Power injected in dissipative systems and the fluctuation theorem

Abstract

Abstract.We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the "fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of "temperature" for a dissipative system out of equilibrium. We consider how this "temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration. PACS

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“…We already noticed that the first situation is much more appropriate to describe realistic systems driven far from equilibrium. Second, formula (44) illustrates well the fact that Fluctuation Theorem seems to hold in so large a number of experimental situations, as explained in [4]: in the vicinity ofε = 0, ρ(ε) must always have a straight line behaviour, as a consequence of the large deviation law; on the other hand, as large negative values ofε are extremely unprobable when τ is large, it becomes practically unpossible even to only measure ρ(ε) for largeε and large τ with enough statistical resolution: possible deviations from the straight line are just even not measurable. In our case, crossover occurs forε = 1/3 and for this value, π ∝ exp(−5γτ /3) which is of order 10 −8 only if γτ = 10 .…”

“…. Our model is thus a good illustration in favour of arguments given in [4] against an universal applicability of conclusions of the Fluctuation Theorem. …”

“…These observations were quite surprising, for the Fluctuation Theorem was established for time-reversal systems, a property which is at the heart of the demonstration of the theorem. A convincing argument was recently proposed [4] to explain this seemingly universal behaviour: if one considers that the signal of I(t) has only finite correlations, large deviation theory predicts that log π(ε) ∼ τ f (ε) for large τ . Consequently, ρ(ε) = f (ε) − f (−ε); but for large τ , it is extremely unprobable to observe large negative occurences of the averaged injected power (but not unpossible), since in average, the injected power in a dissipative system is positive.…”

“…These two distinct "gates" lead to the establishment of a permanent flow of energy throughout the system, and is obviously a primordial feature amongst the stationary properties of the system. Thus, some experimental measurements [1,2,3] and numerical simulations [4] were performed to characterize the injection of energy (more easily reachable than the dissipation), or more precisely the probability density function (pdf) π(ε) of ε = 1 τ τ 0 I(t)dt, the averaged injected power during a time interval of length τ . In particular, in some works [4,5,6], the quantity ρ(ε) = 1 τ log π(ε) π(−ε)…”

Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence

“…We will now discuss some examples [45,69,70] in which one can construct two models, one reversible and the other irreversible, that seems to describe the same physical system. We will see that the result for the large deviation function of the global entropy production rate, ζ ∞ (p), is very different.…”