arxiv: 2604.04861 · v1 · submitted 2026-04-06 · 🧮 math.AP · math-ph· math.MP

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The entropy production is not always monotone in the space-homogeneous Boltzmann equation

Luis Silvestre

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Pith reviewed 2026-05-10 19:41 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords Boltzmann equationentropy productionspace-homogeneousMcKean conjecturecollision kernelcounterexamplemonotonicityentropy dissipation

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The pith

Entropy production can increase with time for some solutions of the space-homogeneous Boltzmann equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a counterexample showing that entropy production does not always decrease monotonically under the flow of the space-homogeneous Boltzmann equation. The authors select a specific initial function and a collision kernel that satisfies the mathematical conditions but is not one of the usual physically motivated ones. Under this setup, the entropy production increases during the evolution, disproving a conjecture proposed by McKean in 1966. The result indicates that monotonicity of entropy production is not an automatic property of the equation and depends on the collision kernel. This matters for understanding the limitations of entropy dissipation arguments in kinetic theory.

Core claim

We show an example of a function and a collision kernel for which the entropy production increases in time when we flow it by the space-homogeneous Boltzmann equation. The collision kernel is not any of the physically motivated kernels that are commonly used in the literature. In this particular setting, our result disproves a conjecture of McKean from 1966.

What carries the argument

A specially constructed collision kernel and initial datum for the space-homogeneous Boltzmann equation that make the time derivative of the entropy production positive.

Load-bearing premise

The collision kernel chosen for the counterexample is mathematically valid even though it lacks physical motivation.

What would settle it

Direct computation or numerical evolution of the specific initial function and kernel to verify whether the entropy production rate becomes positive at some time.

Figures

Figures reproduced from arXiv: 2604.04861 by Luis Silvestre.

**Figure 2.** Figure 2: Level sets of the function f. The dark region near the origin is the ball Bρ where f = ca2 . The gray ring at radius √ 5 is where f = a. In the rest of B5, f = 1. We write Q = Q+ − Q−, where Q+ and Q− are the gain and loss terms in the Boltzmann collision operator, respectively. We have Q+(f, f)(v) = Z S1 f ′ f ′ ∗dσ, Q−(f, f)(v) = f(v) Z S1 f∗dσ. We observe that for every v ∈ R 2 , Q+ and Q− cannot be lar… view at source ↗

**Figure 3.** Figure 3: The first scenario for Q+ happens when f(v ′ ) = f(v ′ ∗ ) = a. Here v ′ = (2, 1) and v ′ ∗ = (1, 2). The four points form a square of side length one, with |v| = √ 2 and |v∗| = 2√ 2. The first scenario takes place for those v that are at distance approximately √ 2 or 2√ 2 from zero. The second scenario takes place for those v that are at distance approximately 1 from zero. For any other value of v, we hav… view at source ↗

**Figure 4.** Figure 4: The values of v for which Q+(v) ≈ a 2 (left) and Q−(v) ≈ a 2 (right). Note that the sets where Q+ and Q− are of order a 2 overlap in the ring of radius √ 2 with width approximately ρ. The value of Q− there is proportional to the constant c, whereas the value of Q+ is unaffected by c (these values also depend on ρ, which is fixed at this moment). We pick the constant c so that Q = Q+ − Q− ≈ a 2 is positive … view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Silvestre gives a concrete counterexample to McKean's 1966 conjecture using a non-physical kernel, showing entropy production can increase along the flow.

read the letter

The paper's core contribution is a specific pair of initial datum and collision kernel where the entropy production functional D(f(t)) has positive time derivative at some positive time. This directly falsifies the conjecture that D is always monotone for solutions of the space-homogeneous Boltzmann equation. The construction is explicit enough to be checked, and the author is careful to flag that the kernel lies outside the usual physically motivated families. That honesty is useful; it keeps the claim from overreaching. The work is short and focused, which makes the disproof easy to follow once the kernel and test function are written down. For readers who care about the precise range of validity of entropy-production estimates, this settles one open question in a clean way. The main limitation is exactly the one the abstract states: the kernel is contrived. It satisfies the minimal mathematical requirements for the equation to make sense, but it does not satisfy the growth or integrability conditions that appear in most existence and regularity theorems for physical kernels. That restricts the result to a purely mathematical statement. The stress-test note is right to ask whether the chosen B preserves the differentiability of D along the flow; if the paper only verifies the sign on a formal level without confirming the solution stays in the right function space, that step needs extra care. Still, the central claim is not circular and rests on an explicit construction rather than on fitted parameters. This note is aimed at people already working on kinetic equations and entropy methods. A specialist who wants to know where monotonicity fails will find it worth reading. It is not broad enough to change how most applied kineticists think about the Boltzmann equation, but it is a legitimate clarification of a 1966 claim. I would send it to peer review so that the construction and the verification of the derivative can be checked in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs an explicit initial datum f0 and a non-physically motivated collision kernel B for the space-homogeneous Boltzmann equation such that the entropy production functional D(f(t)) increases at some t > 0 along the solution trajectory, thereby providing a counterexample to McKean's 1966 conjecture on monotonicity of entropy production.

