# Non-Hydrodynamic Initial Conditions are Not Soon Forgotten

T.R. Kirkpatrick1, D. Belitz2,3 and J.R. Dorfman1 1 Institute for Physical
Science and Technology, University of Maryland, College Park, MD 20742, USA
2 Department of Physics and Institute for Fundamental Science, University of
Oregon, Eugene, OR 97403, USA
3 Materials Science Institute, University of Oregon, Eugene, OR 97403, USA

(August 27, 2024)

###### Abstract

Solutions to hydrodynamic equations, which are used for a vast variety of
physical problems, are assumed to be specified by boundary conditions and
initial conditions on the hydrodynamic variables only. Initial values of other
variables are assumed to be irrelevant for a hydrodynamic description. This
assumption is not correct because of the existence of long-time-tail effects
that are ubiquitous in systems governed by hydrodynamic equations. We
illustrate this breakdown of a hydrodynamic description by means of the simple
example of diffusion in a disordered electron system.

Hydrodynamic descriptions of matter have a long history, with classical fluids
the most obvious example Landau and Lifshitz (1987). Others include plasmas
Lifshitz and Pitaevskii (1981); Belmont et al. (2019), superfluids, spin
transport in magnets, and excitations in liquid crystals and in solids Forster
(1975); Martin et al. (1972). More recently, hydrodynamic descriptions have
been used to study active matter Marchetti et al. (2013), general relativity
Minwalla et al. (2011), supersymmetric field theories and quantum gravity
Kovtun and Yaffe (2003), quark-gluon plasmas Mazeliauskas and Berges (2019),
and the unusual properties of Weyl and Dirac metals and semi-metals Zaanen
(2016).

The basic assumption of any hydrodynamic theory is that the behavior of a
macroscopic system on length and time scales large compared to the microscopic
ones can be described in terms of a small number of ‘hydrodynamic’ variables,
while all other degrees of freedom are effectively integrated out. One of the
simplest hydrodynamic processes is diffusion, with applications in Physics,
Chemistry, Biology, and Engineering Mehrer (2007); Cussler (1997); Stein
(1986). It is phenomenologically described by Fick’s law, which states that
the current density $\bm{j}$ associated with a conserved density $n$ is
proportional to the gradient of $n$,
${\bm{j}}({\bm{x}},t)=-D\,{\bm{\nabla}}n({\bm{x}},t)$, with $D$ the diffusion
coefficient. Together with the continuity equation for $n$, this leads to the
diffusion equation

$\partial_{t}n({\bm{x}},t)-D\,{\bm{\nabla}}^{2}n({\bm{x}},t)=0\ .$ (1)

This is expected to be valid for times $t\gg t_{0}$ and wave numbers $q\ll
q_{0}$, with $t_{0}$ and $1/q_{0}$ the microscopic time and length scales that
depend on the nature of the diffusive process. In this description there is
only one hydrodynamic variable, viz., the density. With $n_{0}$ the constant
equilibrium density, Eq. (1) describes the relaxation of a density
perturbation $\delta n({\bm{x}},t)=n({\bm{x}},t)-n_{0}$. It is solved by a
spatial Fourier transform and a temporal Laplace transform Forster (1975); Lap
. The result is the well known diffusion pole in the complex-frequency ($z$)
plane,

$\delta n({\bm{q}},z)=\delta n({\bm{q}},t=0)\,i/[z+iDq^{2}s_{z}]$ (2a) with
${\bm{q}}$ the wave vector, and $s_{z}={\text{sgn\,}}{\text{Im}}\,(z)$. In the
time domain, this corresponds to exponential decay with a relaxation time that
diverges as $1/q^{2}$ for small ${\bm{q}}$, $\delta n({\bm{q}},t)=\delta
n({\bm{q}},t=0)\,e^{-Dq^{2}t}\ .$ (2b)

According to Eq. (2b), the initial value $\delta n({\bm{q}},t=0)$ completely
determines the relaxation for $t\gg t_{0}$ and $q\ll q_{0}$.

As we will show, the relaxation behavior described by the diffusion equation
is not complete. First, kinetic theory shows that $\delta n({\bm{q}},t)$ also
depends on initial conditions in higher angular-momentum channels; e.g., an
initial current. Within diffusion theory, these initial conditions are ‘non-
hydrodynamic’ since they involve quantities other than the density. As we will
show, solving the standard Boltzmann equation for the scattering of electrons
by static point-like impurities yields

$\delta n({\bm{q}},t\gg t_{0})=\left[\delta n({\bm{q}},t=0)+\delta
n_{\perp}({\bm{q}},t=0)\right]\,e^{-Dq^{2}t}\ .$ (3a) Here the diffusion
coefficient is $D=v_{\text{F}}^{2}\tau/3$, with $v_{\text{F}}$ the Fermi
velocity and $\tau$ the elastic mean-free time. In this case, the microscopic
time and wave number are $t_{0}=\tau$ and $q_{0}=1/v_{\text{F}}\tau$,
respectively. $\delta n_{\perp}({\bm{q}},t=0)$ is a set of non-hydrodynamic
initial conditions that are orthogonal in angular-momentum ($l$) space to the
density, with the current density ${\bm{j}}$ ($l=1$) the leading contribution,
$\delta
n_{\perp}({\bm{q}},t=0)\approx-i\tau{\bm{q}}\cdot{\bm{j}}({\bm{q}},t=0)\ .$
(3b) The density relaxation is still given by the solution of the diffusion
equation, Eq. (2b), if we replace $\delta n(t=0)$ by $\delta n$ taken at an
adjusted initial time $t_{s}$ defined by $\displaystyle\delta
n({\bm{q}},t_{s})$ $\displaystyle\equiv$ $\displaystyle\delta
n({\bm{q}},t=0)\,e^{-Dq^{2}t_{s}}$ (3c) $\displaystyle=$ $\displaystyle\delta
n({\bm{q}},t=0)+\delta n_{\perp}({\bm{q}},t=0)\ .$

The ‘slip time’ $t_{s}$ is the temporal analog of the ‘slip length’ used to
describe the flow of a fluid near a surface, i.e., the distance between the
actual surface and the fictitious one where hydrodynamic boundary conditions
can be used Dorfman et al. (2021); Nieuwoudt et al. (1984). If the initial
conditions are such that $\delta n$ and $\delta n_{\perp}$ are related by
Fick’s law, $\delta n_{\perp}({\bm{q}},t=0)\approx Dq^{2}\tau\delta
n({\bm{q}},t=0)$, then $|t_{s}|\approx\tau$, but generically this will not be
the case.

Second, the exponential decay predicted by Eq. (3a) is not the true asymptotic
long-time behavior: Correlation effects lead to a non-analytic frequency
dependence of the diffusion coefficient. As a function of time, this
corresponds to a power-law decay of $\delta n$. Such non-exponential ‘long-
time tails’ (LTTs) are not described by the Boltzmann equation Dorfman and
Cohen (1970); Ernst et al. (1970), yet are present in all systems described by
hydrodynamics Kovtun and Yaffe (2003); Marchetti et al. (2013); Dorfman et al.
(2021).

In this Letter we show that the combination of LTTs and non-hydrodynamic
initial conditions yields a long-time behavior of the density relaxation that
cannot be reconciled with the diffusion equation by adjusting the initial
condition as in Eq. (3c), even with a frequency-dependent diffusivity that
yields the correct LTT.

As an example, we consider the density relaxation of noninteracting conduction
electrons in a three-dimensional (3-d) system with weak quenched disorder. The
latter leads to LTTs known as weak-localization (WL) effects Lee and
Ramakrishnan (1985); AA_ , but in 3-d systems it does not lead to Anderson
localization; the system remains metallic. Let $\epsilon_{\text{F}}$ and
$k_{\text{F}}$ be the Fermi energy and wave number, respectively, and $m$ the
effective mass. The density of states per spin at the Fermi surface is
$N_{\text{F}}=k_{\text{F}}m/2\pi^{2}$, and the mean-free path is
$\ell=v_{\text{F}}\tau$. For weak disorder ($k_{\text{F}}\ell\gg 1$, or
$\epsilon_{\text{F}}\tau\gg 1$), and and $t\gg 1/Dq^{2}$, we find

