A modified moment method and irreversible thermodynamics

We investigate a method of solving the Boltzmann equation especially with a view to a deeper understanding of irreversible thermodynamics for systems in nonequilibrium state. By modifying the conventional moment method for the Boltzmann equation, we show that there can result irreversible thermodynamics with attendant evolution equations for macroscopic variables consistent with the second law of thermodynamics. An important result is the extended Gibbs relation, $$\mathrm{TdS=dE+pdv-}{\displaystyle \sum _i}{\mathrm{\mu ̂}}_i{\mathrm{dc}}_i+{\displaystyle \sum _l}{\displaystyle \sum _i}{\mathrm{X̂}}_i^{\mathrm{(l)}}{\mathrm{\odot d\Phi }}_i^{\mathrm{(l)}}$$ whereT is the temperature, S the entropy density, E the internal energy density, v the specific volume, μ̂_i the chemical potential, c_i the concentration, X̂_i^(l) the generalized thermodynamic force conjugate to Φ_i^(l) which can be stress tensor, heat flux, material flux, and possibly other higher moments. This relation is valid for any nonequilibrium state and can be used for studying the geometric structure of entropy in the extended Gibbs space {E,v,c_i,Φ_i^(l)}. It is shown that linear irreversible processes correspond to the stationary processes for which (dΦ_i^(l)/dt)=0 among other conditions. The usual equilibrium Gibbs relation is recovered from the extended Gibbs relation in such a case. The evolution equations obtained in this paper can serve as a guide and a model for future phenomenological formulation of irreversible thermodynamics for systems removed far from equilibrium or for nonlinear processes.