ASYMPTOTIC PROBABILITY OF ENERGY INCREASING SOLUTIONS TO THE HOMOGENEOUS BOLTZMANN EQUATION

• 成果类型: Article
• 署名作者: Basile, Giada; Enedetto, Dariob; Bertin, Lorenzo; Caglioti, Emanuele
• 署名单位: Sapienza University Rome
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2057
• 发表日期: 2024
• 页码: 3995-4021
• 关键词: large deviations; entropy

摘要

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 collisions (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.