Violation of the Second Fluctuation-dissipation Relation and Entropy Production in Nonequilibrium Medium

Abstract

We investigate a class of nonequilibrium media described by Langevin dynamics that satisfies the local detailed balance. For the effective dynamics of a probe immersed in the medium, we derive an inequality that bounds the violation of the second fluctuation-dissipation relation (FDR). We also discuss the validity of the effective dynamics. In particular, we show that the effective dynamics obtained from nonequilibrium linear response theory is consistent with that obtained from a singular perturbation method. As an example of these results, we propose a simple model for a nonequilibrium medium in which the particles are subjected to potentials that switch stochastically. For this model, we show that the second FDR is recovered in the fast switching limit, although the particles are out of equilibrium.

Appendices

Appendix A: Derivation of the Effective Dynamics for the Potential Switching Medium

Excess Action

We first confirm that, to first order in $X_s-X_t$, the excess action ${\mathcal {A}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X]))$ is given by

$$\begin{aligned} -{\mathcal {A}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X]) \simeq \dfrac{1}{2k_{\mathrm {B}}T}&\left[ \int ^t_{-\infty }ds(X_s-X_t) \sum _j\dfrac{\partial }{\partial x^j_s}\varPhi ({{\varvec{x}}}_s,X_t)\circ {\dot{x}}^j_s\right. \nonumber \\&\quad \left. -\int ^t_{-\infty }ds(X_s-X_t){\mathcal {L}}^\dagger _s\varPhi ({{\varvec{x}}}_s,X_t)\right] \end{aligned}$$

(A.1)

with the backward operator for the dynamics (80) with $X_t$ held fixed:

$$\begin{aligned} {\mathcal {L}}^\dagger _s:=\sum _j\left[ \dfrac{1}{\gamma }(-\kappa _b (x^j_s-\sigma ^j_sL)-\lambda \kappa _c(x^j_s-X_t))\dfrac{\partial }{\partial x^j_s}+\dfrac{k_{\mathrm {B}}T}{\gamma }\dfrac{\partial ^2}{\partial (x^j_s)^2}\right] . \end{aligned}$$

(A.2)

To this end, we calculate ${\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])$ and ${\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|X_t)$. We first consider a trajectory in the time interval [0, t] and discretized time $t_n=n\varDelta t\in [0,t]$ ($n=0,1,\cdots ,M$) with $t\equiv M\varDelta t$. Correspondingly, let $[{{\varvec{x}}},{{\varvec{\sigma }}}]:=\{({{\varvec{x}}}_0,{{\varvec{\sigma }}}_0),({{\varvec{x}}}_1,{{\varvec{\sigma }}}_1),\cdots ,({{\varvec{x}}}_M,{{\varvec{\sigma }}}_M)\}$ be the discretized trajectory, where $({{\varvec{x}}}_n,{{\varvec{\sigma }}}_n):=({{\varvec{x}}}_{t_n},{{\varvec{\sigma }}}_{t_n})$. Suppose that the state $\sigma ^j$ is switched at time intervals with $n=n^j_1,n^j_2,\cdots ,n^j_{k_j}\in \{0,1,\cdots ,M\}$ as

$$\begin{aligned} \sigma ^j_{n^j_\ell +1}=1-\sigma ^j_{n^j_\ell }. \end{aligned}$$

(A.3)

We denote by $\varSigma ^j_\ell \in \{0,1\}$ the value of $\sigma ^j_n$ for $n^j_\ell <n\le n^j_{\ell +1}$ with $n^j_0:=-1$ and $n^j_{k_j+1}:=M$. For notational simplicity, we rewrite (80) as

$$\begin{aligned} \gamma {\dot{x}}^j_s&=-U'_1(x^j_s,\sigma ^j_s)-\lambda V'_1(x^j_s,X_s) +\sqrt{2\gamma k_{\mathrm {B}}T}\xi ^j_s\nonumber \\&=-U'_1(x^j_s,\sigma ^j_s)-\lambda V'_1(x^j_s,X_t)+h_sf(x^j_s)+\sqrt{2\gamma k_{\mathrm {B}}T}\xi ^j_s, \end{aligned}$$

