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

Subhajit Acharya; Biman Bagchi

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arXiv · 2001.09132

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

Subhajit Acharya·Biman Bagchi

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Abstract

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.

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Subhajit Acharya, Biman Bagchi. 2020-01-24. Study of Entropy-Diffusion Relation in a Deterministic Hamiltonian System through Microscopic Analysis. https://arxiv.org/abs/2001.09132

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