Chapman–Enskog Expansion in Kinetic Theory

• Chapman–Enskog expansion is a systematic perturbative method that derives macroscopic fluid dynamics from the microscopic Boltzmann equation using a small parameter.
• It exposes the factorial divergence of series coefficients in relativistic settings, highlighting intrinsic non-hydrodynamic modes.
• Generalized non-perturbative expansions and resummation techniques improve the modeling of far-from-equilibrium systems beyond classical hydrodynamics.

The Chapman–Enskog expansion is a systematic perturbative method for deriving macroscopic fluid dynamic equations and associated transport coefficients from the microscopic Boltzmann equation in kinetic theory. Fundamental to nonequilibrium statistical mechanics, the method projects the dynamics of dilute gases or similar systems, via an ordering in a small parameter (such as the Knudsen number), onto a sequence of hydrodynamic equations of increasing order. Despite its success in connecting kinetic theory with continuum mechanics, recent research has revealed structural limitations of the expansion in relativistic regimes, specifically in far-from-equilibrium and rapidly expanding systems. These findings underscore the importance of non-perturbative contributions and the necessity of approaches that transcend the classic Chapman–Enskog paradigm.

1. Classical Chapman–Enskog Expansion and Hydrodynamic Gradient Series

In its original, non-relativistic context, the Chapman–Enskog expansion treats the Boltzmann equation by assuming the distribution function can be written as an asymptotic series in powers of a small parameter (typically the Knudsen number, $K_N \sim \lambda / L$). For relativistic kinetic theory, this expansion is frequently formulated for moments of the distribution function $M_{n,\ell}$: $$M_{n,\ell}(\hat\tau) = \sum_{p=0}^\infty \frac{\alpha_p^{(n,\ell)}}{\hat\tau^p},$$ where $\hat\tau \equiv \tau/\tau_R$, with $\tau$ the macroscopic proper time and $\tau_R$ the microscopic relaxation time as specified in the relaxation time approximation (RTA) of the Boltzmann equation. The coefficients $\alpha_p^{(n,\ell)}$ are determined recursively.

Hydrodynamics emerges as an expansion in gradients, with each order representing corrections due to deviations from local equilibrium. In relativistic settings with symmetry reductions (e.g., Bjorken flow geometry), the expansion parameter becomes parametrically large at early times, making the treatment of large gradients essential.

2. Divergence of the Chapman–Enskog Expansion in Relativistic Kinetic Theory

Studies of the Chapman–Enskog series in the context of longitudinally expanding (Bjorken) relativistic systems, particularly for a massless gas in the RTA, demonstrate that the expansion has zero radius of convergence (Noronha et al., 2017, Denicol et al., 2016). The large-order coefficients exhibit factorial growth: $\alpha_p^{(n,\ell)} \sim p!$ as $p \to \infty$, confirmed via both recursive analytical relations and explicit numerical calculation. This divergence stems intrinsically from the equations of motion and is not an artifact of the approximation or the kinetic model.

Mathematically, if the series converged for any finite $K_N$, it would have to yield a meaningful sum for small complex $K_N$; however, the underlying kinetic equation yields non-analyticities and is even ill-defined for negative relaxation time, invalidating such convergence (paralleling Dyson’s argument in quantum field theory).

3. Physical Origin: Non-hydrodynamic Modes and Non-perturbative Contributions

The divergence of the Chapman–Enskog series is fundamentally tied to the existence of non-hydrodynamic modes—transient excitations of the kinetic system that decay on microscopic timescales and are not captured in any finite-order gradient (hydrodynamic) expansion. These modes contribute terms such as

$\sim e^{-1/K_N}$

which are non-perturbative in $K_N$ and remain invisible to all orders of the Chapman–Enskog (gradient) series.

In the analysis of the moments’ evolution, terms analytic in $1/\hat\tau$ correspond to the Chapman–Enskog sector, but full solutions involve additional pieces (e.g., $e^{-\hat\tau}$) that encode initial conditions and non-hydrodynamic relaxation—these rapidly decaying contributions are essential for the accurate description of far-from-equilibrium and early-time physics.

