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 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.
Funding Statement
The work of G. Basile and D. Benedetto has been supported by PRIN 202277WX43 “Emergence of condensation-like phenomena in interacting particle systems: kinetic and lattice models”, founded by the European Union—Next Generation EU.
Acknowledgments
The authors would like to thank the anonymous referees, for their constructive comments that improved the quality of this paper.
D. Benedetto and E. Caglioti would like to thank GNFM—INdAM.
Citation
Giada Basile. Dario Benedetto. Lorenzo Bertini. Emanuele Caglioti. "Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation." Ann. Appl. Probab. 34 (4) 3995 - 4021, August 2024. https://doi.org/10.1214/24-AAP2057
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