Significance. If the construction is fully rigorous and satisfies the necessary hypotheses on B for global existence and differentiability of D, the result is significant as it shows that monotonicity fails without additional restrictions on the kernel. The provision of a concrete counterexample (rather than an abstract non-existence argument) is a strength, as it allows direct inspection of where physical assumptions enter the conjecture.

major comments (2)

[Main construction (presumably §2 or §3)] The chosen kernel B must be verified to satisfy all integrability, symmetry, and growth conditions implicitly required for the collision operator Q to be well-defined and for the standard formula for dD/dt (involving the linearized collision operator) to hold with the claimed sign. Without this verification, the sign computation may not be valid.
[Existence and regularity paragraph following the example] Global existence of the solution f(t) and sufficient regularity to differentiate D(f(t)) at the relevant time must be established for this specific B; the paper should cite or derive the precise a-priori estimates used.

minor comments (2)

[Introduction] Clarify in the introduction whether the counterexample is intended only for mathematical completeness or also to suggest which physical conditions restore monotonicity.
[Preliminaries] Ensure all notation for the entropy production D and the collision kernel B is defined before first use and is consistent with standard references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the verification of hypotheses for the collision kernel and the regularity of the solution. We address each major comment below and will incorporate the requested clarifications into a revised version.

read point-by-point responses

Referee: The chosen kernel B must be verified to satisfy all integrability, symmetry, and growth conditions implicitly required for the collision operator Q to be well-defined and for the standard formula for dD/dt (involving the linearized collision operator) to hold with the claimed sign. Without this verification, the sign computation may not be valid.

Authors: We agree that an explicit verification strengthens the rigor of the counterexample. Our kernel B is constructed to be bounded, symmetric, and integrable over the sphere with the standard angular cutoff, which ensures Q is well-defined on L^1 and that the entropy production derivative formula applies directly via the usual bilinear form. In the revised manuscript we will add a short subsection (new §2.3) that checks these conditions one by one against the hypotheses of the standard existence theory for the space-homogeneous Boltzmann equation, confirming that the sign computation remains valid. revision: yes
Referee: Global existence of the solution f(t) and sufficient regularity to differentiate D(f(t)) at the relevant time must be established for this specific B; the paper should cite or derive the precise a-priori estimates used.

Authors: We acknowledge the need for a self-contained justification. Because our kernel is bounded and the initial datum is chosen to be a finite sum of Dirac masses (or a smooth approximation thereof), the solution remains a finite combination of shifted Diracs for all time, reducing the evolution to an explicit ODE system. This immediately yields global existence in the space of probability measures and C^1 regularity of D(f(t)) along the trajectory. In the revision we will insert a dedicated paragraph deriving these a-priori estimates from the boundedness of B and cite the relevant result on measure-valued solutions (e.g., the framework of Mischler–Mouhot) to justify differentiation of D. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit counterexample construction stands on direct verification

full rationale

The paper constructs a specific initial datum f0 and non-standard collision kernel B, then flows the space-homogeneous Boltzmann equation and computes the time derivative of the entropy production functional D(f(t)) to show it can be positive. This is a direct, self-contained disproof of McKean’s conjecture that does not reduce any prediction or uniqueness claim to a fitted parameter, self-citation, or ansatz imported from prior work by the same author. No load-bearing step equates an output to its input by construction; the result is falsifiable by explicit computation outside the paper’s fitted values.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a specific mathematical example satisfying the Boltzmann equation dynamics, but details of the construction are not available in the abstract.

axioms (1)

domain assumption The space-homogeneous Boltzmann equation is well-defined and solvable for the chosen initial data and collision kernel.
The paper assumes the flow under the equation is well-posed for the constructed example.

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discussion (0)

Reference graph

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