Fluctuations of the total entropy production in stochastic systems

Fluctuations of the excess heat in an out of equilibrium steady state are experimentally investigated in two stochastic systems : an electric circuit with an imposed mean current and a harmonic oscillator driven out of equilibrium by a periodic torque. In these two linear systems, we study excess heat that represents the difference between the dissipated heat out of equilibrium and the dissipated heat at equilibrium. Fluctuation theorem holds for the excess heat in the two experimental systems for all observation times and for all fluctuation magnitudes. Thermodynamics of systems at equilibrium has been developed by defining the internal energy, the injected work, and the dissipated heat. The first law of Thermodynamics describes energy conservation and gives a relation between those three energies. The second law of Thermodynamics imposes that the entropy variation is positive for a closed system. Statistical Physics further gave a microscopic definition of entropy, which allowed analytical results on entropy production. The extension of Thermodynamics to non-equilibrium systems in steady states is an active field of research. Within this context, the first law has been extended for stochastic systems described by a Langevin equation [1–3]. It has been noted that the second law is not verified at all times but only in average over macroscopic times, i.e. entropy production fluctuates can instantly be negative. The probabilities of getting positive and negative entropy production are quantitatively related in nonequilibrium systems by the Fluctuation Theorem (FT). This theorem has been first demonstrated in deterministic systems [4–6] and secondly extended to stochastic dynamics [7–10]. Work and heat fluctuations have been theoretically and experimentally studied in some Brownian systems described by the Langevin equation [2, 3, 11–17]. For these systems in a non-equilibrium steady state, Fluctuation Theorems hold only in the limit of infinite time: