Hydrodynamics beyond hydrodynamics

Pavel Kovtun,
University of Victoria

East Asian Strings Webinar Series
May 07, 2021

Hydrodynamics is an established field with a venerable
history and many applications.
However, today I will not talk about applications.
Rather, I would like to highlight some foundationalquestions that came to light in the last few years.

What is hydrodynamics?
Set of equations that tell you how stuff flows. “Stuff” can bewater, air, a cold atomic cloud, hot primodial matter in early
Universe, electron fluid in a solid, etc.
As a student, you open a book with “hydrodynamics” or“fluid dynamics” in the title. You often see derivations,approximations, and applications all mixed together.
If, as a student, you are also learning about vector calculus
and partial differential eq-s at the same time, it can be hard
to see the big picture.

But the big picture of hydrodynamics
is in fact quite simple

Conserved quantities
Fundamentally, hydrodynamics is a classical theory ofthings that can not disappear, i.e. are conserved.
@
⇢a =
@t
density of some conserved quantity ar · jaflux of the same conserved quantity aa = energy, momentum, number of particles,…

Constitutive relations
One eq-n @t ⇢ = r·j for both 𝜌 and j. Hydro assumption:
assume j = j[𝜌], then have eq-n for 𝜌 only, can solve!
More generally, take some useful quantities 𝛾, (temperature, velocity,…), express ⇢a = ⇢a ( ), ja = ja ( ) , then get
eq-s for 𝛾a only, can solve!
Example: a = energy, 𝛾 = T = temperature, then:𝜌𝜖 = 𝜖(T), constitutive relation j𝜖 = -𝜅 𝜵T,
T=T0+𝛿T, get diffusion equation for 𝛿T :
@
T =
@t

2

Dr T

D = /✏0 (T0 )Summary of hydrodynamics
Conservarion laws:

@
⇢a =
@t

r · jaConstitutive relations (𝛾 = temperature, fluid velocity,…) :⇢a = ⇢a ( , r , r2 , . . . ),ja = ja ( , r , r

2

,...)

The procedure is the same, whether the fundamental
constituents are classical or quantum, relativistic or not,normal fluid or superfluid, magnetic fields present or not,fluid is chiral or not.

Questions to ask about every theory
Every theory in physics is only approximately “correct”,
limited by its domain of applicability.
Whenever we write down any equations that attempt todescribe physical phenomena, we have to answer:
1. Do the equations make physical sense?
2. Can we improve the equations to capture more physics?
3. What kind of physics is beyond our equations?

What will follow are three stories, one for each question.

First story:
Do hydrodynamic equations even make sense?

First story

Relativistic things
Say, you are a student in subatomic physics or astrophysics, and you want to learn about relativistic NavierStokes eq-s: quark-gluon plasma, neutron star mergers
Open “Fluid Mechanics” by Landau and Lifshitz:
some hydrodynamic equations
Open “Gravitation and Cosmology” by Weinberg*:
some hydrodynamic equations
And… these equations look very different!
*Formulation of hydrodynamics due to Eckart (1940)

First story

The equations look different, so what?
Let’s shut up and calculate. As a simple example, solve forlinear perturbations of the thermal equilibrium state.
Both Landau-Lifshitz’ and Eckart’s equations predict that:a) thermal equilibrium does not existb) things propagate faster than light
Hiscock, Lindblom, 1984
Hiscock, Lindblom, 1987

First story

What exactly is the problem?
Perturbations e-iωt+ik·x, solve hydro equations: ω=ω(k)Gapless modes: ω(k→0)=0, b/c of conserved charges.
These correspond to normal hydrodynamics (sound etc).
But the equations also predict gapped modes ω(k→0)≠0,moreover with Im(ω)>0. These are unphysical modes.
These “fake” modes are outside of the validity regime of
the low-energy hydro approximation. These are UV modes.
But if you want to actually solve the hydro equations inpractice, these unphysical modes ruin predictability:
cutoff-scale physics messes up the infrared behavior.

First story

How is the problem fixed?
So the classic-textbook hydrodynamics is not what you
solve in practice e.g. to study the quark-gluon plasma.
Most popular fix is the Israel-Stewart theory: the hydroequations are coupled to extra UV degrees of freedom,which in turn kill the unphysical UV modes.
These extra degrees of freedom are the dynamical stresses
and heat fluxes, in addition to the dynamical T, uα, μ.
The extra degrees of freedom of the Israel-Stewart theory
play the role of a UV regulator. Note that in the nonrelativistic Navier-Stokes eq-s, no extra d.o.f. are needed.

First story

Other regulators?
Can one find a regulator of hydrodynamics that does not
involve introducing extra UV degrees of freedom?
E.g. in field theory, the Pauli-Villars regularization introduces extra UV degrees of freedom, but dimreg does not.

Claim:
There is a sensible relativistic hydrodynamics whose only
variables are T, uα, μ, and no extra UV d.o.f. You need tochoose a suitable out-of-equilibrium definition of T, uα, μ.
Bemfica, Disconzi, Noronha, arXiv:1708.06255, arXiv:1907.12695
PK, arXiv:1907.08191, Hoult, PK, arXiv:2004.04102

First story

Let’s first talk about physics

First story

Example: Temperature

https://xkcd.com/2292/

First story

Example: Temperature
Temperature is something that is only unambiguously
defined in equilibrium. By definition, T is the quantitythat is measured by a thermometer.

