguishable states that a system passes through in time.

Thermodynamics and Statistical Mechanics of Small Systems

“…In [16], a stochastic logistic model with multiplicative noise, which shows a transition for sufficiently strong noise, is studied. Such a transition between different solutions is analyzed in terms of entropy and information length.…”

Information geometry in a reduced model of self-organised shear flows without the uniform coloured noise approximation

“…In this paper, we extend [9,10] to investigate the time-evolution of PDFs to elucidate the effects of different initial conditions and correlation times. A particular interest will be to understand the information change in the relaxation of an initial PDF to a stationary PDF by using the information length L [11][12][13][14][15][16][17][18][19][20][21]. In the case of a stochastic variable x and time-dependent PDF p(x, t), L is defined by…”

Information length in quantum systems

“…A full knowledge of Probability Density Functions (PDFs), especially time-dependent PDFs, becomes essential to describe these systems. Once computed analytically, numerically, or constructed from data, time-dependent PDFs provide a system-independent way of quantifying the change in information during time-evolution by the number of statistically different states that a system passes through in time [4][5][6][7][8][9][10]. (Note that we use information for statistically different states, refraining ourselves from the debate on the exact definition of information (see, e.g.…”

“…On the other hand, doubling/halving PDFs induces a logarithmic increase in information. In order to do this systematically, we first define the dynamical time τ (t) [4][5][6][7][8][9][10],…”