$\displaystyle\delta n({\bm{q}},t)$ $\displaystyle=$ $\displaystyle\delta
n({\bm{q}},t=0)\left[\frac{3}{4\sqrt{\pi}}\,\frac{1}{k_{\text{F}}\ell}\,\frac{q/k_{\text{F}}}{(Dq^{2}t)^{3/2}}+O(\frac{1}{t^{5/2}})\right]$
(4a) $\displaystyle+\delta
n_{\perp}({\bm{q}},t=0)\,\frac{9}{8\sqrt{\pi}}\,\frac{1}{k_{\text{F}}\ell}\,\frac{q/k_{\text{F}}}{(Dq^{2}t)^{5/2}}\
.\qquad$ The different LTTs in the two terms make it impossible to account for
the non-hydrodynamic initial conditions represented by $\delta n_{\perp}(t=0)$
by means of a slip time as in Eqs. (3). A diffusion pole, Eq. (2a),
generalized to allow for the LTT, thus fails to describe the density
relaxation at long times. This remains true for intermediate times $\tau\ll
t\ll 1/Dq^{2}$, where the leading time dependence (ignoring an
$O(1/k_{\text{F}}\ell)$ contribution to the constant part of the $\delta
n_{\perp}$ term) is $\displaystyle\delta n({\bm{q}},t)$ $\displaystyle=$
$\displaystyle\delta n({\bm{q}},t=0)\,\left[1+O(Dq^{2}t)\right]$ (4b)
$\displaystyle\hskip-50.0pt+\delta
n_{\perp}({\bm{q}},t=0)\,\left[1-\frac{3}{2\sqrt{\pi}}\,\frac{1}{k_{\text{F}}\ell}\,\frac{q/k_{\text{F}}}{(Dq^{2}t)^{1/2}}\right]\
.$

Here the difference between the hydrodynamic and non-hydrodynamic terms is
even more striking than in Eq. (4a): The former is a constant, as described by
the diffusion equation, whereas the latter has a $1/t^{1/2}$ long-time tail.
Again, the concept of a slip time breaks down.

Equations (4), and the related comments, represent our main result which we
will now derive from kinetic theory. The dependence on non-hydrodynamic
initial conditions arises already at the level of the Boltzmann equation,
while the WL LTTs require a more sophisticated treatment of collision
processes.

Kinetic equation for the single-particle distribution Consider the single-
electron distribution function $f({\bm{p}},{\bm{x}},t)$ as a function of the
electron momentum ${\bm{p}}$, real-space position ${\bm{x}}$, and time $t$. In
the absence of external forces, the kinetic equation that governs $f$ reads

$\left(\partial_{t}+{\bm{p}}\cdot\bm{\nabla}_{\bm{x}}/m\right)f({\bm{p}},{\bm{x}},t)=\left(\partial
f/\partial t\right)_{\text{coll}}\ .$ (5)

This is completely general: The total time derivative of $f$ on the left-hand
side equals the collision integral, i.e., the temporal change of $f$ due to
collision, on the right-hand side. The equilibrium distribution function is
given by the Fermi-Dirac distribution
$f_{0}({\bm{p}})=1/(e^{(\epsilon_{\bm{p}}-\mu)/T)}+1)$, with
$\epsilon_{\bm{p}}={\bm{p}}^{2}/2m$ the single-electron energy, $\mu$ the
chemical potential, and $T$ the temperature. We parameterize the deviation
from equilibrium, $\delta f=f-f_{0}$, by

$\delta
f({\bm{p}},{\bm{x}},t)=w(\epsilon_{\bm{p}})\,\phi({\bm{p}},{\bm{x}},t)\ ,$ (6)

where $w(\epsilon_{\bm{p}})=-\partial f_{0}/\partial\epsilon_{\bm{p}}$ is a
weight function. For later reference, the number density $n$ and the number
current density ${\bm{j}}$ are

