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Conditional one-particle kinetic entropy production

Physical scope

Conditional prediction of nondecrease of the declared one-particle velocity-reference information entropy for the adopted homogeneous cutoff reflection model and the specified positive regular Gaussian-mixture preparation. It is not a statement about fine-grained N-particle Gibbs entropy, generic gases, a hard-sphere Boltzmann–Grad limit, hydrodynamic limits, experiments, or all physical realizations.

Statement

Assume the physical adoption Selected phenomenological cutoff collision model with a positive collision rate $λ$ , particle mass $m > 0$ , reference speed $v_{0} > 0$ , and $k_{B} > 0$ . Prepare the normalized one-particle velocity density by

$f_{0} (v) = \frac{1}{2} γ_{a I, u} (v) + \frac{1}{2} γ_{b I, u} (v), u \in R^{3}, 0 < a < b,$

where $u \in R^{3}$ is a mean velocity measured in $m s^{- 1}$ , $a, b$ are covariance variances in $(m s^{- 1})^{2}$ , and $γ_{Σ, u}$ is the normalized Gaussian defined in Regularity and invariance of the cutoff Gaussian-mixture class. For the resulting global regular model solution $f_{t}$ , define the one-particle reference information entropy

$S_{B} (t) : = - k_{B} \int_{R^{3}} f_{t} (v) \log (v_{0}^{3} f_{t} (v)) d v .$

Since $v_{0}^{3} f_{t} (v)$ is dimensionless, $S_{B}$ has SI units $J K^{- 1}$ .

Then the model deduction is

$\frac{d S_{B}}{d t} = \frac{k_{B} λ}{4} \int_{R^{3} \times R^{3} \times S^{2}} (B - A) \log (B / A) d μ \geq 0,$

where $A = f_{t} (v) f_{t} (w)$ and $B = f_{t} (v^{'}) f_{t} (w^{'})$ use the exact reflection variables of Selected phenomenological cutoff collision model. The derivative is strictly positive at $t = 0$ . Equality at any time holds exactly when, for some mean velocity $\overset{ˉ}{u} \in R^{3}$ and speed parameter $s > 0$ (in $m s^{- 1}$ ),

$f_{t} (v) = (2 π s^{2})^{- 3 / 2} \exp ⁣ (- \frac{∣ v - \overset{ˉ}{u} ∣^{2}}{2 s^{2}}) .$

This conclusion is conditional on the adopted closed homogeneous model and regular preparation. It identifies a one-particle information functional with entropy units; it does not assert the fine-grained entropy of an $N$ -particle ensemble, a general thermodynamic second law, or empirical entropy increase.

Facts & Assumptions

Given: The postulated kinetic model, ${A C}_{ω}$ , $m > 0$ , $λ > 0$ , $v_{0} > 0$ , $k_{B} > 0$ , and the stated Gaussian-mixture initial preparation.

[A1]

The selected model equation is an equality in $X$ for norm- $C^{1}$ trajectories in the stated admissible domain, with normalized velocity density and the finite angular reflection law (Selected phenomenological cutoff collision model). The Gaussian-mixture branch also supplies the specified pointwise regular representatives by [F1].

[A2]

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) as required by the mathematical measure and change-of-variables suppliers.

[F1]

H theorem and Maxwellian equality for the regular cutoff model: the specified mixture has a global positive smooth solution with $H^{'} (t) = - \frac{λ}{4} \int (B - A) \log (B / A) d μ \leq 0$ , equality exactly for a normalized isotropic Gaussian, and strict production at the non-Gaussian initial mixture.

Proof

Given: [A1], [A2], and [F1].

Proof technique: direct.

1.1A1A2F1algebra

The physical postulate [A1] supplies the trajectory and the selected one-particle entropy convention; the mathematical theorem [F1] applies to this exact reflection measure, rate, reference speed, and initial mixture. Since $S_{B} = - k_{B} H$ and $k_{B} > 0$ , differentiating gives $S_{B}^{'} = - k_{B} H^{'} = \frac{k_{B} λ}{4} \int (B - A) \log (B / A) d μ \geq 0$ .

2.1A1F1step 1.1∎

The strict-production and equality clauses in [F1] transfer with the same positive factor $k_{B}$ : the initial mixture has $S_{B}^{'} (0) > 0$ , and at any later time $S_{B}^{'} = 0$ exactly when the velocity density has the normalized isotropic Gaussian form in the Statement. This is a deduction inside the adopted model and supplies no empirical premise.

Conditional one-particle kinetic entropy production

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources