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Study of Entropy-Diffusion Relation in a Deterministic Hamiltonian System through Microscopic Analysis

Although an intimate relation between entropy and diffusion has been advocated for many years and even seems to have been verified in theory and experiments, a quantitatively reliable study, and any derivation of an algebraic relation between the two does not seem to exist. Here we explore the nature of this entropy-diffusion relation in three deterministic systems where an accurate estimate of both can be carried out. We study three deterministic model systems, (a) the motion of a single point particle with constant energy in a two-dimensional periodic potential energy landscape, (b) the same in regular Lorentz gas where a point particle with constant energy moves between collisions with hard disc scatterers and (c) motion of a point particle among the boxes with small apertures. These models, introduced by Zwanzig, exhibit diffusive motion in the limit where ergodicity is shown to exist. We then explore the diffusion-entropy relation by an accurate calculation of both diffusion and entropy for the aforementioned model systems. We estimate the self-diffusion coefficient of the particle by employing computer simulations and entropy by quadrature using Boltzmann's formula. We observe an interesting crossover in the diffusion-entropy relation in some specific regions which is attributed to the emergence of correlated returns. The crossover could herald a breakdown of the Rosenfeld-like exponential scaling between the two, as observed at low temperatures. Later, we modify the scaling relation to account for the correlated motions and present a detailed analysis of the dynamical entropy obtained via Lyapunov exponent which is rather an important quantity in the study of deterministic systems.