4. Generalized Non-perturbative Expansions Beyond Chapman–Enskog

To resolve the limitations of the Chapman–Enskog approach, a generalized expansion was proposed (Denicol et al., 2016) where the coefficients themselves are made Knudsen-number dependent: $$M_{n,\ell}(\hat\tau) = \sum_{p=0}^{\infty} \frac{\beta_p^{(n,\ell)}(\hat\tau)}{\hat\tau^p}$$ Here, $\beta_p^{(n,\ell)}(\hat\tau)$ are determined by differential equations, not simple recursion, allowing the resummation of certain non-perturbative effects. The solutions for these coefficients manifest explicit contributions of the form $\exp(-j\hat\tau) \sim [\exp(-1/K_N)]^j$, which yield exponentially suppressed, non-analytic terms in the expansion parameter.

Crucially, for suitable initial conditions (such as equilibrium initial data), these generalized expansions rapidly converge to the exact solution of the Boltzmann equation, accurately reproducing both transient and late-time behavior. The approach can absorb the physical influence of the non-hydrodynamic sector and captures the full dynamical evolution, including far-from-equilibrium physics inaccessible to the Chapman–Enskog series.

The table below summarizes the properties of the two expansions:

Expansion Type	Structure	Non-perturbative Physics	Convergent?	Valid for Large Gradients
Chapman–Enskog (CE)	Power series in $K_N$	No	No	No
Generalized (proposed)	Series with $\beta_p(\hat\tau)$ terms	Yes	Yes	Yes

5. Implications for Relativistic Fluid Dynamics and Modeling

The principal consequence of the Chapman–Enskog divergence is the fundamental limitation of gradient expansions for relativistic fluids in scenarios with strong gradients or rapid expansion, as encountered in heavy-ion collisions or early-universe cosmology. In such regimes, classical hydrodynamics (understood as a perturbation series in gradients) is, at best, an asymptotic approximation, and even optimal truncation cannot reconstruct the full behavior of the system.

The findings imply:

• Hydrodynamic descriptions derived from gradient expansions miss crucial non-hydrodynamic (transient) physics, particularly relevant at early times or in small systems.
• For accurate modeling—especially in the context of emergent hydrodynamics in extreme conditions—it is necessary to incorporate non-perturbative terms, either via resummation techniques or via generalized expansions that retain full Knudsen number dependence.
• Approaches such as the generalized expansion (Denicol et al., 2016) or resurgent asymptotics provide paths forward, and motivate the development of frameworks that can seamlessly interpolate between kinetic and hydrodynamic regimes, transparently encoding the impact of transient non-hydrodynamic modes.

This recognition marks a significant conceptual shift in the interpretation of how and when hydrodynamics can emerge from microscopic dynamics in strongly non-equilibrium, relativistic systems.

6. Theoretical Significance and Future Directions

The identification of Chapman–Enskog series divergence in relativistic kinetic theory establishes for the first time that hydrodynamics, taken as a strict series in gradients, lacks general validity in highly dynamical relativistic systems. The precise mapping of the divergence to non-hydrodynamic modes builds a bridge to fields such as resurgent analysis and non-perturbative mathematical physics, suggesting that the macroscopic behavior of complex fluids is governed by a richer structure than previously appreciated.

Consequently, future research will likely emphasize:

• The construction and use of asymptotic and resurgent series that incorporate both hydrodynamic and non-hydrodynamic (e.g., instanton-like, exponentially suppressed) contributions.
• Non-perturbative matching between kinetic theory and effective fluid models, particularly for far-from-equilibrium or rapidly evolving systems.
• The application of these advanced expansions to data modeling in heavy-ion collisions and related areas, where thermalization and hydrodynamization times are key and where deviations from simple fluid dynamics are experimentally accessible.

The lessons from the divergence of the Chapman–Enskog expansion fundamentally alter the landscape of kinetic-to-fluid modeling and set the stage for the next generation of theoretical and phenomenological analyses in high-energy and relativistic fluid dynamics.