Integrable Systems, Analysis And Probability Seminar
Thursday, February 19, 2004, 4:00pm, 120 Physics
Eric Carlen (Georgia Tech)
Sharp Form of the Central Limit Theorem for Maxwellian Molecules
Abstract:- In a form of the Maxwell--Boltzmann equation specifically due to
Maxwell, the so--called Wild summation formula gives an
expression for the solution of the spatially homogeneous Maxwell--Boltzmann
equation in terms of its initial data $F$
as a sum
${\displaystyle f(v,t) = \sum_{n=0}^\infty e^{-t}(1 - e^{-t})^n Q_n^+(F)(v)}$.
Here, $Q_n^+(F)$ is an average over $n$--fold iterated ``Wild convolutions'' of
$F$. If $M$ denotes the Gaussian equilibrium corresponding to $F$, then it is
of
interest to determine bounds on the rate at which $\|Q_n^+(F) - M\|_{L^1(\R)}$
tends to zero. We prove that for every $\epsilon>0$, if $F$
has moments of every order and finite Fisher information, there is a constant
$C$ so that for all $n$, $\|Q_n^+(F) - M\|_{L^1(\R)} \le Cn^{\Lambda+\epsilon}$
where $\Lambda$ is the least negative eigenvalue for the linearized collision
operator. We show that $\Lambda$ is the best
possible exponent by relating this estimate to a sharp estimate for the
rate of relaxation of $f(\cdot,t)$ to $M$.
A key role in the analysis is played by a decomposition of $Q_n^+(F)$ into a
smooth part and a small part.
This depends in an essential way on a probabilistic construction of McKean. It
allows us to circumvent difficulties
stemming from the fact that the evolution does not improve the
qualitative regualrity of the initial data. This is joint work with C.
Carvalho and E. Gabetta.
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