Fluctuations in partitioning systems with few degrees of freedom

Resolving the Debate between Boltzmann and Gibbs Entropy: Relative Energy Window Eliminates Thermodynamic Inconsistencies and Allows Negative Absolute Temperatures

Small systems consisting of a few particles are increasingly technologically relevant. In such systems, an intense debate in microcanonical statistical mechanics has been about the correctness of Boltzmann's surface entropy versus Gibbs' volume entropy. Both entropies have shortcomings─while Boltzmann entropy predicts unphysical negative/infinite absolute temperatures for small systems with an unbounded energy spectrum, Gibbs entropy entirely disallows negative absolute temperatures, in disagreement with experiments. We consider a relative energy window, motivated by the Heisenberg energy-time uncertainty principle and eigenstate thermalization in quantum mechanics. The proposed entropy ensures positive, finite temperatures for systems without a maximum limit on their energy and allows negative absolute temperatures in bounded energy spectrum systems, e.g., with population inversion. It also closely matches canonical ensemble predictions for prototypical systems, thus correctly describing the zero-point energy of an isolated quantum harmonic oscillator. Overall, we enable accurate thermodynamic models for isolated systems with few degrees of freedom.

Linear and nonlinear thermodynamics of a kinetic heat engine with fast transformations

We investigate a kinetic heat engine model composed of 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 ε, respectively, constitute the thermodynamic forces dragging the system out of equilibrium. The analysis of the entropy production of the system allows us to define the conjugated fluxes, which are proportional to the extracted work and the consumed heat. In the limit of small δ and ε 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,ε→0 are not directly related.

Kinetic model for the finite-time thermodynamics of small heat engines

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.