Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

Abstract

Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics,

Response theory and phase transitions for the thermodynamic limit of interacting identical systems

“…In the thermodynamic limit N → ∞, the one particle density satisfies the McKean–Vlasov equation ( 2.2 ) and it has been proven [ 1 , 103 ] that the infinite particle system undergoes a continuous phase transition, with 〈 x 〉 being a suitable order parameter. The Desai–Zwanzig model can be seen as a stochastic model of key importance for elucidating order–disorder phase transitions [ 107 ].…”

Stochastic thermodynamics and fluctuation theorems for non-linear systems

“…In this appendix we show how to derive a non-linear master equation from a Markov chain with probabilitydependent transition rates. More details about non-linear Markov chains and their expression in terms of stochastic processes with probability-dependent transition rates can be found e.g., in [66]. Consider a discrete-time stochastic process whose n-point distribution can be written in a form of non-linear Markov process, i.e., p(x n , t n ; x n−1 , t n−1 ; .…”

“…In the Supplemental material (SM) [65], we show how this non-linear master equation can be derived from Markov chains with probabilitydependent transition probabilities. Overview of physical and life-science applications of Markov chains with probability-dependent transition rates can be found e.g., in review [66]. Note that the antisymmetric form of the summand in Eq.…”

Response Theory and Phase Transitions for the Thermodynamic Limit of Interacting Identical Systems

“…The Desai-Zwanzig model [66] has a paradigmatic value as it features an order-disorder thermodynamic phase transition arising from the interaction between systems [100] and has been used also as a model for systemic risk [3]. Each of the systems can be interpreted as a particle, moving on a one dimensional line (M = 1) in a double well potential V α (x) =…”