Stochastic thermodynamics of holonomic systems

Stochastic thermodynamics is a recently introduced approach to deals with small systems in contact with one or more thermal baths. This theory has been applied to systems of unconstrained particles to investigate the role of the thermodynamics principles in micro- and nano-scale systems and to demonstrate some important fluctuations theorems. Nowadays, the manipulations of small systems with advanced nanotechnologies provided the experimental evidence of most of results based on stochastic thermodynamics. Here, this approach is generalized to consider arbitrary holonomic systems subjected to arbitrary external forces and described by Lagrange and Hamilton equations of motion. In both the underdamped and overdamped cases, the principles of thermodynamics are obtained in the out-of-equilibrium regime by giving microscopic interpretations of heat, energy and entropy. To do this, the Klein-Kramers (for the underdamped case) and Smoluchowski (for the overdamped case) equations are used in covariant form to be consistent with the Brownian motion on smooth manifolds. Moreover, explicit expressions for the entropy production have been obtained and can be applied to the non-equilibrium thermodynamics of holonomic systems.