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)=−∫dxρ(x,t)ln∫dx′ρ(x′,t)ϕΔt(x,x′,t), where ρ(x,t) is the ensemble distribution in phase space and ϕΔt(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 Δt2, and Δt 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.