Geometric structure and information change in phase transitions

Abstract

We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale τ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length L between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation γ of a control parameter from the critical state in BPs while increasing logarithmically with γ in FPs. L scales as | ln D| and D −1/2 in FPs and BPs, respectively, where D measures the strength of the stochastic forcing. These differing scalings with γ and D suggest a great utility of L in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

“…Of course, Z → ∞ cannot be achieved in any numerical solver; some finite upper boundary must always be chosen. In previous work on other Fokker-Planck equations [35,36], the resulting PDFs dropped off sufficiently rapidly for large z (exponentially or even faster) that just imposing p(Z) = 0 yielded excellent results, and conserved the total probability extremely well. Here though this approach was found not to work, and caused the integral Z 0 p dz to decrease in time, even if Z as large as 100 was chosen.…”

“…[39,40]), our recent work [27,28,[33][34][35][36] adapted this concept to a non-equilibrium system to elucidate geometric structure of non-equilibrium processes. Specifically, [35] mapped out the attractor structure L ∞ vs z 0 for linear and cubic processes and showed that a linear damping preserves a linear geometry L ∞ ∝ z 0 whereas a nonlinear damping gives rise to a power-law scaling L ∞ ∝ z n 0 (n ∼ 1.5 − 1.9) of the attractor structure. [28] found interesting geodesic solutions in a non-autonomous Ornstein-Uhlenbeck (O-U) process [32] by modulating model parameters and by including time-dependent external deterministic killing term.…”

“…Notably, the modulation of the model parameters and the killing term were periodic/oscillatory. [35,36] reported the asymmetry in L in order-to-disorder versus disorder-to-order transitions.…”

“…In this case, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time, and is called the information length [27,28,[33][34][35].…”

“…Defining the information length involves two steps [27,28,[33][34][35][36]: First we need to compute the dynamic time unit τ (t), which is the characteristic timescale over which p(z, t) temporally changes on average at time t. Second, we need to compute the total elapsed time in units of this τ (t). As done in [27,28,[33][34][35][36], we compute τ by utilising the following second moment E:…”