Violation of the method of images in non-Markovian processes and its connection to stochastic thermodynamics

TL;DR

The paper investigates why the widely used method of images fails for non-Markovian diffusion and first-passage problems. By constructing a conjugate reflected path and analyzing the path-probability ratio, the authors derive a fluctuation-type relation for a heat-like quantity $Λ$ and show memory breaks reflection symmetry, causing the MI solution to fail. They identify $Λ$ as the heat exchanged during the post-first-passage evolution and establish an integral fluctuation theorem ⟨ e^{2 Λ} ⟩ = 1, yielding ⟨ Λ ⟩ ≤ 0 and a thermodynamic interpretation of MI breakdown. Using fractional Brownian motion as a paradigmatic non-Markovian process, they also obtain an exact expression for the survival probability in terms of the exponential heat average, linking memory, energetics, and first-passage statistics. The results provide a quantitative framework for non-Markovian first-passage problems within stochastic thermodynamics and offer insights into memory-induced forces in complex environments such as polymers.

Abstract

We discuss a failure of the wide-spread method of images solution to describe the time evolution of probability distribution in diffusive processes with memory. For a path that touches a target during stochastic evolution, we define its conjugate twin of reflected path, and show that their path probability ratio obeys a relation analogous to the fluctuation theorem. With a key quantity properly identified as the heat, the resultant thermodynamic interpretation of the processes provides a quantitative basis as well as an intuitive physical picture on how and why the method of images breaks down for non-Markovian processes.

Figures (6)

• Figure 1: (a)-(c) Plots of PDFs $P_D(x,N|x_0)$ (red) and $P_0(x,N|x_0)$ (black dashed) in Eq. \ref{['P_D']} after $N=10^3$ steps starting from $x_0= 300^{\alpha/2}$ for (a) Markovian walker and (b) anti-persistent or (c) persistent non-Markovian walker. Also shown in each panel is $P_0(x,N|-x_0)$ (blue dotted). As a model of non-Markovian walker, we adopt the fBm with anomalous exponent $\alpha ( \neq 1)$ for mean square displacement. $P_0(x,N|\pm x_0)$ is the Gaussian function with the mean $\pm x_0$ and the variance $2 (N \Delta t)^{\alpha}$, while $P_D(x,N|x_0)$ is obtained by numerical simulation (see SI). (d) Schematic spatiotemporal plots of the original and the reflected paths with solid and dashed lines, respectively, for the time period $[0,N]$ along vertical time-axis. The reflected path is constructed by folding $\{ x_i \}_0^n$.
• Figure 2: Examples of first passage paths and resultant memory forces acting on the walker in subsequent process for (a) subdiffusive fBm ($\alpha = 0.5$) and (b) superdiffusive fBm ($\alpha=1.5$). The time axis is shifted to set the moment of the first passage to be origin.
• Figure 3: Average heat flow in the post first passage process. (a) fBm with $\alpha=0.8$ and (b) fBm with $\alpha=1.2$. The starting position is $x_0=300^{\alpha/2}$. Average absorbed heat $\langle \Lambda \rangle_{D,x}$ along the dead paths (red) and $\langle \Lambda \rangle_{{\overline D},x}$ along the reflected paths (blue) are shown as functions of the end point $x\, (\ge 0)$ after $N=10^3$ steps. Note that the relation $\langle \Lambda \rangle_{D,0}= \langle \Lambda \rangle_{{\overline{D}},0}$ holds due to symmetry.
• Figure 4: Numerical plots of the key factors appearing in Eq. (\ref{['DF_2_cal']}) for (a) subdiffusive fBm ($\alpha=0.5$, $x=5$, $x_0=\sqrt{300^{\alpha}}$, $N=1\times 10^3$) and (b) superdiffusive fBm ($\alpha=1.5$, $x=100$, $x_0=\sqrt{300^{\alpha}}$, , $N=1\times 10^3$). Red or blue solid line represents $P_D(x,\Lambda,N|x_0)$ or, $P_{\overline{D}}(x,-\Lambda,N|\overline{x}_0)$, respectively. Black cross symbols indicate $P_{\overline{D}}(x,-\Lambda,N|\overline{x}_0)e^{-2\Lambda}$.
• Figure 5: Sample averages of conditional paths with $x^*=0$ for (a) subdiffusive fBm ($\alpha = 0.5$) with $x_0=\sqrt{300^{\alpha}}$, $n=200$, $N=1\times 10^3$ and (b) superdiffusive fBm ($\alpha=1.5$) with $x_0=\sqrt{300^{\alpha}}$, $n=200$, $N=1\times 10^3$. The averaging is done with 71 sample paths for (a) and 30 sample paths for (b).