Diffusion in a bistable potential: A systematic WKB treatment

Caroli, B.; Caroli, C.; Roulet, B.

doi:10.1007/bf01009609

Cited by 152 publications

(65 citation statements)

References 10 publications

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“…which recovers the previous results [45,46] when α = 1. To understand the evolution of the peak of the PDF, we calculate the location x m where PDF takes its local maximum or minimum from ∂ ∂x ln p(x,t)| x=x m = 0, which satisfies…”

Section: A Forward Process (Fp)supporting

confidence: 79%

“…(45) and (46) are in reasonable agreement with each other. The constant term is somewhat different, but the best fit to this term is also very strongly affected by the best fit to the ln D term, since, e.g., | ln 10 −7 | = 16, which is already as large as the largest L ∞ in Fig.…”

Section: A Forward Processsupporting

confidence: 77%

“…The FP starts with a Gaussian PDF for small D and goes through the following two stages (e.g., see [45][46][47][48]): (i) the initial stage of the broadening of the initial Gaussian PDF due to stochastic noise ξ and instability γ towards the development of the two peaks at x = ± γ μ , and (ii) the final stage (Kramer's regime) of narrowing of the PDF to the final (equilibrium) (double) Gaussian PDF. To understand order formation, it is instructive to examine the evolution of PDF analytically in stage (i) in detail.…”

Section: A Forward Process (Fp)mentioning

confidence: 99%

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Geometric structure and information change in phase transitions

Kim

Hollerbach

2017

Phys. Rev. E

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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).

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Section: A Forward Process (Fp)supporting

confidence: 79%

Section: A Forward Processsupporting

confidence: 77%

Section: A Forward Process (Fp)mentioning

confidence: 99%

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Geometric structure and information change in phase transitions

Kim

Hollerbach

2017

Phys. Rev. E

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“…Hence, in general, the contribution of the paths in the vicinity of a saddle-point becomes important at low temperatures, when the parameter θ is small. This regime corresponds to the semi-classical limit, in the quantum language [7]. In the low temperature limit, it is possible to express the transition probability (10) as a sum over a countable set of contributions i = 1, 2, ..., each of which accounts for the pathways which are close to a corresponding dominant trajectory, denoted as x i (τ ).…”

Section: Statistically Significant Reaction Pathwaysmentioning

confidence: 99%

Dominant reaction pathways in high-dimensional systems

Autieri

Faccioli

Sega

et al. 2009

The Journal of Chemical Physics

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This paper is devoted to the development of a theoretical and computational framework to efficiently sample the statistically significant thermally activated reaction pathways, in multidimensional systems obeying Langevin dynamics. We show how to obtain the set of most probable reaction pathways and compute the corrections due to quadratic thermal fluctuations around such trajectories. We discuss how to obtain predictions for the evolution of arbitrary observables and how to generate conformations which are representative of the transition state ensemble. We present an illustrative implementation of our method by studying the diffusion of a point particle in a 2-dimensional funneled external potential.

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“…(8) is motivated by its norm conservation and long term stability and, as showed hereafter, applicability to ergodicity breaking and bifurcation of relaxation time as a natural extension of Eqs. (1)(2)(3)(4)(5)(6)(7)(8).…”

Section: Fokker-planck and Imaginary Time Schrödinger Equation Schemementioning

confidence: 99%

Symplectic integration approach for metastable systems

Klotins

2006

Eur. Phys. J. B

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Diffusion in a bistable potential: A systematic WKB treatment

Cited by 152 publications

References 10 publications

Geometric structure and information change in phase transitions

Geometric structure and information change in phase transitions

Dominant reaction pathways in high-dimensional systems

Symplectic integration approach for metastable systems

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