The onset of fluid-dynamical behavior in relativistic kinetic theory

TL;DR

This paper investigates the divergence of the Chapman-Enskog expansion in relativistic kinetic theory, revealing that non-hydrodynamic modes cause the series to have zero radius of convergence in a Bjorken expanding fluid.

Contribution

It demonstrates the non-convergence of the Knudsen series due to non-hydrodynamic modes in relativistic kinetic theory.

Findings

The Chapman-Enskog series diverges for a Bjorken expanding fluid.

Non-hydrodynamic modes cause non-perturbative contributions.

The series has zero radius of convergence.

Abstract

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.

Figures

Equations

$${\partial }_{\tau }{f}_{\mathbf{k}}=-\frac{1}{{\tau }_R}\left(f_{\mathbf{k}}-f_{\mathrm{eq}}\right),$$

$$\frac{{\partial }_{\tau }T}{T}+\frac{1}{3\tau }-\frac{\pi }{12\tau }=0,$$

$${\rho }_{n,\ell }=\int \frac{d^3\mathbf{k}}{(2\pi {)}^3\tau }{\left(k^0\right)}^n{\left(\frac{k_{\eta }}{k^0\tau }\right)}^{2\ell }{f}_{\mathbf{k}},$$

$${\partial }_{\tau }{\rho }_{n,\ell }+\frac{1+2\ell }{\tau }{\rho }_{n,\ell }+\frac{n-2\ell }{\tau }{\rho }_{n,\ell +1}=-\frac{1}{{\tau }_R}\left({\rho }_{n,\ell }-{\rho }_{n,\ell }^{\mathrm{eq}}\right),$$

$${\rho }_{n,\ell }^{\mathrm{eq}}=\frac{\left(n+2\right)!}{2\ell +1}\frac{T^{n+3}}{2{\pi }^2}.$$

$$M_{n,\ell }(\tau )\equiv \frac{{\rho }_{n,\ell }-{\rho }_{n,\ell }^{\mathrm{eq}}}{{\rho }_{n,\ell }^{\mathrm{eq}}}$$

$${\partial }_{\tau }{M}_{n,\ell }+\frac{1}{{\tau }_R}{M}_{n,\ell }+\frac{6\ell -n}{3\tau }{M}_{n,\ell }-\frac{n+3}{12\tau }{M}_{1,1}\left(1+M_{n,\ell }\right)+\frac{1}{\tau }\frac{\left(n-2\ell \right)\left(1+2\ell \right)}{2\ell +3}{M}_{n,\ell +1}=-\frac{1}{\tau }\frac{4\ell \left(n+3\right)}{3\left(2\ell +3\right)},$$

$$M_{n,\ell }(\hat{\tau })=\sum _{p=0}^{\infty }\frac{{\alpha }_p^{\left(n,\ell \right)}}{{\hat{\tau }}^p},$$

$${\alpha }_{m+1}^{\left(n,\ell \right)}=-\frac{6\ell -n-3m}{3}{\alpha }_m^{\left(n,\ell \right)}+\frac{n+3}{12}{\alpha }_m^{\left(1,1\right)}-\frac{\left(n-2\ell \right)\left(1+2\ell \right)}{2\ell +3}{\alpha }_m^{\left(n,\ell +1\right)}+\frac{n+3}{12}\sum _{p=0}^m{\alpha }_p^{\left(1,1\right)}{\alpha }_{m-p}^{\left(n,\ell \right)},$$

$$M_{n,\ell }\left(\hat{\tau }\right)=\sum _{p=0}^{\infty }\frac{{\beta }_p^{\left(n,\ell \right)}\left(\hat{\tau }\right)}{{\hat{\tau }}^p}$$

Full text

1 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 $K_{N}\sim\lambda/L$, a ratio between a microscopic scale $\lambda$ (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 $u_{\mu}$. Fluid dynamics is expected to provide a good description of the system’s evolution when there is a large separation of scales such that $K_{N}\ll 1$. 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.

