[Paper Review] Large Deviations of Kac's Conservative Particle System and Energy Non-Conserving Solutions to the Boltzmann Equation: A Counterexample to the Predicted Rate Function

Daniel Heydecker|arXiv (Cornell University)|Mar 26, 2021

Laser-induced spectroscopy and plasma4 citations

TL;DR

This paper establishes a large deviations principle for Kac's conservative particle system, proving an upper bound with a rate function matching prior kinetic theory predictions. However, it presents a counterexample showing that the predicted rate function fails as a global lower bound because energy-nonconserving solutions to the Boltzmann equation—though rare—occur more frequently than the rate function predicts, due to rare macroscopic energy concentration in few particles with probability $e^{-\mathcal{O}(N)}$. The result challenges the universality of the standard rate function in large deviation theory for stochastic particle systems with conservation laws.

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 $\mathfrak{o}(N)$ particles with probability $e^{-\mathcal{O}(N)}$.

Motivation & Objective

• To analyze the dynamic large deviation behavior of Kac’s conservative particle system for a range of initial conditions, including equilibrium.
• To investigate whether the rate function derived for kinetic large deviation problems applies globally to the Kac process.
• To determine whether energy-nonconserving solutions to the Boltzmann equation can emerge in the large deviation regime despite energy conservation in the particle system.
• To construct a counterexample demonstrating a discrepancy between predicted and actual large deviation rates for non-conservative solutions.
• To clarify the role of rare fluctuations in energy concentration among $\mathfrak{o}(N)$ particles in violating the predicted rate function.

Proposed method

• Derive an upper bound for the large deviation rate using a variational formulation of the rate function, consistent with known kinetic large deviation principles.
• Establish a restricted lower bound for paths that are sufficiently regular or well-behaved, using a change of measure technique via Girsanov’s theorem for jump processes.
• Construct a change of measure $\mathbb{Q}^N$ using a tilting function $K(\nu,t,v,v_\star,\sigma)$ to reweight the dynamics and simulate non-conservative behavior.
• Use the martingale property of the Radon-Nikodym derivative $Z^\nu_t$ to verify that the new measure $\mathbb{Q}$ induces a Markov process with time-dependent generator matching the perturbed dynamics.
• Analyze the asymptotic behavior of the process under $\mathbb{Q}$ to show that the empirical measure converges to a solution of the Boltzmann equation that does not conserve energy.
• Demonstrate that the probability of such non-conservative paths is $e^{-\mathcal{O}(N)}$, implying they are rarer than predicted by the standard rate function.

Experimental results

Research questions

• RQ1 Does the standard rate function for kinetic large deviation problems extend to a global lower bound for Kac’s conservative particle system?
• RQ2 Can energy-nonconserving solutions to the Boltzmann equation emerge in the large deviation regime of a system that strictly conserves energy at the microscopic level?
• RQ3 What is the actual large deviation rate for paths that lead to non-conservative Boltzmann solutions in the Kac process?
• RQ4 How does rare macroscopic energy concentration in $\mathfrak{o}(N)$ particles affect the large deviation behavior of the system?
• RQ5 Is the predicted rate function for large deviations in Kac’s process valid for all types of fluctuating paths, including those with non-conservative energy evolution?

Key findings

• An upper bound for the large deviation rate is established with a rate function matching the form previously derived for kinetic systems.
• A restricted lower bound is proven for a class of sufficiently regular paths, confirming consistency with the predicted rate function in the well-behaved regime.
• A counterexample is constructed showing that energy-nonconserving solutions to the Boltzmann equation occur with probability $e^{-\mathcal{O}(N)}$, which is strictly greater than predicted by the standard rate function.
• The counterexample arises from rare fluctuations where a macroscopic proportion of energy concentrates in $\mathfrak{o}(N)$ particles, a phenomenon occurring with probability $e^{-\mathcal{O}(N)}$.
• This implies that the proposed rate function does not provide a global lower bound, invalidating its universality for the Kac process.
• The result reveals a fundamental mismatch between the large deviation rate function and actual rare event statistics in conservative particle systems with conservation laws.