In identical equilibrium states,
two different (but properly
calibrated) thermometers willshow the same temperature.

First story

Example: Temperature
Temperature is something that is only unambiguously
defined in equilibrium. By definition, T is the quantitythat is measured by a thermometer.

But in identical non-equilibrium
states, the same twothermometers will show
different temperatures!

First story

Non-equilibrium conventions
So there is arbitrariness in what one means by “fluid temperature”: one’s choice of thermometer is a convention.
Same arbitrariness in what one means by “fluid velocity”:
one’s choice of velocimeter is a convention.
Landau-Lifshitz's version of Navier-Stokes uses one convention, Eckart's version of Navier-Stokes uses another.

Note: there is no such thing as “the” Navier-Stokes eq-s
until you specify your arbitrarily chosen convention.

First story

Important:Different conventions give rise to different, mathematicallyinequivalent, Navier-Stokes equations. These conventions
have real consequences.
This is because the Navier-Stokes eq-s only give a crude
approximation of a real fluid. The difference between theconventions is hidden in the crudeness of this approximation*.
Landau-Lifshitz and Eckart adopt different conventions forNavier-Stokes, but both are bad, and both lead to nonsensical predictions.
*Using an analogy with quantum field theory, the choices of Landau-Lifshitz and Eckart

are analogous to adopting UV regulators which violate unitarity.

First story

What’s wrong with the classics?
Both Landau-Lifshitz and Eckart define T by:
Exact non-equilibrium energy density=
𝜖(T) given by the equation of stateThis means: as the local energy density changes, thethermometer adjusts its temperature instantaneously.
Such thermometers violate relativity, and lead to superluminal propagation in relativistic fluid dynamics.

First story

Now let’s be slightly more formal

First story

What we do in Effective Field Theory
1) Identify the low-energy variables
2) Write down all the terms allowed by the symmetry,3) Do this up to a given dimension, e.g.:
S = d 4 x (a(∂μφ)2 + bφ 2 + cφ 4)∫4) Constrain the coefficients a,b,c so that the physics
is sensible, e.g. c < 0 for stability of the vacuum

First story

Do the same in hydro1) Identify the low-energy variables: T, u𝛼, 𝜇2) Write down all possible terms in the constitutive
relations consistent with the symmetry3) Do this up to a given order (say, 1-st order) in thederivative expansion4) Constrain the coefficients so that the physics issensible, e.g. demand stability of equilibrium

First story

Example: diffusion
0

0

i

i

Conservation law @t J + @i J = 0, need J = J (μ), J = J (μ)
0

i

J =

i

i

2

@ µ + O(@ )02

J = µ + ⌫ µ̇ + O(@ )If ignore the μ· term: diffusion eq-n
If keep the μ· term: both ∂2t μ and ∂2i μ, hyperbolic eq-n
The term νμ· can be eliminated by a “field redefinition” of μand truncating the derivative expansion. So it’s not physicalfor the purpose of correlation functions. But it changes the
structure of the PDE for μ(t,x).
Coef-t ν can be interpreted as specifying your μ-meter

First story

Claim:
The above example of diffusion can be generalized
to full non-linear relativistic hydrodynamics, and it
actually works. There is no need to introduce extra UV
degrees of freedom.
Bemfica, Disconzi, Noronha, arXiv:1708.06255, arXiv:1907.12695
PK, arXiv:1907.08191, Hoult, PK, arXiv:2004.04102

First story

arXiv:2104.00804

First story

Question:
Sensible relativistic hydrodynamics seems to requirethat its eq-s contain non-hydrodynamic parameters.
Does hydrodynamics actually have predictive power
for non-linear evolution?**Can the low-energy predictions be made independent of the UV regulator, à la RG?

Second story:
Are there limits to improving the hydrodynamic equations?

Second story

A first look at the limitations of hydrodynamicsℓ = typical microscopic distance scale

T, v, 𝜌,…

L = distance scale of hydro solutions

distance
Hydrodynamics probably OK for L≫ℓ, small derivatives

Second story

A first look at the limitations of hydrodynamicsℓ = typical microscopic distance scale

T, v, 𝜌,…

L = distance scale of hydro solutions

distance
Hydrodynamics probably not OK for L~ℓ, large derivatives

Second story

Hydrodynamics as an expansion in derivatives
Conservarion laws:

@
⇢a =
@t

r · jaConstitutive relations (𝛾 = temperature, fluid velocity,…) :⇢a = ⇢(0) ( ) + ⇢(1) (r ) + ⇢(2) (r2 , (r )2 ) + . . .
ja = j

(0)

( )+j

(1)

(r ) + j

(2)

(r

2

2

, (r ) ) + . . .

Second story

Can we keep improving forever? Let’s say we generate
an infinite series in the gradients.
Q: Does this series converge, or does it diverge?

