Monotonicity of Boltzmann entropy production

Does entropy production necessarily decrease along the three-dimensional homogeneous Boltzmann flow with kernel B=|v−v*|^γ/(4π), 0≤γ≤1? The counterexample target starts from a smooth positive radial mixture of two Maxwellians and has dD/dt>0 initially.

Why it matters and what counts as progress

Why it matters: It tests the monotonicity claim within standard Maxwell-molecule and hard-sphere collision settings.

What counts as progress: A smooth positive solution with initially increasing entropy production in the standard collision model, or a monotonicity theorem with the necessary restrictions.

How the problem developed

1966 · Conjecture [1]
Henry P. McKean Jr. Conjectured that entropy production is monotone decreasing along solutions of the space-homogeneous Boltzmann equation.
6 Apr 2026 · Counterexample [1]
Luis Silvestre Disproved the broad conjecture with a space-homogeneous example using a collision kernel outside the usual physically motivated class.
1 Sept 2026 · Counterexample extension [2]
Luis Silvestre Extended the counterexample to hard spheres and Maxwell molecules with constant angular cross section and parameter γ ∈ [0,1], using radial mixtures of Maxwellians.

AI contribution

The entropy production is not always monotone for hard spheres or for Maxwell molecules

Luis Silvestre

For the three-dimensional space-homogeneous Boltzmann equation with collision kernel B=|v−v*|^γ/(4π), 0≤γ≤1, Silvestre constructs smooth positive radial two-Maxwellian initial densities f_R=(1−p)M₁+pM_R. For 0<p<p_γ and sufficiently large R, the entropy production D initially increases: dD(f_R)/dt>0 at t=0, with a positive leading term proportional to R^γ log R. The range includes constant-angle Maxwell molecules and hard spheres.

AI involvementAI-led

ImportanceNotable

Technical verificationReported

Human understandingFull