Emergent second law for non-equilibrium steady states

Abstract

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information $${{{{{{{\mathcal{I}}}}}}}}(x)=-\!\log ({P}_{{{{{{{{\rm{ss}}}}}}}}}(x))$$ I ( x ) = − log ( P ss ( x ) ) of microstate x to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes $${{\Delta }}{{{{{{{\mathcal{I}}}}}}}}={{{{{{{\mathcal{I}}}}}}}}({x}_{t})-{{{{{{{\mathcal{I}}}}}}}}({x}_{0})$$ Δ I = I ( x t ) − I ( x 0 ) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law $${{\Sigma }}+{k}_{b}{{\Delta }}{{{{{{{\mathcal{I}}}}}}}}\,\ge \,0$$ Σ + k b Δ I ≥ 0 , which contains the usual second law Σ ≥ 0 as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods.

Dissipation in hydrodynamics from micro- to macroscale: wisdom from Boltzmann and stochastic thermodynamics

“…If the system is coupled to a single reservoir with no time-dependent driving of T B and p B , then the thermodynamic potential Y(t) would be maximized at the (equilibrium) steady state. Note that, by using the thermodynamic potential (47), equation (46) reads…”

“…This construction reveals substantial differences between global and local thermodynamics because all global conservation laws can be broken at the local level [42]. These macroscopic thermodynamic theories with local equilibrium can be explicitly constructed from an underlying ST by scaling up particle numbers (see [43][44][45], for chemical reaction networks, Potts models and electronic circuits, respectively, and [46,47] for the general arguments). These approaches correspond to overdamped dynamics where inertial effects are neglected, but generalization to underdamped dynamics seems possible and could be used to derive hydrodynamics.…”

On the contribution of work or heat in exchanged energy via interaction in open bipartite quantum systems

“…Extending work and heat from classical thermodynamics to quantum thermodynamics has been one of the major issues in the literature 5 . Irreversibility and entropy production of quantum systems have also been intensively examined in both equilibrium and non-equilibrium processes 1 , 5 , 10 , 11 As is discussed in the following some difficulties appear in identifying work and heat properly that need to be taken care of. In classical thermodynamics a change in the energy of a system is divided into two parts: heat and work 12 – 15 , where is the heat absorbed by system A and the work performed on system A .…”

Large deviations theory for noisy nonlinear electronics: CMOS inverter as a case study

“…The long time fluctuations of these observables can be computed using large deviations theory, taking the time of observation as the scale parameter [24][25][26]. Another application has been to understand the macroscopic fluctuations of these dynamical observables, using the system size as the scale parameter, analogous to the description in equilibrium systems [27,[31][32][33]. The noncommutativity of the long time and the large size limit is an important feature of systems with dynamical phase transitions, and large deviations theory provides a mathematical framework to characterize them [34][35][36][37][38].…”