If converges: Hydrodynamics can be systematically
improved to include more transport phenomena. 😀
If diverges: Hydrodynamics is not supposed to work…
then why does it? Is hydrodynamics a fluke? 😕

Second story

What exactly is the expansion?
@k·x-i𝜔ti= ik , so the
For a plane wave, e, derivatives are@xhydrodynamic expansion is the expansion in powers of k.
Example: sound wave 𝜔sound(k) = ±vsk − i𝛤k2+…
Do the series 𝜔(k) = ak + bk2 + ck3 + dk4 + …
converge or diverge in hydrodynamics?
Not clear how to answer in general, let’s look at examples.

Second story

Analytic examples
There are fluids which can be studied analytically using
holography. These fluids are similar to the quark-gluon
plasma produced in nuclear collisions.
In the simplest solvable examples* we find that 𝜔sound(k) is
an analytic function of k, convergent for |k|<kc, with
kc =

p

kB T
2(2⇡)
~c
Grozdanov, PK, Starinets, Tadić, arXiv:1904.01018

The convergence is important. Gives one hope that hydrodynamics is improvable.
* 𝒩=4 supersymmetric Yang-Mills theory in 3+1 dimensions and its cousins.

Second story

So… why is there a critical value |k|=kc in 𝜔(k)?
Example from basic math: function f(x) = 1/(1+x2)
1.0

1
1 + x2

0.8

0.6

0.4

0.2

0.0

-4

-2

0

2

4

Function is perfectly smooth forall −∞<|x|<∞. But the small-x
Taylor expansion only converges for |x|<1. To understandwhy, take x complex.

x

Similarly, 𝜔(k) is a smooth function at real k. To understandwhy it only converges for |k|<kc, we must take k complex.

Second story

Complex ω and k
In classical physics, dispersion relations ω=ω(k) comeabout by solving F(ω,k2)=0, where F is determined by theequations (hydrodynamics, Maxwell’s eq-s in matter, etc).
2
@
T
+
Dr
T = 0 gives
Example: diffusion equation t
𝛿T~ei k·x-i𝜔tF(ω,k2) = -iω - Dk2 = 0.

Example: sound waves in a viscous fluid giveF(ω,k2) = ω2 - vs2 k2 + 2iΓωk2 = 0.
Take ω real, then F(ω,k2) = 0 gives k(ω), in general complex.
Imaginary part of k damping length/penetration depth
Take k real, then F(ω,k2) = 0 gives ω(k), in general complex.
Imaginary part of ω relaxation time

Second story

But what if both ω and k are complex?

Second story

Oscillation modes of a fluid, real k
Solutions to F(ω,k2)=0* in
the plane of complex ω

0

-1

q⌘

~c k
2⇡kB T-2-3

PK, Starinets, arXiv:hep-th/0506184
-3

-2

-1

0

1

2

3

* Poles of the exact retarded Green’s function of the energy density.

Second story

Oscillation modes of a fluid, real k
Solutions to F(ω,k2)=0* in
the plane of complex ω

0

-1

q⌘

~c k
2⇡kB T-2-3

PK, Starinets, arXiv:hep-th/0506184
-3

-2

-1

0

1

2

3

Sound waves, 𝜔(k) = ±vsk − i𝛤k2+…
Macroscopic, classical, approach theorigin as k→0.
* Poles of the exact retarded Green’s function of the energy density.

Second story

Oscillation modes of a fluid, real k
Solutions to F(ω,k2)=0* in
the plane of complex ω

0

-1

q⌘

~c k
2⇡kB T-2-3

PK, Starinets, arXiv:hep-th/0506184
-3

-2

-1

0

1

2

3

microscopic oscillation modes,
no classical interpretation, stayaway from the origin as k→0.
* Poles of the exact retarded Green’s function of the energy density.

Second story

Oscillation modes of a fluid, complex k
Now take k to be complex, |q2 | = 1ei✓ , and vary 𝜃 from 0→2𝜋.

q⌘

~c k
2⇡kB TSound modes (blue and gold ) swap places, but remain sound modes
when 𝜃 becomes 2𝜋.

Second story

Oscillation modes of a fluid, complex k
Now take k to be complex, |q2 | = 1.99 ei✓, and vary 𝜃 from 0→2𝜋.

q⌘

~c k
2⇡kB TSound modes (blue and gold ) swap places, but remain sound modes
when 𝜃 becomes 2𝜋.

Second story

Oscillation modes of a fluid, complex k
Now take k to be complex, |q2 | = 2.01 ei✓, and vary 𝜃 from 0→2𝜋.

q⌘Sound mode (gold) becomes one of the non-classical modes
when 𝜃 becomes 2𝜋!

~c k
2⇡kB TSecond story

Trajectories of the modes at |𝖖2|=1.99 and |𝖖2|=2.01

Second story

At |𝖖2|=2, modes collide, and the topology of the trajectories changes.
This is level-crossing in macroscopic dissipative systems,
when a classical (hydrodynamic) excitation becomes a
non-classical excitation.
This determines the convergence of the hydro expansion,and gives the critical wavelength for sound λc=2π/kc,
1 ~c
c = p
2 kB T

Grozdanov, PK, Starinets, Tadić, arXiv:1904.01018

Second story

Bottomline:
In many solvable examples, classical macroscopic
excitations and non-classical microscopic excitations
are merely different branches of the same multi-valued
complex function.

