*Article* **Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes**

**Carlos Medel-Portugal 1, Juan Manuel Solano-Altamirano <sup>2</sup> and José Luis E. Carrillo-Estrada 1,\***


**Abstract:** We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell *H*-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship *dH*/*dt* ≤ 0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the *H*-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the *H*-functionals are in agreement with the correspondence principle. We discuss how the *H*-functionals can be identified with the system's entropy and analyze the relaxation processes of out-of-equilibrium systems.

**Keywords:** non-equilibrium thermodynamics; entropy; variational entropy
