Heterogeneous diffusion in an harmonic potential: the role of the interpretation

TL;DR

This work analyzes a Brownian particle in a harmonic potential within a heterogeneous environment, focusing on how the interpretation of multiplicative noise, encoded by $α$, shapes diffusion when $D(x)$ is spatially varying. By deriving and solving the corresponding Fokker–Planck equations for two paradigms—constant damping with broken FD relations and FD-consistent, position-dependent damping—the authors obtain analytical forms for the time-dependent and stationary PDFs, including a generalised two-piece Gaussian whose weights depend on $α$ (and on the interface location at $x=0$). They validate these results with numerical Langevin simulations across Itô, Stratonovich, and Hänggi–Klimontovich interpretations, and highlight how the $α$-dependence leads to qualitatively different interface behavior, with HK often recovering Boltzmann statistics even in heterogeneous environments. The study connects to broader themes in quenched/annealed disorder and spurs avenues for extending to arbitrary potentials or correlated noise processes, with practical relevance to interpreting diffusion in complex biological media.

Abstract

Diffusion in heterogeneous energy and diffusivity landscapes is widespread in biological systems. However, solving the Langevin equation in such environments introduces ambiguity due to the interpretation parameter $α$, which depends on the underlying physics and can take values in the range $0<α<1$. The typical interpretations are Itô ($α=0$), Stratonovich ($α=1/2$), and Hänggi-Klimontovich ($α=1$). Here, we analyse the motion of a particle in an harmonic potential -- modelled as an Ornstein-Uhlenbeck process -- with diffusivity that varies in space. Our focus is on two-phase systems with a discontinuity in environmental properties at $x=0$. We derive the probability density of the particle position for the process, and consider two paradigmatic situations. In the first one, the damping coefficient remains constant, and fluctuation-dissipation relations are not satisfied. In the second one, these relations are enforced, leading to a position-dependent damping coefficient. In both cases, we provide solutions as a function of the interpretation parameter $α$, with particular attention to the Itô, Stratonovich, and Hänggi-Klimontovich interpretations, revealing fundamentally different behaviours, in particular with respect to an interface located at the potential minimum.

Figures (5)

• Figure 1: PDFs of the heterogeneous OU process with $\tau=25$ and piecewise diffusion coefficient with $D_-=2$ and $D_+=1$ for the interpretation parameters $\alpha=0$, $1/2,$ and $1$, corresponding to the Itô, Stratonovich, and Hänggi-Klimontovich interpretations, respectively. (a-c) PDFs at time $t=1$ so that $t/\tau=0.04\ll1$. The dotted lines are the PDFs of Brownian motion, Eq. (\ref{['eq:BM_sol']}). (d-f) PDFs at time $t=25$ so that $t/\tau=1$. The dashed lines are analytical solutions, Eq. (\ref{['eq:gen_PDF']}). (g-i) PDFs at time $t=1000$ so that $t/\tau=40\gg1$. The solid lines are the steady state solutions, Eq. (\ref{['eq:stat_sol']}).
• Figure 2: PDFs of the heterogeneous OU process with a piecewise diffusion coefficient with $D_-=2$ and $D_+=1$ for the Itô interpretation with initial condition $x_0=5$. In the simulations we set $\tau=25$. PDF at times (a) $t=1$, (b) $t=25$, and (c) $t=1000$. The dotted line is the PDF of the homogeneous OU process, Eq. (\ref{['eq:free_OU']}), with $D_+=1$ at $t=1$. The solid line shows the steady state PDF, Eq. (\ref{['eq:stat_sol']}).
• Figure 3: Temporal evolution of the probability of finding a particle in the negative part of the $x$-axis, $P(x<0)$, for the heterogeneous OU process with a piecewise constant diffusion coefficient with $D_-=2$ and $D_+=1$ with Itô, Stratonovich, and Hänggi-Klimontovich (HK) interpretations with initial condition $x_0=5$, and $\tau=25$. The dashed lines corresponds to $\beta(\alpha)$ for $x_0=0$ (Eq. (\ref{['eq:beta']})) for the three interpretations.
• Figure 4: PDFs of the heterogeneous OU process with a piecewise diffusion coefficient, constant temperature, and fluctuation-dissipation relations. The diffusivities are $D_-=2$ and $D_+=1$. In the simulations, we set $D^*\tau^*=k_BT/k=100$. (a-c) PDFs at time $t=1$ so that $t/D^*\tau^* =0.01\ll1$. The dotted lines are the PDF (\ref{['eq:BM_sol']}) of Brownian motion. (d-f) PDFs at time $t=100$ so that $t/D^*\tau^*=1$. The dashed lines are the analytical solutions, Eq. (\ref{['eq:gen_PDF_2']}). (g-i) PDFs at time $t=1000$ so that $t/D^*\tau^*=10>1$. The solid lines represent the steady state PDF (\ref{['eq:PDF_HK_2']}).
• Figure 5: PDFs of the heterogeneous OU process with continuous diffusion coefficient defined in Eq. (\ref{['eq:cont_D']}), shown for three values of the parameter $k$: $0.5$, $1$, and $10$ corresponding to panels (a), (b), and (c), respectively. Each panel also displays the corresponding diffusion coefficient used in the simulations.