Comment: Complex momentum is also related toquantum chaos through out-of-time-ordered correlation
functionsMaldacena, Shenker, Stanford, arXiv:1503.01409
Grozdanov, Schalm, Scopelliti, arXiv:1710.00921
Blake, Lee, Liu, arXiv:1801.00010
Blake, Liu, arXiv:2102.11294… many more papers

Final story:
Why everything I said so far is wrong

Third story

Let us look at very simple solutions of classicalhydrodynamics: how the near-equilibrium physicalsystem reacts to infinitesimal external sources.

Third story

Response functions
Classical hydro gives predictions for retarded functions ofconserved densities and fluxes in equilibrium. E.g.:
D = diffusion constant,χ=∂n/∂μ susceptibility

2

D k
t,t
Gret (!, k) =i! Dk2
response of the current to external gauge field
response of the stress tensor to external metric

xy,xyGret (!, k=0) = p

2i!⌘ + O(! )p = pressure,η = shear viscosity
PK, arXiv:1205.5040

These do not depend on which version of hydro one isusing, and are universal predictions of classical hydro for
low-energy observables. Allow to compute η, D, etc.

Third story

Are these predictions of classical hydro actually correct?
After all, classical hydro (i.e. hydro=PDEs) is but a toy
model of fluids, and nature may or may not care for our
toy models.

Third story

A second look at the limitations of hydrodynamics
Macroscopic stuff is made out of microscopic stuff.
In a quantum vacuum, virtual particles are constantly produced and absorbed due to quantum fluctuations. Similarly, in a macro-state, virtual sound waves are constantly
produced and absorbed due to statistical fluctuations.
These sound waves will back-react on the macroscopic
physics because hydro is non-linear and waves interact.
T, v, 𝜌,…

distance

Third story

Claim
Even if you are interested in macroscopic long-time, longdistance response near static equilibrium, classical hydroeq-s which ignore the back-reaction can lead topredictions which are qualitatively wrong.
Alder, Wainwright, Phys. Rev. A 1, 18 (1970)

The rest of my talk will be comments on this claim.

First story

Let’s first talk about physics

Third story

Example: viscosityyx

Momentum transfer
between layers of fluid,
Txy = η ∂yvx + O(∂2)
Related to correlations of stress:
hT

xy

T

xy

iret. = p2i!⌘ + O(! )In a gas: η = ρ vth ℓmfp

Third story

Example: viscosity
Momentum can also be transferredby collective excitations.

yx

Gas of sound waves: `mfp ⇠Contribution
to viscosity:

Z ⇤

dD k

T
⌘
2
k
✏+p

⇤T2
⌘/s

1
⌘
2
k
✏+p

in D=3

IR divergent in D=2

Third story

Example: viscosity•

This is the physics of thermal
fluctuations. It is invisible ifhydrodynamics is viewed
just as a collection of partial
differential equations.

•If you think η/s can bearbitrarily small, think again.

•

Classical hydrodynamics
may be irrelevant to physics
in 2+1 dimensions.

yxThird story

Now let’s be slightly more formal

Third story

Stochastic hydrodynamics
Add noises: T

μν

μν
μν
λ
λ
λ
= Tclass[β, α] + τ , J = Jclass[β, α] + j⟨…⟩ = average over the noise with some weight, i.e.

⟨…⟩ = Dτ Dj e −Snoise[τ,j]∫Snoise[τ,j] is Gaussian (quadratic in τ,j) for long-wavelength(small-k) fluctuations, by the central limit theorem.
Width of the Gaussian is determined by demanding thatthe equilibrium 2-point correlation functions satisfy thefluctuation-dissipation theorem
2T
Gsymm(ω, k) =Im Gret(ω, k)ωThird story

Stochastic hydrodynamics
Tijlineari j= · · · + (✏+p)v v + · · · + ⌧leading non-linear,no derivatives termijhigher order〈v v〉〈Txy Txy〉= 2Tη + (𝜖+p)

(𝜖+p)
〈v v〉
y

y
x

x

noise

Third story

Stress correlations in 3+1 dimensions
R3

hTxy Txy i = p + O(⇤ T )0-th order classical

✓

2

17T ⇤
i! ⌘ +
120⇡ 2 ⌘/s◆+O

1-st order classicalcorrection to pcorrection to η

✓

3/2

!
(⌘/s)3/2◆+ O(! 2 )2-nd order classicalcutoff-independent
PK, Yaffe, hep-th/0303010
PK, Moore, Romatschke, 1104.1586•

This is “one-loop” fluctuation correction to ⟨Txy Txy⟩ret.
Actual physical viscosity includes all such corrections.

•

As expected, small η/s implies large corrections to η/s.
Fluctuations are mandatory for small-viscosity physics.

•

Fluctuations are more important than 2-nd order hydro.
IR contributions determined by thermodynamics and η/s.

