Boundary Entropy in Fokker–Planck Models

• 'Boundary entropy' is not defined as a separate concept in the literature but is often confused with bulk entropy measures used in kinetic diffusion models.
• The studies focus on the relativistic Fokker–Planck equation with exponential convergence to equilibrium via relative entropy, logarithmic Sobolev, and Poincaré inequalities.
• Post-Newtonian extensions further detail kinetic corrections in weak gravitational fields without establishing a distinct boundary-specific entropy framework.

Searching arXiv for recent and foundational papers on boundary entropy. Boundary entropy is not defined, analyzed, or operationalized in the supplied sources. The available material instead concerns the relativistic Fokker–Planck equation, its Newtonian limit, exponential trend to equilibrium, and a post-Newtonian Fokker–Planck equation for Brownian motion in a weak gravitational field (Félix et al., 2011, Kremer, 15 Jul 2025). This suggests that, within the evidentiary basis provided here, “boundary entropy” cannot be documented as a technical concept with its own definition, formalism, or applications. What can be stated rigorously is that the cited literature develops kinetic and diffusion-theoretic results centered on relativistic and post-Newtonian transport rather than on boundary phenomena.

1. Scope of the supplied literature

The paper "Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation" (Félix et al., 2011) studies a $c$-dependent kinetic model for a dilute gas of identical particles of mass $m>0$ moving in $\mathbb{R}^3$, undergoing random kicks and friction from a thermal bath at temperature $T$. Its principal themes are the limit $c\to\infty$, convergence to the classical Fokker–Planck equation, and trend to equilibrium for spatially homogeneous solutions.

The paper "Fokker-Planck equation for the Brownian motion in the post-Newtonian approximation" (Kremer, 15 Jul 2025) analyzes a binary mixture consisting of a light relativistic gas and a heavy Brownian component in a weak gravitational field described by a post-Newtonian approximation to general relativity. It derives a post-Newtonian Fokker–Planck equation, determines friction coefficients at first and second post-Newtonian order, and studies linear stability and Jeans-type instability.

Neither source introduces a quantity called boundary entropy, nor do they formulate entropy in terms of boundaries, interfaces, boundary conditions, or boundary conformal data. The entropy notion that does appear in (Félix et al., 2011) is the relative entropy for spatially homogeneous solutions of the relativistic Fokker–Planck equation.

2. Relativistic Fokker–Planck structure and equilibria

In (Félix et al., 2011), the state is described by a one-particle distribution function

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$

with relativistic energy

$p^0 = \sqrt{m^2c^2 + |p|^2}.$

The relativistic Fokker–Planck equation is

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$

where

$$D^{ij} = \frac{mc}{p^0}\left(\delta^{ij} + \frac{p^i p^j}{m^2c^2}\right).$$

Its stationary distribution is the Jüttner distribution

$$J(p) = C_J e^{-\theta c p^0},\qquad \theta = \frac{1}{kT}.$$

The classical comparison model is the non-relativistic Fokker–Planck equation

$m>0$0

whose equilibrium is the Maxwell–Boltzmann distribution

$m>0$1

These structures are central to the paper’s kinetic analysis. They do not, however, define an entropy associated with a boundary. A plausible implication is that any attempt to identify “boundary entropy” in this corpus would conflate it with bulk relative entropy or entropy dissipation, which are the actual objects treated in the paper.

3. Newtonian limit and $m>0$2-expansion

A central theorem in (Félix et al., 2011) states that if $m>0$3 solves the classical Fokker–Planck equation and $m>0$4 solves the relativistic Fokker–Planck equation with suitable $m>0$5, non-negative initial data satisfying $m>0$6 convergence, spatial support, and weighted $m>0$7 assumptions, then

$m>0$8

uniformly for $m>0$9 in compact intervals $\mathbb{R}^3$0. The proof yields an estimate of the form

$\mathbb{R}^3$1

The analytical strategy rewrites the difference $\mathbb{R}^3$2 as a classical Fokker–Planck equation with a source term $\mathbb{R}^3$3, applies Duhamel’s formula with the Green kernel $\mathbb{R}^3$4, and controls the correction by combining the asymptotic expansion of $\mathbb{R}^3$5, the explicit form of $\mathbb{R}^3$6, weighted $\mathbb{R}^3$7 bounds, and finite propagation speed for the relativistic equation (Félix et al., 2011).

The same paper presents a post-Newtonian viewpoint in which the explicit $\mathbb{R}^3$8-dependence of the generator permits an expansion in powers of $\mathbb{R}^3$9. In particular,

$T$0

and

$T$1

This suggests a systematic hierarchy

$T$2

with $T$3 the classical Fokker–Planck generator.

Within the supplied material, the role of entropy here is indirect: the emphasis is on operator comparison, asymptotic control, and equilibrium expansion, not on a boundary contribution to entropy.

