[Paper Review] Unified approach to reset processes and application to coupling between process and reset

G. J. Lapeyre, Marco Dentz|arXiv (Cornell University)|Mar 19, 2019

Diffusion and Search DynamicsBiochemistry, Genetics and Molecular Biology64 references3 citations

TL;DR

This paper presents a unified analytical framework for observables in stochastic processes under reset, focusing on functionals dependent on the most recent renewal period. It reveals that coupling diffusion and reset via microscopic disorder drastically alters the effective decay rate, contradicting the naive expectation that reset counterbalances decay, and derives exact asymptotic expressions for power-law scaling observables in subdiffusive systems with disordered reset.

ABSTRACT

We present a unified approach to those observables of stochastic processes under reset that take the form of averages of functionals depending on the most recent renewal period. We derive solutions for the observables, and determine the conditions for existence and equality of their stationary values with and without reset. For intermittent reset times, we derive exact asymptotic expressions for observables that vary asymptotically as a power of time. We illustrate the general approach with general and particular results for the power spectral density, and moments of subdiffusive processes. We focus on coupling of the process and reset via a diffusion-decay process with microscopic dependence between transport and decay. In contrast to the uncoupled case, we find that restarting the particle upon decay does not produce a probability current equal to the decay rate, but instead drastically alters the time dependence of the decay rate and the resulting current.

Motivation & Objective

• To develop a general analytical approach for observables in reset processes that depend on the most recent renewal period.
• To investigate the impact of microscopic disorder on the coupling between transport (diffusion) and reset dynamics.
• To determine conditions under which stationary distributions exist and are equal with and without reset.
• To derive exact asymptotic expressions for observables that scale as a power of time under intermittent reset.
• To analyze the interplay between diffusion, decay, and reset in disordered media, challenging the assumption of independent reset mechanisms.

Proposed method

• Derives general solutions for observables in reset processes using Laplace and Fourier transforms of the probability density.
• Introduces a formalism based on the renewal structure of reset processes, focusing on functionals of the most recent renewal interval.
• Applies the framework to subdiffusive processes by incorporating a time-fractional diffusion coefficient $\mathcal{D}_\alpha$.
• Uses the generating function $\langle(\lambda + \rho)^\alpha\rangle$ to model the distribution of reset times $\rho$.
• Derives the stationary density by taking the $\lambda \to 0$ limit in the Fourier-Laplace domain, revealing the role of $\langle\rho^\alpha\rangle$.
• Establishes equivalence between the long-time normalized density under decay and the stationary density under reset, under specific conditions on $\psi_\rho(r)$.

Experimental results

Research questions

• RQ1 How do observables in reset processes depend on the renewal history when reset times are stochastic?
• RQ2 What conditions ensure the existence and equality of stationary distributions with and without reset?
• RQ3 How does microscopic disorder coupling diffusion and reset alter the effective decay rate compared to independent reset?
• RQ4 What are the exact asymptotic forms of observables like power spectral density and moments in subdiffusive systems with reset?
• RQ5 In what way does the coupling between transport and reset dynamics invalidate the assumption that reset rate equals decay rate?

Key findings

• The stationary density under disordered reset is $\hat{\tilde{p}}_{\text{r}}(k,\lambda \to 0) = \frac{\tilde{c}_0(k)\langle\rho^\alpha\rangle}{\langle\rho^\alpha\rangle + k^2\mathcal{D}_\alpha}$, showing a non-equilibrium steady state distinct from equilibrium.
• The long-time normalized density under decay converges to the same form as the stationary density under reset, implying asymptotic equivalence in distribution shape.
• The effective decay rate is not simply the reset rate; instead, the coupling induces a non-trivial time dependence that alters the emergent current and relaxation dynamics.
• For power-law scaling observables, the framework yields exact asymptotic expressions that depend on the tail behavior of the reset time distribution $\psi_\rho(r) \sim r^{\nu-1}$ near $r=0$.
• The Laplace transform of the reset rate kernel $\hat{k}_{R_d}(\lambda)$ and the memory kernel $\hat{K}(\lambda)$ are derived, enabling a generalized Fokker-Planck-type equation with memory.
• The solution of the generalized Fokker-Planck equation (Eq. S59) shows that the system's dynamics are governed by a convolution with memory kernels, reflecting the history dependence of reset.