Coarse Graining, Nonmaximal Entropy, and Power Laws.

Lorenzo Resca,Fernando C Pérez-Cárdenas,Ian L Pegg

doi:10.3390/e20100737

Abstract

We show that coarse graining produces significant and predictable effects on the entropy of states of equilibrium when the scale of coarse graining becomes comparable to that of density fluctuations. We demonstrate that a coarse-grained entropy typically evolves toward a state of effective equilibrium with a lower value than that of the state of maximum entropy theoretically possible. The finer the coarse graining, the greater the drop in effective entropy, and the more relevant the fluctuations around that. Fundamental considerations allow us to derive a remarkable power law that relates coarse graining to the effective entropy gap. Another power law is found that precisely relates the noise range of effective entropy fluctuations to coarse graining. We test both power laws with numerical simulations based on a well-studied two-dimensional lattice gas model. As expected, the effects of these power laws diminish as our description approaches a macroscopic level, eventually disappearing in the thermodynamic limit, where the maximum entropy principle is reasserted.

Highlights

The discovery and formulation of entropy and its maximization principles, in relation to the second law of thermodynamics and the arrow of time, stand among the greatest experimental and intellectual achievements of science
We have demonstrated that a finely coarse-grained entropy of a system in a state of effective equilibrium fluctuates over time narrowentropy range below theoretically possible, We have demonstrated that awithin finelya relatively coarse-grained of a the system in a state of effective but practically unattainable, maximum entropy
This fact does not contradict general statements made equilibrium fluctuates over time within a relatively narrow range below the theoretically possible, to the effect that entropy is expected to maintain its maximum value in equilibrium and only quite but practically unattainable, maximum entropy

Summary

Introduction

The discovery and formulation of entropy and its maximization principles, in relation to the second law of thermodynamics and the arrow of time, stand among the greatest experimental and intellectual achievements of science. We will show that the formation of an entropy gap between the possible maximum entropy, Smax , and the effective equilibrium average entropy, Sequil , obeys a precise power law depending on coarse graining. Another coarse-graining-dependent power law rules the narrow range of the entropy fluctuations around Sequil. That multitude of distributions produces a large number of equal or comparable entropies that fluctuate remarkably below Smax. In retrospect, such phenomena should have been expected, but they have not been unequivocally demonstrated heretofore, to the best of our knowledge. Power law effects diminish with larger coarse graining, and they eventually disappear in the thermodynamic limit, where the maximum entropy principle is reasserted

Nonmaximal Equilibrium Entropy and Power Laws

Entropy Gap and Range of Fluctuations in a Lattice Gas

Conclusions