An action principle for dissipative fluid dynamics

An action principle for dissipative fluid dynamics

by

Michael James Crossley

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Physics

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 2016

@

Massachusetts Institute of Technology 2016. All rights reserved.

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A uthor ...

Department of Physics

Jan 29, 2016

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Certified by...

Hong Liu

Professor of Physics

Thesis Supervisor

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r

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An action principle for dissipative fluid dynamics

by

Michael James Crossley

Submitted to the Department of Physics on Jan 29, 2016, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy in Physics

Abstract

Fluid dynamics is the universal theory of low-energy excitations around equilibrium states, governing the physics of long-lived modes associated with conserved charges. Historically, fluid dynamics has been formulated at the level of equations of mo-tion, in terms of a local fluid velocity and thermodynamic quantities. In this thesis, we describe a new formulation of fluid dynamics in terms of a path integral, which systematically encodes the effects of thermal and quantum fluctuations. In our for-mulation, the dynamical degrees of freedom are Stuckelberg-type fields associated to the conserved quantities, which are subject to natural symmetry considerations, and the time evolution of the path integral is along the closed-time contour. Our formu-lation recovers the standard hydrodynamics, including the expected constraints from thermodynamics and the fluctuation-dissipation theorem, as well as an additional non-linear generalization of the Onsager relations. We demonstrate an emergent su-persymmetry in the "classical statistical" limit of our theory. For the non-linear fluid, the formalism is encoded in a trivial differential geometric structure, with a non-vanishing torsion tensor required to recover the correct physics of the most genbral fluid. Finally, we discuss progress in obtaining a holographic derivation of the action formulation at the ideal level, in which the low energy degrees of freedom emerge naturally as the relative embedding of the boundary and horizon hypersurfaces. Thesis Supervisor: Hong Liu

Acknowledgments

I would like to thank my advisor Hong Liu for his invaluable mentorship and

sup-port, and for allowing me the opportunity to work on such important and exciting unsolved problems in physics. I would also like to thank Paolo Glorioso for working so tirelessly with me, and for being available on Skype at any hour of the day or night. I would like to thank my other collaborators: Ethan Dyer, Dan Freedman, Julian Sonner and Yifan Wang, each of whom has taught me a great deal, and all of the students, postdocs, faculty and staff of the CTP, past and present: it was a privilege to work alongside so many great people. I would also like to thank everyone I've known at MIT, and in particular Jeremy Scott and Charles Mackin, for being such good friends to me over the past few years.

Thanks to Chloe, to both of our families, and in particular, to my mum and dad: your loving support has been a constant source of inspiration and happiness for me.

Contents

1 Introduction

1.1 The universality of fluid dynamics 1.2 Historical formulation . . . .

1.2.1 Constitutive relations . . . .

1.2.2 Equations of motion . . . . 1.2.3 Physical constraints . . . . .

1.3 Motivations for an action principle

1.3.1 A thermodynamic analogy 1.3.2 For hydrodynamics . . . . .

1.3.3 An example: long-time tails 1.4 Plan for this thesis . . . .

2 Action formulation of fluid dynamics

2.1 Schwinger-Keldysh contour . . . . 2.1.1 Nonlinear response functions . . . . 2.1.2 Time reversed process and discrete symmetries .

2.2 Effective field theory . . . .

2.3 Degrees of freedom . . . . 2.4 Symmetry considerations . . . . 2.4.1 Thermal density matrix: KMS condition . . . . 2.5 Loop corrections: Ghost fields and BRST . . . .

2.5.1 KMS conditions and SUSY . . . .

2.6 Various limits and expansion schemes . . . .

13 13 14 15 19 20 23 23 25 26 29 31

31

. . . . 34 . . . . 36 . . . . 38 . . . . 39 . . . . 45 . . . . 50 . . . . 53 . . . . 56 . . . . 60

3 Standard hydrodynamics and beyond 3.1 Hydrodynamic equations of motion ...

3.2 Constraints on hydrodynamics . . . .

3.2.1 Thermal equilibrium and the KMS condition

3.2.2 The classical statistical limit . . . .

3.2.3 Constraints on response functions from KMS

3.3 Beyond the standard formulation . . . .

4 Stochastic charge diffusion

4.1 Quadratic order . . . . 4.1.1 The quadratic action . . . . 4.1.2 Off-shell currents and constitutive relations . 4.1.3 BRST invariance and supersymmetry . . . .

4.1.4 The full generating functional . . . .

4.2 Cubic order . . . .

4.2.1 The cubic action . . . . 4.2.2 BRST invariance and supersymmetry . . . .

4.2.3 Multiplet of currents . . . . 4.3 A minimal model for stochastic diffusion . . . .

4.3.1 Linear stochastic diffusion . . . . 4.3.2 Action for a variation of the stochastic Ka equation . . . . 63 65 69 69 .. 71 72 74 conditions. 77 . . . . 78 . . . . 78 . . . . 80 . . . . 82 . . . . 83 . . . . 86 . . . . 86 . . . . 88 . . . . 90 . . . . 9 1 . . . . 92 rdar-Parisi-Zhang 93 5 Non-linear charged fluids 5.1 Preparations . . . . 5.1.1 Organization of variables 5.1.2 Covariant derivatives . . 5.1.3 Torsion and curvature . 5.2 The bosonic action . . . . 5.2.1 General structure . . . . 5.3 Stress tensor and current operators 97 . . . . 97 . . . . 97 . . . . 100 . . . . 102 . . . 103 . . . . . ...103 . . . . 105

. .

