Mean-squared displacement and variance for confined Brownian motion

TL;DR

This paper derives an analytical expression for the mean-squared displacement of confined Brownian motion, revealing how it transitions from diffusive to confined behavior, and extends the analysis to higher dimensions and different initial conditions.

Contribution

The paper provides a quantitative solution for MSD in confined Brownian motion using EMA, including the intermediate regime and extension to multi-dimensional systems with various initial PDFs.

Findings

MSD transitions from 2Dt to cL^2 over time

Derived explicit formulas for MSD and power α(t) in confinement

Analyzed effects of initial conditions on MSD and PV

Abstract

For one-dimension Brownian motion in the confined system with the size $L$, the mean-squared displacement(MSD) defined by $\left \langle (x-x_0)^2 \right\rangle$ should be proportional to $t^{\alpha(t)}$. The power $\alpha(t)$ should range from $1$ to $0$ over time, and the MSD turns from $2Dt$ to $c L^2$, here the coefficient $c$ independent of $t$, $D$ being the diffusion coefficient. The paper aims to quantitatively solve the MSD in the intermediate confinement regime. The key to this problem is how to deal with the propagator and the normalization factor of the Fokker-Planck equation(FPE) with the Dirichlet Boundaries. Applying the Euler-Maclaurin approximation(EMA) and integration by parts for the small $t$, we obtain the MSD being $2Dt(1-\frac{2\sqrt{\xi} }{3\pi\sqrt{\pi}})$, with $t_{ch}=\frac{L^2}{4\pi^2D},\xi\equiv \frac{t}{t_{ch}}$, and the power $\alpha(t)$ being $\frac{1-0.18\sqrt{\xi}}{1-0.12\sqrt{\xi}}$. Further, we analysis the MSD and the power for the $d$-dimension system with $\gamma$-dimension confinement. In the case of $\gamma< d$, there exists the sub-diffusive behavior in the intermediate time. The universal description is consistent with the recent experiments and simulations in the micro-nano systems. Finally, we calculate the position variance(PV) meaning $\left\langle (x-\left\langle x \right\rangle)^2 \right\rangle$. Under the initial condition referring to the different probability density function(PDF) being $p_{0}(x)$, MSD and PV should exhibit different dependencies on time, which reflect corresponding diffusion behaviors.As examples, the paper discusses the representative initial PDFs reading $p_{0}(x)=\delta(x-x_0)$, with the midpoint $x_0=\frac{L}{2}$ and the endpoint $x_0=\epsilon$(or $0^+$).The MSD(equal to PV) reads $2Dt(1-\frac{5\pi^3 Dt}{L^2})$,and $\frac{4}{\pi}(2Dt)[1+\frac{2\sqrt{\pi Dt}}{L}]$for the small $t$,respectively.