QCD Kinetic Theory Simulations

• QCD kinetic theory simulations are computational frameworks that employ the relativistic Boltzmann equation and gradient expansions to model nonequilibrium quark-gluon plasma dynamics.
• They integrate perturbative QCD techniques with advanced numerical algorithms, ensuring accurate conservation laws and reliable hydrodynamic transport coefficients.
• These simulations bridge early Glasma dynamics to hydrodynamization, offering quantitative insights into jet quenching, equilibration, and transport in high-energy nuclear collisions.

Quantum Chromodynamics (QCD) kinetic theory simulations constitute a foundational computational and conceptual tool for describing the nonequilibrium evolution, equilibration, and transport properties of the quark–gluon plasma (QGP) in high-energy nuclear collisions. These frameworks, grounded in the relativistic Boltzmann equation with QCD-specific collision kernels, enable the systematic paper of both macroscopic hydrodynamization and microscopic processes such as jet quenching, chemical and kinetic equilibration, and the early-time generation of electromagnetic probes. The field integrates perturbative techniques, advanced numerical algorithms, analytic gradient expansions, and machine learning, as well as quantum computational methods, to address the challenges of simulating the highly anisotropic, transient, and strongly-interacting matter produced in heavy-ion collisions.

1. Kinetic Theory Formalism and Hierarchy of Expansions

QCD kinetic theory is formulated within the weak-coupling regime ($g \ll 1$), where quasiparticles (quarks, gluons) with well-defined dispersion relations dominate. The evolution is governed by a covariant Boltzmann equation,

$2P^\mu \partial_\mu f^a(x, p) = -C[f],$

with $f^a(x, p)$ the phase-space distribution for species $a$, $P^\mu$ the on-shell momentum, and $C[f]$ the collision operator incorporating elastic (2↔2) and inelastic (1↔2) scatterings (0811.0729). Within gradient expansions, the distribution is expressed as $f = f_0 + f_1 + f_2 + \cdots$, with $f_0$ the equilibrium distribution, and subsequent terms $f_1, f_2$ encoding successively higher orders in flow gradients.

The operator $C[f]$ is linearized around equilibrium at first order (viscous shear corrections) and includes quadratic (e.g., $C_{11}[f_1]$) and plasma screening corrections ($C_{1;1}[f_1]$) at second order, as required for a consistent hydrodynamic expansion. The resulting framework is conformal and massless at leading order, yielding vanishing bulk viscosity and ensuring that transport is governed exclusively by shear-driven processes (0811.0729, Kurkela et al., 2018).

2. Hydrodynamic Transport Coefficients and Gradient Expansion

A principal outcome of QCD kinetic theory is the derivation of first- and second-order dissipative hydrodynamic coefficients. The stress-energy tensor correction to second order in gradients in conformal QCD reads

$$\Pi^{\mu\nu}_{2\textrm{ order}} = \eta \tau_\Pi [u^\alpha \partial_\alpha \sigma^{\mu\nu} + \sigma^{\mu\nu} (\partial \cdot u)/3] + \lambda_1 [\sigma^{\mu}_{\ \alpha} \sigma^{\nu\alpha} - (1/3)\Delta^{\mu\nu} \sigma_{\alpha\beta} \sigma^{\alpha\beta}] + \lambda_2 \left[\frac{1}{2}(\sigma^{\mu}_{\ \alpha} \Omega^{\nu\alpha} + \sigma^{\nu}_{\ \alpha} \Omega^{\mu\alpha}) - \frac{1}{3}\Delta^{\mu\nu} \sigma_{\alpha\beta} \Omega^{\alpha\beta}\right],$$

with $\eta$ the shear viscosity, $\tau_\Pi$ the relaxation time, and $\lambda_{1,2}$ nonlinear coefficients; $\Delta^{\mu\nu}$ is the projector orthogonal to $u^\mu$, and $\sigma^{\mu\nu}$ and $\Omega^{\mu\nu}$ the shear and vorticity tensors, respectively (0811.0729).

Explicit values for dimensionless ratios of second-order coefficients in weak-coupling QCD (and their relations, such as $\lambda_2 = -2\eta\tau_\Pi$) are calculated:

• $(\epsilon+P)\,\eta\tau_\Pi/\eta^2 \simeq 5.0-5.9$
• $(\epsilon+P)\,\lambda_1 /\eta^2 \simeq 4.1-5.2$
• $\kappa = 0$, $\lambda_3 = 0$ (to leading order in conformal kinetic theory).

Comparison with scalar and QED kinetic theories shows similar structure but numerically distinct ratios; weak-coupling QCD coefficients exceed those in strongly-coupled SYM by about a factor of two (0811.0729).

3. Conservation Laws, Discretization, and Algorithmic Implementations

Accurate QCD kinetic simulations must rigorously enforce number, energy, and momentum conservation, especially in discretized phase-space implementations. State-of-the-art numerical schemes utilize weighted-integral formalisms for phase-space (e.g., $n(i_p, j_\theta, k_\phi)$ as sums over basis functions), guaranteeing that integrals of the discretized distribution reconstruct continuum moments to machine precision (Du et al., 2020). Collision integrals for both elastic (2↔2) and inelastic (1↔2) processes are projected onto these weights with extensive Monte Carlo sampling, preserving conservation laws channel-by-channel.