5.3.1 General discussion . . . . 105

5.3.2 Lowest order in a-field expansion . . . . 107

5.4 Formulation in the physical spacetime . . . . 110

5.5 The source action . . . . 114

5.6 Constraints on constitutive relations from local KMS conditions . . . 115

5.6.1 Spatial partition function condition . . . . 116

5.6.2 Generalized Onsager relations . . . . 119

5.7 Constraints from fluctuation-dissipation relations . . . . 120

5.8 Non-negativity of transport coefficients . . . . 120

5.9 Two-point functions . . . . 122

5.10 Stochastic hydrodynamics . . . . 125

5.11 Entropy current . . . . 127

6 Progress on a holographic derivation 129 6.1 Setup. ... ... .... ... ... 133

6.1.1 Isolating hydrodynamical degrees of freedom . . . . 133

6.1.2 Saddle point evaluation . . . . 138

6.1.3 Einstein gravity . . . . 140

6.2 Action for an ideal fluid . . . . 141

6.2.1 Solving the dynamical equations . . . . 142

6.2.2 Effective action for T and X. . . . . 144

6.2.3 Horizon lim it . . . . 146

6.2.4 Entropy current . . . . 147

6.2.5 Hydrodynamical action and volume-preserving diffeomorphism 149 6.2.6 More on the off-shell gravity solution . . . . 150

6.3 Generalization to higher orders . . . . 153

6.3.1 Structure of derivative expansions to general orders . . . . 153

6.3.2 Non-dissipative action at second order? . . . . 155

B Fluctuation-dissipation theorem at general orders 161

B.1 Properties of various Green functions . . . 161 B.2 KMS conditions in terms of correlation functions . . . . 163

B.3 Implications for response functions . . . 164

C KMS conditions for tree-level generating functional 167

D SUSY Ward identities and KMS conditions 169

E Derivative expansion for vector theory at cubic order 173

F Useful formulas 175

F.1 Integrability conditions . . . . 175

F.2 Variations with respect to background metric and gauge field . . . . . 176 G Structure of stress tensor and current at order 0(a0) 179

H Conformal neutral fluids to second order in derivatives 183

I Boundary term 187

1.1 Boundary compatible with foliation . . . . 187

1.2 Boundary incompatible with foliation . . . . 188

List of Figures

1-1 A generic excitation relaxes to equilibrium locally. . . . . 16

1-2 For a locally conserved quantity, local excesses can only relax away

diffusively. . . . . 17

2-1 (a) Evolution of a general initial density matrix po. (b) Closed time path contour from taking the trace. Inserted operators should be path ordered, as indicated by the arrows. . . . . 31

2-2 Relations between the fluid spacetime and two copies of physical space-times. The red straight line in the fluid spacetime with constant U is mapped by Xi'2(-0 oa) to physical spacetime trajectories (also in red) of the corresponding fluid element. In the holographic context, the fluid spacetime corresponds to the horizon hypersurface, and the two copies of physical spacetimes correspond to two asymptotic boundaries of AdS. X"2 describe relative embeddings of these hypersurfaces and T1,2 are the proper distances between the horizon and boundaries. . 44

6-1 (a) Complex bulk manifold M, consisting of two copies of asymp-totic AdS spacetimes patched together at a horizon hypersurface. Also labeled are stretched horizons El, E2 discussed around (6.2). (b) A

boundary theory Schwinger-Keldysh contour used to describe non-equilibrium physics. The two AdS regions map to the two horizontal legs of the Schwinger-IKeldysh contour, while the analytic continuation around the horizon corresponds to the vertical leg which defines the initial thermal density matrix. . . . . 130

6-2 The gapless degrees of freedom in the path integrals (6.4) are the relative embedding coordinates Xl(a-) between the horizon and the boundary. X' can be understood geometrically as follows: start with

o-' at El, shooting a congruence of geodesics orthogonal to E toward the boundary, the intersections of these geodesics with the boundary

define X . . . . . 131

Chapter 1

Introduction

1.1

The universality of fluid dynamics

Fluid dynamics is experimentally validated as the correct description of low-energy excitations around thermal equilibrium in a vast number of physical systems ranging in scale from the sub-nuclear quark gluon plasma observed in heavy ion collisions [1] to disk galaxy formation [2]. Amongst countless modern physical applications at in-termediate scales, atmospheric, oceanic and geodynamical physics concern the study of large-scale fluid motions [3], magnetohydrodynamics, in which a gauge field is in a Coulomb phase and there is an additional long-wavelength mode associated to the magnetic field is applied to important engineering problems such as plasma confine-ment [4] and nuclear reactor cooling [5], and superfluid hydrodynamics [6] represents the natural extension of the fluid dynamics of conserved charges to include an addi-tional gapless U(1) mode associated with the spontaneous breaking of a continuous phase symmetry, responsible for the so-called second sound.

Recently, interest in the theoretical formulation of fluid dynamics has been reignited in the high energy theory community by studies of gauge-gravity duality [7, 8, 9], and in particular the fluid-gravity correspondence [10, 11, 12, 13]. The thermodynamics of black holes is a long-studied subject (reviewed in [14]), which has been contextualized in the AdS-CFT correspondence as reflecting a thermal state of the dual boundary theory. One heuristic way of motivating this is that a state in the strongly-interacting

boundary theory should approach thermalization after sufficient time, while in the

AdS bulk theory, generic initial states eventually collapse into a black hole. Going

beyo[d this, [111 showed that the physics of low energy excitations of black brane con-figurations in the bulk theory maps precisely onto the hydrodynamics of the boundary

theory.

The theoretical motivation for fluid dynamics makes its ubiquity inevitable: on length and time scales greater than the thermal relaxation scales of a given mi-croscopic mechanical system, all modes relax to thermal equilibrium, barring the locally-conserved quantities, as well as possible gapless excitations, for example in the case of superfluids, for which low-wavelength excitations decay away arbitrarily

slowly. In essence, fluid dynamics is the most general low-energy effective field

the-ory description of these modes and their interactions, with free parameters encoding physical quantities such as the shear and bulk viscosity, which in principle could be calculated explicitly for a given microscopic theory by integrating out the remaining, fast-decaying modes according to the usual Wilsonian procedure. For the purposes of making contact with a given physical system, one need only input the values of the most relevant coefficients in terms of a derivative expansion to a given order to completely determine the dynamics of the system in the fluid dynamical regime.

