Statistical mechanics and thermodynamics of small systems

In this thesis many aspects of the statistical mechanics and thermodynamics of small systems are studied. The very same possibility of defining a thermodynamics for this class of systems, for which the usual properties of the thermodynamic limit do not apply, is discussed by means of general considerations and specific examples. We show that it is possible to preserve most of the features of thermodynamics for a specific class of systems which are, at the same time, far enough from the infinite-N limit to be small, but large enough to be studied with a statistical approach. A review of the necessary mathematical and physical tools to study this particular class of systems is included. Eventually, a specific system is studied, both from an equilibrium and a non- equilibrium perspective: it is found that this system, composed by a gas included in a container with a moving wall (the piston), has an highly non-trivial dynamics caused by the interplay of the different degrees of freedom of the system, which cannot be easily reproduced by means of coarse-grained equations. At the same time, the smallness of the system is responsible for large fluctuations that strongly characterize the system. We show that this system reproduces the behavior of an heat engine, when the external parameters vary in time: in particular we show that different working regimes (engine, refrigerator, heat pump) can be obtained depending upon the total time of a cycle of the external parameters. We also derive some analytical results reproducing, with a fair degree of approximation, the behavior of the system.