We review two definitions of temperature in statistical mechanics, TB and TG, corresponding to two possible definitions of entropy, SB and SG, known as surface and volume entropy respectively. We restrict our attention to a class of systems with bounded energy and such that the second derivative of SB with respect to energy is always negative: the second request is quite natural and holds in systems of obvious relevance, i.e. with a number N of degrees of freedom sufficiently large (examples are shown where N ∼ 100 is sufficient) and without long-range interactions. We first discuss the basic role of TB, even when negative, as the parameter describing fluctuations of observables in a sub-system. Then, we focus on how TB can be measured dynamically, i.e. averaging over a single long experimental trajectory. On the contrary, the same approach cannot be used in a generic system for TG, since the equipartition theorem may be spoiled by boundary effects due to the limited energy. These general results are substantiated by the numerical study of a Hamiltonian model of interacting rotators with bounded kinetic energy. The numerical results confirm that the kind of configurational order realized in the regions at small SB, or equivalently at small |TB|, depends on the sign of TB.

We study a molecular engine constituted by a gas of N∼102 molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period τ. In the adiabatic limit τ→∞, even for finite size N, the average work and heat reproduce the thermodynamic values, recovering the Carnot result for the efficiency. The system exhibits a stall time τ* where the net work is zero: for τ<τ* it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At τ>τ* the efficiency at maximum power is close to the Curzorn-Ahlborn limit. The fluctuations of work and heat display approximatively a Gaussian behavior. Based upon kinetic theory, we develop a three-variables Langevin model in which the piston's position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system's features, such as the inversion of the work's sign, the efficiency at maximum power, and the approximate shape of the fluctuations. A further simplification in the model allows us to compute analytically the average work, explaining its nontrivial dependence on τ.

We investigate a kinetic heat engine model constituted by particles enclosed in a box where one side acts as a thermostat and the opposite side is a piston exerting a given pressure. Pressure and temperature are varied in a cyclical protocol of period τ : their relative excursions, δ and respectively, constitute the thermodynamic forces dragging the system out-of-equilibrium. The analysis of the entropy production of the system allows to define the conjugated fluxes, which are proportional to the extracted work and the consumed heat. In the limit of small δ and the fluxes are linear in the forces through a τ -dependent Onsager matrix whose off-diagonal elements satisfy a reciprocal relation. The dynamics of the piston can be approximated, through a coarse-graining procedure, by a Klein-Kramers equation which -in the linear regime -yields analytic expressions for the Onsager coefficients and the entropy production. A study of the efficiency at maximum power shows that the Curzon-Ahlborn formula is always an upper limit which is approached at increasing values of the thermodynamic forces, i.e. outside of the linear regime. In all our analysis the adiabatic limit τ → ∞ and the the small force limit δ, → 0 are not directly related.

This article aims at revisiting, with the aid of simple and neat numerical examples, some of the basic features of macroscopic irreversibility, and, thus, of the mechanical foundation of the second principle of thermodynamics as drawn by Boltzmann. Emphasis will be put on the fact that, in systems characterized by a very large number of degrees of freedom, irreversibility is already manifest at a single-trajectory level for the vast majority of the far-from-equilibrium initial conditions -a property often referred to as typicality. We also discuss the importance of the interaction among the microscopic constituents of the system and the irrelevance of chaos to irreversibility, showing that the same irreversible behaviours can be observed both in chaotic and non-chaotic systems.

We study the behavior of a moving wall in contact with a particle gas and subjected to an external force. We compare the fluctuations of the system observed in the microcanonical and canonical ensembles, by varying the number of particles. Static and dynamic correlations signal significant differences between the two ensembles. Furthermore, velocity-velocity correlations of the moving wall present a complex two-time relaxation that cannot be reproduced by a standard Langevin-like description. Quite remarkably, increasing the number of gas particles in an elongated geometry, we find a typical time scale, related to the interaction between the partitioning wall and the particles, which grows macroscopically.