Positive definiteness of entropy production in the nonlinear Robertson formalism

The Robertson formalism, which derives from the Liouville equation via a projection operator a kinetic equation obeyed exactly by the information-theoretic phase space distribution, is used to calculate Ṡ, the time-derivative of the Jaynesian model of entropy. The Robertson kinetic equation involves an operator T̂ for which Robertson gives an expression whose terms must be regrouped to consider times t≫τ, with τ the relaxation time of a variable η(t) having properties typical of fast variables of extended thermodynamics. When driving forces are applied, causing the system to approach a steady state with constant η, one can show that Ṡ≧0 as t→∞ for experimentally-accessible states. In practice, this holds so long as t≫τ, i.e., for the time scale of most measurements in simple fluids. It is also found that Ṡ=S̈(0)t>0 as t→0 for an arbitrary nonequilibrium initial state.