Abstract
We propose a toy model for a cyclic orderdisorder 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 timedependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Timedependent 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 selfregulation 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 lnD and D1/2 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 orderdisorder transitions (e.g., fitness).
Original language  English 

Article number  062107 
Journal  Physical Review E 
Volume  95 
Issue number  6 
DOIs  
Publication status  Published  6 Jun 2017 
Externally published  Yes 
ASJC Scopus subject areas
 Statistical and Nonlinear Physics
 Statistics and Probability
 Condensed Matter Physics
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Profiles

Eunjin Kim
Person: Teaching and Research