Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems Benjamin Doyon1Gabriele Perfetto2Tomohiro Sasamoto3and Takato Yoshimura4 5

2025-05-03 0 0 505.85KB 6 页 10玖币

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Emergence of hydrodynamic spatial long-range correlations in nonequilibrium

many-body systems

Benjamin Doyon,1Gabriele Perfetto,2Tomohiro Sasamoto,3and Takato Yoshimura4, 5

1Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K.

2Institut f¨ur Theoretische Physik, Eberhard Karls Universit¨at T¨ubingen,

Auf der Morgenstelle 14, 72076 T¨ubingen, Germany.

3Department of Physics, Tokyo Institute of Technology, Ookayama 2-12-1, Tokyo 152-8551, Japan

4All Souls College, Oxford OX1 4AL, U.K.

5Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K.

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive

many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that

each mesoscopic region independently reaches a state of maximal entropy under the constraints

given by the available conservation laws. Away from phase transitions, maximal entropy states

show exponential correlation decay, and independence of ﬂuid cells might be assumed to subsist

during the course of time evolution. We show that this picture is incorrect: under ballistic scaling,

regions separated by macroscopic distances develop long-range correlations as time passes. These

correlations take a universal form that only depends on the Euler hydrodynamics of the model. They

are rooted in the large-scale motion of interacting ﬂuid modes, and are the dominant long-range

correlations developing in time from long-wavelength, entropy-maximised states. They require the

presence of interaction and at least two diﬀerent ﬂuid modes, and are of a fundamentally diﬀerent

nature from well-known long-range correlations occurring from diﬀusive spreading or from quasi-

particle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical

framework to exactly evaluate them, an adaptation of the macroscopic ﬂuctuation theory to the

Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical

simulations and ﬁnding excellent agreement.

Introduction.— Finding universal laws that govern

many-body extended systems [1–5] at large scales away

from equilibrium is a fundamental problem in physics.

Hydrodynamics is arguably the most far-reaching and

successful such set of laws [6]. The largest scale of hy-

drodynamics, the Euler scale, exists and is nontrivial as

soon as the system admits ballistic transport and inter-

actions are on short enough distances; in particular, the

system must possess at least a few extensive conserved

quantities, and hydrodynamic modes are in one-to-one

correspondence with these. Euler hydrodynamics applies

to a wide array of many-body systems, including gases

and ﬂuids of interacting particles. A prominent example,

which came to the fore recently, is the theory of gen-

eralised hydrodynamics (GHD) [7–12], which proposes a

universal structure for the Euler hydrodynamics of many-

body integrable systems, and which has been shown to

correctly describe cold atomic gases constrained to one

dimension [13–15] and experimentally accessible gases of

solitons [16,17]. The Euler hydrodynamics of a micro-

scopic model only requires the knowledge of basic aspects

of the emergent degrees of freedom, such as their ﬂuid ve-

locities and their static correlations in stationary states.

From these data, it makes a range of nontrivial physical

predictions, including the large-scale motion of local ob-

servables, correlations at large separations in space-time

[18–20], and the large-deviation theory for long-time bal-

listic transport [21–24].

Euler hydrodynamics is based on a simple extension

of equilibrium thermodynamics [6,25]: in every meso-

scopic region, or “ﬂuid cell”, the many-body system is

assumed to maximise its entropy, under the local con-

straints provided by the extensive conserved charges

Qi=Rdxˆqi(x, t). Here ˆqi(x, t)’s are the microscopic

densities, related to the currents ˆ

ji(x, t) via continu-

ity equations ∂tˆqi+∂xˆ

ji= 0; we concentrate on one-

dimensional systems for simplicity. Thus, in every ﬂuid

cell, a diﬀerent Gibbs state, or generalised Gibbs en-

semble (in integrable models) [26], is reached: ⟨•⟩β=

Tr[exp(−Piβiˆ

Qi)•]/Z characterized by Lagrange pa-

rameters β. “Mesoscopic” refers to a size, L, which

is much greater than microscopic sizes ℓmicro – such as

inter-particle distances and interaction ranges – but much

smaller than the size ℓwhere ﬁnite variations of local ob-

servables can be seen, ℓmicro ≪L≪ℓ. Perhaps the

most natural setup where these principles apply is when

an initial state is prepared in the presence of external,

long-wavelength ﬁelds. Maximising entropy with ﬁelds

βi(x/ℓ) coupled to the conserved densities and varying

on large scales ℓ, the state is

⟨•⟩ℓ= Trexph−X

iZdx βi

ini(x/ℓ)ˆqi(x)i•/Z. (1)