(A.4)

where $U_1(x^j,\sigma ^j):=\kappa _b(x^j-\sigma ^jL)^2/2$, $V_1(x^j,X):=\kappa _c(x^j-X)^2/2$, and the prime denotes the derivative with respect to $x^j_t$. In the second line, we have introduced a time-dependent amplitude $h_s:=X_s-X_t$ and $f(x^j_s):=\partial _j\varPhi ({{\varvec{x}}}_s,X_t)=\lambda \kappa _c$ to explicitly represent the deviation from the dynamics with $X_t$ held fixed. Then, the probability density of a trajectory ${\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])$ starting from $({{\varvec{x}}}_0,{{\varvec{\sigma }}}_0)$ reads [6]

$$\begin{aligned}&{\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])\nonumber \\&=\prod _j\prod ^{n^j_1-1}_{n=0}\sqrt{\dfrac{\gamma }{4\pi k_{\mathrm {B}}T \varDelta t}}\nonumber \\&\quad e^{-\frac{\varDelta t}{4\gamma k_{\mathrm {B}}T} \left[ \gamma \frac{x^j_{n+1}-x^j_n}{\varDelta t}+U'_1({\bar{x}}^j_n,\varSigma ^j_0) +\lambda V'_1({\bar{x}}^j_n,X_t)-{\bar{h}}_nf({\bar{x}}^j_n)\right] ^2 +\frac{\varDelta t}{2\gamma }\left[ U''_1({\bar{x}}^j_n,\varSigma ^j_0) +\lambda V''_1({\bar{x}}^j_n,X_t)-{\bar{h}}_nf'({\bar{x}}^j_n)\right] -r\varDelta t} \nonumber \\&\times \prod ^{k_j}_{\ell =1}r\varDelta t\delta (x^j_{n^j_\ell +1}-x^j_{n^j_\ell })\nonumber \\&\times \prod ^{n^j_{\ell +1}-1}_{n=n^j_\ell +1}\sqrt{\dfrac{\gamma }{4\pi k_{\mathrm {B}}T\varDelta t}}\nonumber \\&\quad e^{-\frac{\varDelta t}{4\gamma k_{\mathrm {B}}T}\left[ \gamma \frac{x^j_{n+1}-x^j_n}{\varDelta t}+U'_1({\bar{x}}^j_n,\varSigma ^j_\ell )+\lambda V'_1({\bar{x}}^j_n,X_t)-{\bar{h}}_nf({\bar{x}}^j_n)\right] ^2 +\frac{\varDelta t}{2\gamma }\left[ U''_1({\bar{x}}^j_n,\varSigma ^j_\ell )+\lambda V''_1({\bar{x}}^j_n,X_t)-{\bar{h}}_nf'({\bar{x}}^j_n)\right] -r\varDelta t}. \end{aligned}$$

(A.5)

Here, ${\bar{x}}^j_n:=(x^j_{n+1}+x^j_n)/2$ and ${\bar{h}}_n:=(h_{n+1}+h_n)/2$. We note that ${\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|X_t)$ is immediately obtained from (A.5) by setting ${\bar{h}}_n=0$. From these expressions, it follows that

$$\begin{aligned}&\quad \ln \dfrac{\mathcal {{\mathbb {P}}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])}{{\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|X_t)}=\sum _j\left[ \dfrac{1}{2k_{\mathrm {B}}T}\sum ^{M-1}_{n=0}{\bar{h}}_nf({\bar{x}}^j_n) (x^j_{n+1}-x^j_n) \right. \nonumber \\&\quad \left. -\dfrac{1}{2\gamma k_{\mathrm {B}}T}\left\{ \sum ^{n^j_1-1}_{n=0} {\bar{h}}_nf({\bar{x}}^j_n)[-U'_1({\bar{x}}^j_n,\varSigma ^j_0)-\lambda V'_1 ({\bar{x}}^j_n,X_t)]\varDelta t\right. \right. \nonumber \\&\quad \left. \left. +\sum ^{k_j}_{\ell =1}\sum ^{n^j_{\ell +1}-1}_{n=n^j_\ell +1} {\bar{h}}_nf({\bar{x}}^j_n)[-U'_1({\bar{x}}^j_n,\varSigma ^j_\ell )-\lambda V'_1({\bar{x}}^j_n,X_t)]\varDelta t\right\} \right. \nonumber \\&\quad \left. -\dfrac{1}{2\gamma }\sum ^{M-1}_{n=0}{\bar{h}}_nf'({\bar{x}}^j_n)\varDelta t\right] +O({\bar{h}}^2_n). \end{aligned}$$

(A.6)

By taking the continuum limit and replacing the time interval from [0, t] to $[-\infty ,t]$, we obtain the excess action ${\mathcal {A}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])$:

$$\begin{aligned} -{\mathcal {A}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|[X])&=\ln \dfrac{{\mathbb {P}}([{{\varvec{x}}}, {{\varvec{\sigma }}}]|[X])}{{\mathbb {P}}([{{\varvec{x}}},{{\varvec{\sigma }}}]|X_t)}=\sum _j\left[ \dfrac{1}{2k_{\mathrm {B}}T}\int ^t_{-\infty }dsh_sf(x^j_s) \circ {\dot{x}}^j_s\right. \nonumber \\&\quad \left. -\dfrac{1}{2\gamma k_{\mathrm {B}}T}\int ^t_{-\infty }dsh_sf(x^j_s) (-U'_1(x^j_s,\sigma ^j_s)-\lambda V'_1(x^j_s,X_t))\right. \nonumber \\&\quad \left. -\dfrac{1}{2\gamma } \int ^t_{-\infty }dsh_s\dfrac{\partial }{\partial x^j_s}f(x^j_s)\right] +O(h^2_s)\nonumber \\&\simeq \dfrac{1}{2k_{\mathrm {B}}T}\left[ \int ^t_{-\infty }dsh_s\sum _j \dfrac{\partial }{\partial x^j_s}\varPhi ({{\varvec{x}}}_s,X_t)\circ {\dot{x}}^j_s-\int ^t_{-\infty }dsh_s{\mathcal {L}}^\dagger _s\varPhi ({{\varvec{x}}}_s,X_t)\right] , \end{aligned}$$

