DETERMINISTIC AND STOCHASTIC ASYMPTOTIC-BEHAVIOR OF NONEQUILIBRIUM CHEMICAL-SYSTEMS WITH TIME-DEPENDENT RATE COEFFICIENTS

We analyze the long time limit of the nonequilibrium solutions of a sy stem of multivariable nonlinear kinetic equations with time-dependent rate coefficients, as achieved, for example, by temporal variation of the temperature. If the characteristic time scale attached to the chan ge of rate coefficients is smaller than the relaxation time to equilib rium, then the system is constrained to evolve away, possibly far, fro m equilibrium. However, after a sufficiently large time the system for gets its past: in the long run all solutions of the kinetic equations tend toward a special (normal) solution which depends on the previous values of the rate coefficients, but it is independent of the initial state of the system. The normal solution may be very different from th e equilibrium solution. The occurrence of this type of time-dependent normal regime for the deterministic kinetic equations is intimately co nnected to a similar behavior of the stochastic evolution equation of the system. In the long run all solutions of the stochastic equation f or the state probability also tend toward a normal form which is indep endent of the initial preparation of the system. The logarithm of the normal form of the state probability is a Lyapunov function of the sys tem of deterministic kinetic equations and plays the role of a general ized thermodynamic potential which may be used for developing a thermo dynamic description of the chemical process, A Gibbsian ensemble descr iption is introduced in terms of a multireplica stochastic master equa tion. The logarithm of the large time solution of the multireplica sto chastic master equation is also a Lyapunov function, one for the stoch astic evolution equations of the system. If the time dependence of the rate coefficients is generated by the variation of an external constr aint, then the deterministic normal regime can be considered as repres enting the response of the system to an excitation, The coupling betwe en the excitation and the system is nonlinear and multiplicative and g enerates interesting effects. A kinetic experiment is suggested for ch ecking the existence of normal solutions for chemical systems with tim e-dependent rate coefficients.