The emergent, slow dynamics from the initial state (1)

is that induced by the continuity equations for the

mesoscopic densities qi(x, t) = ⟨ˆqi⟩β(x,t)and currents

ji(x, t) = ⟨ˆȷi⟩β(x,t)as measured in the ﬂuid cells,

∂tqi(x, t) + Aj

i(x, t)∂xqj(x, t)=0,(2)

with Aj

i=∂ji/∂qjthe ﬂux Jacobian. The mesoscopic

densities are related to the generalized inverse tempera-

arXiv:2210.10009v2 [cond-mat.stat-mech] 18 Jul 2023

tures q↔βbijectively thanks to positivity of the static

covariance matrix Cij =−∂qi/∂βj.

Euler hydrodynamics asserts that the slow variation in

space of the state (1) induces a corresponding slow vari-

ation in time, such that the state keeps the local equi-

librium form; one may propose that at all times (1) cor-

rectly describes the states, with βi

ini(x)→βi(x, t) (see,

e.g., [6,25,27]). In (1), correlations vanish exponentially

with the distance, under very broad conditions including

non-zero temperature (ﬁnite βi’s) and the lack of phase

transition [28,29] (which we assume here). Thus, spa-

tial correlations should vanish exponentially, even during

time evolution; indeed each ﬂuid cell is at “local equilib-

rium”, and entropy maximisation should occur indepen-

dently in every ﬂuid cell [19,23,30,31].

In fact, certain long-range, algebraic correlations are

known to emerge in non-equilibrium situations when con-

servation laws are present. This is well studied for dif-

fusive systems in non-equilibrium steady states (NESS)

[32–34]: unbalanced thermostats at the system’s bound-

aries lead to nonzero gradients, and correlations between

conserved densities at macroscopic distances decay as

(system size)−1. This is due to the breaking of detailed

balance at the diﬀusive scale and determined by viscous

coeﬃcients, and may be quantitatively described by ﬂuc-

tuating hydrodynamics and macroscopic ﬂuctuation the-

ory (see, e.g., [33–35]). But what happens at the Euler

scale, where viscous eﬀects are scaled down to zero size?

In NESS emerging from the partitioning protocol in

systems of inﬁnite size [36], gradients vanish and corre-

lations are weaker. The strongest are found in integrable

systems, including free particles, where conserved density

correlations decay as (distance)−2because of discontinu-

ities in the occupation function of hydrodynamic modes

[37,38]. But this decay is too quick to correlate Euler-

scale ﬂuid cells (see below).

We note that a similar situation occurs at zero temper-

ature, under the diﬀerent physics of quantum ﬂuctuations

at Fermi points, and that a theory for the transport of

such weak algebraic correlations on top of moving ﬂuids

is proposed in [30,39] (in GHD). Very far from equilib-

rium, stronger long-range correlations may develop: for

instance, global quantum quenches generate ﬁnite den-

sities of entangled particles that may propagate (diﬀu-

sively or ballistically) and carry nontrivial entanglement

[40–42] and correlations [43,44]. But entangled particle

production is not expected to occur in long-wavelength

states, Eq. (1).

Up to now, there has been no prediction, observa-

tion or theory for eventual long-range correlations emerg-

ing under ballistic scaling from (1). The assumption of

uncorrelated Euler-scale ﬂuid cells, and that the form

(1) stays valid in time, has remained, and appears to

play an important role in recent studies of the evolu-

tion of correlations and ﬂuctuations under inhomoge-

neous conditions and nonlinear hydrodynamic response

theory [19,23,30,39,45].