(A.7)

where

$$\begin{aligned} {\mathcal {L}}^\dagger _s:=\sum _j\left[ \dfrac{1}{\gamma }(-U'_1(x^j_s,\sigma ^j_s)-\lambda V'_1(x^j_s,X_t))\dfrac{\partial }{\partial x^j_s}+\dfrac{k_{\mathrm {B}}T}{\gamma }\dfrac{\partial ^2}{\partial (x^j_s)^2}\right] . \end{aligned}$$

(A.8)

1.1 Explicit Calculation of $G(X_t)$, $\gamma (t-s)$, and $\gamma _{\mathrm {ex}}(t-s)$

Here, we calculate $G(X_t)$, $\gamma (t-s)$, and $\gamma _{\mathrm {ex}}(t-s)$ explicitly. The starting point is the stationary solution of (80) with $X_t$ held fixed (93):

$$\begin{aligned} x^j_t&=\dfrac{\lambda \kappa _c}{\kappa _b+\lambda \kappa _c}X_t +\dfrac{\kappa _b}{\kappa _b+\lambda \kappa _c}\dfrac{L}{2} +\int ^t_{-\infty }dse^{-\frac{t-s}{\tau _x}}\left[ \dfrac{\kappa _b}{\gamma } \left( \sigma ^j_s-\dfrac{1}{2}\right) L+\sqrt{\dfrac{2k_{\mathrm {B}}T}{\gamma }} \xi ^j_s\right] \nonumber \\&=\langle x^j_t\rangle ^{X_t}+\int ^t_{-\infty }dse^{-\frac{t-s}{\tau _x}} \left[ \dfrac{\kappa _b}{\gamma }\left( \sigma ^j_s-\dfrac{1}{2}\right) L +\sqrt{\dfrac{2k_{\mathrm {B}}T}{\gamma }}\xi ^j_s\right] . \end{aligned}$$

(A.9)

The statistical force is immediately obtained by substituting (A.9) into its definition:

$$\begin{aligned} G(X_t)&:=\langle \varPhi ({{\varvec{x}}}_t,X_t)\rangle ^{X_t}\nonumber \\&=\left\langle -\lambda \kappa _c\sum _j(X_t-x^j_t)\right\rangle ^{X_t}\nonumber \\&=-\dfrac{N\lambda \kappa _c\kappa _b}{\kappa _b+\lambda \kappa _c} \left( X_t-\dfrac{L}{2}\right) . \end{aligned}$$

(A.10)

To calculate the friction kernel, we first calculate the response function $R_{\varPhi \varPhi }(t-s)$. The response function $R_{\varPhi \varPhi }(t-s)$ is expressed as

$$\begin{aligned}&R_{\varPhi \varPhi }(t-s)\nonumber \\&=\dfrac{1}{2k_{\mathrm {B}}T}\left[ \dfrac{d}{ds}\langle \varPhi ({{\varvec{x}}}_s,X_t); \varPhi ({{\varvec{x}}}_t,X_t)\rangle ^{X_t}-\langle {\mathcal {L}}^\dagger _s\varPhi ({{\varvec{x}}}_s,X_t); \varPhi ({{\varvec{x}}}_t,X_t)\rangle ^{X_t}\right] \nonumber \\&=\dfrac{\lambda ^2\kappa ^2_c}{2k_{\mathrm {B}}T}\sum _i\sum _j\left[ \dfrac{d}{ds} \langle x^i_s-X_t;x^j_t-X_t\rangle ^{X_t}+\dfrac{\kappa _b}{\gamma }\langle x^i_s -\sigma ^i_sL;x^j_t-X_t\rangle ^{X_t}\right. \nonumber \\&\quad \left. +\dfrac{\lambda \kappa _c}{\gamma }\langle x^i_s -X_t;x^j_t-X_t\rangle ^{X_t}\right] \nonumber \\&=\dfrac{\lambda ^2\kappa ^2_c}{2k_{\mathrm {B}}T}\sum _i\sum _j\left[ \dfrac{d}{ds} \langle x^i_s;x^j_t\rangle ^{X_t}+\dfrac{\kappa _b+\lambda \kappa _c}{\gamma }\langle x^i_s;x^j_t\rangle ^{X_t}-\dfrac{\kappa _b}{\gamma }L\langle \sigma ^i_s;x^j_t\rangle ^{X_t}\right] . \end{aligned}$$