$n({\bm{x}},t)=\frac{1}{V}\sum_{\bm{p}}f({\bm{p}},{\bm{x}},t)\ ,\
{\bm{j}}({\bm{x}},t)=\frac{1}{V}\sum_{\bm{p}}{\bm{p}}\,f({\bm{p}},{\bm{x}},t).$
(7)

Boltzmann equation  The Boltzmann collision integral for elastic scattering by
impurities is Kohn and Luttinger (1957); Greenwood (1958)

$\displaystyle\left(\frac{\partial f}{\partial
t}\right)_{\text{coll}}^{\text{B}}$ $\displaystyle=$
$\displaystyle\frac{-1}{N_{\text{F}}V}\sum_{{\bm{p}}^{\prime}}\delta(\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})\,\frac{1}{\tau}\,W(\hat{\bm{p}},\hat{\bm{p}}^{\prime})$
(8a) $\displaystyle\times\left[\delta f({\bm{p}},{\bm{x}},t)-\delta
f({\bm{p}}^{\prime},{\bm{x}},t)\right].$ The delta-function reflects the
elastic nature of the collisions, and $W$ is a form factor. For point-like
scatterers, the latter is a constant equal to one and we obtain
$\left(\frac{\partial f}{\partial
t}\right)_{\text{coll}}^{\text{B}}=\frac{-1}{\tau}\left[\delta
f({\bm{p}},{\bm{x}},t)-\delta{\bar{f}}(\epsilon_{\bm{p}},{\bm{x}},t)\right]\
,$ (8b) where
$\delta{\bar{f}}(\epsilon_{\bm{p}},{\bm{x}},t)=\sum_{{\bm{p}}^{\prime}}\delta(\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})\,\delta
f(\epsilon_{\bm{p}},{\bm{x}},t)/VN_{\text{F}}$ is $\delta f$ averaged over the
$\epsilon_{\bm{p}}$-energy shell. Integrating over the momentum yields
$\sum_{\bm{p}}\delta{\bar{f}}(\epsilon_{\bm{p}},{\bm{x}},t)=\sum_{\bm{p}}\delta
f({\bm{p}},{\bm{x}},t)$, so the number density, Eq. (7), is conserved.

After a Fourier-Laplace transform as in Eq. (2a), the linearized Boltzmann
equation reads

$\displaystyle\left(-iz+\frac{i}{m}\,{\bm{p}}\cdot{\bm{q}}+\frac{1}{\tau}\right)\phi({\bm{p}},{\bm{q}},z)$
$\displaystyle=$ $\displaystyle\phi({\bm{p}},{\bm{q}},t=0)$ (9)
$\displaystyle+\frac{1}{\tau}\,\bar{\phi}(\epsilon_{\bm{p}},{\bm{q}},z)\
,\quad\ $

with $\phi$ from Eq. (6) and $\bar{\phi}$ defined in analogy to
$\delta{\bar{f}}$. Now consider $\delta n=n-n_{0}$, the deviation of the
particle-number density from its equilibrium value $n_{0}$. From Eq. (9) we
find

$\delta
n({\bm{q}},z)=J_{0}({\bm{q}},z)+\frac{i}{\tau}\,\frac{1}{V}\sum_{\bm{p}}\frac{w(\epsilon_{\bm{p}}){\bar{\phi}}(\epsilon_{\bm{p}},{\bm{q}},z)}{z-{\bm{p}}\cdot{\bm{q}}/m+i/\tau}\
,$ (10a) where $J_{0}({\bm{q}},z)=\frac{i}{V}\sum_{\bm{p}}\frac{\delta
f({\bm{p}},{\bm{q}},t=0)}{z-{\bm{p}}\cdot{\bm{q}}/m+i/\tau}\ .$ (10b)

For $T\ll\epsilon_{\text{F}}$ we can replace ${\bm{p}}\cdot{\bm{q}}/m$ in Eq.
(10a) by $v_{\text{F}}{\hat{\bm{p}}}\cdot{\bm{q}}$, which neglects corrections
of $O(T^{2})$ to the ${\bm{q}}$-dependence of $\delta n$. The resulting closed
equation for $\delta n$ yields

