Geometric structure and information change in phase transitions

We propose a toy model for a cyclic order-disorder transition and introduce a new geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of Forward and Backward Processes (FP and BP) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent 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 behaviour in FP and BP. In particular, FP driven by instability undergoes the broadening of the PDF with 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, BP is 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 BP than in FP, increasing linearly with the deviation γ of a control parameter from the critical state in BP while increasing logarithmically with γ in FP. L scales as | ln D| and D‾½ in FP and BP, 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).