Phase Transition and Slowing Down in Non-Equilibrium Stochastic Processes

Abstract

A general criterion of appearance of slowing down in non·equilibrium stochastic processes is proposed. Many examples for this general criterion are shown, in which a phenomenon of slowing down occurs at a certain value of the relevant parameter. In particular, a generalized scaling treatment of transient phenomena is effectively applied to deriving the relaxation spectra of some multiplicative stochastic processes. The concept of asymptotic slowing down for finite systems is also proposed.

Noncritical Slowing Down of Photonic Condensation

“…2). The mode [0, 0] couples strongly in the region C0 between those maxima, and the mode [0,2] couples strongly between and outside those two maxima as indicated by C2. There are thus regions in which the competition between mode [0, 1] and the modes [0, 0] and [0, 2] for access to the excitations of the dye molecules is particularly strong.…”

“…By contrast, systems with many degrees of freedom can exhibit emergent behavior, with dynamics on timescales that have no clear origin in the microscopic equations of motion. A prominent example is the relaxation towards a steady state which is known to slow down when a system is close to a critical point [1,2]. This critical slowing down is present in the statistical mechanics of systems from atomic quantum gases [3] or magnetic metamaterials [4] to entire ecosystems or human societies [5].…”

“…Crucially, unlike the photonic cavity modes, the excitation profiles are not mutually orthogonal, and their dynamics are coupled as described by Eq. (2). This means that a cavity mode can clamp other excitation profiles as well as its own.…”

“…3. After a quench from phase B, however, mode [0, 0] is already macroscopically occupied and immediately clamps mode [0, 1], which only attains a population of 10 7 before being clamped by modes [0, 0] and [0,2]. This reduction in the population of mode [0, 1] reduces the time taken to decay to the steady state in E, but increases the time taken to reach the macroscopic occupation of region D.…”

“…Top: time taken for the system to equilibrate after a quench in pump power by 1%, as a function of the pump power after the quench. Bottom: steady state populations n i of cavity modes i ranging from [0, 0] to[1,2] and [0, 3]. The mode populations feature sharp increases and drops under increase of pump power, and the equilibration times show clear peaks around those phase transitions.…”