On the Generalized Langevin Equation for a Rouse Bead in a Nonequilibrium Bath

Abstract

We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a steady state after some time. Secondly, we consider the case of stochastic active forces which will drive the system to a nonequilibrium state. Including these active forces results in an extra contribution to the second fluctuation–dissipation relation. The form of this active contribution is analysed for the specific case of Gaussian, exponentially correlated active forces. We also discuss the resulting rich dynamics of the middle bead in which various regimes of normal diffusion, subdiffusion and superdiffusion can be present.

Appendix

In this appendix we give a brief outline of our calculations. For more details we refer to chapter 6 of [24]. We consider a Rouse chain with $N=2M+1$ beads. Their position vectors obey the equation of motion (1). We will denote the position of the middle bead by $\mathbf {r}(t)=\mathbf {R}_{M+1}(t)$.

In order to derive the equation of motion for the middle bead, we introduce left and right normal coordinates $\mathbf {X}_p^L(t)$ and $\mathbf {X}_p^R(t)$ ($p=1,2,\ldots ,M$), which are defined as

$$\begin{aligned} \mathbf {X}_p^L(t)= & {} \frac{2}{\sqrt{N}} \sum _{n=1}^M C_n^p \, \mathbf {R}_n(t), \end{aligned}$$

(21)

$$\begin{aligned} \mathbf {X}_p^R(t)= & {} - \frac{2}{\sqrt{N}} \sum _{n=M+2}^N C_n^p \, \mathbf {R}_n(t), \end{aligned}$$

(22)

where the transformation coefficients $C_n^p$ are given by

$$\begin{aligned} C_n^p = \cos \left( \frac{2p-1}{N} \left( n-\frac{1}{2}\right) \pi \right) . \end{aligned}$$

A long but straightforward calculation shows that these normal coordinates obey the following equations of motion

$$\begin{aligned} \gamma \, \dot{\mathbf {X}}_p^L(t)= & {} -k \lambda _p \mathbf {X}_p^L(t) - (-1)^p \frac{2k}{\sqrt{N}} S_p\, \mathbf {r}(t) + \mathbf {f}_p^L(t), \end{aligned}$$

(23)

$$\begin{aligned} \gamma \, \dot{\mathbf {X}}_p^R(t)= & {} -k \lambda _p \mathbf {X}_p^R(t) - (-1)^p \frac{2k}{\sqrt{N}} S_p\, \mathbf {r}(t) + \mathbf {f}_p^R(t), \end{aligned}$$

(24)

with $\lambda _p = 4 \sin ^2 \left( (2p-1)\pi /2N\right)$, $S_p =\sin ((2p-1)\pi /N)$ and

$$\begin{aligned} \mathbf {f}_p^L(t)= & {} \frac{2}{\sqrt{N}} \sum _{n=1}^M C_n^p\, \mathbf {\xi }_{T,n} (t), \end{aligned}$$

(25)

$$\begin{aligned} \mathbf {f}_p^R(t)= & {} -\frac{2}{\sqrt{N}} \sum _{n=M+2}^N C_n^p\, \mathbf {\xi }_{T,n}(t). \end{aligned}$$

(26)

The differential Eqs. (23) and (24) are uncoupled, linear and nonhomogeneous. They can therefore be solved immediately. Inverting (21) and (22), we can write the positions of the beads in terms of the normal coordinates. This gives (for $n=1,2,\ldots ,M$ and $m=M+2,M+3,\ldots ,N$)

$$\begin{aligned} \mathbf {R}_n(t)= & {} \frac{2}{\sqrt{N}} \sum _{p=1}^M C_n^p\, \mathbf {X}_p^L(t), \end{aligned}$$

(27)

$$\begin{aligned} \mathbf {R}_m(t)= & {} -\frac{2}{\sqrt{N}} \sum _{p=1}^M C_m^p\, \mathbf {X}_p^R(t). \end{aligned}$$

(28)

Using these expressions one can easily show that the position of the middle bead evolves according to

$$\begin{aligned} \gamma \, \dot{\mathbf {r}}(t) = \mathbf {\xi }_T (t) - 2k \mathbf {r}(t) - \frac{2k}{\sqrt{N}} \sum _{p=1}^M (-1)^p S_p \Big ( \mathbf {X}_p^L (t) + \mathbf {X}_p^R (t)\Big ). \end{aligned}$$

(29)

