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Coarse Graining, Nonmaximal Entropy, and Power Laws
Fernando C Pérez-Cárdenas1, Lorenzo Resca2, Ian L Pegg2
1Vitreous State Laboratory, The Catholic University of America, Washington, DC 20064, USA.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
Coarse graining significantly impacts equilibrium state entropy, reducing it below theoretical maximums. Finer scales amplify this entropy drop and associated fluctuations, revealing predictable power-law relationships.
Area of Science:
- Statistical Mechanics
- Thermodynamics
- Computational Physics
Background:
- Entropy is a fundamental concept in thermodynamics, quantifying disorder and information.
- Coarse-graining is a common technique to simplify complex systems by reducing their degrees of freedom.
- Understanding the impact of coarse-graining on equilibrium entropy is crucial for theoretical and computational studies.
Purpose of the Study:
- To investigate the effects of coarse-graining on the entropy of equilibrium states.
- To demonstrate how coarse-graining leads to a predictable reduction in effective entropy.
- To derive and validate power-law relationships governing coarse-graining, entropy gap, and fluctuations.
Main Methods:
- Theoretical derivation of power-law relationships.
- Numerical simulations using a two-dimensional lattice gas model.
- Analysis of entropy evolution and fluctuations at different coarse-graining scales.
Main Results:
- Coarse-graining introduces significant, predictable effects on equilibrium entropy.
- Effective entropy typically decreases with increasing coarse-graining, deviating from the maximum entropy principle.
- Two distinct power laws were derived and numerically validated, relating coarse-graining to entropy gap and fluctuation noise range.
Conclusions:
- Coarse-graining introduces an 'effective entropy gap' that scales predictably with the graining level.
- The observed power laws highlight the fundamental interplay between system scale and entropy in equilibrium states.
- These effects vanish in the thermodynamic limit, reasserting the maximum entropy principle at macroscopic scales.
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