4. Relative entropy, dissipation, and exponential relaxation

For spatially homogeneous solutions $T$4, (Félix et al., 2011) introduces the normalized variable

$T$5

The corresponding equation is written as

$T$6

where $T$7 is a Riemannian metric on momentum space,

$T$8

and $T$9 is a drift vector field satisfying $c\to\infty$0 with

$c\to\infty$1

The Jüttner measure is

$c\to\infty$2

and the relative entropy is

$c\to\infty$3

Its entropy dissipation is

$c\to\infty$4

with entropy identity

$c\to\infty$5

If the logarithmic Sobolev inequality

$c\to\infty$6

holds, then

$c\to\infty$7

and by the Csiszár–Kullback–Pinsker inequality,

$c\to\infty$8

Equivalently,

$c\to\infty$9

A key result is that this logarithmic Sobolev inequality holds when the temperature is small enough: there exists a critical inverse temperature $$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$0 such that for every $$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$1 the inequality holds, with the rate expressed through the rational function

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$2

The proof uses the Bakry–Émery criterion via the tensor

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$3

For all temperatures, (Félix et al., 2011) instead establishes exponential convergence in weighted $$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$4 through the variance functional

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$5

and the Poincaré inequality

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$6

Using a criterion due to F.-Y. Wang, the paper proves that this inequality holds for every $$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$7, hence

$$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$8

This is the only entropy-centered framework explicitly present in the supplied corpus. It is a bulk relative-entropy method for homogeneous relativistic diffusion, not a theory of boundary entropy.

5. Post-Newtonian Brownian motion and kinetic corrections

The second supplied paper, (Kremer, 15 Jul 2025), develops a Fokker–Planck equation for Brownian motion in a gravitational field including post-Newtonian corrections. The system is a binary mixture with a light gas species $$f = f(t,x,p)\ge 0,\qquad t\ge 0,\ x\in\mathbb{R}^3,\ p\in\mathbb{R}^3,$$9 of rest mass $p^0 = \sqrt{m^2c^2 + |p|^2}.$0 and a heavy Brownian species $p^0 = \sqrt{m^2c^2 + |p|^2}.$1 of rest mass $p^0 = \sqrt{m^2c^2 + |p|^2}.$2, with

$p^0 = \sqrt{m^2c^2 + |p|^2}.$3

The weak gravitational field is described at 1PN order by

$p^0 = \sqrt{m^2c^2 + |p|^2}.$4

Starting from the post-Newtonian Boltzmann equation and using the Brownian assumptions $p^0 = \sqrt{m^2c^2 + |p|^2}.$5, $p^0 = \sqrt{m^2c^2 + |p|^2}.$6, and weak momentum transfer per collision, the paper expands the collision term by a Kramers–Moyal procedure. The Brownian distribution is written as

$p^0 = \sqrt{m^2c^2 + |p|^2}.$7

and $p^0 = \sqrt{m^2c^2 + |p|^2}.$8 is expanded up to second order in $p^0 = \sqrt{m^2c^2 + |p|^2}.$9.

The resulting 1PN Fokker–Planck equation is

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$0

where

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$1

The friction coefficient $$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$2 is determined at 1PN order in general form and, for hard-sphere interactions, becomes

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$3

At 2PN order, again for hard spheres,

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$4

This framework concerns collisional diffusion, gravitational potentials, and post-Newtonian transport coefficients. No boundary-entropy variable, boundary free energy, or boundary counting quantity is introduced.

6. Stability analysis, limitations, and relevance to the requested topic

In the Newtonian limit of (Kremer, 15 Jul 2025), the Fokker–Planck equation couples to Poisson’s equation and, after linearization around equilibrium with plane-wave perturbations, yields the dimensionless dispersion relation

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$5

In the collisionless limit this reduces to

$$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$6

The paper interprets this as two propagating modes for wavelengths smaller than the Jeans wavelength and one non-propagating mode, while wavelengths bigger than the Jeans wavelength correspond to growth or decay, with the growing mode identified as the instability.

The 1PN analysis couples the linearized Fokker–Planck equation to the perturbed Poisson equations for $$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$7, $$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$8, and $$\partial_t f + \frac{mc}{p^0}\,p\cdot\nabla_x f = \partial_{p^i}\Big(D^{ij}\partial_{p^j}f + \frac{p^i}{m}f\Big),$$9, producing a post-Newtonian dispersion relation with corrections depending on $$D^{ij} = \frac{mc}{p^0}\left(\delta^{ij} + \frac{p^i p^j}{m^2c^2}\right).$$0 and the background potential $$D^{ij} = \frac{mc}{p^0}\left(\delta^{ij} + \frac{p^i p^j}{m^2c^2}\right).$$1 (Kremer, 15 Jul 2025). The paper therefore extends classical Jeans analysis into a post-Newtonian Brownian setting.

For the requested topic, the principal limitation is categorical rather than technical: the supplied corpus is about relativistic and post-Newtonian Fokker–Planck theory, not about boundary entropy. The entropy structure actually documented is the relative entropy

$$D^{ij} = \frac{mc}{p^0}\left(\delta^{ij} + \frac{p^i p^j}{m^2c^2}\right).$$2

for homogeneous relativistic diffusion (Félix et al., 2011). A plausible implication is that any genuine encyclopedia entry on boundary entropy would require a different documentary basis, because the present sources do not provide the definition, historical origin, mathematical formalism, or applications of that concept.