The onset of fluid-dynamical behavior in relativistic kinetic theory

In this proceedings we discuss recent findings regarding the large order behavior of the Chapman-Enskog expansion in relativistic kinetic theory. It is shown that this series in powers of the Knudsen number has zero radius of convergence in the case of a Bjorken expanding fluid described by the Boltzmann equation in the relaxation time approximation. This divergence stems from the presence of non-hydrodynamic modes, which give non-perturbative contributions to the Knudsen series.


Introduction
Relativistic hydrodynamics is the main tool used in the description of the spacetime evolution of the quark-gluon plasma formed in ultrarelativistic heavy ion collisions [1].Following the seminal work by Hilbert [2], Chapman and Enskog [3] in the non-relativistic limit, hydrodynamics has since then been understood [4] as an effective theory whose accuracy is controlled by the Knudsen number K N ∼ λ/L, a ratio between a microscopic scale λ (e.g., the mean free path in gases) and a macroscopic scale that characterizes the spatial gradients of hydrodynamic fields such as the temperature T and flow velocity u µ .Fluid dynamics is expected to provide a good description of the system's evolution when there is a large separation of scales such that K N 1.In this case, one expects that the equations of hydrodynamics may be organized order by order in powers of K N : at 0th-order one finds ideal fluid dynamics while at 1st order in K N one obtains the relativistic version of Navier-Stokes equations [4].In principle, one may continue this expansion (though in practice this is very hard to pursue in the absence of powerful symmetry constraints such as Weyl invariance [5]), with the hope that it provides a better description of the fluid.
The large spatial gradients expected to occur in the early stages of the quark-gluon plasma formed in heavy ion collisions [6,7,8], and the collective behavior present in small collision systems such as pA [9] where such a large separation of scales may not take place, motivate one to understand how relativistic hydrodynamic behavior emerges as an effective description of rapidly evolving kinetic systems [10,11].
In this proceedings we tackle this question using a simple "toy model" of the quark-gluon plasma defined within kinetic theory and study the large order behavior of the Chapman-Enskog gradient series.

Kinetic model
We consider a longitudinally expanding system of massless particles undergoing Bjorken flow [12] in Minkowski spacetime described using the coordinates x µ = (τ, x, y, η) with τ = √ t 2 − z 2 > 0 and η = tanh −1 (z/t).In this case all the quantities depend only on τ and, even though in these coordinates the system is homogeneous and u µ = (1, 0, 0, 0), the fluid experiences a nonzero expansion rate θ ≡ ∂ µ u µ = 1/τ.We employ the Boltzmann equation in the relaxation time approximation (RTA) [10,11,13] where is the relaxation time (assumed to be constant), and f eq = exp (−k 0 /T ) is the local equilibrium distribution function.We use the Landau frame [4], the system's energy density is ε = 3T 4 /π 2 whereas the pressure P = ε/3.The dynamics of the gas is described by Boltzmann RTA equation above supplemented by the equation for the temperature where π ≡ π η η is a component of the shear stress tensor of the fluid defined by the following moment of the distribution function π η η = This type of kinetic system has been previously studied in [14] and, more recently, in [15,16].
Here we solve ( 1) and ( 2) using the method of moments [17].In this formalism, the underlying kinetic dynamics of the Boltzmann equation is described by an infinite set of moments of f k that obey coupled differential equations.Given the symmetries of Bjorken flow [11], we first define the following moments whose exact evolution equations can be found directly from (1) where A nice feature of the method of moments is that one may explore different truncations of the infinite set of moments to systematically improve the description of the fluid [17] and directly reconstruct the solution f k of the kinetic model [18,19].To better understand how these moments deviate from their equilibrium values, we further define the dimensionless moments whose equations of motion are where M 1,1 = −π/P.This hierarchy of nonlinear equations for the moments may then be numerically solved to determine how the kinetic model evolves in τ.The powerful constraints from Bjorken symmetry, and the simplification made in the collision kernel by employing the RTA, made it possible to determine this general set of equations for arbitrary n and .

Divergence of the Chapman-Enskog expansion
The gradient expansion procedure, implemented via the Chapman-Enskog approach, consists in expanding the solution of the Boltzmann equation order by order in powers of the Knudsen number, which for the Bjorken expanding gas considered here corresponds to a series in powers of K N ∼ τ R /τ when τ/τ R 1.Given that f k may be reconstructed using linear combinations of M n, , here we perform the gradient expansion directly in these variables.Therefore, we write where τ = τ/τ R .The series for each moment is characterized by the dimensionless coefficients α (n, ) p , which do not depend on τ, and can be determined order by order by the following algebraic equation for m > 1 while α (n, ) 0 = 0 and α (n, ) A quick look at first term on the right-hand side of the equation above already suggests that α (1, )   m ≈ m!, independently of the value of .This can be confirmed by explicit calculation, as depicted in Fig. (1), which shows that the Chapman-Enskog expansion in this case indeed diverges.Other examples where the gradient expansion was found to diverge, though in strongly coupled plasmas that do not admit a kinetic description, can be found in [20,21].In hindsight, a simpler argument may be used to show that the gradient expansion considered here diverges.Consider the expansion around K N → 0. If such an expansion has a nonzero radius of convergence, this means that the series converges in an open region centered at K N = 0 in the "complex" K N = τ R /τ plane.However, it is clear from (1) that the solution is not well defined when τ R → −τ R or, in other words, when K N → −K N for any nonzero τ R .This indicates that the series cannot be well behaved when K N → 0, having zero radius of convergence (this is similar to Dyson's classical argument for the divergence of perturbative expansions in quantum field theory [22]).
In Ref. [23] a new type of expansion was proposed to replace the Chapman-Enskog divergent series.This new method involves a generalization of the standard series by simply considering that the coefficients of the expansion may possess a nontrivial τ (or K N ) dependence.More specifically, one assumes (10) and determines the differential equations obeyed by the coefficients β (n, ) p (τ) extracted order by order in the expansion.While β (n, ) p (τ) asymptotes to α (n, ) p at large τ, one can show that β (n, ) p (τ) obeys nonlinear relaxation type equations and, thus, the solutions contain terms such as e −τ ∼ e −1/K N , which display an essential singularity in K N → 0. These terms carry the information about the initial condition and they describe the contribution to the dynamics from non-hydrodynamic modes [24], which are not contained in the original Chapman-Enskog expansion.The generalized equations were numerically solved in [23] and compared to the exact solution of the Boltzmann equation [15,16] and an excellent agreement was found already at the lowest orders in the expansion.

Conclusions
In this work we briefly discussed the recent progress towards understanding the emergence of fluid dynamic behavior in rapidly evolving kinetic systems.We focused on the simple example of a Bjorken expanding fluid described by the RTA Boltzmann equation.In this case the Chapman-Enskog series, the well-known expansion defined by powers of the Knudsen number, was shown to diverge.A simple argument involving the behavior of the series in the complex K N plane was given to explain why such a series has zero radius of convergence.A generalized series, proposed in [23], reveals that there are non-perturbative contributions to the series, corresponding to non-hydrodynamic degrees of freedom, which cannot be expanded in powers of K N .It would be interesting to consider more realistic collision kernels and investigate the fate of the Chapman-Enskog series and the role played by non-hydrodynamic modes in this case.