OFFPRINT A thermodynamical formalism describing mechanical interactions

The dynamical behavior of an overdamped mechanical model devoid of any usual thermal effects is analyzed by a formalism that is similar to usual thermodynamics, and completely independent of any ad hoc assumption of a probability distribution of states in phase space of the mechanical model. It leads to the definition of a new entropy function, which does not coincide with the usual thermodynamical entropy. The new step making the difference to previous studies of this system is the identification of two non-equivalent mechanical interaction mechanisms, which are defined and identified as work and pseudo-heat. Together with the introduced effective temperature θ, they make it possible to characterize the equivalent to isothermal, adiabatic, isobaric, and isochoric processes. Three statements, formally analogous to the zeroth, first, and second law of thermodynamics, are issued. The statement of the second law results from the asymmetry in the way energy can be exchanged along the two processes. A Carnot cycle is defined, for which the efficiency is expressed in terms of θ in the operating pseudo-heat reservoirs. The analogous Clausius theorem for the system operating an arbitrary reversible cycle is proved, leading to the new entropy function. Consequences of the extension of thermodynamic formalism to mechanical models with different processes of transferring energy are discussed. Copyright c © EPLA, 2014 The entropy function was introduced into thermodynamics as a result of Clausius theorem, which is itself a mathematical consequence of the statement of the second law according to Kelvin or to Clausius [1]. Rather than being an ad hoc concept, the thermodynamic entropy follows from phenomenological statements that certain events have never been observed in nature. Some time later, the statistical entropy based on the microscopic distribution probability was introduced into physics in the pioneering works by Boltzmann and Gibbs [2]. The entropy measures both the ability of the system to deliver mechanical work as well as its microscopic disorder. Additional efforts to define entropy along a mathematically rigorous way lead to axiomatic formulations, among which that due first to Shannon and afterwards to Khinchin is most acknowledged in the literature [3,4]. In the latter case, four postulates requiring basic entropy properties like the dependence on the probability distribution only, maximum at equiprobability, addition of a new zeroprobability state, and conditional additivity, lead to a specific functional dependence that is generally referred to as the celebrated Boltzmann-Gibbs (BG) entropy. If the fourth postulate is modified or excluded from the axiomatic formulation, other functional forms can be used to define statistical entropies [5–7]. For physical systems, the BG entropy is generally accepted as the actual statistical definition, mainly for the fact that, for a very large number of systems, the derived results agree with those obtained by the thermodynamical formalism and by highly precise experimental records. However, experimental deviations from the predicted theoretical results based on the BG distribution are reported. Such failures are explained by the incompleteness of the data, or the presence of out-of-equilibrium states, or the inadequacy of the BG formalism (see, e.g., ref. [8] for a list of examples). The search for a better physical explanation