Entropy and chemical change. III. The maximal entropy (subject to constraints) procedure as a dynamical theory

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Abstract
An equivalence between the dynamical (equations of motion) and the information theoretic (maximal entropy) approaches to collision phenomena is established. The connection is demonstrated in both directions. The variational procedure of maximal entropy is shown to converge to an exact solution of the equations of motion (be they classical or quantal) throughout the collision. In particular, a stationary precollision state is proved to be a state of maximal entropy (subject to constants of the unperturbed motion) and to remain a state of maximal entropy throughout the collision. Conversely, the exact solution of the equations of motion is shown to be of maximal entropy. In this fashion one obtains an algebraic procedure for the specification of the constraints which determine (via the procedure of maximal entropy) an exact solution of the equations of motion. Surprisal analysis does not require the solution of differential equations. These must be solved to determine the magnitude of the Lagrange parameters (i.e., for surprisal synthesis). The number of coupled differential equations that have to be solved (to obtain an exact solution) equals the number of constraints. Sum rules (which express the mean value of any constraint as a linear function of the initial mean values of the constraints) are derived and offer an alternative route to surprisal synthesis. The information theoretic result for the branching ratio (as the ratio of two partition functions) is shown to be a rigorous result of the present formalism. As an illustration, a simple Hamiltonian for a collinear reactive collision is analytically treated in detail (for a classical motion along the reaction coordinate). The constraints are identified (with special reference to reactive collisions, where the Hamiltonians for the reactants and products do differ); the time dependence of the Lagrange parameters is established and the vibrational state distribution (both during and after the collision) is determined. A sum rule for the mean products vibrational energy is discussed. For a stationary initial state (e.g., a particular vibrational state or a thermal distribution), 〈E′vib〉 is linearly dependent on 〈Evib〉 alone. The slope and intercept are determined only by the dynamics and are essentially proportional to the ratio of the final to initial vibrational frequencies. For a family of related reactions where this ratio is nearly unchanged, the vibrational energy disposal would be quite similar. Inefficient products vibrational excitation is expected when this ratio is low.