Inserting the solutions of (23) and (24) gives for the equation of motion of the tagged middle bead

$$\begin{aligned} \gamma \, \dot{\mathbf {r}}(t)= & {} \mathbf {\xi }_T(t) - 2k\mathbf {r}(t) - \frac{2k}{\sqrt{N}} \sum _{p=1}^M (-1)^p S_p \Bigg [ \Big ( \mathbf {X}_p^L(0) + \mathbf {X}_p^R(0)\Big ) e^{-t/\tau _p} \nonumber \\&\quad +\,\frac{1}{\gamma } \int _0^t d\tau \Big ( \mathbf {f}^L_p(\tau ) + \mathbf {f}^R_p(\tau ) \Big ) e^{(\tau -t)/\tau _p} - (-1)^p \frac{4k}{\gamma \sqrt{N}} S_p \int _0^t d\tau \ \mathbf {r}(\tau )e^{(\tau -t)/\tau _p} \Bigg ]. \nonumber \\ \end{aligned}$$

(30)

The last integral can be rewritten using integration by parts

$$\begin{aligned} \int _0^t d\tau \ \mathbf {r}(\tau )e^{(\tau -t)/\tau _p}=\tau _p \mathbf {r}(t) - \tau _p \mathbf {r}(0)e^{-t/\tau _p} - \tau _p \int _0^t d\tau \ e^{(\tau -t)/\tau _p} \,\dot{\mathbf {r}}(\tau ). \end{aligned}$$

When we put this back into the equation of motion, we can group some terms and identify them in the context of the generalized Langevin equation. We finally get

$$\begin{aligned} \gamma \, \dot{\mathbf {r}}(t) = \mathbf {\xi }_T(t) - \int _0^t d\tau \, K(t-\tau )\, \dot{\mathbf {r}}(\tau ) + \mathbf {\Phi }(t), \end{aligned}$$

(31)

where the memory kernel equals

$$\begin{aligned} K(t) = \frac{8 k^2}{\gamma N} \sum _{p=1}^M \tau _p S_p^2\, e^{-t/\tau _p}, \end{aligned}$$

(32)

and the effective noise is given by

$$\begin{aligned} \mathbf {\Phi }(t) = - \frac{2k}{\sqrt{N}} \sum _{p=1}^N (-1)^p S_p\, e^{-t/\tau _p} \Bigg [ \mathbf {X}_p^L(0) + \mathbf {X}_p^R(0)+ & {} (-1)^p \frac{4k}{\sqrt{N}} \tau _p S_p\, \mathbf {r}(0) \nonumber \\+ & {} \frac{1}{\gamma } \int _0^t d \tau \, \Big ( \mathbf {f}_p^L(\tau ) + \mathbf {f}_p^R (\tau ) \Big ) e^{\tau /\tau _p} \Bigg ].\nonumber \\ \end{aligned}$$

(33)

From this definition and the relations (25) and (26), it can be seen $\mathbf {\Phi }(t)$ is a Gaussian random variable. Long yet straightforward calculations can show that the average of this random variable is zero and that its correlation is given by relation (4).

When active forces $\mathbf {\xi }_{A,n}(t)$ are added, the calculation proceeds along the same line. The main difference is that the functions $\mathbf {f}_p^L(t)$ and $\mathbf {f}_p^R(t)$ are modified to include the active forces. The definitions (25) and (26) are now replaced by

$$\begin{aligned} \mathbf {f}_p^L(t)= & {} \frac{2}{\sqrt{N}} \sum _{n=1}^M C_n^p\, \Big ( \mathbf {\xi }_{T,n} (t) + \mathbf {\xi }_{A,n} (t)\Big ), \end{aligned}$$

(34)

$$\begin{aligned} \mathbf {f}_p^R(t)= & {} -\frac{2}{\sqrt{N}} \sum _{n=M+2}^N C_n^p\, \Big ( \mathbf {\xi }_{T,n}(t) + \mathbf {\xi }_{A,n} (t)\Big ). \end{aligned}$$

(35)

With this modified definition, the equations of motion of the normal coordinates are still given by (23) and (24) and the equation of motion of the tagged particles is still given by (30). As a result the memory kernel K(t) is not modified. The only difference is that the extra terms due to the active forces in (34) and (35) will lead to a different form for the effective noise correlation. Another calculation then leads to the results for the noise correlation given in the main text, i.e. the expressions (9) and (10).