Cercignani's conjecture was proposed in 1982 by an Italian mathematician and
kinetic theorist for the
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G ...
. It assumes a
linear inequality
In mathematics a linear inequality is an inequality (mathematics), inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:
* greater than
* ≤ less than or equal to
* ≥ greater than or equal ...
between the entropy and entropy production functionals for Boltzmann's nonlinear
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms involvi ...
, 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 medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
ist
Cédric Villani
Cédric Patrice Thierry Villani (; born 5 October 1973) is a French politician and mathematician working primarily on partial differential equations, Riemannian geometry and mathematical physics. He was awarded the Fields Medal in 2010, and he ...
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 (typically the Maxwell-Boltzmann 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.
[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
{{Reflist
Conjectures
Statistical mechanics
Thermodynamics
Gases