Abstract
We consider the dynamic large deviation behaviour of Kac’s collisional process for a range of initial conditions including equilibrium. We prove an upper bound with a rate function of the type which has previously been found for kinetic large deviation problems, and a matching lower bound restricted to a class of sufficiently good paths. However, we are able to show by an explicit counterexample that the predicted rate function does not extend to a global lower bound: even though the particle system almost surely conserves energy, large deviation behaviour includes solutions to the Boltzmann equation which do not conserve energy, as found by Lu and Wennberg, and these occur strictly more rarely than predicted by the proposed rate function. At the level of the particle system, this occurs because a macroscopic proportion of energy can concentrate in particles with probability .
Funding Statement
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.
Acknowledgements
I would like to thank Robert Patterson and Michel Renger, conversations with whom at various points sparked and renewed my interest in the topic, as well as Sergio Simonella for an interesting discussion of the problem. I would also like to thank my doctoral supervisor, Prof. James Norris, who pointed out ways in which the counterexample in Theorem 1.3 could be extended into its current form.
Citation
Daniel Heydecker. "Large deviations of Kac’s conservative particle system and energy nonconserving solutions to the Boltzmann equation: A counterexample to the predicted rate function." Ann. Appl. Probab. 33 (3) 1758 - 1826, June 2023. https://doi.org/10.1214/22-AAP1852
Information
