Causal first-order hydrodynamics from kinetic theory and holography

The distribution function (12) with arbitrary non-constant ${\bm{\beta}}_{\mu}(x)$ and ${\bm{\alpha}}(x)$ does not satisfy the Boltzmann equation. Approximate solutions to the Boltzmann equation may be constructed in the derivative expansion. To do so, we write the Boltzmann equation as

with an auxiliary parameter $\varepsilon$ (to be set to one at the end), and aim to find the solution as a power series in $\varepsilon$:

This is sometimes called the Hilbert expansion Cercignani-Kremer . The energy-momentum tensor (7) and the current (6) then take the form

These expansions for $T^{\mu\nu}$ and $J^{\mu}$ are however not necessarily the derivative expansions of the hydrodynamic constitutive relations. In order to talk about the constitutive relations, we need the hydrodynamic variables $T$, $u^{\lambda}$, and $\mu$, or equivalently $\beta^{\lambda}=u^{\lambda}/T$ and $\alpha=\mu/T$. In kinetic theory, the hydrodynamic variables arise as arbitrary functions of $x$ (or “integration constants” in momentum space) in the solutions of the Boltzmann equation. The $x$-dependence of these functions is then fixed by the consistency conditions for the Boltzmann equation at each order in the expansion. These consistency conditions are exactly the hydrodynamic conservation laws. Each order in the $\varepsilon$-expansion generates its own arbitrary functions, namely

where the leading-order hydrodynamic variables $\beta^{\mu}_{(0)}={\bm{\beta}}^{\mu}$ and $\alpha_{(0)}={\bm{\alpha}}$ are the free functions that appear in the equilibrium distribution (12), and the corrections $\beta^{\mu}_{(n)}$ and $\alpha_{(n)}$ appear as undetermined functions in the solution for $f^{(n)}(x,p)$. Connecting to the earlier discussion of the derivative expansion in hydrodynamics, we expect to find in the Hilbert expansion

etc., with analogous expressions for the current $J^{\mu}$. We expect the conservation equations to hold at each order in the expansion,

This is indeed what happens.

At first order in the expansion we have

We expand the collision operator to linear order in $\varepsilon$. Denoting $\phi_{p}(x)\equiv f^{(1)}/{\bf f}(1\pm{\bf f})$, the Boltzmann equation (28) becomes

where ${\cal L}$ is the linearized collision operator. Its explicit form depends on the details of the full collision operator ${\cal C}$, and in general one has

with arbitrary $a(x)$, $b_{\mu}(x)$. The existence of these zero modes is a consequence of ${\cal C}[{\bf f}]=0$, reflecting the microscopic conservation laws of energy, momentum, and particle number. For 2-to-2 elastic collisions (9), the explicit form is

Given two functions $g_{p}\equiv g(x,p)$, $h_{p}\equiv h(x,p)$, the linearized collision operator satisfies

If we take $h_{p}=a(x)+b_{\mu}(x)p^{\mu}$ with arbitrary $a(x)$, $b_{\mu}(x)$, take $g_{p}=\phi_{p}$, and use the linearized Boltzmann equation (29), we immediately find

In other words, at first order in the $\varepsilon$-expansion, the functions ${\bm{\beta}}_{\mu}(x)$ and ${\bm{\alpha}}(x)$ that appear in the local-equilibrium distribution function (12) must obey

These are the perfect-fluid conservation equations. The above ${\bf T}^{\mu\nu}$ and ${\bf J}^{\mu}$ can be written as

where ${\bm{\Delta}}^{\mu\nu}\equiv g^{\mu\nu}+{\bm{u}}^{\mu}{\bm{u}}^{\nu}$, and the coefficients are

corresponding to the ideal-gas particle number density, energy density, and pressure. In the notation of eq. (26a), ${\bf T}^{\mu\nu}=T^{\mu\nu}_{0}$, ${\bf J}^{\mu}=J^{\mu}_{0}$. The conservation equations (34) are

where all quantities are of order $O(\varepsilon^{0})$. The dot stands for $u^{\lambda}\partial_{\lambda}$, and $\partial_{\mu}^{\perp}\equiv\Delta_{\mu\nu}\partial^{\nu}$. The vector conservation equation can be rewritten as $\dot{u}_{\mu}+\partial_{\mu}^{\perp}T/T+\frac{nT}{\epsilon+p}\partial_{\mu}^{\perp}\alpha=0$.

