Linear response in the nonequilibrium zero range process

Abstract

We explore a number of explicit response formulae around the boundary driven zero range process to changes in the exit and entrance rates. In such a nonequilibrium regime kinetic (and not only thermodynamic) aspects make a difference in the response. Apart from a number of formal approaches, we illustrate a general decomposition of the linear response into entropic and frenetic contributions, the latter being realized from changes in the dynamical activity at the boundaries. In particular, in this way one obtains nonlinear modifications to the Green-Kubo relation. We end by bringing some general remarks about the situation where that nonequilibrium response remains given by the (equilibrium) Kubo formula such as for the density profile in the boundary driven Lorentz gas

Response Theory: A Trajectory-Based Approach

“…The merit of response theory is indeed not its formal appearance-in the end we are all doing Taylor expansion assuming (and sometimes proving) convergence of certain integrals. In particular, for non-equilibrium purposes, we emphasize the importance of the frenetic contribution in response; for different details and discussions, we refer to [33][34][35][36][37][38][39].…”

“…It is important here to recall that the separation between timesymmetric (frenesy D) and time-antisymmetric (entropy flux S) contributions is obtained via the time-reversal operation [θ in (20)] which should include the perturbation protocol; i.e., we also reverse the time-dependence in the perturbation, cf. the dynamics (36). Let us take an observable, i.e., a function O of the trajectory ω, always in the window [0, t].…”

A Nonequilibrium Extension of the Clausius Heat Theorem

“…That is the more important second difference with equilibrium. We believe that kinetic aspects such as those summarized under the name of dynamical activity will become important, and indeed some of that is visible in linear response around nonequilibria, [13]. That will also be taken up in Section IV B.…”

Nonequilibrium Response and Frenesy

“…Note that there the variance of N appears in the frenetic term. That was seen already in the context of linear response for the boundary driven zero range process [20]. In fact, since N is time-symmetric the role of entropic and frenetic contribution gets reversed; even for E = 0 we get only the frenetic part.…”