A Conceptual Consideration of Irreversible Phenomena Based on an Information-Theoretic Measure of Correlations

Considering two systems in thermodynamic equilibrium, A and B, described byPhase space density phase space density \({\rho }^{(\text{A})}\) and \({\rho }^{\left(\text{B}\right)}\) , the correlations between the two systems are measured by the mutual information: MI =  \(\left\{\text{SMI}[{\rho }^{\left(\text{A}\right)}] +\text{SMI}\left[{\rho }^{\left(\text{B}\right)}\right] \right\} -\) SMI[ \({\rho }^{(\text{A}\&\text{B})}\) ]. When the two systems are allowed to interact, it is easy to understand that the measure of the correlations between the initial state and the final equilibrium state increases. Since SMI[ \({\rho }^{(\text{A}\&\text{B})}\) ] is constant (Liouville theorem), the increase in correlations is expressed as ΔMI = \({\left\{\text{SMI}\left[{\rho }^{\left(\text{A}\right)}\right]+\text{SMI}\left[{\rho }^{\left(\text{B}\right)}\right] \right\}}^{\text{final}}- {\left\{\text{ SMI}\left[{\rho }^{\left(\text{A}\right)}\right]+\text{SMI}\left[{\rho }^{\left(\text{B}\right)}\right] \right\}}^{\text{initial}} \ge\)  0. Except for a multiplicative constant, this is Planck's statement of the second principle of thermodynamics. The same kind of interpretation in terms of correlations also appears in the simple and short proof we give for the H-theoremH-theorem (i.e., the increase of Boltzmann entropyBoltzmann entropy). Using the same assumptions as for the Boltzmann equation, we show that the increase of the correlations is a direct consequence of the Boltzmann assumption of molecular chaos (i.e., the assumption that the random variables associated with the particles are not correlated at some initial time). Also in this case, the increase in entropy is equal to the increase in correlations (MI), so that the H-theoremH-theorem appears as a consequence of the increase of the correlations and also a secondary consequence of theMolecular chaos assumption molecular chaos assumption. Finally, a simple model is considered to show that a system whose components evolve according to a reversible law of motion, which generates a “mixing” flow in phase space, has an irreversible behavior when described by univariate distribution functions.