Another way to arrive at eqs. (34) is to note that in the Boltzmann equation (29) the linearized collision operator ${\cal L}$ has zero modes, and therefore is not invertible. In general, the linear equation $H={\cal L}[\phi]$ can only be solved for $\phi$ if the left-hand side $H$ is orthogonal to the zero-modes of the operator ${\cal L}$ in the right-hand side. For the linearized Boltzmann equation, the zero-modes are 1 and $p^{\lambda}$, and the consistency conditions amount to

with $H=p^{\mu}\partial_{\mu}{\bf f}_{p}/{\bf f}_{p}$. This again gives eq. (34). In other words, the equations of $0^{\rm th}$-order (perfect-fluid) hydrodynamics arise as constraint equations at $1^{\rm st}$-order in the expansion.

At the first order in the expansion we have to solve eq. (29) which we write as $H={\cal L}[\phi]$, with $H=p^{\mu}\partial_{\mu}{\bf f}_{p}/{\bf f}_{p}$. The solution can be written as $\phi_{p}(x)=a(x)+b_{\mu}(x)p^{\mu}+\Phi(x,p)$ where $a(x)$ and $b_{\mu}(x)$ are arbitrary, and the inhomogeneous solution $\Phi$ satisfies $\Phi|_{H\to 0}=0$. Remembering the definition $\phi_{p}=f^{(1)}/{\bf f}(1\pm{\bf f})$, the distribution function to first order in $\varepsilon$ is

From here, it is clear that the functions $b_{\mu}(x)$ and $a(x)$ can be understood as $O(\varepsilon)$ redefinitions of the functions $\beta_{\mu}(x)$ and $\alpha(x)$ which sit in ${\bf f}_{p}$. Indeed, for $\beta_{\mu}(x)=\beta_{\mu}^{(0)}(x)+\varepsilon\beta_{\mu}^{(1)}(x)$, $\alpha(x)=\alpha^{(0)}(x)+\varepsilon\alpha^{(1)}(x)$ we have in terms of ${\bf f}_{p}^{(0)}\equiv{\bf f}_{p}(\beta^{(0)},\alpha^{(0)})$:

Alternatively, when we evaluate the energy-momentum tensor $T^{\mu\nu}$ and the current $J^{\mu}$ using the distribution function (41), the only effect of the “integration constants” $b_{\mu}(x)$ and $a(x)$ is a linearized redefinition of ${\bm{\beta}}_{\mu}(x)$ and ${\bm{\alpha}}(x)$ in the perfect-fluid ${\bf T}^{\mu\nu}$ and ${\bf J}^{\mu}$. We thus identify the correction to the hydrodynamic variables in (24) and (25) as $\beta^{\mu}_{(1)}=b^{\mu}$, $\alpha_{(1)}=a$, keeping in mind that $b^{\mu}$ and $a$ are arbitrary, hence the fluid velocity, temperature, and the chemical potential at $O(\varepsilon)$ are intrinsically ambiguous quantities. Explicitly, the function (42) leads to the shift of ${\bm{T}}=T_{(0)}$, ${\bm{\alpha}}=\alpha_{(0)}$, and ${\bm{u}}^{\mu}=u^{\mu}_{(0)}$ in the perfect-fluid expressions (35) by

The resulting energy-momentum tensor and the current evaluated with the first-order distribution function (41) are:

where the corrections $T^{\mu\nu}_{1}$, $J^{\mu}_{1}$ are due to the inhomogeneous solution $\Phi(x,p)$ in eq. (41).

