Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

2022/02/15 by Giada Basile, Basile, Giada, Dario Benedetto +5 · 1 citation
Mathematics · Physics and Astronomy · #35Q20 60F10 82C40 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #FOS: Physical sciences #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2202.07311

openalex publication_date 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collision (Kac's model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov's theorem to the microcanonical ensemble and large deviations for the Kac's model in the microcanonical setting.

Cited by

• Entropy dissipation inequality for general binary collision models

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