1.2

Historical formulation

To formalize these ideas, let us begin by considering the strict thermodynamic limit of a system, described at a microscopic level in terms of some degrees of freedom interacting and evolving in time according to a Hamiltonian H. The most general mixed state of a system is described by a density matrix p, and in order for this to

be a steady state, we require

[Hp] = 0. (1.1)

Hence, p must be some operator constructed only from the conserved quantities of a system; that is, the operators commuting with the Hamiltonian. To be explicit, let us suppose that the system under consideration is relativistic in a flat spacetime

of dimension d, with a conserved stress tensor operator Th'', as well as a conserved current J" , where p C (1,. .. , d). In this case,

p = p (H, p', N) , (1.2)

where

H

=J

dda

YTO,

p =

J

dd-i TOi, N =Jdd--J

are the conserved charges, and i E (1 .... , d - 1). Amongst the possible p constructed

in this fashion, the particular dependence

p (W !, U) = exp

(J

d ~XPTOP- + /[JO), (1.4)

where

/3,

y and up , which satisfies uP '" = -1, are constants, and

Z = Tr exp

(J

dd-iX3u,T"

+

[tp Jo ) (1.5)

enjoys the privileged status of being the steady-state distribution function for a system for which the conserved quantities are weakly coupled to a reservoir (see for

exam-ple [15] for a classical argument). This is of course familiar as the grand canonical

ensemble, usually considered in the rest frame u (1,

1.2.1

Constitutive relations

We can now make the jump to fluid dynamics1 . We promote the thermodynamic parameters of our density matrix to be functions of F:

It =

V"(i),

/

=O

t_yi

₌

_pM.

(1.6)

'A

clear recent review of this construction is given in [16], which we will follow in parts here.

We will refer to this set of variables as the "standard hydro variables". Explicitly,

p3

()'

,[t),

u (i)) =

exp

dd-F/3(5)u,(5)TI

+

(

)

) ,

(1.7)

At a given time, we are free to define this state. The assumption of fluid dynamics is essentially that this form will be preserved under time evolution generated by the microscopic Hamiltonian2

. In general, this will not be the case; indeed, the density matrix can be expected to evolve generally in such a way as to be only expressible in terms of the microscopic degrees of freedom, rather than maintaining this restricted dependence on TOI' and J0. However, when considering correlators on sufficiently large

time and distance scales, this assumption should be valid by the usual Wilsonian argument, as long as these conserved currents are the only long-ranged modes in the theory. What we have described here amounts to a restatement of the basic assumption of hydrodynamics [171: Changes in the fluid take place sufficiently slowly that the system can be considered to be in a state of local thermodynamic equilibrium.

Figure 1-1: A generic excitation relaxes to equilibrium locally.

That conserved quantities are the long-lived modes in the low-wavelength limit of excitations in the theory is due to the diffusive character of their equation of motion

2

[18]. Considering a generic spatially-dependent excitation

60(y)

(1.8)

of some non-conserved quantity around its equilibrium value at time t - 0, this

quan-tity is free to decay through the many channels of microscopic interactions available, and does so with some characteristic thermal collision timescale -r, which will be short relative to the macroscopic scales of the system (see Figure 1-1).

That is, such a quantity obeys a relaxation equation of the form

c~o()=

-i

(7).

(1.9)

For a conserved quantity, however, considering charge density for definiteness and exciting around an equilibrium with vanishing chemical potential:

6n(x) =

(J

(y)), (1.10)

local conservation prohibits a local excess or deficit from simply decaying away: the only method of relaxation is for the local excess to flow into regions of local deficit

(see Figure 1-2).

6n(Y)

Figure 1-2: For a locally conserved quantity, local excesses can only relax away diffu-sively.

In terms of equations, this mode instead obeys a low-energy equation of the form

,2 _(1.11)

BOtn(Y)

=

DC ,fn(x'), _(.1

where D is the diffusion constant. The term on the right-hand side must contain spatial derivatives, as a homogeneous excitation of the charge density corresponds to putting the system in a new steady state, and so there is no relaxation without spatial derivatives. The term on the right-hand side of (1.11) is the lowest order term, assuming rotation invariance, and so is the most relevant term for long-wavelength hydrodynamic excitations. Fourier transforming the spatial profile,

DOtn(k) =- . n(k), (1.12)

Td (k)

where

Td(k) (DP)>, (1.13)

is the momentum-dependent diffusion timescale. Therefore, low-wavelength distur-bances take an increasingly long time to decay. In particular, for k -+ 0, the relaxation

time for this mode is infinite, as expected.

Let us observe a strong property of the density matrix (1.7): if the microscopic operators T' and J are themselves local operators, i.e. constructed from finitely many spatial derivatives acting on local fields and their conjugate momenta, then the density matrix (1.7) is also a local operator. Furthermore, the one point functions of physical observables can then be expressed as functionals of B(Y), p('), a (i) with

a finite number of spatial derivatives acting. Expanding in derivatives, then, and including terms up to first order in derivatives, the most general stress tensor and current can be written3 in a covariant way in terms of the standard hydro variables

3

When defining thermodynamic variables and the velocity field outside of equilibrium, there is some freedom in the definitions, known as the "frame choice". Here we have chosen Landau frame. Further details are given in [161

as:

T" =uiN'" + (p + (OpuP) A"" - rA'Po-t ", (1.14)

J= nu- - o-TA""& (M/T) + XTAI&"OT. (1.15)

For hydrodynamics, this is the full operator spectrum: all other modes are integrated out. We have introduced the composite tensors

All" = ?J" + u!'U, (1.16)

(Od -1

Furthermore,

E(T, p.), p(T, p), n(T, p) (1.18)

are algebraic functions of the temperature and chemical potential, representing the thermodynamic energy, pressure and number densities, respectively. As can be seen from the constitutive relations (1.14, 1.15), these can be observed experimentally by measuring the stress tensor and current in an equilibrium state for given T, y and

UA = (1,G). Finally,

(, ', -, XT (1.19)

are transport coefficients, parameterizing the first-derivative fluxes of energy, mo-mentum and current. Again from (1.14,1.15), these can be found experimentally by measuring the momentum flux tensor T"' and current flux J' in a state with small fluid velocity, temperature and chemical potential gradients.

1.2.2

Equations of motion

By assumption, T" and JP are conserved currents at the microscopic level, and so the operator equations

are satisfied. As these quantities are in turn expressed in terms of the thermodynamic variables /3(x), yt(x), &u(x) (with derivatives acting), we have a consistent system of

d+1 equations in d+ 1 unknowns, and so, once suitable initial conditions are imposed,

the evolution is completely determined. This gives precisely the fluid equations of motion: in particular, at first order, this system of equations is charged relativistic Navier-Stokes equations.