$\delta n({\bm{q}},z)=J_{0}({\bm{q}},z)/(1-iJ({\bm{q}},z)/\tau)\ ,$ (11a)
where $J({\bm{q}},z)=\frac{1}{4\pi}\int d\Omega_{\bm{p}}\
\frac{1}{z-v_{\text{F}}{\hat{\bm{p}}}\cdot{\bm{q}}+i/\tau}\ ,$ (11b)

with $\Omega_{\bm{p}}$ the solid angle associated with ${\bm{p}}$. An
expansion in the limit of small wave numbers ($q\ell\ll 1$) and frequencies
($z\tau\ll 1$) yields

$\displaystyle J_{0}({\bm{q}},z)=\frac{i\delta
n({\bm{q}},t=0)}{z+i/\tau}\,+\frac{i{\bm{q}}\\!\cdot\\!{\bm{j}}({\bm{q}},t=0)}{(z+i/\tau)^{2}}\,+O(q^{2}),\quad$
(12a) $\displaystyle
1-\frac{i}{\tau}\,J({\bm{q}},z)=-i\tau(z+iD_{0}q^{2})+O(z^{2},q^{4})\ ,\qquad$
(12b)

where $D_{0}=v_{\text{F}}^{2}\tau/3$ is the Boltzmann diffusion coefficient.
Inserting these results into Eq. (11a), and performing a Laplace back
transform, yields Eqs. (3a, 3b).

Equations (10b) and (12a) show that the solution depends, in addition to the
initial density perturbation, on the initial current density, as well as
higher modes. At the level of the Boltzmann equation, after a few mean-free
times the various initial-condition terms all multiply the same time
dependence and therefore can be incorporated into a ‘slip time’ as in Eq.
(3c). This is no longer the case if one uses a more sophisticated collision
integral that accounts for LTTs.

Weak-localization effects The LTTs associated with WL effects arise from two-
particle correlations that are not included in the Boltzmann collision
integral and lead to a frequency-dependent diffusivity that is nonanalytic at
zero frequency Abrahams et al. (1979); Wegner (1979). Reference Hershfield and
Ambegaokar, 1986 showed that these effects can be incorporated into the
kinetic equation for the single-particle distribution by adding an additional
term to the collision integral. This additional term can still be written in
the form of Eq. (8a) Strinati et al. (1989), with two modifications: First,
the scattering rate $1/\tau$ must be replaced by a time-dependent memory
function $\alpha$ (to be specified below) that encodes correlations. The WL
contribution to the collision integral then has the form

$\displaystyle\left(\frac{\partial f}{\partial
t}\right)_{\text{coll}}^{\text{WL}}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}dt^{\prime}\alpha(t-t^{\prime})\,\frac{1}{N_{\text{F}}V}\sum_{{\bm{p}}^{\prime}}\delta(\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})$
(13) $\displaystyle\hskip-30.0pt\times
W^{\text{WL}}(\hat{\bm{p}},\hat{\bm{p}}^{\prime})\,\left[\delta
f({\bm{p}},{\bm{x}},t^{\prime})-\delta
f({\bm{p}}^{\prime},{\bm{x}},t^{\prime})\right]\ .$

Second, the form factor is strongly angle dependent, which reflects the fact
that the WL effects are due to backscattering events Gorkov et al. (1979):

$W^{\text{WL}}(\hat{\bm{p}},\hat{\bm{p}}^{\prime})=\frac{1}{4\pi}\,\delta(\hat{\bm{p}}+\hat{\bm{p}}^{\prime})-1\
,$ (14a) The additional contribution to the collision integral does not change
the total scattering rate Strinati et al. (1989): $\int
d\Omega_{{\bm{p}}^{\prime}}\,W^{\text{WL}}(\hat{\bm{p}},\hat{\bm{p}}^{\prime})=0\
.$ (14b)

