Stochastic approach and irreversible chemical reactions in small systems

Anton Amann

J. P. Badiali

We investigate the irreversible evolution of a small system in which a chemical reaction takes place.We have two main goals: the first requires to find an equation to produce a time-irreversible behavior,the second consists in introducing a simple exactly solvable model in order to understand basic facts in chemical kinetics.Our basic tool is the transition function counting the number of paths joining two points in the reactive coordinates system.An exact quantum Smoluchowski equation is derived for the reactive system in vacuum, in the presence of a solvent in equilibrium at any time with the reactive system a new Smoluchowski equation is obtained.The transition from a quantum regime to a classical one is discussed.The case of a reactive system not in equilibrium with its neighborhood is investigated in terms of path integral and via a partial differential function.Memory effects and closure assumptions are discussed.Using a simple potential model, the chemical rate constant is exactly calculated and questions such as the meaning of the activation energy or the physical content of the so-called prefactor are investigated.

V. P. Belavkin

George H. Booth

Ronald F. Fox

Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.

John C. Baez, Brendan Fong

Anderson, Craciun, and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits a stationary distribution when the deterministic model of the same system with mass-action kinetics admits an equilibrium solution obeying a certain complex balance condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators, and coherent states. This is an example of stochastic mechanics, where we take techniques from quantum mechanics and replace amplitudes by probabilities.