Bounds of the entropy production of the irreversible thermodynamics and its relation with the non-equilibrium temperature

Based on the entropy balance equation for a multidimensional linear Fokker-Planck equation we have developed connections between the several thermodynamically inspired quantities such as the total entropy production, the entropy production of the irreversible thermodynamics and the non-equilibrium temperature. Then we have calculated both upper and lower bounds of these properties with the help of the entropy balance equation and the Schwartz inequality which has been applied to the differential equation for the time evolution of the entropy. With these we have shown that there is a lower bound of time derivative of the entropy in addition to its upper limit. The present formalism has been applied for the coupling of a Brownian particle with the Markovian and the non-Markovian thermal baths, respectively. (C) 2019 Elsevier B.V. All rights reserved.