In this manuscript, we show that the assumption of

uncorrelated Euler-scale ﬂuid cells is generically incor-

rect. We show that correlations of strength ∝ℓ−1develop

dynamically from (1), at macroscopic (∝ℓ) times and

distances, under generic conditions for systems exhibit-

ing ballistic transport. In particular, if QR

i(ℓt), QR′

j(ℓt)

are total charges lying on ﬁnite but macroscopically

large regions R, R′that are separated by a macro-

scopic distance, |R|,|R′| ∝ ℓ, dist(R, R′)∝ℓ, evaluated

at macroscopic time ℓt, then their covariance is large,

⟨QR

i(ℓt)QR′

j(ℓt)⟩c∝ℓ. This shows strong correlations

between separated cells. The appearance of ballistically

scaled long-range correlations at all macroscopic times is

a general phenomenon which, to our knowledge, has not

been discussed before. It holds no matter the nature of

the system, quantum or classical, integrable or not, and

is solely controlled by its Euler hydrodynamics.

This phenomenon is not explained by the theories for

diﬀusive long-range correlations recalled above, as it does

not depend on viscous coeﬃcients or phenomenological

noise, and occurs in ballistic times t∝x. It gives the

dominant correlations on large distances, beyond diﬀu-

sive broadening and of larger strength than the 1/x2cor-

relations due to occupation discontinuities. It is not due

to quasi-particle excitations, as it is a universal hydrody-

namic eﬀect. By contrast, we show that the phenomenon

occurs in long-wavelength inhomogeneous state (as in

(1)), only if the Euler hydrodynamic theory is interact-

ing, and only if it admits at least two diﬀerent ﬂuid modes

(with diﬀerent velocities). Euler-scale long-range correla-

tions invalidate the assumption that on every time-slice a

state such as (1) is found. This thus calls for a new under-

standing of the principles of Euler hydrodynamics, and a

re-think of recent studies of hydrodynamic nonlinear re-

sponse and the evolution of correlations and ﬂuctuations.

We quantify this phenomenon by proposing that the

principle replacing independent local entropy maximisa-

tion of ﬂuid cells is that of relaxation of ﬂuctuations: lo-

cal observables relax to ﬁxed, non-ﬂuctuating functions

of conserved densities, which themselves ﬂuctuate. This

is developed into a universal theory, the ballistic macro-

scopic ﬂuctuation theory (BMFT). The BMFT is a hy-

drodynamic large-deviation theory, solely based on the

emergent Euler hydrodynamic data of the model, which

characterises all ﬂuctuations and correlations at the bal-

listic hydrodynamic scale, including under ﬂuid motion.

For illustration, we study the paradigmatic hard-rod

model of statistical physics, which is simple enough to

be amenable to high-accuracy numerical simulations, yet

truly interacting. We ﬁnd that the model does indeed

develop long-range correlations, which are quantitatively

in excellent agreement with our theory.

Ballistic long-range correlations.— We show that

correlations in the initial state (1), between macroscopi-

cally separated observables ˆo1(ℓx1, ℓt) and ˆo2(ℓx2, ℓt) at

macroscopic times, generically has strength ℓ−1. That

is, the connected correlation function ⟨ˆo1ˆo2⟩c:= ⟨ˆo1ˆo2⟩ −

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摘要：

Emergenceofhydrodynamicspatiallong-rangecorrelationsinnonequilibriummany-bodysystemsBenjaminDoyon,1GabrielePerfetto,2TomohiroSasamoto,3andTakatoYoshimura4,51DepartmentofMathematics,King’sCollegeLondon,Strand,LondonWC2R2LS,U.K.2Institutf¨urTheoretischePhysik,EberhardKarlsUniversit¨atT¨ubingen,AufderM...

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Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems Benjamin Doyon1Gabriele Perfetto2Tomohiro Sasamoto3and Takato Yoshimura4 5

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