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

$H(f(t))-H(f_\infty)\le^$,

where $H(f)$ is the entropy of distribution $f$, $C$ and $\lambda$ are constants >0 and $\lambda$ 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.Borsoni, T. Extending Cercignani’s Conjecture Results from Boltzmann to Boltzmann–Fermi–Dirac Equation. J Stat Phys 191, 52 (2024). https://doi.org/10.1007/s10955-024-03262-3

References

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Conjectures
Statistical mechanics
Thermodynamics
Gases