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We demonstrate that the Gibbs-Shannon entropy is applicable tonon-equilibrium systems of any size and boundary conditions. The change inmicroscopic entropy can be attributed to the stochastic nature of dynamicprocesses and to the inherent uncertainties of thermodynamic systems. Thelatter predicts that entropy production is nonnegative on average and varieswith different trajectories according to the fluctuation theorem. By contrast,heat is affiliated with stochastic processes underlying particle motions andthe ensemble average over all possible trajectories leads to the Clausiusinequality. The Jarzynski/Crooks equations can be readily derived by applyingthe fluctuation theorem to heat variation over different trajectories linkingequilibrium states.

On microscopic entropy production, heat and fluctuation theorem