(A.11)

By using (A.9) and the relation

$$\begin{aligned} \langle \sigma ^i_t\sigma ^j_s\rangle = {\left\{ \begin{array}{ll} (1+e^{-2r|t-s|})/4\quad \text {for}\quad i=j\\ 1/4\quad \text {for}\quad i\ne j, \end{array}\right. } \end{aligned}$$

$\langle x^i_s;x^j_t\rangle ^{X_t}$ and $\langle \sigma ^i_s;x^j_t\rangle ^{X_t}$ are calculated as

$$\begin{aligned} \langle x^i_s;x^j_t\rangle ^{X_t}&=\delta _{ij}\left[ \dfrac{\kappa ^2_b/\gamma ^2}{(\kappa _b+\lambda \kappa _c)^2/ \gamma ^2-4r^2}\dfrac{L^2}{4}\left( e^{-2r|t-s|}-\dfrac{2r\gamma }{\kappa _b +\lambda \kappa _c}e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|}\right) \right. \nonumber \\&\quad \left. +\dfrac{k_{\mathrm {B}}T}{\kappa _b+\lambda \kappa _c}e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|}\right] , \end{aligned}$$

(A.12)

$$\begin{aligned} \langle \sigma ^i_s;x^j_t\rangle ^{X_t}&=\delta _{ij} \left\{ \begin{array}{llll} \dfrac{\kappa _b/\gamma }{(\kappa _b+\lambda \kappa _c)/\gamma +2r} \dfrac{L}{4}e^{-2r|t-s|},\quad \text {for}\quad t<s\\ \dfrac{\kappa _b/\gamma }{(\kappa _b+\lambda \kappa _c)/\gamma -2r} \dfrac{L}{4}\left( e^{-2r|t-s|}-e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|} \right) \\ +\dfrac{\kappa _b/\gamma }{(\kappa _b+\lambda \kappa _c)/\gamma +2r}\dfrac{L}{4} e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|},\quad \text {for}\quad t\ge s. \end{array}\right. \end{aligned}$$

(A.13)

Therefore, for $t\ge s$, the response function is

$$\begin{aligned} R_{\varPhi \varPhi }(t-s)=\dfrac{N\lambda ^2\kappa ^2_c}{\gamma }e^{-\frac{\kappa _b +\lambda \kappa _c}{\gamma }(t-s)}. \end{aligned}$$

(A.14)

From this result, it follows that

$$\begin{aligned} \gamma (t-s)&:=\int ^s_{-\infty }du R_{\varPhi \varPhi }(t-u)\nonumber \\&=\dfrac{N\lambda ^2\kappa ^2_c}{\kappa _b+\lambda \kappa _c}e^{-\frac{\kappa _b +\lambda \kappa _c}{\gamma }(t-s)},\quad \text {for}\quad t\ge s. \end{aligned}$$

(A.15)

To obtain the explicit expression of $\gamma _{\mathrm {ex}}(t-s)$, we calculate the noise correlation $\langle \eta _t\eta _s\rangle ^{X_t}$. By using (A.12), the noise correlation is calculated as

$$\begin{aligned} \langle \eta _t\eta _s\rangle ^{X_t}&=\lambda ^2\kappa ^2_c\sum _i\sum _j \langle (x^i_t-X_t);(x^j_s-X_t)\rangle ^{X_t}\nonumber \\&=k_{\mathrm {B}}T\gamma (t-s)+\dfrac{N\lambda ^2\kappa ^2_c\kappa ^2_b/ \gamma ^2}{(\kappa _b+\lambda \kappa _c)^2/\gamma ^2-4r^2}\dfrac{L^2}{4} \left( e^{-2r|t-s|}-\dfrac{2r\gamma }{\kappa _b+\lambda \kappa _c} e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|}\right) . \end{aligned}$$

(A.16)

Thus, the excess friction kernel $\gamma _{\mathrm {ex}}(t-s)$ is given by

$$\begin{aligned} \gamma _{\mathrm {ex}}(t-s)=\dfrac{1}{k_{\mathrm {B}}T}\dfrac{N\lambda ^2\kappa ^2_c \kappa ^2_b/\gamma ^2}{(\kappa _b+\lambda \kappa _c)^2/\gamma ^2-4r^2}\dfrac{L^2}{4} \left( e^{-2r|t-s|}-\dfrac{2r\gamma }{\kappa _b+\lambda \kappa _c} e^{-\frac{\kappa _b+\lambda \kappa _c}{\gamma }|t-s|}\right) . \end{aligned}$$

(A.17)