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Complexity and Regime Transitions in Simple Master Equations
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: Received: 10 July 2024 / Approved: 11 July 2024 / Online: 11 July 2024 (04:52:33 CEST)
How to cite: Plastino, A.; Monteoliva, D. Complexity and Regime Transitions in Simple Master Equations. Preprints 2024, 2024070920. https://doi.org/10.20944/preprints202407.0920.v1 Plastino, A.; Monteoliva, D. Complexity and Regime Transitions in Simple Master Equations. Preprints 2024, 2024070920. https://doi.org/10.20944/preprints202407.0920.v1
Abstract
We investigated the dynamics of a simple master equation describing the evolution of a probability distribution between two states, \( p_1 \) and \( p_2 \) (with \( p_1 + p_2 = 1 \)). Our focus is on the behavior of the entropy \( S \), the distance \( D \) from the uniform distribution (\( p_1 = p_2 = 1/2 \)), and the free energy \( F \). We define two ratios, \( \eta_S = \frac{dS/dt}{dF/dt} \) and \( \eta_D = \frac{dD/dt}{dF/dt} \), and find that both diverge to plus-minus infinity as the system evolves towards the uniform distribution. This divergence marks a critical transition point where the master equation temporarily loses physical meaning. However, after this divergence, the system settles into a new, well-behaved regime where the solutions stabilize at finite values, representing a new equilibrium or steady state. This two-regime behavior underscores the intricate dynamics of simple probabilistic systems as they approach and surpass equilibrium, providing insights into the complex interplay of entropy, distance, and free energy in statistical mechanics.
Keywords
Master equation; order; disorder; Entropy; Disequilibrium
Subject
Physical Sciences, Thermodynamics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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