Stochastic thermodynamics

We develop the stochastic approach to thermodynamics based on
the stochastic dynamics, which can be discrete or continuous,
and on two assumptions concerning entropy.
In the discrete case, the system is governed by master equation
and in the continuous case, by a Fokker-Planck equation.
The first assumption concerns the definition of entropy, which is
taken to be the Boltzmann-Gibbs entropy, and thus has the same
form as the equilibrium definition. The difference is that
the probability distribution may be time-dependent.
The second assumption has to do with the definition of entropy
production rate, which is taken to be the expression introduced
by Schnakenberg. This expression is nonnegative by definition
and vanishes in the thermodynamic equilibrium.
Based on these assumptions we study interacting systems
with many degrees of freedom in equilibrium or out
of thermodynamic equilibrium, and how the macroscopic laws are
derived from the stochastic dynamics.
In particular, we will discuss the quasi-static processes defined
as the ones in which the thermodynamic fields are varied slowly.
We show that the rates of energy, entropy and number
of particle, are linear in the rate of the thermodynamics fields whereas
the production of entropy is quadratic in these rates so
that it may be negected and the system may be considered to be
in thermodynamic equilibrium. We show that along a quasi-static process
the representative point in the space energy, entropy and number of
particle, remains on a surface and that this surface
has the property of convexity.