1.2.3

Physical constraints

Entropy current

Although we have a consistent set of local thermodynamic variables and equations of motion, thermodynamics in fact imposes further constraints on the free coefficients in our theory. Motivated by the second law of thermodynamics, we may anticipate the existence of a local entropy current, the divergence of which should be non-decreasing. We begin by introducing the entropy density of a thermodynamic system

s(p, T). (1.21)

In terms of the total entropy S = sV, internal energy U = EV, and particle number

N =

nV,

the first law for a system of volume V with a single conserved charge in its rest frame,

TdS = dU + pdV - pdN, (1.22)

in combination with the extensivity of entropy,

S (AU, AV, AN) = AS (U, V, N), (1.23)

immediately implies

Ts = E + p - pn, (1.24)

so that s(p, T) can be expressed in terms of thermodynamic functions which have already been defined. This relation can be written more covariantly in terms of the

standard hydro variables and the conserved currents as

TS" = -- T "u + pu" - pJ", (1.25)

where we have introduced the entropy current

S = sul. (1.26)

The local second law then requires

>ts"

> 0. (1.27)

The justification for imposing this is the notion of local thermal equilibrium. Each fluid element is considered to be a thermodynamic system. Adjacent fluid elements can exchange heat as well as conserved quantities. For a conserved quantity such as the charge, any charge lost by a fluid element flows into an adjacent element, which is captured by the operator equation 0,J" = 0. Similarly, any heat which flows out of one element flows into an adjacent element, but these elements may differ in temperature by an infinitesimal 6T. As heat only flows from a hot body to a cold body, the entropy 6S = 6Q/T lost by the hot element is therefore always less than the entropy gained by the cold element, hence the equation satisfied by the entropy

current is (1.27).

Substituting the explicit form of T" and J in terms of the hydro variables, and requiring the divergence of the resulting expression to be greater than or equal to zero for arbitrary values of these variables, imposes the following constraints on the free coefficients of the theory 4 :

ri 0,( 0, >O- 0, XT = 0 (1.28)

4_{One might question whether the frame ambiguity mentioned earlier affects these results. The}

Onsager reciprocal relations

Going beyond this, there are further relations which should be satisfied by of a sensible fluid when considering excitations around a thermal equilibrium. The hydrodynamic equations allow us to immediately compute a subset of higher order correlators. In particular, at two-point level, the retarded Green's function, again considering the

charge density operator J0

= n(t, -) for simplicity,

GS(t, y) -iO(t)( n(t, 7), n(0,

6)]),

(1.29)

satisfies the equation of motion for the charge density operator, with simple initial conditions given by the canonical commutation relations at t = 0. When one considers multiple hydrodynamic variables labelled by i and

j

(precise definitions are given in Appendix A), if the theory is microscopically time-reversal invariant, the Onsager reciprocal relations [19, 20] are satisfied; in Fourier space,

G (t, 7) G (t, -7). (1.30)

5

Fluctuation-dissipation theorem

Additional physics at the two-point level is encoded in the symmetric Green's function

GSt X) T?(n#,X-), n(, of). (1.31)

Further information is required at the two-point level to evaluate this expression, for which the initial conditions are related to the statistical fluctuations of the theory under consideration. However, for excitations around a thermal density matrix, the

5

1f the fields pick up non-trivial phase factors rj under time-reversal, there is an additional factor of r;2ijj on the right-hand side of this equation.

fluctuation-dissipation theorem [211,

ImG (w,k) = 2 tanh2G (w,k) ._{2 ijPk} (1.32)

can be shown to hold, which fixes the symmetric Green's function in terms of the retarded Green's function. We will discuss these conditions and extensions thereof in later chapters.

1.3

Motivations for an action principle

The minimum requirement for any reasonable theory of fluid dynamics, then, is to reproduce the above formalism. That is, to imply the constitutive relations for the stress tensor and conserved currents in terms of the standard hydro variables and their derivatives, the equations of motion (conservation equations), and the standard con-straints. But this standard hydrodynamics clearly does not encode the full statistical physics of the system under consideration.

1.3.1

A thermodynamic analogy

As an analogy, in thermodynamics, the variables under consideration are the expec-tation values of thermodynamic variables, i.e. one-point functions, but there is a spectrum of higher moments, which encode fluctuative physics. Although this flue-tuative physics is neglected within thermodynamics, there are a range of important physical phenomena, such as the Brownian motion of particles [22], which are essential to understand and only take place because of fluctuations.

As a concrete example of the fluctuations of thermodynamic variables, we may have access to a thermodynamic energy variable

U = (H) (1.33)

encoded, for instance, in the two-point function

((H - U)2). (1.34)

The justification for this is the thermodynamic limit: that the relative fluctuations for a system with a large number of degrees of freedom N

>

1 are negligible: specifically,

((H - U)2₎

1

~ -. (1.35)

E2 N

But there are many circumstances in which the fluctuative physics is important to know, as we will detail below. In a microscopic statistical mechanical description, which gives an explicit probability for the system being given in a microstate, for instance

P(Ei)C< eEi-pNi

p(Es) O( e kBT (1.36)

for the grand canonical ensemble, we can explicitly evaluate general n-point functions,

by taking suitable derivatives of

Ei -pNi

Ze kB , (1.37)

statesi

which serves as the moment-generating functional for the thermodynamic variables. For instance,

((H -U) 2_{) =}

. (1.38)

Statistical mechanics also elucidates relations between the fluctuation spectrum and thermodynamic response functions: for instance, the above expression can be

rewrit-ten as

DE

((H -- U)2) = O

kBT

2CV (1.39)

where CV is the heat capacity of the system at constant volume. This is an equilibrium version of the fluctuation-dissipation theorem introduced in the previous section: the fluctuations of the system and its response to external sources are related.

Further-more, the left-hand side is non-negative by definition, and so we obtain the inequality constraint

Cv > 0 (1.40)

for any thermodynamic system which can be described by the canonical ensemble, i.e. placed in equilibrium with a thermal bath. This constraint can be motivated

by thermodynamic arguments, without appeal to statistical mechanics, as a negative

heat capacity implies an instability when putting the system in contact with a heat bath (c.f. ([23]), but the statistical mechanical formulation succeeds in putting this

constraint, as well as other thermodynamic arguments, on a much firmer footing.

