  The present work provides a definitive answer to the problem of quantifying
relaxation to equilibrium of the solution to the spatially homogeneous
Boltzmann equation for Maxwellian molecules. The beginning of the story dates
back to a pioneering work by Hilbert, who first formalized the concept of
linearization of the collision operator and pointed out the importance of its
eigenvalues with respect to a certain asymptotic behavior of the Boltzmann
equation. Under really mild conditions on initial data - close to being
necessary - and a weak, physically consistent, angular cutoff hypothesis, our
main result (Theorem 1.1) contains the first precise statement that the total
variation distance between the solution and the limiting Maxwellian
distribution admits an upper bound of the form C e^{Lambda_b t}, Lambda_b being
the least negative of the aforesaid eigenvalues and C a constant which depends
only on a few simple numerical characteristics (e.g. moments) of the initial
datum. The validity of this quantification was conjectured, about fifty years
ago, in a paper by Henry P. McKean but, in spite of several attempts, the best
answer known up to now consists in a bound with a rate which can be made
arbitrarily close to Lambda_b, to the cost of the "explosion" of the constant
C. Moreover, its deduction is subject to restrictive hypotheses on the initial
datum, besides the Grad angular cutoff condition. As to the proof of our
results, we have taken as point of reference an analogy between the problem of
convergence to equilibrium and the central limit theorem of probability theory,
highlighted by McKean. Our work represents in fact a confirmation of this
analogy, since the techniques we develop here crucially rely on certain
formulations of the central limit theorem.