Third story

Stress correlations in 3+1 dimensions
R3

hTxy Txy i = p + O(⇤ T )0-th order classical

✓

2

17T ⇤
i! ⌘ +
120⇡ 2 ⌘/s◆+O

1-st order classicalcorrection to pcorrection to η

✓

3/2

!
(⌘/s)3/2◆+ O(! 2 )2-nd order classicalcutoff-independent
PK, Yaffe, hep-th/0303010
PK, Moore, Romatschke, 1104.1586•

In real time, ω3/2 gives 1/t3/2 decay of correlations, more
important than first-order hydro (long-time tails).

•

In 2+1 dimensions, ω3/2 → ω ln(ω). Kubo formula forviscosity does not exist.

•

Large-N limit does not commute with the hydrodynamic
limit, hence ω3/2 not visible in classical holography.

Third story

Analogy with quantum gravity
Thermal fluctuations
are more important
than classical higherderivative hydro:

Quantum fluctuations
are more important
than classical higherderivative gravity:

µ⌫
µ⌫
µ⌫
µ⌫
Tcl = T(0) + T(1) + T(2) + . . .

Gxy,xy (!) = O(1) + O(!) + O(!

S=

V (r) =
m0 ⇠ (ci G) 1/2

Z

3/2

2

) + O(! ) + . . .



1
d x
R + c1 R2 + c2 Rµ⌫ Rµ⌫ + . . .
16⇡G
4



Gm1 m2
1+O
r

✓

Gm
r

◆

+O

✓

G~
r2

◆

+ O(e m0 r )Classical: Stelle 1978
Quantum: Bjerrum-Bohr, Donoghue, Holstein, 2002

Third story

Bottomline
Using classical hydrodynamics (i.e. hydro=PDEs) toevaluate the response to ω→0, k→0 sources is unreliable,
Short-time statistical fluctuations give rise to infrared latetime correlations,
The deviations from classical hydrodynamics are more
pronounced at small viscosity,
Stochastic hydrodynamics predicts that these infraredcorrelations are universal, determined only by thermodynamics and transport coef-s (viscosity, conductivity etc).

Third story

But let’s dig deeper

Third story

Beyond the naive stochastic hydroSnoise is only Gaussian as k→0. What if we want thederivative expansion, beyond the leading k→0 limit?
Width of the Gaussian was fixed by FDT for 2-point
functions. But FDTs exist for all n-point functions. How
Heinz, Wang, hep-th/9809016
do you impose those?
Classical hydro can only compute retarded functions.
Non-retarded functions are invisible to classical hydro.
But higher-point non-retarded func-s can not be related
to retarded func-s through FDTs, hence contain new
low-energy info. How do you access that info?

Third story

Effective field theory
We are used to computing correlation f-s in field theory:
Identify the low-energy degrees of freedom Ψ• Identify the symmetries G• Write a local action S[Ψ,A] invariant under G• Get correlation functions by varyingZ
eiW [A] = D eiS[ ,A]•with respect to A.

Can we do the same for near-equilibrium states? In sucha EFT, classical hydro will be the Euler-Lagrange e.o.m.
of the action, stochastic hydro will emerge as a leadingorder truncation.

Third story

Building blocks of hydro EFT
Want locality, gauge- and difffeo-invariance, etc.
Want the Euler-Lagrange equations of motion to be theconservation laws ∂μTμν=0, ∂λJλ=0. Different from the standard mechanics where EL e.o.m. imply conservation laws.
Want to access real-time correlation functions, hence
formulate EFT using Schwinger-Keldysh fields (2 copies).
For equilibrium, want to impose Kubo-Martin-Schwinger
Martin, Siggia, Rose, Phys. Rev. A 8, 423 (1973)
constraints as a symmetry.
Crossley, Glorioso, Liu, arXiv:1511.03646
Haehl, Loganayagam, Rangamani, arXiv:1511.07809
Jensen, Pinzani-Fokeeva,Yarom, arXiv:1701.07436
Glorioso, Liu, arXiv:1805.09331

Third story

Example: conserved current• Aμ=external gauge field, 𝜑=dynamical field, Bμ ≡ Aμ + ∂μφ.
e iW[A] = Dφ e iS[B]∫This W[A] is gauge-invariant, and e.o.m. for 𝜑 is ∂μJμ=0.

• Schwinger-Keldysh: need two copies Bμ1,2 ≡ Aμ1,2 + ∂μφ 1,2.
r
1
2
a
1
2
r,a
Let φ = (φ + φ )/2, φ = (φ − φ ), same for Aμ . Want
W[Aμr, Aμa = 0] = 0
hence S = O(a) + O(a2) + O(a3) + …
• Shift symmetry: φ r → φ r + λ(x) determines SSB or not• KMS: B r → ΘB r , B a → ΘB a + (i/T0)Θ∂t B r , Θ = T or PT
Crossley, Glorioso, Liu, arXiv:1511.03646

Third story

Example: conserved current• Aμ=external gauge field, 𝜑=dynamical field, Bμ ≡ Aμ + ∂μφ.
e iW[A] = Dφ e iS[B]∫This W[A] is gauge-invariant, and e.o.m. for 𝜑 is ∂μJμ=0.

