Fully symmetric nonequilibrium response of stochastic systems

Nonequilibrium response theory has long been studied near equilibrium, where general results such as the fluctuation-dissipation theorem or the Onsager reciprocity relations hold. Far from equilibrium, relations for the response coefficients have been derived as consequences of the fluctuation theorem for currents [1].

However, these nonequilibrium relations are much more intricate than their near equilibrium counterparts. In particular, they are not fully symmetric and involve combinations of multiple response coefficients.

Recently, the author derived a mapping between nonequilibrium fluctuations and an associated equilibrium dynamics [2, 3]. Building on this result, we demonstrate that the nonequilibrium response of currents is symmetric within an equivalence class of stochastic dynamics. In addition, the nonequilibrium response coefficients can be expressed in terms of equilibrium correlation functions. To this end, we show that:

1. 1.

   The space of stochastic dynamics can be structured into equivalence classes. Each equivalence class is determined by the composition of an equilibrium dynamics and all posssible affinities

2. 2.

   The equilibrium dynamics of an equivalence class determines the fluctuations of all (nonequilibrium) dynamics in that class

3. 3.

   Within an equivalence class, the nonequilibrium reponse of currents is symmetric at all orders and for any values of the affinities (i.e., both near and far from equilibrium)

We consider a Markov chain characterized by a transition matrix $P=\left(P_{ij}\right)\in\mathbb{R}^{N\times N}$ on a finite state space [4]. We assume that the Markov chain is primitive, i.e., there exists an $n_{0}$ such that $P^{n_{0}}$ has all positive entries.

The space of stochastic dynamics can be partitioned into equivalence classes. We define the operator

$\displaystyle\bar{P}_{ij}=\sqrt{P_{ij}P_{ji}}$

(1)

and the corresponding equivalence relation

$\displaystyle P\sim H\quad\text{if there exists a factor}\ \gamma\ \text{such that}\quad\bar{P}=\gamma\ \bar{H}\,.$

(2)

This relation is reflexive, symmetric, and transitive. It thus forms a partition of the space of stochastic dynamics: every dynamics belongs to one and only one equivalence class.

An equivalence class $[P]=\{H:H\sim P\}$ is determined by the composition $(E,\boldsymbol{A})$ of an equilibrium dynamics $E$ and the set of all possible affinities $\boldsymbol{A}$ (Fig. 1, see Ref. [2] for a detailed construction). In particular, to any $P$ we can associate an equilibrium dynamics $E[P]$ as follows. Let $\rho$ denote the Perron root of $\bar{P}$ and $x$ its right Perron eigenvector. If $D={\rm diag}(x_{1},\ldots,x_{N})$ we have that

$\displaystyle E[P]=\frac{1}{\rho}D^{-1}\ \bar{P}\ D$

(3)

defines an equilibrium stochastic dynamics $E\sim P$.

We consider the thermodynamic currents $\boldsymbol{J}=\{J_{\alpha}\}$ and their fluctuations (see Ref. [5, 1] for a definition). The current fluctuations for a dynamics $P$ are characterized by the generating function

$\displaystyle Q_{P}(\boldsymbol{\lambda})=\lim_{n\rightarrow\infty}\frac{1}{n}\ln\left\langle\exp\left(\boldsymbol{\lambda}\cdot\sum_{k=1}^{n}\boldsymbol{J}(k)\right)\right\rangle_{P}\,.$

(4)

All cumulants can then be obtained by successive derivations of $Q$ with respect to the counting parameters $\boldsymbol{\lambda}$:

$\displaystyle Q_{P}(\boldsymbol{\lambda})=\sum_{m=1}^{\infty}\frac{1}{m!}K_{\alpha_{1}...\alpha_{m}}[P]\,\lambda_{\alpha_{1}}\cdot\cdot\cdot\ \lambda_{\alpha_{m}}\,,$

(5)

where we sum over repeated indices. In particular, the first-order cumulant equals the mean current, $K_{\alpha}=\left\langle J_{\alpha}\right\rangle$. Note that the cumulants $K_{\alpha_{1}...\alpha_{m}}$ are symmetric in $(\alpha_{1},...,\alpha_{m})$.

