To exploit the fact that the correlation time τ^* of the fluctuating force F^' is very short, co

Step 1: Let's start by recalling the Langevin equation, which describes the motion of a particle

To exploit the fact that the correlation time τ^* of the...

To exploit the fact that the correlation time $\tau^*$ of the fluctuating force $F^{\prime}$ is very short, consider a time $\tau$ such that $\boldsymbol{\tau} \gg \boldsymbol{\tau}^*$, but which is macroscopically very short in the sense that $\mathrm{r} \ll \gamma^{-1}$. The force $F^{\prime}$ is then not correlated in successive intervals of length $\tau$ (sll correlations due to the slowly varying interaction force having already been explicitly absorbed in the term $-\gamma v$ of the Langevin equation). By dividing the time interval $t$ into $N$ successive intervals s so that $t=N_{\mathrm{T}}$, ahow that in the preceding problem the solution of the Langevin equation can be written in the form $$ \begin{gathered} v-v_k e^{-y^{\prime}}-Y=\sum_{k=0}^{x_j-1} y_k \\ y_k=e^{-\gamma t} e^{\gamma k} G_k=e^{-\gamma r(N-k)} G_k \\ G_k=\frac{1}{m} \int_0^r F^{\prime}(k r+\varepsilon) d z \end{gathered} $$ Since $\tau \gg r^*$, the statistical properties of $G_k$ are the same in each interval of length $r$. Furthermore, the quantities $b_k$ (or $G_k$ ) are statistically independent of each other.

To exploit the fact that the correlation time $\tau^*$ of the fluctuating force $F^{\prime}$ is very short, consider a time $\tau$ such that $\boldsymbol{\tau} \gg \boldsymbol{\tau}^*$, but which is macroscopically very short in the sense that $\mathrm{r} \ll \gamma^{-1}$. The force $F^{\prime}$ is then not correlated in successive intervals of length $\tau$ (sll correlations due to the slowly varying interaction force having already been explicitly absorbed in the term $-\gamma v$ of the Langevin equation). By dividing the time interval $t$ into $N$ successive intervals s so that $t=N_{\mathrm{T}}$, ahow that in the preceding problem the solution of the Langevin equation can be written in the form

$$
\begin{gathered}
v-v_k e^{-y^{\prime}}-Y=\sum_{k=0}^{x_j-1} y_k \\
y_k=e^{-\gamma t} e^{\gamma k} G_k=e^{-\gamma r(N-k)} G_k \\
G_k=\frac{1}{m} \int_0^r F^{\prime}(k r+\varepsilon) d z
\end{gathered}
$$

Since $\tau \gg r^*$, the statistical properties of $G_k$ are the same in each interval of length $r$. Furthermore, the quantities $b_k$ (or $G_k$ ) are statistically independent of each other.

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