1.3.2

For hydrodynamics

We have an analogous situation in hydrodynamics. Firstly, in fluids, fluctuations, both quantum and statistical in nature, occur spontaneously and continuously. Un-derstanding such fluctuations is necessary for a wide range of physical problems: examples include equilibrium time correlation functions (c.f. [17, 24]), dynamical

critical phenomena in classical and quantum phase transitions (c.f. [25, 26]),

non-equilibrium steady states (c.f. [27]), and possibly turbulence (c.f. [28]). Furthermore, at a theoretical level, hydrodynamical fluctuations can help probe quantum gravi-tational fluctuations of a black hole in holographic duality. The current framework for incorporating fluctuations in hydrodynamics is to couple a stochastic force in the fluid equations at a phenomenological level (c.f. [16]). This is sufficient at a linearized level, but at nonlinear level, the noise and dynamical variables inevitably couple and interact, and a path integral formulation is necessary to treat these interactions sys-tematically. A path integral formulation would go beyond this, systematically encod-ing the statistleal fluctuations of the theory in a way in which the dynamics can be straight-forwardly extracted.

Secondly, such a formulation provides a systematic procedure and firmer footing to impose the requisite constraints. For instance, the entropy current formulation of constraints is technically cumbersome, in that the constraints are obtained by

requir-ing certain relations to hold for the most general solution to the equations of motion of the theory. It is also not entirely theoretically satisfactory, in that the definition of thermodynamic quantities outside of equillibrium is ambiguous, and so the physical meaning of the requirement of a locally conserved entropy current is slightly unclear. For this reason, a microscopic formulation of constraints would be more satisfac-tory. Recent progress was made in the observation [29, 30, 31] that a subset of the entropy current constraints, the equality constraints, are equivalent to the require-inent that in stationary equilibrium, the stress tensor and other conserved currents in the theory can be derived from a partition function, but the physical origin of this equivalence remained unclear. Considering fluctuative physics, there are additional relations which should be satisfied such as the fluctuation-dissipation theorem and Onsager reciprocal relationships, as mentioned in the previous section. These, too, are imposed in a phenomenological way in hydrodynamics. There may foresecably be further constraints required at higher orders in fluctuations, and beyond the linear level in amplitude. For these reasons, it would be preferable to have a systematic approach to incorporating the requisite constraints, and a path integral, subject to a fixed set of symmetry considerations, achieves this.

1.3.3

An example: long-time tails

A classic example of fluctuative hydrodynamical physics which goes beyond results

found in linear response theory is the long-time tails. First discovered in numer-ical simulations of hard-sphere scattering in [32], the velocity auto-correlation was observed to decay with a slow power-law dependence as opposed to the expected exponential decay predicted by linear response. We will again consider a conserved

U(1) charge for illustration, following the discussion in [33], and working in three

spa-tial dimensions. The spaspa-tial current for long-wavelength at first order in derivatives6

6

the subleading terms are third order in derivatives in higher, and always negligible for sufficiently long wavelengths

predicted by linear response theory in this case is

J -DO~n (1.41)

where D is again the diffusion constant7

. In the limit k -+ 0, then, the factor of Do implies that J' should relax to zero exponentially on a microscopic timescale. How-ever, at a non-linear level, there is already another possible term in the constitutive relation at quadratic order, with no spatial derivatives. The constitutive expression for the spatial current with this term included is given by

1

J' = -Din + nT0s. (1.42)

E + p

As the non-linear coupling contains no derivatives, it will always dominate the first term at sufficiently long wavelengths. Note that the coefficient is fixed in terms of thermodynamic quantities, E and p, as it must be, as the relation is determined by

thermodynamics alone with no derivatives of quantities involved.

If one now computes the Wightman correlator for the current due to this

non-linear contribution, pow averaging over all space f d3z = V for convenience, one finds

J d3(i(t, X)Ji (0, 0)) =

Jda_

(

i(t,)Ji(0,0)})

(1.43)

V1 2 = p

J

dj({n(t,

)n(0, 0)})({T2(t, Y)TZ(0, 0)})

(E + P)2 (1.44) = -(1.45)

f

+p

12 ((D

+2

)7r lti)3/2'

where y is the charge susceptibility. In the first line, we have replaced the operator product with its symmetrization, which is the dominant term, as [Ji, Ji] is smaller by a factor of hw/T, which is a small quantity in the hydrodynamic limit. In the second

7

The earlier diffusion equation (1.11) of course follows from combining the above with the

line, we inserted the explicit non-linear term from the constitutive relation (1.42). We have also factorizated the four-point correlator into two two-point functions, with the intention of using the results of linear response, as the key non-linear effects are already captured from the non-linear constitutive relation for J (1.42). We insert the explicit linear response results in the third line. The details of this calculation are worked through illuminatingly, including the explicit linear response expressions for

(n (t, Y) n(0, 0)}) (T04(t, x-)To" (0, 0)}) (1.46) in [33].

A similar result is obtained for the spatial stress-tensor two-point function:

J,(, (t,)t /2 Hi, (1.47)

where H" is the transverse traceless projector, defined explicitly in [33].

We have seen the importance of these contributions, as the power law decay in t implies that they are the dominant terms in the relaxation of the spatial current and spatial stress tensor at long times. These processes contribute additional diagrams which correct the two-point functions of the spatial current and stress tensor. There-fore, they correspond to corrections to the transport coefficients, and lead to a running coupling for these quantities, e.g. one can define q- q(p), o- = o-(t) etc., where [t is some energy scale8, which should in fact receive contributions from diagrams at all loops when one includes all quantum and statistical fluctuations at the full non-linear level. An action principle gives a convenient way to sum these corrections and so to calculate the expected running correction to the transport coefficients in a systematic way at any given order in fluctuations.

8

1.4

Plan for this thesis

In this thesis, we discuss our proposal for an effective field theory for dissipative fluids, based on the original work in [34]. This formalism provides a systematic treatment of statistical and quantum hydrodynamical fluctuations at the full nonlinear level.

