Generalizing response theory of open systems far from equilibrium is a central quest of nonequilibrium statistical physics. Using stochastic thermodynamics, we develop an algebraic method to study the static response of nonequilibrium steady state to arbitrary perturbations. This allows us to derive explicit expressions for the response of edge currents as well as traffic to perturbations in kinetic barriers and driving forces. We also show that these responses satisfy very simple bounds. For the response to energy perturbations, we straightforwardly recover results obtained using nontrivial graph-theoretical methods.

In nonequilibrium steady states of Markov jump processes, we derive exact fluctuation-response relations (FRRs) that express the covariance between any pair of currents in terms of static responses in a notably simple form, thus generalizing the fluctuation-dissipation theorem far from equilibrium. We begin by considering perturbations in the symmetric part of the rates. We demonstrate that FRRs imply a hierarchy of thermodynamic bounds. These hierarchies prove the recently conjectured response thermodynamic uncertainty relation, which bounds the ratio between any current's response and its variance by the entropy production rate (EPR). We furthermore strengthen this bound in two distinct ways, using a partial EPR in one case and a pseudo-EPR in the other. For perturbations in the antisymmetric part of the rates, we show that the ratio between any current's response and its variance is bounded by traffic, a metric representing the total number of transitions per unit time in the system. As an application, we use FRRs to explain the origin of positive correlations between currents in Coulomb-blockaded systems previously observed in experiments.

The precision of currents in Markov networks is bounded by dissipation via the so-called thermodynamic uncertainty relation (TUR). In our Letter, we demonstrate a similar inequality that bounds the precision of the static current response to perturbations of kinetic barriers. Perturbations of such type, which affect only the system kinetics but not the thermodynamic forces, are highly important in biochemistry and nanoelectronics. We prove that our inequality cannot be derived from the standard TUR. Instead, it implies the standard TUR and provides an even tighter bound for dissipation. We also provide a procedure for obtaining the optimal response precision for a given model.

We consider Markov jump processes on a graph described by a rate matrix that depends on various control parameters. We derive explicit expressions for the static responses of edge currents and steady-state probabilities. We show that they are constrained by the graph topology (i.e., the incidence matrix) by deriving response relations (i.e., linear constraints linking the different responses) and topology-dependent bounds. For unicyclic networks, all scaled current responses are between zero and one and must sum to one. Applying these results to stochastic thermodynamics, we derive explicit expressions for the static response of fundamental currents (which carry the full dissipation) to fundamental thermodynamic forces (which drive the system away from equilibrium).

Batteries, supercapacitors, and several other electrochemical devices charge by accumulating ions in the pores of electrolyte-immersed porous electrodes. The charging of such devices has long been interpreted using equivalent circuits and the partial differential equations these give rise to. Here, we discuss the validity of the transmission line (TL) circuit and equation for modeling a single electrolyte-filled pore in contact with a reservoir of resistance $R_r$. The textbook derivation of the pore-reservoir impedance, $R_r+Z_p$, from the TL equation does not correctly account for ionic current conservation at the pore-reservoir interface. However, correcting this shortcoming leads to the same impedance. We also show that the pore impedance, $Z_p$, can be derived directly from the TL circuit, bypassing the TL equation completely. The TL circuit assumes equipotential lines in an electrolyte-filled pore to be straight, which is not the case near the pore entrance and end. To determine the importance of these regions, we numerically simulated the charging of pores of different lengths, ${\ell }_p$, and radii, ${\varrho }_p$, through the Poisson-Nernst-Planck equations. We find that pores with aspect ratios beyond ${\ell }_p/{\varrho }_p⪆5$ have impedances in good agreement with $Z_p$.

We develop a general classification of the nature of the instabilities yielding spatial organization in open nonideal reaction-diffusion systems, based on linear stability analysis. This encompasses dynamics where chemical species diffuse, interact with each other, and undergo chemical reactions driven out of equilibrium by external chemostats. We find analytically that these instabilities can be of two types: instabilities caused by intermolecular energetic interactions (E type), and instabilities caused by multimolecular out-of-equilibrium chemical reactions (R type). Furthermore, we identify a class of chemical reaction networks, containing unimolecular networks but also extending beyond them, that can only undergo E-type instabilities. We illustrate our analytical findings with numerical simulations on two reaction-diffusion models, each displaying one of the two types of instability and generating stable patterns.