Anomalous Fluctuations in the Motion of Partitioning Objects

Abstract

We study how an object that partitions in two its thermally agitated environment relaxes to its equilibrium fluctuations. Event-driven molecular dynamics indicate that a conventional diffusive-like bound-Brownian motion is reached after a long-lived sub-diffusive regime. The two-stage behavior is not eliminated increasing the number of agitated particles, suggesting that the origin of the effect is a topologically frustrated exploration of phase-space.

Appendix

We note that our study of fluctuations in partitioning object position $x$ gives us direct information on its velocity correlation function. In fact, starting from $x(0)=0$, we have

$$\begin{aligned} x(t)=\int \limits _0^tdt'v(t'), \end{aligned}$$

(2)

that is,

$$\begin{aligned} \langle x^2(t) \rangle =\int \limits _0^tdt'\int _0^tdt''\langle v(t')v(t'') \rangle , \end{aligned}$$

(3)

from which we obtain

$$\begin{aligned} \frac{d}{dt}\langle x^2(t) \rangle =\int \limits _0^tdt''\langle v(t)v(t'')\rangle \nonumber \\ +\int \limits _0^tdt'\langle v(t')v(t)\rangle \nonumber \\ =2\int \limits _0^tdt'\langle v(t')v(t)\rangle , \end{aligned}$$

(4)

i.e., after setting $\theta =t'-t$,

$$\begin{aligned} \frac{d}{dt}\langle x^2(t) \rangle =2\int \limits _{-t}^0d\theta \langle v(t)v(t+\theta )\rangle . \end{aligned}$$

(5)

Since we are interesting in the behavior for large values of $t$ and, as we will see, the velocity correlation time tends to vanish as $t$ increases, Eq. (5) can be approximated as

$$\begin{aligned} \frac{d}{dt}\langle x^2(t) \rangle =2\int \limits _{-\infty }^0d\theta \langle v(t)v(t+\theta )\rangle . \end{aligned}$$

(6)

If we reasonably assume a Gaussian decay $\langle v(t)v(t+\theta )\rangle \sim \exp {(-C(t)\theta ^2)}$ (for $\theta <0$) and recall that, after a short initial transient, $\langle v^2(t) \rangle$ reaches its equilibrium value $kT/M$, Eq. (6) gives

$$\begin{aligned} \frac{d}{dt}\langle x^2(t) \rangle&= 2 \frac{kT}{M}\int \limits _{-\infty }^0d\theta \exp {(-C(t)\theta ^2)} \nonumber \\&= \sqrt{\frac{\pi }{C(t)}}\frac{kT}{M}. \end{aligned}$$

(7)

We now consider the two situations characterized by exponential and power law behavior of $\langle x^2(t) \rangle$, i.e., in particular, diathermal and adiabatic partitioning objects,

$$\begin{aligned} \langle x^2(t)\rangle = \langle x^2\rangle _{\text {EQ}}[1-\exp {(-t/\tau )}], \end{aligned}$$

(8)

$$\begin{aligned} \langle x^2(t) \rangle =\langle x^2 \rangle _{\text {EQ}}[1-(\tau '/t)^{\alpha }], 0<\alpha <1. \end{aligned}$$

(9)

In the first case, from Eq. (8), we have

$$\begin{aligned} \frac{d}{dt}\langle x^2 \rangle =\frac{\langle x^2 \rangle _{\text {EQ}}}{\tau }e^{-t/\tau }, \end{aligned}$$

(10)

while in the second case, Eq. (9) leads to

$$\begin{aligned} \frac{d}{dt}\langle x^2 \rangle =\langle x^2 \rangle _{\text {EQ}}\alpha (\tau )'^{\alpha }t^{-\alpha -1}. \end{aligned}$$

(11)

Comparing Eqs. (10,11) with Eq. (7), we obtain, respectively,

$$\begin{aligned} C(t)&= \pi \frac{k^2T^2\tau ^2}{M^2 \langle x^2 \rangle _{\text {EQ}}^2}e^{\frac{2t}{\tau }},\end{aligned}$$

(12)

$$\begin{aligned} C(t)&= \pi \frac{k^2T^2}{M^2 (\tau ')^{2\alpha }\alpha ^2\langle x^2 \rangle ^2_{\text {EQ}}}t^{2\alpha +2}. \end{aligned}$$

(13)

For a Gaussian behavior $\exp {(-C(t)\theta ^2)}$ the typical decay time is given by $C(t)^{-1/2}$, so that the velocity self-correlation times $\tau _{c,exp}$ and $\tau _{c,pl}$ are

$$\begin{aligned} \tau _{c,exp}(t)&= \frac{M\langle x^2\rangle _{\text {EQ}}}{\sqrt{\pi }kT\tau }e^{-t/\tau }.\end{aligned}$$

(14)

$$\begin{aligned} \tau _{c,pl}(t)&= \frac{M\langle x^2 \rangle _{\text {EQ}}}{\sqrt{\pi }kT}\tau _{pl}^{\alpha }\alpha t^{-\alpha -1}. \end{aligned}$$

(15)

Interestingly, the characteristic velocity memory times tend to vanish as $t$ increases either following an exponential or a power-law depending on whether $\langle x^2 \rangle$ approaches its equilibrium value $\langle x^2\rangle _{\text {EQ}}$ with an exponential or a power-law behavior.