On the Rigorous Derivation of a Kinetic Equation for a Chemical Reaction Taking Place in a Simple Mechanical Model System, following Boltzmann’s Ideas Using the ‘Stosszahlansatz’
Berlin University of Technology
The aim of this work is to validate Boltzmann’s ideas concerning the derivation of “irreversible” kinetic equations from “reversible” mechanical laws by reformulating them in a rigorous, modern mathematical context and applying them to a simple mechanical model.
First, Boltzmann’s ideas, developed as a reaction to the objections raised against the Boltzmann equation, are carefully restated in a modern way. It turns out that Boltzmann’s view was far more “mechanical” than the perspectives usually adopted in the later literature. In particular, “molecular chaos”—a crucial argument in the derivation of the Boltzmann equation—was a property of single microstates (not, e.g., of a probability measure) and therefore a purely mechanical term. Probability only entered the theory in a way that does not question its basically mechanical nature. Unfortunately, Boltzmann was not able to work out his ideas in the necessary rigor (which is probably impossible for real mechanical systems because of their complexity). Therefore, the power of persuasion of his ideas has been diminished.
In the second part of the work, a very simple model system is constructed. It shows a “chemical reaction” for which an irreversible kinetic equation can be obtained in a heuristic way, completely analogous to the derivation of the Boltzmann equation. Since the underlying microscopic dynamics is reversible, we are in the same dilemma as Boltzmann was. Then, Boltzmann’s ideas are applied, but in a mathematically rigorous manner. In particular, “molecular chaos” is exactly defined as a property of single microstates. As long as the microstate possesses this property, the kinetic equation holds. It can be shown that—with respect to a reasonable probability measure—the microstate is and will remain “molecular chaotic” with probability one. As a consequence, the kinetic equation holds with probability one.
Finally, the implications of the work are discussed. The opinion is given that the work contributes to the clarification of Boltzmann’s ideas and may be used as a pedagogical attempt to teach these ideas.
(April 20, 2004)