Effective Fluctuation and Response Theory

Abstract

The response of thermodynamic systems slightly perturbed out of an equilibrium steady-state is described by two milestones of early nonequilibrium statistical mechanics: the reciprocal and the fluctuation-dissipation relations. At the turn of this century, the so-called fluctuation theorems extended the study of fluctuations far beyond equilibrium. All these results rely on the crucial assumption that the observer has complete information about the system: there is no hidden leakage to the environment, and every process is assigned its due thermodynamic cost. Such a precise control is difficult to attain, hence the following questions are compelling: Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only be able to establish local equilibrium amidst the whirling of hidden degrees of freedom, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations nevertheless? We address these two fundamental problems, providing a broad theory of the statistical behavior of some out of many currents that flow across a thermodynamic system.We model the dynamics of open systems as Markov jump processes on finite networks. Configurationspace currents count the net number of transitions between pairs of configurations; conjugate forces quantify their thermodynamic cost. Phenomenological currents are linear combinations of configuration currents, and only ensue when affinities enjoy appropriate symmetries, granting thermodynamic consistency. A complete thermodynamic description is achieved when the set of currents under consideration covers all cycles in the network, otherwise the set is marginal.Within this formalism, we establish that: 1) While marginal currents do not obey a full-fledged fluctuation relation, there exist effective affinities for which an integral fluctuation relation holds; 2) Under reasonable assumptions on the parametrization of the rates, effective and "real" affinities only differ by a constant; 3) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds while reciprocity does not; 4) There exists a notion of marginal time-reversal that plays a role akin to that played by time-reversal for complete systems, which restores the fluctuation relation and reciprocity; 5) The effective affinity is the putative affinity of an observer who only has marginal information about a system and formulates a minimal model accounting for his/her steady-state observations; 6) There exist fluctuation relations across different levels in the hierarchy of more and more "complete" theories. The above results hold for configurationspace currents, and for phenomenological currents provided that certain symmetries of the effective affinities are respected -a condition that we call marginal thermodynamic consistency, which is stricter thermodynamic consistency and whose range of validity we deem the most interesting question left open to future inquiry. Our results are construc...

Stochastic thermodynamics with arbitrary interventions

“…We still allow for imprecise measurements happening at arbitrary discrete times, thus we still have to deal with incomplete information similar to the situations considered in Refs. [7][8][9][10][11][12][13]. As incomplete information can be handled in many very different ways, we do not make an attempt to compare our framework in detail with any of those proposed in those references.…”

“…[7,8,12,13], within one common unified framework. In that respect it would be very important to extend the present theory to multiple heat reservoirs too [12,13]. In addition, for practical applications it would be desirable to gain further insights into the physical nature of the rather abstract ancillas introduced by us.…”

Operational approach to quantum stochastic thermodynamics

“…Obviously, the very definition of a trajectory dependent quantity is nontrivial as any measurement disturbs the system and the meaning of a 'trajectory' is a priori not clear. We note that incomplete and disturbing measurements are also prevalent in classical systems [14], but exploring their consequences for classical stochastic thermodynamics has raised relatively little attention so far [15][16][17][18][19][20][21].…”

Effective thermodynamics of two interacting underdamped Brownian particles

“…In this context, it would also be interesting to explore if other coarse-graining schemes that were applied to jump processes, for instance as proposed in Refs. [63,64], can also be utilized in underdamped Fokker-Planck systems. These studies could give rise to new strategies for systematically and thermodynamically consistently coarse-graining many-body systems.…”