Our formulation should, yield a complete set of constraints on hydrodynamical equations, and elucidate the microscopic origin of the phenomenological constraints mentioned in the previous sections. As illustrations, we derive actions for a vari-ation of the stochastic Kardar-Parisi-Zhang equvari-ation and the relativistic stochastic Navier-Stokes equations. Furthermore, we find a new set of constraints on the hydro-dynamical equations of motion, which may be considered as nonlinear generalizations of Onsager relations. Our discussion also reveals connections between thermal equi-librium and supersymmetry at a level much more general than that in the context of the Langevin equation.9 In particular, we find hints of the existence of a "quantum deformed" supersymmetry involving an infinite number of time derivatives.

In the next chapter, we review the formulation of our effective field theory. In particular, the closed time path contour integral, the dynamical degrees of freedom, and the requisite symmetry principles our theory should satisfy.

In chapter 3, we first explain how the standard formulation of hydrodynamics arises in our formulation, and then discuss aspects of our theory going beyond it. Finally, we discuss constraints on the standard hydrodynamical equations of motion following from our symmetry principles. In particular, in addition to recovering all the currently known constraints, we will find a set of new constraints to which we refer as generalized Onsager conditions.

In chapter 4, we consider a first illustrative example: the hydrodynamics of a conserved current in a thermal medium, working to all orders in the derivative ex-pansion. We discuss an emergent supersymmetry in detail at quadratic and cubic level in the small amplitude expansion. We give an explicit example in which the generalized Onsager conditions give new constraints at second derivative order at

cu-9_See

e.g. [35, 36, 37, 38]. See also Chap. 16 and 17 of [39] for a nice review on supersymmetry

bic level (details in Appendix E). Finally, We derive a minimal truncation of our theory, which provides a path integral formulation for a variation of the stochastic Kardar-Parisi-Zhang equation.

In chapter 5, we apply the formalism to full dissipative charged fluids as a second example, considering a double expansion in derivatives and a-fields. We prove that our formulation reproduces the standard formulation of non-linear hydrodynamics as its equations of motion. We also use our formalism to derive the two-point functions of a neutral fluid, and provide a path integral formulation of the relativistic stochastic Navier-Stokes equations. Finally we show that a conserved entropy current arises at the ideal fluid level from an accidental symmetry.

In chapter 6, we describe progress on a holographic derivation of an action principle for fluids. We recover the expected ideal fluid action for a conformal system, with the low energy degrees of freedom arising naturally as the relative embedding between the horizon and bounday hypersurfaces. We describe how this procedure should generalize at higher orders, with a Schwinger-Keldysh doubled bulk required to recover the correct dissipative physics.

The search for an action principle for fluids dates back at least as far as [40], with subsequent work including [41, 42], mostly limited to the consideration of ideal fluids

i'. Recent investigations include [46, 47, 48, 49, 50., 51, 52, 53, 54, 55, 56, 57, 58, 59,

60, 61, 62, 63].

Chapter 2

Action formulation of fluid

dynamics

2.1

Schwinger-Keldysh contour

We begin by reviewing the closed time path integral, or Schwinger-Keldysh formalism, at an abstract level (for more details, c.f. [64, 65, 66, 67]), which is the natural contour when evaluating time-dependent expectation values or correlation functions in a general time-dependent density matrix.

(a) t1 -4 -00 P0 ! U(tf, ti) Ut(tf, _ti) -4 00 (b) t -4 -00 U(t1, tj) Ut(tf, ti)

Figure 2-1: (a) Evolution of a general initial density matrix po. (b) Closed time

path contour from taking the trace. Inserted operators should be path ordered, as indicated by the arrows.

Exponentiating (1.1), the evolution of a system with an initial state po at some ti --+ -oo to some final state tf -+ oc can be written as

(2.1)

tf -* 00

where we have defined the evolution operator U(tf, ti), which takes the form

U(t

1 ,

ti)

= e-iH(tf -ti) (2.2)

for a Hamiltonian without explicit time dependence. As usual, a leg of evolution can

be expressed as a path integral between the initial and final times, with each path

weighted by a phase proportional to the associated classical action. p(tf), therefore, can be seen as a path integral running from tj to ti, with an insertion of the initial density matrix po, and a second path integral running from tj to tf (see 2-1a). In general, we will be intersted in taking ensemble averages of quantities, and therefore taking the trace of p(tj). _{Of course, this is equivalent to closing the contour at tf,}

so that the fields are continuous there (see 2-1b). Explicitly, if we consider a single degree of freedom x(t), a matrix element of p(tf) is given by

/

fX1(tf)=X" X X2(tf)=x'

(x"|y|' = dx'odxoDxD2 si1-s2

1(ti)=.'" x2(ti)=-X'

(2.3)

and the trace of p(tf), which follows from setting x" = x' = x and integrating over x, is given by

rjy (pop

.

..

(P

) J

dxJ

D(f)

Dx1Dx' eiS[x1-iS[iX2] .(. . (X

1(tf)=x2(tf)=X

(2.4)

where we have written explicitly the continuity of the field at tj x1 (tf) = x2(tf) = x.

We have allowed for possible additional operator insertions - , which will be naturally

contour-ordered in the path integral by the usual arguments, where we have denoted the path-ordering operator by P. More precisely, with po always in the left of the cyclic trace by convention, operators on the second, i.e. lower segment always lie to the left of those on the first, i.e. upper segment, with operators inserted on the first segment time-ordered, while those on the second segment are anti-time-ordered.

given operator 0,

Z[01, 02J - eW[1021 = Tr [poP exp

(if

dt

(0 1

(t

1

(t)

- 02(t) 2

(t)))

, (2.5)

where the subscript 1, 2 in

0

denotes whether the operator is inserted on the first or second segment of the contour (note that 01 and 02 are the same operator), and

#1, #2

are sources for the operator 0 along each segment, which we can take to be independent in order to encapsulate a more general set of possible operator orderings, including time-ordered and anti-time-ordered as a subset. The - sign before terms with subscript 2 arises from the reversed direction of time integration 1.

