From Gibbs-Shannon entropy and microscopic reversibility to entropy production, heat, and fluctuation theorems

This work provides a self-contained derivation of several fundamental results in stochastic thermodynamics, including the Jarzynski equality, Crooks fluctuation theorem, and the Clausius inequality. Although the principal theoretical conclusions are well established in the literature, the present approach departs from conventional formulations of stochastic entropy by employing a trajectory-independent probability density constructed as the marginal of the full path-space distribution. This construction establishes a direct connection between microscopic path statistics and macroscopic state probabilities, thereby providing a natural framework for quantifying the relative statistical weights of distinct dynamical histories leading to the same microstate. We demonstrate that entropy production and the fluctuation theorems emerge directly from the Gibbs inequality under the assumption of microscopic reversibility of trajectories along with the extension of the Gibbs-Shannon entropy to time-evolving ensembles. For reduced stochastic descriptions such as Langevin dynamics, the counterpart of microscopic reversibility is local detailed balance, which is inherited from the underlying time-reversal symmetry of the full system-reservoir dynamics and is the condition under which the present framework applies to effective open-system descriptions. We show that the change in microscopic entropy can be attributed to both underlying stochastic fluctuations and the statistical uncertainties inherent in thermodynamic ensembles. This dual perspective ensures that the average entropy production remains non-negative, providing a consistent microscopic basis for thermodynamic irreversibility through the statistical preference of forward over time-reversed trajectories. The connection between microscopic entropy flow and heat, as well as the subsequent derivation of the Clausius inequality and Jarzynski-Crooks relations, requires the additional assumption of an ideal thermal reservoir at a fixed temperature. While this formulation introduces no new physical or mathematical insights, it illuminates the universality of fluctuation theorems across diverse systems, offering a unified perspective that may serve a pedagogical purpose for those studying the statistical-mechanical foundations of non-equilibrium thermodynamics.