Ensembles, turbulence and fluctuation theorem
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Ensembles, turbulence and fluctuation theorem Giovanni Gallavottia INFN Roma1, and Accademia dei Lincei, Roma, Italy Received 12 March 2020 and Received in final form 8 May 2020 Published online: 12 June 2020 c EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The fluctuation theorem is considered intrinsically linked to reversibility and therefore its phenomenological consequence, the fluctuation relation, is sometimes considered not applicable. Nevertheless here is considered the paradigmatic example of irreversible evolution, the 2D Navier-Stokes incompressible flow, to show how universal properties of fluctuations in systems evolving irreversibily could be predicted in a general context. Together with a formulation of the theoretical framework several open questions are formulated and a few more simulations are provided to illustrate the results and to stimulate further checks.
1 Introduction Many macroscopic systems are modeled by equations of motion which are not reversible due to the action of viscous forces, like Navier Stokes fluid equations. And often the equations can be derived from reversible microscopic models (e.g., see [1]). The question that is addressed in this work is whether stationary states of such systems could also be described by reversible equations. In the ‘80s transport properties of interacting particles systems have been studied by introducing “thermostat forces” [2–4]: the idea behind the introduction of “non-Newtonian” forces was that the important stationary properties of the system depend on stationarity and not on the way it is achieved. A feature of resulting equations is that often they are reversible, i.e. on phase-space a map x → Ix, independent of the forces (typically I is the change of the velocities sign), exists with the property that I 2 x = x anticommutes with the time evolution x → x(t) = St x (i.e., ISt x = S−t x). In the case of equilibrium ergodicity is the basis of the theory of equilibrium statistical properties and of their independence of initial conditions; likewise in systems in stationary states (equilibrium or not) the chaoticity of their evolutions takes the role of ergodicity and it is the key to understand the typical initial state independence of the stationary states properties (with the possibility of a few stationary states, just as in equilibrium at phase transitions at most a handful of different states arise depending on the initial conditions) [5, 6]. a Present address: Dipartimento di Fisica, Universit` a di Roma “La Sapienza” and INFN, Roma, Italy; e-mail: [email protected]
This did show that strongly chaotic evolutions may be described equivalently by different equations: a striking example is in [7] in the case of the NS equations and gave rise to several studies in which strong chaoticity was present [8–10]. Here a different view will be presented, pursuing the ideas of the latter references: it is exposed in recent pu
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