Taking functional derivatives of W gives path ordered connected correlation func-tions. For example

1 W

1~

i (t)6 2 4W( (3 ) (P (01(t1)02(t2)01(t3)02(t4)))

i4 6#1 (t1)602 (t2)601 (63)602 (t4) 01=02=

=

(

O(t

₂

)o(t

₄

))T(O(t)O(t))

(2.6)

In the second line, T and

t

denote the time-ordering and anti-time ordering operations respectively. In this notation, equation (2.5) can thus be written as

CwV[012]

=Tr [p (tei f t (t)02(t)) (Teifdt0(t)01(t))] .

(2.7)

We will take all operators

Oi

under consideration to be Hermitian and bosonic, and the sourcesli, 02i to be real. Taking the complex conjugate of (2.7), we then

find the property

W* [01, #2] = W [02, 01 .(2.8)

Equation (2.5) can also be written as

e

V0,2 =-Tr [Ui(+oo, -o; #1)poU2(+oo, -o; 2)] , (2.9)

where U, is the evolution operator for the system obtained from the original system

under the deformation f dt#10, and similarly for U2. From (2.9), we have

W[0, 0] = 0,

k1 = #2= .

2.1.1

Nonlinear response functions

It is convenient to introduce the so-called r - a variables with

0

a = 01 - 02,

1

Or - 1(01+02), 2

for which (2.5) becomes

Cet1.;,

= Tr

[poP exp

(2

_j

dt

(#0a(t)Or(t) +

#r(0)a(t))

From (2.12), one obtains a set of correlation functions (in the absence of sources) with specific orderings:

Gal...a.(ti, trj) = _ _ - =i('P0 (t₁) -Oa,(tn))

(2.13)

where a1,- .. ,

a

E (a, r) and = r, a for a = a, r. 7,,a are the number of r and

a-index in { a,.. , a} respectively (7a + n, = 'T). The r - a representation

(2.11)-(2.13) is convenient as (2.11)-(2.13) is directly related to (nonlinear) response and fluctuation

functions, which we will review momentarily.

Equations (2.8) and (2.10) can also be written as

Wi, [Oa - 0, .r} = 0, (2.14) and

W/*[#,1 =,p] IV[-Oa, 0,] . (2.15)

Equation (2.14) implies that

(2.10)

2 (#1 + 02),

2 (2.11)

(2.12)

To understand the physical meaning of correlation functions introduced in (2.13), let us first expand W in terms of 0a's:

W[#a, 01] = i dt1 D(ti)#a(ti) + -

J

dtidt2 Drr(ti, t2)0a(ti)#a(t2) + - - , (2.17) where

Dr...r(ti, , t") 6W - ('P0,(t₁) ... 0r(tn)), . (2.18)

Ill 60a ft1) ... 6#a(tn) _'=

For

#a

0, we have 01 =

#2

= Or = 0. Writing the last expression of (2.18) explicitly in terms of orderings of O's, we find that

1

Dr(t) = (0(t)) , Drr(ti, t2) = {0(t1), 0(t2)}), , (2.19)

and Dr...(ti,--- , t,) is the fully symmetric n-point fluctuation functions of 0, in the

presence of external source

#.

They are referred to as non-equilibrium fluctuation

functions [68, 69] (see also [66]).

One can further expand these non-equilibrium fluctuations functions in the exter-nal source 0(t), for example,

Dr(t1)

=

(0) =

G.(t

1)

+

dt2 Gra(t, t2 #(t2) + - dt2dt3 Graa.(tl, t2, t3)#(t2)#(t3) +

-(2.20)

D,,(tI, t2) = (1 t1), 0(t,)) = (t1, t2) + dt3 Grra.(ti, t2, t)0(t6) +

-(2.21)

where Ga .... were introduced in (2.13). From (2.20), it follows that G, is the

one-point function in the absence of source, and Gra, Graa, - - are respectively linear, quadratic and high order response functions of 0 to the external source. Similarly, G,, is the symmetric two-point function in the absence of source, and Grra, Graa, *

-are response functions for the second order fluctuations. Indeed, writing the last

are the fully retarded n-point Green functions of [70], while G.... is the symmetric n-point fluctuation function [68, 69]. Other Ga... . involve some combinations of symmetrizations and antisymmetrizations.

Note that, by definition, for hermitian operators, all of these functions are real in coordinate space. At the level of two-point functions, one has

Gra(t1, t2) = GR(t1, t2), Gar (ti, t2) = GA(tI, t2), Grr(ti,t2) = Gs(ti,t2),

(2.22) where GR, GA and GS are retarded, advanced and symmetric Green functions respec-tively. Explicit forms of various three-point functions are given in Appendix A.

2.1.2

Time reversed process and discrete symmetries

Let us now consider constraints on the connected generating functional W when Po invariant under certain discrete symmetries. We will now consider a discrete set of fields

#i(t,

7) where i labels distinct fields, and take spacetime dimension to be d.

Suppose that po is invariant under parity P or charge conjugation C, i.e.

PpoPt = po, or CpoCt = po. (2.23)

Then, from (2.7)

W [0, Ii 2i] W [P

,

i(X) ='q[Ufri(PX),

(2.24)

W[#U , 02i] = W [7#1i, Y212i], (2.25)

where we have taken

Po0(x)Pt =rf Oi(P.), COi(x)Ct = 77 Oi(x) . (2.26)

For even spacetime dimensions, Px changes the signs of all spatial directions, while for odd dimensions, it changes the sign of a single spatial direction.

and evolve the system backward in time with the same external perturbations:

eTr

_U

_{(+oo, -00; #1i)poU}

2(+oo, -00; 02i)]

= Tr [p0 (Tei

f

d 2i(t 2i(t)) (teCif dt'Z'e M" (2.27)

It should be stressed that W is a definition and we have not assumed time reversal

symmetry. At quadratic order in

#'s,

we can write W as

W

=

if

ddxiddX₂ 9Gi (x1 - X2)#ai(X1)#aj(X2) + Ki(xI - X2)#ai(Xi)#r(X2),

(2.28)

with symmetric, retarded and advanced Green functions given respectively by

G.'(x) = Gij(x) = Gji(-x), G'(x) - Kij(x), _{G'(x) = k'ij(x) -_ KgCi(-x)}

(2.29)

From (2.27), WT can be written as

WT =

if

ddx1ddx2 Gig(x1- X2)0ai(X1)#aj(X2) -

Ki

(x1 - X2)Oai(x1>)#rj(X2))

(2.30)

but for higher point functions, WT can no longer be directly obtained from W Now let us suppose that po is invariant under time-reversal symmetry, i.e.

