Fluctuation theorem and chaos

Abstract

The heat theorem (i.e. the second law of thermodynamics or the existence of entropy) is a manifestation of a general property of hamilto- Boltzmann's Heat TheoremIn equilibrium statistical mechanics states are identificed with time invariant probability distributions µ on phase space. Thermodynamic functions, identified with time averages of mechanical observables, are expressed as integrals F of suitable mechanical obeservables F . The averages depend on control parameters α, like volume, energy, kinetic energy. Under changes b dα of the control parameters the thermodynamic quantities change so that the variation of the average energy and the variation of the volume are dU and dV and are related to the time averages of the kinetic energy

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

“…There is by now a large body of literature on fluctuation theorems, including reviews and pedagogical treatments. [21,22,23,24,25,26,3,27,28,29,30,31,32,33,34,35,36,37,38] In my view these are not results that one might naturally have obtained, by starting with a solid understanding of macroscopic thermodynamics and extrapolating down to small system size. Rather, they reveal genuinely new, nanoscale features of the second law.…”

“…5,24,28,35) can be rewritten as equalities (Eqs. 15,25,30,31). Among the "take-home messages" that emerge from these developments are the following :…”

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

“…54 -56, and Eqs. 30,15,19,20, respectively, should be obvious, although viewed as mathematical statements they are different. The microscopic definition of entropy production in Eqs.…”

Refined Second Law of Thermodynamics for Fast Random Processes

“…The last term does not contribute to the expectation value due to the martingale property of the Itô integral so that 16) or, upon integration by parts over space,…”

“…The result of the junction will be a refinement, relevant for fast processes, of the Second Law of Stochastic Thermodynamics. Our improvement of the Second Law does not go in the direction of a better control of fluctuations of thermodynamical quantities [35], as do various Fluctuation Relations studied intensively in last years, see [14,16,23]. Instead, it establishes the optimal lower bound on the total entropy production in non-equilibrium processes of fixed duration.…”