The hard part is to find the inhomogeneous solution $\Phi$ which satisfies

In general, for any timelike $\beta^{\mu}(x)$ and $\alpha(x)$ we have the identity

where $\sigma^{\mu\nu}[u]=(\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\nu\alpha}\Delta^{\mu\beta}-\frac{2}{d}\Delta^{\mu\nu}\Delta^{\alpha\beta})\partial_{\alpha}u_{\beta}$ is the shear tensor, $d$ is the number of space dimensions, $p^{\mu}_{\perp}\equiv\Delta^{\mu\lambda}p_{\lambda}$, and the hydrodynamic variables are $T=\frac{1}{\sqrt{-{\beta}^{2}}}$, $u^{\mu}=\frac{{\beta}^{\mu}}{\sqrt{-{\beta}^{2}}}$.

Based on eqs. (48) and (2), it is difficult to guess how the solution $\Phi$ depends on the derivatives of the hydrodynamic variables, e.g. the relative contributions of $\dot{\bm{T}}$ and $\partial{\cdot}{\bm{u}}$ to $\Phi$. The standard approach is to use the constraints (39) to eliminate the zeroth-order $\dot{\bm{u}}_{\mu}$, $\dot{\bm{T}}$, and $\dot{\bm{\alpha}}$ in terms of the zeroth-order $\partial{\cdot}{\bm{u}}$, $\partial_{\mu}^{\perp}{\bm{\alpha}}$, $\partial_{\mu}^{\perp}{\bm{T}}$. Then the linearized Boltzmann equation (48) becomes

where the functions $F_{\sigma}$, $F_{u}$, $F_{\alpha}$ depend on ${\bm{T}}$, ${\bm{\alpha}}$, $(p{\cdot}{\bm{u}})$, and are fixed by the ideal-gas equation of state. In particular, $F_{\sigma}=\frac{1}{2{\bm{T}}}$, $F_{\alpha}=1+\frac{{\bm{n}}}{{\bm{\epsilon}}+{\bm{p}}}(p{\cdot}{\bm{u}})$. For massless particles, $F_{u}$ would vanish (at order $\varepsilon$), as a consequence of scale-invariant thermodynamics, ${\bm{p}}({\bm{T}},{\bm{\alpha}})={\bm{T}}^{d+1}g({\bm{\alpha}})$. Note that $\partial_{\mu}^{\perp}{\bm{T}}$ does not appear in the left-hand side of eq. (50), once $\dot{\bm{u}}_{\mu}$ has been eliminated. Now from eq. (50), the unknown $\Phi$ can be parametrized as

where the coefficients $K_{\eta}$, $K_{\zeta}$, $K_{\alpha}$ in general depend on ${\bm{T}}$, ${\bm{\alpha}}$, and $p{\cdot}{\bm{u}}$, and can in principle be found by solving the linearized Boltzmann equation (50). Let us write the first-order distribution function in terms of the first-order hydrodynamic variables $\beta_{\mu}=\beta_{\mu}^{(0)}+\varepsilon\beta_{\mu}^{(1)}$, $\alpha=\alpha^{(0)}+\varepsilon\alpha^{(1)}$,

The first term has both $O(1)$ and $O(\varepsilon)$ contributions. In the second term, the $O(\varepsilon)$ contributions in $u$, $T$, and $\alpha$ give $O(\varepsilon^{2})$ contributions to $f_{p}$ which can be neglected at first order. We can now use the distribution function (52) to evaluate the corrections $T^{\mu\nu}_{1}$, $J^{\mu}_{1}$ in (46), (47).

Beyond leading (perfect-fluid) order, the energy-momentum tensor and the current will no longer have the simple form (35). For any normalized timelike vector $u^{\mu}$, the energy-momentum tensor and the current may be decomposed as Kovtun:2012rj

