The diffusion equation is compatible with special relativity

Abstract: Due to its parabolic character, the diffusion equation exhibits instantaneous spatial spreading, and becomes unstable when Lorentz-boosted. According to the conventional interpretation, these features reflect a fundamental incompatibility with special relativity. In this Letter, we show that this interpretation is incorrect by demonstrating that any smooth and sufficiently localized solution of the diffusion equation is the particle density of an exact solution of the relativistic Vlasov-Fokker-Planck equation. This establishes the existence of a causal, stable, and thermodynamically consistent relativistic kinetic theory whose hydrodynamic sector is governed exactly by diffusion at all wavelengths. We further demonstrate that the standard arguments for instability arise from considering solutions that admit no counterpart in kinetic theory, and that apparent violations of causality disappear once signals are defined in terms of the underlying microscopic data.

• The paper demonstrates that a causal, stable relativistic kinetic framework exactly embeds the classical diffusion equation.
• It provides an explicit mapping from well-behaved hydrodynamic initial data to a positive microscopic solution, ensuring stability.
• It clarifies that apparent acausality and instability stem from unphysical extrapolations rather than fundamental relativistic constraints.

Special Relativistic Consistency of the Diffusion Equation

Introduction

The paper "The diffusion equation is compatible with special relativity" (2601.19464) systematically revisits the widespread assertion that the (parabolic) diffusion equation is intrinsically incompatible with the principles of special relativity due to its apparent acausality and pathological stability features under Lorentz boosts. Contrary to the standard perspective found in the relativistic hydrodynamics literature, the work demonstrates that the diffusion equation emerges as an exact hydrodynamic sector of a causal, stable, and thermodynamically consistent relativistic kinetic theory. The implications are significant for both the foundations of out-of-equilibrium relativistic fluid dynamics and the proper formulation of constitutive laws.

Parabolic Diffusion and Standard Objections

The canonical diffusion equation,

$\partial_t n = \mathfrak{D}\,\partial_x^2 n,$

describes conserved density evolution under diffusion. However, its retarded Green’s function exhibits instantaneous spatial spread, violating the relativistic speed limit at the level of macroscopic densities. Additionally, a Lorentz boost renders the equation ill-posed, with spatially homogeneous perturbations growing exponentially, signifying an instability for certain modes. Consequently, a substantial body of work has developed hyperbolic, manifestly causal extensions—Cattaneo-type, Israel-Stewart, BDNK, and divergence-type frameworks—to circumvent these perceived pathologies. However, alternative views posit that the issue arises only when these equations are naively extrapolated beyond the hydrodynamic sector justified by kinetic theory.

Relativistic Embedding via Kinetic Theory

The central result is an explicit construction that embeds smooth, localized solutions of the diffusion equation in the one-particle distribution function of the relativistic Vlasov-Fokker-Planck (VFP) equation. The VFP equation describes Brownian motion of massive relativistic particles with stochastic momentum exchange in a thermal bath and is given by: $$(\partial_t + v \partial_x) f = \frac{1}{\beta^2 \mathfrak{D}}\partial_p(\partial_p f + \beta v f),$$ where $v = p/\varepsilon$, $\varepsilon = \sqrt{m^2+p^2}$, and $f(t,x,p)$ is manifestly causal ($|v|\leq 1$), Lyapunov stable in all inertial frames, and obeys the second law of thermodynamics.

The main theorem states that for any sufficiently regular (Schwartz-class) initial density $n(0,x)$—with perturbations localized in space—there exists a positive distribution $f(t,x,p)$ solving the VFP equation such that the associated density, after momentum integration, precisely evolves under the diffusion equation for all $t\geq0$. The construction is explicit, with the mapping from hydrodynamic initial data to microscopic $f$ given by a prescribed kernel in Fourier space involving the modified Bessel function $K_1$.

Key claim: Within the admissible sector of kinetic theory, the hydrodynamic evolution of densities is exactly diffusive at all wavelengths, without gradient corrections, UV regularization, or finite-propagation-speed modifications.

Stability, Causality, and Physical Admissibility

Stability Revisited

The standard argument for instability in boosted frames invokes solutions that violate spatial locality and correspond to initial data that are nonnormalizable or singular and thus cannot arise from the underlying relativistic kinetic theory. The authors prove that within the VFP solution space, only perturbations with well-behaved phase-space localization are allowed, and these are always stable. Modes responsible for instability, such as $e^{\Gamma\gamma(t-Vx)}$, cannot be embedded in the kinetic sector; their existence reflects an unphysical extension of macroscopic equations.

Causality Resolution

The apparent acausality arises from considering the macroscopic density $n(t,x)$ in isolation: even if initially compactly supported, it instantaneously acquires global support at $t>0$. However, microscopic inspection reveals that such macroscopic initial localization is impossible when the full phase-space distribution $f(0,x,p)$ is generated by admissible microscopic processes. Each far-field density tail at $t>0$ is completely determined by nonzero $f(0,x,p)$ at large $p$ and $x$, already set at $t=0$, meaning no information has traveled superluminally. True causal propagation is thus preserved at the fundamental level.

Explicit Examples and Higher Dimensions

In the massless (ultrarelativistic) limit, the embedding becomes especially transparent. The explicit kinetic solution aligns with the diffusive density for arbitrary (but appropriately regular) $n(t,x)$. The construction generalizes trivially to higher spatial dimensions, with inner products replacing scalar products and the momentum-space operators promoted accordingly; the proof structure remains unaffected.

Implications for Relativistic Hydrodynamics

The paper’s result represents a contradictory claim to a foundational dogma in the field: relativity does not mandate a finite speed of signal propagation at the level of macroscopic diffusion. Rather, it is the microphysical structure and the sufficient regularity/locality of admissible initial data that enforces causality. Hyperbolic extensions remain essential for numerical well-posedness, treatment of arbitrary frames, and for describing strongly non-equilibrium physics, but not due to fundamental relativistic requirements at the hydrodynamic level.

In effect, Cattaneo-type and Israel-Stewart-like equations are not strictly "more relativistic" than parabolic diffusion in the physically valid regime—they simply offer well-defined standalone PDEs which can extend beyond the strictly diffusive sector.

Theoretical Extensions and Future Directions

These results necessitate careful specification of the hydrodynamic sector’s domain of validity and motivate reevaluation of diffusion-limited processes in relativistic systems, including neutron star cooling, heavy-ion collision phenomenology, and radiative transfer, potentially requiring reinterpretation of boundary and initial conditions. Extension to nonlinear regimes, fluctuating hydrodynamics, and general-relativistic backgrounds are natural future directions.

Conclusion

The paper conclusively demonstrates that the parabolic diffusion equation, underpinned by a consistent relativistic microphysical theory, is stable and causal for physically realizable states—resolving a longstanding conceptual tension in the literature. The findings clarify the theoretical structure of relativistic dissipation, establish rigorous criteria for hydrodynamic admissibility, and recommend revisiting traditional arguments on the necessity of hyperbolicity in dissipative relativistic theories. This foundational advance has broad relevance for constructing and interpreting effective theories in relativistic out-of-equilibrium physics (2601.19464).