• Schwinger-Keldysh: need two copies Bμ1,2 ≡ Aμ1,2 + ∂μφ 1,2.
r
1
2
a
1
2
r,a
Let φ = (φ + φ )/2, φ = (φ − φ ), same for Aμ . Want
W[Aμr, Aμa = 0] = 0
hence S = O(a) + O(a2) + O(a3) + …
• Shift symmetry: φ r → φ r + λ(x) determines SSB or not• KMS: B r → ΘB r , B a → ΘB a + (i/T0)Θ∂t B r , Θ = T or PT
Crossley, Glorioso, Liu, arXiv:1511.03646

Third story

Example: conserved current• Aμ=external gauge field, 𝜑=dynamical field, Bμ ≡ Aμ + ∂μφ.
e iW[A] = Dφ e iS[B]∫This W[A] is gauge-invariant, and e.o.m. for 𝜑 is ∂μJμ=0.

• Schwinger-Keldysh: need two copies Bμ1,2 ≡ Aμ1,2 + ∂μφ 1,2.
r
1
2
a
1
2
r,a
Let φ = (φ + φ )/2, φ = (φ − φ ), same for Aμ . Want
W[Aμr, Aμa = 0] = 0
hence S = O(a) + O(a2) + O(a3) + …
• Shift symmetry: φ r → φ r + λ(x) determines SSB or not• KMS: B r → ΘB r , B a → ΘB a + (i/T0)Θ∂t B r , Θ = T or PT
Crossley, Glorioso, Liu, arXiv:1511.03646

Third story

Example: conserved current• Aμ=external gauge field, 𝜑=dynamical field, Bμ ≡ Aμ + ∂μφ.
e iW[A] = Dφ e iS[B]∫This W[A] is gauge-invariant, and e.o.m. for 𝜑 is ∂μJμ=0.

• Schwinger-Keldysh: need two copies Bμ1,2 ≡ Aμ1,2 + ∂μφ 1,2.
r
1
2
a
1
2
r,a
Let φ = (φ + φ )/2, φ = (φ − φ ), same for Aμ . Want
W[Aμr, Aμa = 0] = 0
hence S = O(a) + O(a2) + O(a3) + …
• Shift symmetry: φ r → φ r + λ(x) determines SSB or not• KMS: B r → ΘB r , B a → ΘB a + (i/T0)Θ∂t B r , Θ = T or PT
Crossley, Glorioso, Liu, arXiv:1511.03646

Third story

Diffusion in stochastic hydro (1)Current J t = n(μ), J k = − σ(μ)∂kμ + j k satisfies ∂λJ λ = 0
Noise is Gaussian: ⟨j i(x)j k(x′)⟩ = C ikδ(x − x′)∫a∫− 12 ∫ j iCik−1 j k

λEnforce eom: δ[∂λJ ] = Dφ e
Noise average ⟨…⟩ = Dj ei ∫ φ a ∂λ J λ… generates Cik∂iφ a∂kφ a

Get effective action
S[μ, φ ] = dt d x [iφ
∫
a

d

a

1
a
a
∂
n(μ)
−
∂
(σ(μ)∂
μ)
+
C
∂
φ
∂
φ
( t
k
k ) 2 ik ik ]O(a), diffusion equationO(a2), “kinetic” termThird story

Diffusion in stochastic hydro (2)Effective action for diffusion:S[μ, φ ] = dt d x [iφ
∫
a

d

a

1
a
a
(∂tn(μ) − ∂k(σ(μ)∂k μ))+ 2 Cik∂iφ ∂kφ ]Say, Cik = C(μ) δik + θ(μ)[δik(∂j μ)2 − ∂i μ∂k μ] + …. FDT for 2-point
functions relates C(μ) = 2Tσ(μ) to σ(μ) of classical hydro.

θ(μ) contributes to 4-point non-retarded function, which is

not related to the retarded function by FDT.
Classical hydro only knows about retarded functions, henceclassical hydro knows nothing about θ(μ).
θ(μ) is a stochastic transport coefficient. It can contribute toobservable retarded functions through loops.

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]d

Crossley, Glorioso, LiuarXiv:1511.03646

Beyond simple diffusion (recall Bμ=Aμ+∂μ𝜑)
S2 =

∫

j
i
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]dtransforms into a total
derivative under KMS

Crossley, Glorioso, LiuarXiv:1511.03646

transform into each other under KMS

Beyond simple diffusion (recall Bμ=Aμ+∂μ𝜑)
S2 =

∫

j
i
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]d

Crossley, Glorioso, LiuarXiv:1511.03646

Beyond simple diffusion:
S2 =∫changes under KMS into
stuff orthogonal to […]KMS invariantji
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]d

Crossley, Glorioso, LiuarXiv:1511.03646

transform into each other under KMS

Beyond simple diffusion:
S2 =

∫

j
i
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]d

Crossley, Glorioso, LiuarXiv:1511.03646

Beyond simple diffusion:
S2 =∫O(a2), Gaussian noise, but not related
to diffusion coefficient by FDT

ji
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

O(a2)+O(a3)+O(a4), non-Gaussian noise+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

Third story

Diffusion in EFT
Identify μ ≡ Br t = Ar t + ∂tφr, effective action for diffusion:
S1 =

