A deductive proof of the second law of thermodynamics in its most general form

We provide a deductive proof for the second law of thermodynamics by demonstrating that the Gibbs-Shannon entropy is applicable to non-equilibrium systems of any size and boundary conditions. The change in microscopic entropy can be attributed to the stochastic nature of dynamic processes and to the inherent uncertainties of thermodynamic systems. The latter predicts that entropy production is nonnegative on average and varies with different trajectories according to the fluctuation theorem. By contrast, heat is affiliated with stochastic processes underlying particle motions and its ensemble average over all possible trajectories lead to the Clausius inequality. The Jarzynski/Crooks equations can be readily derived by applying the fluctuation theorem to heat variation over different trajectories linking equilibrium states.