TpoT

= P0,

TO(x)T

= ,T) Tx

-t

)

then from (2.7) and (2.27) we find

#T (X) qoi(Tx) (2.32)

For po invariant under some products of C, P, T, the results can be readily obtained

from (2.24)-(2.25) and (2.32). For example, suppose that po is invariant under PT, i.e.

EpoEt = po, (2.33)

(2.31)

then

W[&l, 02i] =W [i, #k], #V(x) 2iTM-X mT .qrP r[* (2.34)

From (2.28) and (2.30), for a system with PT symmetry, (2.34) implies that

Gi (x) = fr Tyr'T Gij (-x), Ky (x) = I4TR7Kji(x) . (2.35)

For higher point functions, (2.34) does not impose any direct constraints on W itself, only relating W to WT.

2.2

Effective field theory

Our goal is to formulate an effective field theory at low energies for a quantum many-body system in some macroscopic state defined by a density matrix po. Our starting point is the microscopic closed-time path integral described in the previous section

Tr (po

.

-) =j

D@'

1D- 2 eiS[V'1-iS[2] --... , (2.36)

where 01,2 is now used to collectively denote the collection of dynamical fields param-eterizing perturbations around the background mixed state po for the two legs of the path.

Following the usual Wilsonian approach, it should be possible to integrate out the gapped modes of the theory to obtain an effective low energy theory for long-ranged degrees of freedom. We assume for our purposes that the only gapless modes of the system in po are those associated with conserved quantities such as the stress tensor and conserved currents for some global symmetries, which we will refer to as the hydrodynamical fields. This assumption is well-motivated for general systems, as detailed in section 1.2.1 - the incorporation of additional gapless modes, for instance in the case of the superfluid, should also proceed straight-forwardly.

energy effective theory for hydrodynamical modes only, given up to an irrelevant normalization by:

Tr (PO-) =

DX

1

DX

2 eiShydo[x'x2P;PO -.. , (2.37)

where X1,2 _{collectively denote the hydrodynamical fields for the two legs of the path,}

and Shydro is the low energy effective action (hydrodynamical action) for them. The

first contentful feature of (2.37) are that the low energy effective action Shydro no longer has the factorized structure of the original microscopic action (2.36) - couplings are generated between the hydrodynamic modes on the two legs of the contour, which is essential to incorporate dissipative effects. Furthermore, the dependence of the background state po will affect the form of Shydro only implicitly through the values of the effective theory coupling constants. In particular, certain symmetries obeyed by the density matrix should manifest as symmetries in the low-energy effective theory. While the integrating-out procedure cannot be performed explicitly, following the usual philosophy of effective field theories, we should be able to write down Shydro in a derivative expansion based on general symmetry principles.

The two basic questions to be resolved, then, are (i) what is the appropriate formulation of the hydrodynamical modes X1,2, i.e. what are the low energy degrees of freedom? It is clear from previous attempts that the standard hydrodynamical variables such as the velocity field and local chemical potential are not suited for writing down an action. (ii) what are the symmetries? We will address these questions in the following two sections.

2.3

Degrees of freedom

As described in (2.5), it is useful to include sources for certain operators, so as to write the generating functional for the operators' correlation functions. In particular, we would like to couple the conserved U(1) current to an external gauge field. Similarly, we can source the stress tensor by considering the theory in a curved background

spacetime. Because these operators are conserved, the generating functional will be invariant under gauge transformations of the external gauge field and diffeomorphisms of the spacetime. This, in turn, motivates a natural definition of the hydrodynamical fields as Stueckelberg-like fields associated with these symmetries.

Considering the case in which we only switch on an external gauge field coupled to the conserved current, the microscopic generating functional is written as (2.7)

e W[A4,A2p] = Tr [Po (T'e~ddxJ-4/A2v) (Te' fd^dXJA1j . (2.38)

The conservation of JP, 0,JP = 0, immediately implies

W[Al,, A2 ] = W[A1, + OAl, A2p + 01,A21 (2.39)

for arbitrary functions A, \2. That is, W is separately invariant under independent

gauge transformations of A1, and A2,. We do expect that W is a non-trivial functional

even at zeroth derivative order, and hence W should not be a local functional of

Ai,, A2,. Of course, this is to be anticipated, as in evaluating W we have integrated

over all dynamical modes, including the gapless hydrodynamic mode associated with the conserved current.

In order to obtain a local action, we should not integrate out the hydrodynamic mode. From (2.39), one can readily guess the answer: we can rewrite W as

ewlA1,,A2p] =

J

DWiD P₂ eiI[BjB2/] , (2.40)

source A, in the specific Stueckelberg combination 2

B, = Ai + 0,W. (2.42)

(2.39) is automatically satisfied by an action taking the specific dependence in (2.40),

as the derivative transformation in the external fields A1,2,, can be compensated by a

shift in the integration variables

%,2-This discussion can be generalized immediately to allow the inclusion of the stress tensor T", turning on the source of which corresponds to putting the system in a curved spacetime. As the background spacetime couples to the stress tensor non-linearly, it is most convenient to use the form (2.9), with the source dependence impleit in the evolution operator:

eW[gpv,A1p;92L,,A2pI Tr [U (+oc, -- oo; gi t,, A1

/,)poU

(+oo, -cc; g2,v, A2 )] (2.43)

where U1 is the evolution operator for the system in a curved spacetime with metric

gi, and external field A,,, and similarly with U2. Due to the covariant conservation of

the stress tensor and the current, W is invariant under independent diffeomorphisms

of 91,2 and "gauge transformations" of A1,2:

W[gi,

A

1; 92 -42] =

W[ ,

A,;

2, A2], (2.44)

where

i

t

t(X) =aPx"ga(y(X))

A, ,(x)

=

4,A(y(x)) + ,A(x),

(2.45)

2_{Here and in future equations, for notational convenience we will suppress the contour leg label}

1, 2 when an equation applies on each leg separately with the leg index appearing on each instance of the dynamical fields and external sources. For example, (2.40) is shorthand for