Cercignani conjecture
Cercignani's conjecture was proposed in 1982 by an Italian mathematician and kinetic theorist for the Boltzmann equation. It assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator, describing the statistical distribution of particles in a gas. Cercignani conjectured that the rate of convergence to the entropical equilibrium for solutions of the Boltzmann equation is time-exponential, i.e. the entropy difference between the current state and the equilibrium state decreases exponentially fast as time progresses. A Fields medalist Cédric Villani proved that the conjecture "is sometimes true and always almost true"
Mathematically:
Let be the distribution function of particles at time
, position and velocity, and the equilibrium distribution, then our conjecture is:
where is the entropy of distribution, and are constants >0 and is related to the convergence rate.
Thus the conjecture provides us with insight into how quickly a gas approaches its thermodynamic equilibrium.
In 2024, the result was extended from the Botzmann to the Boltzmann-Fermi-Dirac equation.