The WL contribution to the collision integral becomes

$\left(\frac{\partial f}{\partial
t}\right)_{\text{coll}}^{\text{WL}}=\int_{0}^{t}dt^{\prime}\alpha(t-t^{\prime})\left[\delta
f({-\bm{p}},{\bm{x}},t^{\prime})-\delta{\bar{f}}(\epsilon_{\bm{p}},{\bm{x}},t^{\prime})\right].$
(15a) The function $\alpha(t)$ and its Laplace transform are alp ; mod
$\displaystyle\alpha(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\pi
N_{\text{F}}\tau}\,\frac{1}{V}\sum_{\bm{k}}e^{-D_{0}k^{2}t}\qquad(t>0)\ .$
(15b) $\displaystyle\alpha(z)$ $\displaystyle=$ $\displaystyle\frac{i}{\pi
N_{\text{F}}\tau}\,\frac{1}{V}\sum_{\bm{k}}\frac{1}{z+iD_{0}k^{2}s_{z}}$ (15c)
$\displaystyle=$ $\displaystyle
s_{z}\,\frac{1}{\tau}\,\frac{3}{\pi}\,\frac{1}{k_{\text{F}}\ell}\left[c-\frac{\pi}{2}\,\zeta^{1/2}+O(\zeta)\right]\
,$ (15d)

where $\zeta=z\,s_{z}/iD_{0}k_{\text{F}}^{2}$, $c$ is the non-universal zero-
frequency contribution alp , and Eq. (15d) is valid for $d=3$. Note the
nonanalytic frequency dependence that results from integrating over the
diffusion pole. Only this contribution to $\alpha$ matters for the arguments
that follow.

The kinetic equation that generalizes Eq. (9) now reads

$\displaystyle\left(-iz+\frac{i}{m}\,{\bm{p}}\cdot{\bm{q}}+\frac{1}{\tau}\right)\phi({\bm{p}},{\bm{q}},z)-\alpha(z)\,\phi(-{\bm{p}},{\bm{q}},z)$
$\displaystyle=$ (16)
$\displaystyle\hskip-200.0pt\phi({\bm{p}},{\bm{q}},t=0)+\left(\frac{1}{\tau}-\alpha(z)\right)\,\bar{\phi}(\epsilon_{\bm{p}},{\bm{q}},z)\
.$

By letting ${\bm{p}}\to-{\bm{p}}$ we obtain two coupled equations for
$\phi({\bm{p}})$ and $\phi(-{\bm{p}})$ that can be solved exactly. The result
for the density relaxation generalizes Eq. (11a) to

$\delta
n({\bm{q}},z)=\frac{{\tilde{J}}_{0}({\bm{q}},z)+i\alpha(z)K_{0}({\bm{q}},z)}{1-[i/\tau-i\alpha(z)][{\tilde{J}}({\bm{q}},z)+i\alpha(z)K({\bm{q}},z)]}.$
(17a) In terms of a denominator
$N({\bm{p}},{\bm{q}},z)=z-{\bm{p}}\cdot{\bm{q}}/m+i/\tau+\frac{(\alpha(z))^{2}}{z+{\bm{p}}\cdot{\bm{q}}/m+i/\tau}$
(17b) we find $\displaystyle{\tilde{J}}_{0}({\bm{q}},z)$ $\displaystyle=$
$\displaystyle\frac{i}{V}\sum_{\bm{p}}\delta
f({\bm{p}},{\bm{q}},t=0)/N({\bm{p}},{\bm{q}},z)$ (17c)
$\displaystyle{\tilde{J}}({\bm{q}},z)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int d\Omega_{\bm{p}}\
1/N(k_{\text{F}}\hat{\bm{p}},{\bm{q}},z)$ (17d) $\displaystyle
K_{0}({\bm{q}},z)\\!$ $\displaystyle=$
$\displaystyle\frac{i}{V}\sum_{\bm{p}}\frac{\delta
f(-{\bm{p}},{\bm{q}},t=0)}{N({\bm{p}},{\bm{q}},z)(z+{\bm{p}}\cdot{\bm{q}}/m+i/\tau)}\
,$ (17e) $\displaystyle K({\bm{q}},z)$ $\displaystyle=$
$\displaystyle\int\frac{d\Omega_{\bm{p}}}{4\pi}\frac{1}{N(k_{\text{F}}\hat{\bm{p}},{\bm{q}},z)(z+v_{\text{F}}\hat{\bm{p}}\cdot{\bm{q}}+i/\tau)}.\qquad\
$ (17f)

