Compatibility of linear-response theory with the second law of thermodynamics and the emergence of negative entropy production rates

Abstract

The reliability of physical theories depends on whether they agree with well established physical laws. In this work, we address the compatibility of the Hamiltonian formulation of linear-response theory with the Second Law of Thermodynamics. In order to do so, we verify three complementary aspects often understood as statements of the Second Law: 1. No dissipation for quasistatic process; 2. Dissipation for finite-time processes; 3. Positive entropy production rate. Our analysis focus on two classes of nonequilibrium isothermal processes: slowly-varying and finite-time but weak ones. For the former, we show that these aspects are easily verified. For the later, we present conditions for the achievement of the first two aspects. We also show that the third one is not always verified, presenting an example based on Brownian motion in which we observe negative values in the entropy production rate. In particular, we compare linear-response and exact results for this example.

Fluctuation theorem for irreversible entropy production in electrical conduction

“…The natural question arises, whether the two paradigms are consistent with each other, or rather whether the two "versions" of entropy production are equivalent. In particular, the occurrence of negative rates in irreversible thermodynamics [5,6,14] appears incompatible with the strict positivity of the stochastic entropy production rate often discussed in the literature [19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

“…Interestingly, negative entropy production rates in open systems undergoing non-Markovian dynamics are somewhat commonplace [8][9][10][11][12]. However, our analysis is entirely based on the lag of response [13,41], which also gives rise to negative rates in Markovian settings [5,14].…”

“…Even more remarkably, negative entropy production rates are not indicative of scenarios operating far from thermal equilibrium, but they can be observed fully within the regime of linear, irreversible thermodynamics [14]. Since this approach fully rests on phenomenological and macroscopic arguments, the considered entropy production is often considered as genuinely thermodynamic.…”

Linear response theory in stock markets

“…This is a practical extension of Onsager’s regression hypothesis, namely, a system relaxes to equilibrium after an external perturbation in a similar manner as from fluctuations 2 . This powerful tool and the formalism of time correlation functions have been applied to study several physical systems, such as soft matter 3 – 6 , spin glasses 7 , or magnetism 8 , but also it has been used to derive a conceptual basis for equilibrium and non-equilibrium thermodynamics 9 , 10 . The drawback is that only the first order in the perturbation is retained, which might not be sufficient in some cases 11 .…”

Measuring Entropy Production at the Mesoscale

“…As such, entropy production quantifies the "cost" of keeping a system in a nonequilibrium steady state. Complementary facets of the second law are the necessity of energy dissipation in finite-time processes and the lack of dissipation in quasistatic systems, two effects that are recovered by linear response theory [3].…”