Abstract
We reverse engineer entropy formulas from entropic divergence, optimized to given classes of probability distribution function (PDF) evolution dynamical equation. For linear dynamics of the distribution function, the traditional Kullback–Leibler formula follows from using the logarithm function in the Csiszár’s f-divergence construction, while for nonlinear master equations more general formulas emerge. As applications, we review a local growth and global reset (LGGR) model for citation distributions, income distribution models and hadron number fluctuations in high energy collisions.
Highlights
Given an evolution dynamics for the microstate probabilities, we first seek for an expedient formula for the entropic divergence [28]: a non-negative measure between two probability distributions (in a continuous model probability distribution functions (PDFs) which shrinks during the dynamical evolution
In the framework of local growth and global reset model (LGGR) we demonstrate that far from the detailed balance state stationary distributions show a thermodynamical behavior, akin to the fluctuation–dissipation theorem [29]
We present some details on why, in the generalized Markovian dynamics, a similar proof based on the convexity property of the definition of the entropic divergence suffices only for an approach to the stationary distribution
Summary
The entropic convergence is being intensively studied alongside functional inequalities. Given an evolution dynamics for the microstate probabilities, we first seek for an expedient formula for the entropic divergence [28]: a non-negative measure between two probability distributions (in a continuous model probability distribution functions (PDFs) which shrinks during the dynamical evolution. Popularity measured by citations and hadron multiplicities in high-energy collisions demonstrate the flexibility and usefulness of the LGGR model As part of this presentation, we repeat some formulas published already earlier [26,28]. We present some details on why, in the generalized Markovian dynamics, a similar proof based on the convexity property of the definition of the entropic divergence suffices only for an approach to the stationary distribution. Arguments for the generalization to nonlinear probability dynamics are shortly mentioned, but the applications in the framework of our LGGR model is again linear, concentrating on the processes with a stationary state without detailed balance
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