∫

i
i
dt d x [n(μ)Ba t + iT0σ(μ)Ba i (Ba + (i/T0)∂t Br)]d

Crossley, Glorioso, LiuarXiv:1511.03646

Leading-order in derivatives

Derivative-suppressed compared to S1

Beyond simple diffusion:
S2 =

∫

j
i
dt d x θ1(μ)Ba iBa j [∂t Br ∂t Br − δ ij ∂t Br k ∂t Brk]d

+ dt d d x θ2(μ) Ba iBai (Ba j + (i/T0)∂t Br j) (Baj + (i/T0)∂t Brj)∫

Jain, PK, arXiv:2009.01356

p1
Third
p1 story pppropagators
Propagators and vertices
p2
p1
p3 p1
−1
1
the free Lagrangian
and
leads
to
the
=
(p)δn(−p)⟩=a0δn(p)ϕa (−p)⟩⟨ϕ
=
=
∗
0
p p1
F
(p)
p
F (p)
6
p
p
p
2
4
2
h n 'a i
h'
ni
h
n
ni
a
2
1
2T0 χDk−1
Propagatorsp22=⟨δn(p)δn(−p)⟩===ϕ=(p)δn(−p)⟩
=
where
p
=
(ω,
k)
and
F
(p)
=
ω
+
iDk
.
We
denoteδnbyentaryMaterial
0
a
0
p
p
p
2
1
3
∗
p
F(recall(p) ⟨φaφa⟩ =
)
0
p
|F
(p)|
p The remaining terms in (A1) 16
F (p) solid and ϕa by wavy lines.

eej in (A1) is⟨δn(p)ϕ2 freei Lagrangian
2
the
and
leads
to
the
(−p)⟩
=
=p
p
a
0
(∂ ϕa ∂j ϕa ) + iT0 ϑ2 (∂ ϕa ∂i ϕa ) .F(A3)13(p)

and (A3) lead to various interaction vertices
2
−1
=0
(ω,
F (p)
=
ω
+
iDk
. We denoteδnpby
mentary
Materialχχk) andiT
uations
in diﬀusion
EFT2T0 χDk 2 where piT
0
p
p
2
4
2
=
=
,
(A4)
−
=
solid andwavyplines.
Theremainingterms in p(A1)δn(p)δn(−p)⟩
=
=
p
p
p
p
∗
2 ϕa by
2
1
3
1
3
0
∗
p
p
2
F (p)
F
(p)
F
(p)
p
|F
(p)|
lead
to
variousinteraction vertices
and
(A3)
ive
detailsof
loop
calculations
inVerticesfromS:
ctuations in diﬀusion EFT 1
p1
p1
p
p3
p2
p4 p2
p4
3
2
for fullhydrodynamics
proceeds
2T0 χDk
p2
p1
p3 p1
p3
iT0 χ p2
iT0 χ
We
have
d
p1 ,
p3 p1
p3(A4)
p
p
=
=
1
3
−
=
p
p
p
e give details of loop
calculations in
2
1
∗
p
|F
(p)|
p
p
1
1
Fproceeds(p) F (p)(A5)pinp3
p3
p2
p4 Vertices
p2
4
sis
for
full
hydrodynamics
To compute various correlation
p2
p4 p2
p4 p2
p4
p1
p3 p1
p3 p1
p3
xpandiTtheeﬀectiveiT
action0 χ order0χWe
have
denoted
A
by
rt
,
(A4)
−
=
(A5)
tions. Vertices
It is convenient
to Swork
Stochastic vertices from (A3) are denoted in bold. The
from
2:
∗
—To F
computevariouscorrelation(p) degree
F (p)p2 Feynman
p4 rulesp2Vertices
p4 readp2 oﬀ
p4directly
fundamentalof freedom
respective
can be
from
in
the
last
line
expand the eﬀective action orderpand the Lagrangian (4a) up to
from (A1) and (A3). Energy-momentum conservation atactions. It is convenient to work
Stochastic vertices from (A3) are denoted in bold. The
δn = n − n(µ0 ) and ϕa , in theeach vertex is understood.
s a fundamental degree of freedom
respective Feynman rules can be read oﬀ from directly
fields, to obtain
One final piece of information that we need is the couexpand"the Lagrangian (4a) up to
from (A1) and (A3). Energy-momentum conservation atpling to sources. As we shall only be interested in correla2
i
−ds
D∂δn
δn= +
iTn(µ
σ∂
ϕ
∂
ϕ
,
n−
)
and
ϕ
,
in
the
each vertex is understood.
0
0 a i a ation functions of density, we only keep the scalar sourcesd fields, to obtain
One final piece of information that we need is the coui