2 Kinetic model

We consider a longitudinally expanding system of massless particles undergoing Bjorken flow [12] in Minkowski spacetime described using the coordinates $x^{\mu}=(\tau,x,y,\eta)$ with $\tau=\sqrt{t^{2}-z^{2}}>0$ and $\eta=\tanh^{-1}(z/t)$. In this case all the quantities depend only on $\tau$ and, even though in these coordinates the system is homogeneous and $u^{\mu}=\left(1,0,0,0\right)$, the fluid experiences a nonzero expansion rate $\theta\equiv\partial_{\mu}u^{\mu}=1/\tau$. We employ the Boltzmann equation in the relaxation time approximation (RTA) [10, 11, 13]

[TABLE]

where $f_{\mathbf{k}}\equiv f\left(\tau,k_{\eta},k_{0}\right)$ is the single particle distribution function, the 4-momentum is $k_{\mu}=(k_{0},\mathbf{k})$ and $k_{0}=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{\eta}^{2}/\tau^{2}}$, $\tau_{R}$ is the relaxation time (assumed to be constant), and $f_{\mathrm{eq}}=\exp\left(-k_{0}/T\right)$ is the local equilibrium distribution function. We use the Landau frame [4], the system’s energy density is $\varepsilon=3T^{4}/\pi^{2}$ whereas the pressure $P=\varepsilon/3$. The dynamics of the gas is described by Boltzmann RTA equation above supplemented by the equation for the temperature

[TABLE]

where $\pi\equiv\pi^{\eta}_{\eta}$ is a component of the shear stress tensor of the fluid defined by the following moment of the distribution function $\pi_{\eta}^{\eta}=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3}\tau}k_{0}\left[\frac{1}{3}-\left(\frac{k_{\eta}}{k_{0}\tau}\right)^{2}\right]f_{\mathbf{k}}$. 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 $f_{\mathbf{k}}$ that obey coupled differential equations. Given the symmetries of Bjorken flow [11], we first define the following moments

[TABLE]

whose exact evolution equations can be found directly from (1)

[TABLE]

where

[TABLE]

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 $f_{\mathbf{k}}$ of the kinetic model [18, 19]. To better understand how these moments deviate from their equilibrium values, we further define the dimensionless moments

[TABLE]

whose equations of motion are

[TABLE]

where $M_{1,1}=-\pi/P$. This hierarchy of nonlinear equations for the moments may then be numerically solved to determine how the kinetic model evolves in $\tau$. 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 $\ell$.

3 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 $K_{N}\sim\tau_{R}/\tau$ when $\tau/\tau_{R}\gg 1$. Given that $f_{\mathbf{k}}$ may be reconstructed using linear combinations of $M_{n,\ell}$, here we perform the gradient expansion directly in these variables. Therefore, we write

[TABLE]

where $\hat{\tau}=\tau/\tau_{R}$. The series for each moment is characterized by the dimensionless coefficients $\alpha_{p}^{\left(n,\ell\right)}$, which do not depend on $\hat{\tau}$, and can be determined order by order by the following algebraic equation for $m>1$

[TABLE]

while $\alpha_{0}^{\left(n,\ell\right)}=0$ and $\alpha_{1}^{\left(n,\ell\right)}=-\frac{4\ell\left(n+3\right)}{3\left(2\ell+3\right)}$. A quick look at first term on the right-hand side of the equation above already suggests that $\alpha_{m}^{\left(1,\ell\right)}\approx m!$, independently of the value of $\ell$. 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 $K_{N}\to 0$. 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” $K_{N}=\tau_{R}/\tau$ plane. However, it is clear from (1) that the solution is not well defined when $\tau_{R}\to-\tau_{R}$ or, in other words, when $K_{N}\to-K_{N}$ for any nonzero $\tau_{R}$. This indicates that the series cannot be well behaved when $K_{N}\to 0$, 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 $\hat{\tau}$ (or $K_{N}$) dependence. More specifically, one assumes

[TABLE]

and determines the differential equations obeyed by the coefficients $\beta_{p}^{\left(n,\ell\right)}\left(\hat{\tau}\right)$ extracted order by order in the expansion. While $\beta_{p}^{\left(n,\ell\right)}\left(\hat{\tau}\right)$ asymptotes to $\alpha_{p}^{(n,\ell)}$ at large $\hat{\tau}$, one can show that $\beta_{p}^{\left(n,\ell\right)}\left(\hat{\tau}\right)$ obeys nonlinear relaxation type equations and, thus, the solutions contain terms such as $e^{-\hat{\tau}}\sim e^{-1/K_{N}}$, which display an essential singularity in $K_{N}\to 0$. 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.

4 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.

Acknowledgements

J. N. thanks FAPESP and CNPq for support.