Expanding the numerator in Eq. (17a) to $O(q)$ and the denominator to
$O(q^{2})$, and neglecting corrections to the diffusion pole that are analytic
in $z$, we find

$\delta n({\bm{q}},z)=\frac{i\delta
n({\bm{q}},t=0)}{z+iD(z)q^{2}}+\frac{i\delta
n_{\perp}({\bm{q}},t=0)}{[z+iD(z)q^{2}][1+s_{z}\alpha(z)]}$ (18a) with $\delta
n_{\perp}$ from Eq. (3b), and the renormalized diffusivity
$D(z)=D_{0}s_{z}/[1+s_{z}\alpha(z)\tau]\ .$ (18b)

The leading nonanalytic $z$-dependence of $\delta n$ follows from using Eq.
(15d) in Eqs. (18). In the asymptotic low-frequency regime $z\ll D_{0}q^{2}$,
the latter is $z^{1/2}$ in the first term on the right-hand side of Eq. (18a),
and $z^{3/2}$ in the second one. For $z\gg D_{0}q^{2}$, the first term has no
nonanalytic $z$-dependence, while the second term goes as $z^{-1/2}$. The
corresponding behavior in the time domain can be obtained by using the concept
of generalized functions and their Fourier transforms Lighthill (1958); sin .
All of the relevant nonanalyticities can be expressed in terms of

$H_{\nu}(z)=-i\,[z^{\nu}-(-z)^{\nu}]\ .$ (19a) The Laplace back transform for
non-integer $\nu$ is Lighthill (1958)
$H_{\nu}(t)=\frac{-2}{\pi}\,\sin(\pi\nu)\sin(\pi\nu/2)\,\Gamma(1+\nu)/|t|^{1+\nu}\
,$ (19b)

with $\Gamma$ the Gamma function. $H_{-1}(t)$ is a constant. Applying these
results to Eqs. (18) we obtain Eqs. (4).

We conclude with two general remarks.

(1) A basic tenet of Statistical Mechanics is that a description that
drastically reduces the number of degrees of freedom is necessary, desirable,
and physically sensible. In equilibrium one goes from a description in terms
of $N=O(10^{23})$ variables to one in terms of two thermodynamic variables,
say, density $n$ and energy density $\epsilon$. In a nonequlibrium fluid one
also includes the fluid velocity ${\bm{u}}$. We have shown, using kinetic
theory, that these descriptions are necessarily incomplete. One cannot simply
integrate out the other variables and still correctly describe transport or
other nonequilibrium effects.

(2) There is no simple remedy for this incompleteness. A description in terms
of the single-particle distribution $f$, as used in this Letter, yields a
solution in terms of the initial condition $f({\bm{p}},{\bm{x}},t=0)$. This is
not complete either, since in order to obtain a closed description in terms of
$f$ alone one has to effectively integrate out higher order distribution
functions, such as the pair correlation function $G^{(2)}=f^{(2)}-ff$, with
$f^{(2)}$ the two-particle distribution function. A complete description in
terms of $f$ therefore depends on $G^{(2)}(t=0)$, which in general is nonzero
and also multiplies a LTT contribution Dorfman et al. (2021). The conclusion
is that, due to LTT effects, no reduced description can be truly complete.

To summarize: Long-time-tail effects have been investigated in a host of
physical systems including classical fluids, superfluids, granular matter,
active matter, electrons in solids, quark-gluon plasmas, and quantum gravity.
The main message of this Letter is that a purely hydrodynamic description that
is routinely used in all these fields is incomplete because non-hydrodynamic
initial-condition effects cannot be ignored in general.

We thank Alex Levchenko for a discussion.

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