We now examine the fluctuations of dynamics within the equivalence class $[P]$. We denote by $Q_{[P]}(\lambda,\boldsymbol{A})$ the generating functions of the elements $(E[P],\boldsymbol{A})$ within $[P]$. We then have the

Theorem. Let $[P]$ be an equivalence class and $E[P]$ its associated equilibrium dynamics. Then their current generating functions are related as

$\displaystyle Q_{[P]}(\boldsymbol{\lambda},\boldsymbol{A})=Q_{E}(\boldsymbol{\lambda}+\boldsymbol{A}/2)-Q_{E}(\boldsymbol{A}/2)\,.$

(6)

DEMONSTRATION: In Ref. [3], we showed that the nonequilibrium current fluctuations of an arbitrary nonequilibrium dynamics $P$ are determined by its corresponding equilibrium dynamics $E[P]$ according to

$\displaystyle Q_{P}(\boldsymbol{\lambda})=Q_{E}(\boldsymbol{\lambda}+\boldsymbol{A}/2)-Q_{E}(\boldsymbol{A}/2)\,.$

(7)

By construction, all dynamics in the same equivalence class $P^{\prime}\sim E$ will satisfy the relation (6) with the same function $Q_{E}$. As a result, relation (6) holds for all dynamics $P^{\prime}\sim P$ with affinities $\boldsymbol{A}$. Conversely, for any affinity $\boldsymbol{A}$ there exists a corresponding dynamics $P^{\prime}$ in the equivalence class. The relation (6) therefore holds for all $\boldsymbol{A}$ within the same equivalence class. $\Box$

In other words, the generating function of the current $Q_{P}$ is entirely determined by the equilibrium fluctuations $Q_{E}$ (Fig 2) [6]. Therefore, $E[P]$ acts as a reference equilibrium dynamics that is intrinsic to $P$. We also note that the generating functions $Q_{P}$ is directly related to the factor $\gamma$ defining the equivalence relation (2). If $\bar{P}=\gamma\bar{E}$ then $Q_{P}=-2\log\gamma$ [2].

We now explore the consequences of Eq. (6) on the response theory. As the right-hand side of Eq. (6) only depends on the affinities through the translation $\boldsymbol{\lambda}+\boldsymbol{A}/2$, the first order response of the current $\left\langle J_{\alpha}\right\rangle=K_{\alpha}(\boldsymbol{A})$ reads

$\displaystyle\frac{\partial\left\langle J_{\alpha}\right\rangle}{\partial A_{\beta}}(\boldsymbol{A})=\frac{1}{2}\frac{\partial^{2}Q_{E}}{\partial\lambda_{\alpha}\partial\lambda_{\beta}}(\boldsymbol{A}/2)=\frac{\partial\left\langle J_{\beta}\right\rangle}{\partial A_{\alpha}}(\boldsymbol{A})\,$

(8)

which is symmetric in ($\alpha,\beta$) for any $\boldsymbol{A}$, i.e. both near and far from equilibrium (Fig. 3). Also, since $Q_{E}(\boldsymbol{\lambda})=Q_{E}(-\boldsymbol{\lambda})$, the current is antisymmetric with respect to $\boldsymbol{A}$, $\left\langle J_{\alpha}\right\rangle(-\boldsymbol{A})=-\left\langle J_{\alpha}\right\rangle(\boldsymbol{A})$, while its response $\partial\left\langle J_{\alpha}\right\rangle/\partial A_{\beta}$ is symmetric.