Auxiliary terms, such as boost-invariant longitudinal expansion ($-(p_z/\tau)\partial_{p_z}$), are recast as effective collision-like operators for integration into adaptive time-stepping schemes. The adaptive choice of time steps is based on monitoring variations in phase-space moments, ensuring stability even as physical scales (e.g., soft screening masses, effective temperatures) evolve rapidly during thermalization.

4. Modeling Pre-Equilibrium and Hydrodynamization

Recent QCD kinetic theory has established powerful frameworks bridging the initial Glasma dynamics (from classical field theory) to hydrodynamics. This multistage evolution is achieved by propagating the energy-momentum and current perturbations, decomposed into background and fluctuations, via nonequilibrium Green's functions from early times ($\tau_{ekt}$) to hydrodynamization ($\tau_{hydro}$) (Kurkela et al., 2018).

Universal attractor behavior is observed for energy and pressure evolution: when time is measured in units of the kinetic relaxation time $\tau_R = (\eta/s)/T$, both the background and fluctuations follow a universal "hydrodynamic attractor". Initial spatial inhomogeneities are evolved through linear kinetic response functions, and the switching of evolution from kinetic theory to viscous hydrodynamics is achieved with minimal sensitivity to the precise matching time. This reduces longstanding uncertainties in setting initial conditions for hydrodynamics and directly constrains the admissible early-time energy densities and transport coefficients in high-energy nuclear collision simulations (Kurkela et al., 2018, Du et al., 2020).

5. Technical Assumptions, Screening, and Limitations

The reliability of kinetic theory simulations is subject to key technical and physical assumptions:

• Validity is restricted to weak coupling (absence of non-perturbative effects such as magnetic screening or strong turbulent fields).
• The quasiparticle picture is assumed, excluding regimes where colored collective excitations or infrared instabilities dominate.
• All leading-order processes (elastic $2\leftrightarrow2$, inelastic $1\leftrightarrow2$), including HTL-subtracted matrix elements, are required for quantitative fidelity.
• The collision kernel is truncated to dominant processes; sub-leading corrections or non-conformal effects (e.g. bulk viscosity near $T_c$) are not included at leading order (0811.0729, Du et al., 2020).
• The spatial and temporal gradients must be small compared to microscopic timescales to justify the gradient expansion underpinning hydrodynamics.

Kubo formulae, as cross-checked in the kinetic framework, provide independent evaluations of key coefficients (e.g., $\tau_\Pi$, $\kappa$), with agreement confirming that certain transport parameters (notably $\kappa$) vanish at leading order in QCD kinetic theory.

6. Application to QCD and Other Theories

Though most advanced for QCD, the formalism generalizes to gauge (QED), and scalar theories. Explicit numerical results, including for equilibrium and out-of-equilibrium transport ratios, are found to differ predominantly in normalization rather than form; e.g., scalar $\lambda\phi^4$ yields $(\epsilon + P)\eta\tau_\Pi/\eta^2 \simeq 6.1$, compared to QCD's $5.0-5.9$ (0811.0729).

In strongly-coupled $\mathcal{N}=4$ SYM (AdS/CFT), second-order coefficients are about half as large, indicating that the kinetic regime is less dissipative and, thus, hydrodynamizes more slowly. Realistic heavy-ion collision modeling often interpolates between these extremes, but the kinetic theory values provide robust microscopic input for the weak-coupling limit.

7. Phenomenological Impact and Simulation Practices

For practical hydrodynamic modeling in heavy-ion collisions, kinetic theory not only supplies accurate values for the relaxation time ($\eta\tau_\Pi$) and higher-order coefficients $\lambda_{1,2}$, but also underpins a quantitative description of how deviations from first-order (Navier–Stokes) hydrodynamics decay in time and space (0811.0729, Kurkela et al., 2018). This is essential for event-by-event simulations where gradients, anisotropies, and fluctuations are large, particularly close to the QCD phase transition.

While the kinetic theory predictions are strictly applicable only in the weakly-coupled, high-temperature, conformal regime, the dimensionless ratios computed remain broadly useful as both qualitative and semi-quantitative guides even away from these limits. Extensions to include mean field effects near $T_c$ (cf. EQPM models (Mitra et al., 2018)) or for systems with significant electromagnetic or chiral anomaly dynamics are active areas of research.

---

In summary, QCD kinetic theory simulations, anchored in systematic weak-coupling expansions and realized via advanced numerical algorithms, yield precise predictions for energy-momentum relaxation and the transport of QGP. The computed second-order hydrodynamic coefficients, conservation-preserving discretization schemes, and attractor-based frameworks for connecting nonequilibrium to hydrodynamic evolution are central to interpreting heavy-ion collision data and to the continued refinement of dynamical modeling of hot QCD matter.