where ${{\cal Q}}{\cdot}u={{\cal J}}{\cdot}u={{\cal T}}{\cdot}u=0$, and ${\cal T}^{\mu\nu}$ is symmetric and traceless. These decompositions define ${\cal E}$, ${\cal P}$, ${\cal Q}$, ${\cal T}$, ${\cal N}$ and ${\cal J}$, for a given $u^{\mu}$. At first order in the $\varepsilon$-expansion, $u^{\mu}={\beta}^{\mu}/{\sqrt{-{\beta}^{2}}}$, where $\beta^{\mu}=\beta^{\mu}_{(0)}+\varepsilon\beta^{\mu}_{(1)}$, as in Eq. (45). Similarly, at first order $T=T_{(0)}+\varepsilon T_{(1)}$, and $\alpha=\alpha_{(0)}+\varepsilon\alpha_{(1)}$. The first-order corrections to $\beta^{\mu}$ and $\alpha$ are arbitrary, and one can always redefine $\beta^{\mu}_{(1)}\to\beta^{\mu}_{(1)}+b^{\prime\mu}$, $\alpha_{(1)}\to\alpha_{(1)}+a^{\prime}$. At zeroth order in the expansion, ${\cal E}={\bm{\epsilon}}+O(\varepsilon)$, ${\cal P}={\bm{p}}+O(\varepsilon)$, ${\cal N}={\bm{n}}+O(\varepsilon)$, while ${\cal Q}^{\mu}$, ${\cal T}^{\mu\nu}$ and ${\cal J}^{\mu}$ are $O(\varepsilon)$. Substituting the distribution function (52) into the general expressions (6), (7), we find the following coefficients of the decomposition (53), (54) in terms of first-order $u^{\mu}$, $T$, and $\alpha$:

Here $\epsilon$, $p$, and $n$ are functions of ($\varepsilon$-corrected) $T$ and $\alpha$. The angular brackets stand for $\langle\cdots\rangle=\int_{p}{\bf f}_{p}\cdots$. These are the constitutive relations for a viscous relativistic fluid at first order in the derivative expansion. The energy-momentum tensor and the current given by these constitutive relations (in terms of $O(\varepsilon)$-corrected hydrodynamic variables) must obey the standard conservation equations (11), which are true for any distribution function.

One might be tempted to ignore the “integration constants” $b_{\mu}^{\prime}(x)$ and $a^{\prime}(x)$ altogether. However, they have a simple physical meaning: the hydrodynamic variables $T$, $u^{\lambda}$, and $\alpha$ that appear in the $O(\varepsilon)$ (i.e. Navier-Stokes) hydrodynamic equations can differ from the hydrodynamic variables that appear in the distribution function (12) by derivative corrections, reflecting the ambiguity in what one chooses to mean by “fluid velocity”, “fluid temperature” and “fluid chemical potential” beyond the perfect-fluid approximation. The most general parametrization of such arbitrary one-derivative corrections is

with arbitrary coefficients $b_{n}(T,\alpha)$ and $a_{n}(T,\alpha)$. In relativistic hydrodynamics, one’s choice of a particular form of these derivative corrections is often called a choice of “frame”.

The parametrization (56) contains the most general one-derivative corrections with arbitrary coefficients $b_{n}$ and $a_{n}$. One could further demand that the redefinitions of $T$, $\alpha$ and $u^{\lambda}$ (provided by $b^{\prime}_{\mu}$ and $a^{\prime}$) are such that they vanish in equilibrium, even when the fluid is subject to a static external gravitational field. In equilibrium, one can choose the fluid velocity as the normalized timelike Killing vector. In zero-derivative hydrodynamics (perfect fluids) this is manifested by eq. (13), however such a choice of the fluid velocity in equilibruim of course extends beyond zero-derivative hydrodynamics, and has non-trivial consequences Jensen:2012jh . The Killing equation (13) for $\beta_{\mu}$ implies $\dot{u}_{\mu}+\partial_{\mu}^{\perp}T/T=0$, even though $\dot{u}_{\mu}$ and $\partial_{\mu}^{\perp}T$ may separately be non-zero in external gravitational field. Thus demanding that (56a), (56b) vanish in equilibrium, we have $b_{4}=b_{5}$. Such a choice was called a “thermodynamic frame” in ref. Jensen:2012jh . The choice amounts to demanding that the hydrostatic limit of the constitutive relations (55) follows by varying the equilibrium grand canonical free energy with respect to the external metric (for $T^{\mu\nu}$), or with respect to the external gauge field (for $J^{\mu}$).

The popular frame adopted by Landau and Lifshitz LL6 is obtained in the following way. One chooses $b_{1}=b_{3}=b_{4}=b_{5}=a_{1}=a_{3}=0$, leaving one with