More generally, the response of any cumulant is expressed as

$\displaystyle\frac{\partial^{l}K_{\alpha_{1}...\alpha_{m}}(\boldsymbol{A})}{\partial A_{\beta_{1}}...\partial A_{\beta_{l}}}=\left(\frac{1}{2}\right)^{l}\frac{\partial^{m+l}Q_{E}}{\partial\lambda_{\beta_{1}}...\partial\lambda_{\beta_{l}}\partial\lambda_{\alpha_{1}}...\partial\lambda_{\alpha_{m}}}(\boldsymbol{A}/2)\,,$

(9)

which is fully symmetric in ($\alpha_{1},...,\alpha_{m},\beta_{1},...,\beta_{l}$) for all $\boldsymbol{A}$. The cumulant response is symmetric in $\boldsymbol{A}$ if $m+l$ is even, and antisymmetric otherwise. Alternatively, expression (9) reveals that the current response and cumulants are directly related:

$\displaystyle K_{\alpha_{1}...\alpha_{m}}(\boldsymbol{A})=2^{m-1}\,\frac{\partial^{m-1}\left\langle J_{\alpha_{1}}\right\rangle(\boldsymbol{A})}{\partial A_{\alpha_{2}}...\partial A_{\alpha_{m}}}\,.$

(10)

In other words, knowledge of the currents $\left\langle J_{\alpha}\right\rangle(\boldsymbol{A})$ is sufficient to obtain all cumulants.

We recover the traditional response theory by expanding the currents as functions of the affinities around $\boldsymbol{A}=0$:

$\displaystyle\left\langle J_{\alpha}\right\rangle=\sum_{l=1}^{\infty}\frac{1}{l!}L_{\alpha,\beta_{1}...\beta_{l}}\,A_{\beta_{1}}\cdot\cdot\cdot\ A_{\beta_{l}}\,,$

(11)

where we sum over repeated indices. Using Eq. (9) and the symmetry $Q_{E}(\boldsymbol{\lambda})=Q_{E}(-\boldsymbol{\lambda})$, the response coefficients can be expressed in terms of equilibrium correlations:

	$\displaystyle L_{\alpha,\beta_{1}...\beta_{l}}$	$\displaystyle=$	$\displaystyle 0\quad\quad\quad\quad\quad\quad\quad\quad\text{if {\it l} is even}$		(12)
	$\displaystyle L_{\alpha,\beta_{1}...\beta_{l}}$	$\displaystyle=$	$\displaystyle\left(\frac{1}{2}\right)^{l}K_{\alpha\beta_{1}...\beta_{l}}(\boldsymbol{0})\quad\text{if {\it l} is odd.}$		(13)

This shows that all response coefficients $L_{\alpha,\beta_{1}...\beta_{l}}$ are fully symmetric in ($\alpha,\beta_{1},...,\beta_{l}$). In particular, we recover the Onsager symmetry $L_{\alpha,\beta}=L_{\beta,\alpha}$ and the corresponding Green-Kubo formula, $L_{\alpha,\beta}=(1/2)K_{\alpha\beta}(\boldsymbol{0}).$ Similar expressions can be derived for the higher-order cumulants and their nonequilibrium response [7].

Remarkably, expressions (12) and (13) take a simpler form than the ones derived in Ref. [1], which involve combinations of multiple cumulants and their nonequilibrium responses. Here the coefficients either vanish or can be expressed as equilibrium correlations [8].

The results (8) and (9) complement the traditional response theory. In the traditional response theory, the dynamics shifts across equivalence classes as the affinities are varied (Fig. 4a). As a result, the shape of the generating function changes as a function of the affinities in addition to being translated. This leads to additional contributions of the form $\partial K/\partial A_{\alpha}...\partial A_{\beta}$ in the response theory, breaking the symmetry (9). In contrast, here the dynamics remains in its original equivalence class when varying the affinities (Fig. 4b). This leads to symmetric response coefficients that can be expressed as equilibrium correlation functions.

Note that varying affinities while staying in a given equivalence class typically requires changing multiple transition probabilities at the same time (Fig 1b). As experimental setups improve their ability to manipulate mesoscopic degrees of freedom, it will become possible to use equivalence classes to control the nonequilbrium transport properties of fluctuating systems. In any case, the present results offer a new way to explore nonequilibrium systems by leveraging their intrinsic dynamical characteristics.

Disclaimer. This paper is not intended for journal publication.