Time-dependent probability density function in cubic stochastic processes

2017

Phys. Rev. E

Self Cite

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).

Section: Modelsmentioning

confidence: 99%

Geometric structure and information change in phase transitions

2017

Phys. Rev. E

Self Cite

“…That is, our system is subcritical for λ < 0, supercritical for λ > 0 and critical at λ = 0. Equation (3) is the extension of our recent work [28,42] to a cyclic transition. The Fokker-Planck equation [43,44] corresponding to Equations (3) and (4) is as follows:…”

Section: Modelsmentioning

confidence: 99%

“…To estimate the PDF width R in Equation (17), we use the variance of the quartic exponential function (e.g., [42]):…”

Section: Pdf Of Forward Processmentioning

confidence: 99%

Information Geometry of Non-Equilibrium Processes in a Bistable System with a Cubic Damping

2017

Entropy

Self Cite

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic systems and information geometry. For a non-equilibrium process, 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. By using this concept, we investigate the information geometry of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system. Specifically, we compute time-dependent PDFs, information length, the rate of change in information length, entropy change and Fisher information in disorder-to-order and order-to-disorder transitions and discuss similarities and disparities between the two transitions. In particular, we show that the total information length in order-to-disorder transition is much larger than that in disorder-to-order transition and elucidate the link to the drastically different evolution of entropy in both transitions. We also provide the comparison of the results with those in the case of the transition between the subcritical and supercritical states and discuss implications for fitness.

“…In order to isolate the key effect of nonlinear interaction, we consider stochastic driven-dissipative systems with linear and nonlinear damping, respectively, which correspond to the Ornstein-Uhlenbeck (O-U) process and a nonlinear diffusion model with a cubic damping [27][28][29][30][31] and investigate similarities and differences during their relaxation processes in statistical space. Specifically, we quantify the change in timedependent PDFs during relaxation by using the information length [23][24][25][26] and examine the difference in geometric structure associated with the linear O-U process and nonlinear processes.…”

Section: Introductionmentioning

confidence: 99%

Signature of nonlinear damping in geometric structure of a nonequilibrium process

2017

Phys. Rev. E

Self Cite

We investigate the effect of nonlinear interaction on the geometric structure of a nonequilibrium process. Specifically, by considering a driven-dissipative system where a stochastic variable x is damped either linearly (∝x) or nonlinearly (∝x 3 ) while driven by a white noise, we compute the time-dependent probability density functions (PDFs) during the relaxation towards equilibrium from an initial nonequilibrium state. From these PDFs, we quantify the information change by the information length L, which is the total number of statistically distinguishable states which the system passes through from the initial state to the final state. By exploiting different initial PDFs and the strength D of the white-noise forcing, we show that for a linear system, L increases essentially linearly with an initial mean value y 0 of x as L ∝ y 0 , demonstrating the preservation of a linear geometry. In comparison, in the case of a cubic damping, L has a power-law scaling as L ∝ y m 0 , with the exponent m depending on D and the width of the initial PDF. The rate at which information changes also exhibits a robust power-law scaling with time for the cubic damping.