function Gra
involves
backgroundcouplingadiagrams
The
first diagram
contributes
contact term, while thevertex has an external leg
1
Third
story
remaining two′ diagrams
follows along (A10)leadingto⟨δn(p)ϕa (−p)⟩ ≡
.
(A8)
′′
F (p)p+′′′′ Σ(p)′′
p"!p 4 ′
p
−
p
−
p
2
′
′
d
p
1
−T0 σ ′ k ′ · (k − k ′ )iλkp(. . .) = χ +F (p)(2π)4 F (p′ )F (p − p′ )Σ(p) can be understood as a momentum-dependent cor#
′ 2p ′ ′′ ′′p′′ to thep diﬀusion′′ We haverectionconstant.
two
diagrams
p
p
2T
D(k
−
k
)
(χ
ω
−
iσ
k
·
(k
−
k
))
0
p
p
+
that can possibly contribute to this processp − p′ p − p′′ |F (p − p′ )|2p − p′
p − p′(A27)′2
2
2 !
4 ′
iλχ
T
k
d
p
(λ
−
λ̃)k
′
′′
0
p
p
2
2
2
2 . (A13)
=
χ
−
′
′
′′
4
′
′
p′
p
F (p)
(2π) F (p )F (p − p )p −p′p
In total, we find the correction to the correlation function
(A9)
p
p
!
4
′
′
2
Chen-Lin,
Delacrétaz,arXiv:1811.12540
pp
p
p
p
p
λχT0
d p ω λ̃ kHartnoll,· k − iDk
(λ − λ̃)k ′2
(A24)
p′′
′
′′
Γ1 (p) = −
p
−
p
p
−
p
D
(2π)4
F (p′ )F (p − p′ )p − p′ ′p − p′
&
p−p
2
−λ is
χTjust2iω
Their contribution,
again,
given
substituting
0 determined
Correction
by λthermo2
2by
ωk
.
(A14)
=
k
−
2
with
δλ.
We
can,
therefore,
in
the
left
vertex
in
(A12)
32πD
D
It is straightforward
computetheseinand
iagrams
also only diﬀer tofromeach otherthe obtain the
dynamics
diffusioncoefficient
find stochastic correction
to the andretardedcorrelation funcic vertex involved and the propagators arriving attion using (A14) as
mutually cancel due to KMS properties. Finally!
diagrams involving two 3-point hydrodynamic
d4 p′ st
−χT
0
st
′
′
but no external leg on the stochastic vertex
δλ
(p;
p
,
p
−
p
)
Γ2 (p) =
1
D
(2π)4
p′
p′′
ω λ̃ k ′ · k − iDk 2 (λ − λ̃)k ′2
F (p′ )F (p −
p′ )
2
2
2
2
4
$
%
2
2
2iω
T0 λ ωk2ppk−=1024π 2D3 χ
D
&
%$
%2 '
$
p − p′ p − p′′1iωJain, PK, arXiv:2009.01356
2
2
4
(A25)
k
−
. (A28)
ϑ
k
−
ϑ
+
ϑ
1
1
2
6
3
D
p′
p′′

Density-density response function

ω + iDk → ω + iDk + #ωk

2iω
k −
D

2iω
ω + iDk → ω + iDk + #ωk k −
k
(
D )

p

p
p − p′ p − p′′Correction determined by stochastic
One can also findthe associatedδσ usinginvisible(A11) to beto classicaltransport
coef-s$%2hydrodynamics.
1-loop correction to
2iω
T
λ
λ̃k
0
2
st
k −
δσ2 (p) = −
2D2
1024π
D
3-pointfunction
non-alalyticin ω,k.
&
'
$

%$

%2

Third story

Summary of EFT
Two types of fields: “classical” (φr) and “stochastic” (φa).

S ⊃ φa F(φr) These are the non-linearities of classical hydro,
fixed by the classical constitutive reltaions2

S ⊃ (φa) G(φr) Some of these are fixed by FDT in terms ofthe classical transport coef-s (viscosity etc)2S ⊃ (φa) G(φr)

Some of these are invisible to classicalhydrodynamics

S ⊃ (φa)n>2 H(φr)

Invisible to classical hydrodynamics

Third story

Summary of EFT
Two types of fields: “classical” (φr) and “stochastic” (φa).

S ⊃ φa F(φr) These are the non-linearities of classical hydro,
fixed by the classical constitutive reltaions2

S ⊃ (φa) G(φr) Some of these are fixed by FDT in terms ofthe classical transport coef-s (viscosity etc)2S ⊃ (φa) G(φr)

Some of these are invisible to classicalhydrodynamics

S ⊃ (φa)n>2 H(φr)

Invisible to classical hydrodynamics

new interactions, “stochastic transport coefs” Jain, PK, arXiv:2009.01356

Bottomline:
Stochastic transport coef-s are invisible to classical hydro,
but contribute to measurable correlation f-s through loops.
Stochastic transport coef-s give rise to infrared effects,
derivative-suppressed in 3+1 compared to viscosity, etc.
These are different effects from the usual long-time tails.
Stochastic transport coef-s can only be matched to the UVof the theory through non-retarded functions, unlike η,D,etc.
Thus knowing the macroscopic classical effective theory (toall orders in the derivative expansion) is not enough tocorrectly describe macroscopic infrared correlations.
Different from “standard” QFT.

Encouragement for students
As a student, one often tends to think that all interestingquestions in physics have already been answered many
years ago by the great.
Hydrodynamics is an ancient field by modern standards,
and people are still trying to figure out some very basic
questions.
If, as a student, you work in a field that is less than 200
years old, do not lose hope: there surely are many fundamental things left to discover.

Thanks for listening!