Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas (original) (raw)
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Journal of Statistical Physics, 1995
This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+ε, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then ∫∣v∣>R f(v, t) ∣v∣2dv→0 asR→∞, and this convergence is uniform in time.
2011
Consider the homogeneous Boltzmann equation for Maxwellian molecules. We provide a new representation for its solution in the form of expectation of a random probability measure M. We also prove that the Fourier transform of M is a conditional characteristic function of a sum of independent random variables, given a suitable sigma-algebra. These facts are then used to prove a CLT for Maxwellian molecules, that is the statement of a necessary and sufficient condition for the weak convergence of the solution of the equation. Such a condition reduces to the finiteness of the second moment of the initial distribution \mu_0. As a further application, we give a refinement of some inequalities, due to Elmroth, concerning the evolution of the moments of the solution.
A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules
2001
Tanaka [18], showed a way to relate the measure solution {P t } t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.
A Simple Mathematical Proof of Boltzmann's Equal a priori Probability Hypothesis
2009
Using the Fluctuation Theorem (FT), we give a first-principles derivation of Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble. Using a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.
Annali di Matematica Pura ed Applicata (1923 -), 2014
This paper proves the existence of weak solutions to the the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial data are chosen from the space of all Borel probability measures on R 3 with finite second moments and the (angular) collision kernel satisfies a very weak cutoff condition, namely 1 −1 x 2 b(x)dx < +∞. Conservation of momentum and energy is also proved for these weak solutions, without resorting to any boundedness of the entropy.
The Maxwell-Boltzmann distribution has been a very useful statistical distribution for understanding the molecular motion of ideal gases. In this work, it will be shown that it is also possible to obtain a generalized Maxwell-Boltzmann distribution valid for any macroscopic system of any composition and having any arbitrary state of aggregation. This distribution is the result of the large number of collisions and molecular interactions taking place in such macroscopic system. On the other hand, in order to better understand the concept of thermal equilibrium, a mathematical interpretation of the zeroth law of Thermodynamics is included.
Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation
Entropy, 2015
Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We analyse the Boltzmann operator in the geometric setting from the point of view of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts i.e., the H-theorem. In a second part of the paper we discuss a generalization of the Orlicz setting to include spatial derivatives and apply it to the Hyvärinen divergence.