Generalization of Gibbs entropy and thermodynamic relation

In this paper, we extend Gibbs’s approach for quasi-equilibrium thermodynamic processes, and show that, in general non-equilibrium thermodynamic processes, the microscopic expression of entropy is given as S ( t ) = − ∫ d x ρ ( x , t ) ln ∫ d x ′ ρ ( x ′ , t ) ϕ Δ t ( x , x ′ , t ) , where ρ ( x , t ) is the ensemble distribution in phase space and ϕ Δ t ( x , x ′ , t ) is the probability density to obtain that, in macroscopic observation, the system with initial value x ′ in phase space at time t is found at state x after time elapse Δ t 2 , and Δ t is the maximum value of the time interval for which any macroscopic thermodynamic variables increase linearly. Also, we analyze the formal structure of thermodynamic relation in non-equilibrium thermodynamic processes.