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 ${\displaystyle f(t,x,v)}$be the distribution function of particles at time ${\displaystyle t}$, position ${\displaystyle x}$and velocity ${\displaystyle v}$, and ${\displaystyle f_{\infty }(v)}$the equilibrium distribution (typically the Maxwell-Boltzmann distribution), then our conjecture is:

${\displaystyle H(f(t))-H(f_{\infty })\leq {C_{e}}^{-{\lambda }t}}$,

where ${\displaystyle H(f)}$is the entropy of distribution ${\displaystyle f}$, ${\displaystyle C}$and ${\displaystyle \lambda }$are constants >0 and ${\displaystyle \lambda }$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.