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

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.

Adiabatic invariance along the reaction coordinate

“…Then, the dynamical constraints might again be described by a simple model that makes them few in number and readily identifiable if expanded in another set of appropriate operators. Nevertheless, as shown by Levine and co-workers, 29,30 the surprisal and entropy deficiency are constants of the motion, i.e., the lack of energy randomization persists during the entire lifetime of the reacting molecule. Thus, in contradistinction to summary sketches of statistical theories, the system never forgets its previous history.…”

“…As shown by Levine and co-workers, 2,29,30,35 information theory can be used to define measures of the lack of energy randomization in an activated molecule. A nonzero value of them indicates that energy exchange among degrees of freedom is restricted by dynamical constraints.…”

“…Two opposite concepts are confronted: piecewise invariance 19,20 and entropy ͑as defined in information theory͒ conservation. 29,30 …”

Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

“…The Hamiltonians (2) exhibit some advantages that make them easy to tackle by means of the Maximum Entropy Principle approach (MEP). Indeed, 1) as the classical degrees of freedom act as they were parameters in the quantum commutation operation [3], then Equation (2), for adequate operators ˆj O , closes a partial Lie algebra under commutation with the generators of some Lie algebra, 2) as a consequence of the algebra's closure, it is always possible to obtain the Maximum Entropy statistical operator ρ as prescribed by Alhassid & Levine in [4], which enables us to integrate the system's quantum degrees of freedom in the fashion…”

Statistical inference and crystallite size distributions

“…Information theory (IT), after the pioneer work of Shannon (1948), has been extensively applied to a wide range of problems (see Brillouin, 1962;Alhassid & Levine, 1977, 1979Levine, Steadman, Karp & Alhassid, 1978;Grandy, 1980;and references therein;Otero, Proto & Plastino, 1981). Statistical mechanics has received fruitful contributions from this standpoint (e.g.…”