Thermodynamic formalism and linear response theory for nonequilibrium steady states

We study the linear response in systems driven away from thermal equilibrium into a nonequilibrium steady state with nonvanishing entropy production rate. A simple derivation of a general response formula is presented under the condition that the generating function describes a transformation that (to lowest order) preserves normalization and thus describes a physical stochastic process. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are discussed.

Figure 1

Examples for affinities and distances. (a) Colloidal particle in a ring trap with radius $R$. The particle is driven by a constant force $f$ (the affinity) while $X_{\tau }=R\varphi$ is the total distance traveled during time $\tau$. (b) Sketch of an enzyme driving the reaction $\circ \to ▹$. The generalized “distance” $X_{\tau }=N_{\tau }^{▹}=-N_{\tau }^{\circ }$ now corresponds to the number of $▹$ molecules produced during time $\tau$. The affinity $f=-({\mu }^{▹}-{\mu }^{\circ })$ is given by the difference of chemical potential, which we assume to be fixed by chemiostats.

Figure 2

Illustration of the duality of $s$ and current $x=X_{\tau }/\tau$ for the asymmetric random walk (for details see the Appendix, with $f=\frac{1}{2}$ and $k^{+}=1$): (a) Large deviation function ${\varphi }_f^{*}\left(s\right)$ (thick line). The slope at $s=0$ (steady state I) corresponds to the mean current $x_0$. (b) Rate function ${\varphi }_f\left(x\right)$ of the current (thick line) with minimum at $x_0$. Conditioning currents to a value $x_s>x_0$ leads to the steady state (II) with $s>0$ determined by the slope. (c) The unbiased average current as a function of driving force $f$ (thick line) and the current $x_s={\partial }_s{\varphi }_f^{*}$ (thin line) from the generating function, where $s=(f_{\text{eff}}-f)/2$ quantifies a perturbation of the steady state with effective force $f_{\text{eff}}$. Both currents agree for $s=0$ but deviate for increasing values of the perturbation $s$. For fixed $k^{+}$ the unbiased current is bounded (dashed black line).