Stochastic thermodynamics based on an Einstein-Boltzmann definition of fluctuating entropy

Stochastic thermodynamics is an extension of classical nonequilibrium thermodynamics to small systems, where fluctuations are expected to play an important role. A central difficulty met when developing such an extension is how to define a nonequilibrium fluctuating entropy. Typically, the expression used is based on Gibbs' formula for entropy at equilibrium. In this work, we show that one can construct an alternative framework for stochastic thermodynamics based on an extension of Einstein's formula connecting the probability of fluctuations and entropy around equilibrium states. We compare the two approaches and discuss, in particular, how they lead to different interpretations of what a stochastic entropy and entropy production represent.