What the Boltzmann Equation Does — and Does Not — Explain in Cosmology

1. Introduction: Why the Boltzmann Equation Is Often Misapplied

The Boltzmann equation occupies a central place in statistical mechanics and cosmology. It underpins our understanding of particle distributions, thermal histories, and the emergence of macroscopic behaviour from microscopic dynamics. Because of this prominence, it is often treated—implicitly or explicitly—as a universal descriptor of entropy and evolution.

This assumption becomes problematic when cosmological frameworks that operate outside the equation’s natural domain are criticised as “violating thermodynamics.” Such critiques typically reflect not a failure of the framework, but a misunderstanding of what the Boltzmann equation is actually designed to describe.

This article clarifies that distinction. Its purpose is to show that Crowton’s Cosmogenic Field Theory (CCFT) does not conflict with statistical mechanics, but instead operates upstream of where Boltzmann-style descriptions remain valid. The issue is not whether thermodynamics applies, but which form of thermodynamics applies, and where.

2. What the Boltzmann Equation Is Designed For

At its core, the Boltzmann equation governs the time evolution of a particle distribution function in phase space. It assumes:

• identifiable particles,

• well-defined microstates,

• local interactions,

• and statistical behaviour close to equilibrium or describable as perturbations away from it.

In cosmology, this framework is indispensable. It successfully describes:

• thermal decoupling in the early universe,

• particle freeze-out,

• radiation and matter distributions,

• and large portions of cosmic background evolution.

Crucially, the Boltzmann equation does not describe spacetime itself. It presupposes a background on which particles move and interact. Entropy, within this framework, is a statistical quantity derived from microstate counting under stable geometric and causal assumptions.

Within these limits, the equation is both rigorous and powerful.

3. Why Extreme Gravitational Regimes Exceed Its Domain

The difficulty arises when the Boltzmann framework is implicitly extended into regimes it was never designed to handle.

Extreme gravitational environments—such as black hole interiors, horizon-scale dynamics, or early-universe curvature-dominated phases—violate several of the equation’s foundational assumptions:

• spacetime is no longer passive or slowly varying,

• particle identities may not remain well-defined,

• local equilibrium may not exist even approximately,

• and entropy production may be dominated by geometry rather than collisions.

In these regimes, entropy is not merely redistributed among particles; it is generated, regulated, or constrained by spacetime dynamics themselves. The Boltzmann equation does not fail here—it simply ceases to be the correct descriptive tool.

Confusion often arises when this limitation is interpreted as a rejection of thermodynamics, rather than a recognition of domain boundaries.

4. Entropy as a Regulatory Variable, Not Just a Statistical Count

Standard statistical mechanics treats entropy as an emergent quantity derived from microstate multiplicity. This view is both correct and sufficient within its domain.

However, in gravitational physics, entropy acquires an additional role. Black hole thermodynamics, horizon entropy, and holographic bounds already indicate that entropy can act as a constraint on geometry, not merely a bookkeeping device.

CCFT extends this insight by treating entropy as a regulatory variable that couples directly to curvature. In this view:

• entropy constrains how curvature can evolve,

• curvature regulates how entropy can be produced or redistributed,

• and extreme collapse is prevented by entropy–curvature feedback.

This does not replace statistical entropy; it governs the conditions under which statistical descriptions remain valid. Once a system exits those conditions, a higher-level regulatory description becomes necessary.

5. Why CCFT Complements — Not Replaces — Kinetic Theory

A common mischaracterisation is that CCFT competes with or overrides kinetic theory. This is incorrect.

CCFT does not attempt to describe particle distributions, collision terms, or transport phenomena. These remain firmly within the domain of the Boltzmann equation wherever its assumptions hold.

Instead, CCFT addresses regime structure:

• when kinetic descriptions apply,

• when they break down,

• and what governs system behaviour beyond that point.

In practical terms:

• Boltzmann governs behaviour within a regime,

• CCFT governs transitions between regimes.

This layered structure mirrors how physics already operates across scales. Fluid dynamics does not invalidate molecular theory; general relativity does not replace quantum mechanics. Each applies where its assumptions are satisfied.

6. Where Boltzmann Ends and Regime Dynamics Begin

The boundary between Boltzmann-governed behaviour and regime-level dynamics is not arbitrary. It is defined by the breakdown of near-equilibrium assumptions and the emergence of geometry-dominated entropy production.

CCFT formalises this boundary through entropy–curvature thresholds. Below these thresholds, standard kinetic and statistical methods apply. Beyond them, gravitational regulation dominates, and different qualitative behaviour emerges.

This does not weaken thermodynamics; it preserves it. By preventing the inappropriate extension of statistical tools into domains where they lose meaning, regime-based frameworks protect the internal consistency of physical theory.

7. Conclusion: Thermodynamics Has Domains, Not Enemies

The Boltzmann equation remains one of the most successful constructs in physics. Its power lies in its precision and in the clarity of its assumptions.

CCFT does not challenge this legacy. It situates it.

By recognising that entropy can act as both a statistical quantity and a regulatory constraint, CCFT aligns with the broader trajectory of gravitational thermodynamics rather than opposing it. The question is not whether thermodynamics applies to cosmology, but which thermodynamic description applies, and where.

Understanding this distinction prevents false conflicts and allows genuinely complementary frameworks to be evaluated on their actual merits.

Doi: https://doi.org/10.5281/zenodo.15352974