Using modern mathematical techniques from the fields of partial differential equations and harmonic analysis - many of which were developed during the last five to 50 years, and thus relatively new to mathematics - the mathematicians proved the global existence of classical solutions and rapid time decay to equilibrium for the

Boltzmann equation with long-range interactions.

The discrete

Boltzmann equation (1) for the distribution function [f.

That last step was "really a breakthrough from my point of view," comments Roberto Benzi of the University of Rome Tor Vergata, a pioneer in the use of the

Boltzmann equation for fluid flows.

In particular, the

Boltzmann equation is well known to model the transport of neutrons/photons.

Classical

Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations.

After obtaining the stochastic Kac dynamics in the momentum space from the stochastic dynamics in the phase space, he shows that the spatially homogenous

Boltzmann equation can be derived from the stochastic Boltzmann hierarchy in the phase space without using mean-field approximation.

Starting from the

Boltzmann equation, Guyer and Krumhansl (1966) derived an equation of the form (3.