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Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is extremely well described by hydrodynamics. At the Euler scale, it is typically assumed that each mesoscopic region of the system independently reaches a state of maximal entropy under the constraints given by the available conservation laws: locally, in each"fluid cell", one finds a Gibbs or generalised Gibbs state. 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. In this manuscript, we show that this picture is incorrect: regions separated by macroscopic distances develop long-range correlations as time passes. We provide a universal theoretical framework to exactly evaluate these correlations, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in a paradigmatic interacting model of many-body physics, the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is extremely well described by hydrodynamics. At the Euler scale, it is typically assumed that each mesoscopic region of the system independently reaches a state of maximal entropy under the constraints given by the available conservation laws: locally, in each "fluid cell", one finds a Gibbs or generalised Gibbs state. 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. In this manuscript, we show that this picture is incorrect: regions separated by macroscopic distances develop long-range correlations as time passes. We provide a universal theoretical framework to exactly evaluate these correlations, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in a paradigmatic interacting model of many-body physics, the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.
Introduction.-The dynamics of many-body extended systems has been the focus of much recent theoretical and experimental research [1][2][3][4][5]. Finding universal laws that govern them at large scales away from equilibrium is a fundamental problem in this area. Hydrodynamics is arguably the most far-reaching and successful such set of laws [6]. Crucially, recent research has gone much beyond the conventional hydrodynamic equations of Navier-Stokes, applying the hydrodynamic principles to a large variety of systems. The largest scale of hydrodynamics, the Euler scale, exists and is nontrivial as soon as the system admits ballistic transport and interactions are on short enough distances; in particular, the system must possess at least a few extensive conserved quantities. A prominent example is the theory of generalised hydrodynamics (GHD) [7][8][9][10][11][12], which proposes a universal structure for the Euler hydrodynamics of manybody integrable systems, and which has been shown to correctly describes cold atomic gases constrained to one dimension [13][14][15] and experimentally accessible gases of solitons [16,17]. In general, Euler hydrodynamics only necessitates basic aspects of the emergent degrees of freedom, such as their fluid velocities 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 observables when the system is initiated in a state with long-wavelength variations, and correlations of local observables at large separations in space-time [18][19][20], and the large-deviation theory for long-time ballistic transport [21][22][23][24].
The principles behind Euler hydrodynamics are a simple extension of those of equilibrium [6,25]: the main assumption is simply that in every mesoscopic region, or "fluid cell", the many-body system tends to maximise its entropy, under the local constraints provided by the extensive conserved chargesQ i = dxq i (x, t), withq i (x, t) the associated microscopic densities related to the microscopic currentsĵ i (x, t) via continuity equations ∂ tqi + ∂ xĵi = 0; we concentrate henceforth on onedimensional systems for the sake of simplicity. Thus, in every fluid cell, a different Gibbs state, or generalised [26] characterized by the "generalized inverse temperatures" β in integrable models, is reached. Here, "mesoscopic" refers to a size, L, which is much greater than microscopic sizes micro -such as inter-particle distances and interaction ranges -but much smaller than the size where finite variations of local observables can be seen: A typical initial state with variations on long wavelengths is The emergent, slow dynamics, from the initial state (2) is then simply that induced by the continuity equations for the mesoscopic densities q i (x, t) = q i β(x,t) and currents j i (x, t) =  i β(x,t) as measured in the locally equilibrated fluid cells with A j i = ∂j i /∂q j the flux Jacobian. The mesoscopic densities are related to the generalized inverse temperatures q ↔ β bijectively thanks to positivity of the static covariance matrix C ij and C ij = −∂q i /∂β j .
As part of the general wisdom of Euler hydrodynamics, it may naturally be assumed that, at all times, equal-time correlations between local observables separated by large distances must vanish quickly. Indeed, at equilibrium, correlations are known to vanish exponentially with the distance, under very broad conditions (including the lack of phase transition) [27,28]. As each fluid cell is at "local equilibrium", and, a fortiori, as entropy maximisation should occur independently in every individual fluid cell, correlations between cells that are macroscopically separated should vanish exponentially, even during time evolution.
In this manuscript, we show that the assumption of exponentially decaying correlations between fluid cells is generically incorrect. We show that long-range correlations, on macroscopic distances ∝ , which decay as −1 , [which decay as −1 on macroscopic distance of scale ] develop dynamically at all macroscopic times, from the large wavelength initial state (2), under generic conditions for many-body systems exhibiting ballistic transport. The appearance of ballistically scaled long-range correlations at all positive macroscopic times is a new phenomenon of general nature, which, to our knowledge, has not been discussed before. It holds no matter the nature of the system, quantum or classical, integrable or not. Algebraic decay of correlations has been observed, indeed, only in different nonequilibrium steady-state scenarios, as 1/x in driven-diffusive systems [29,30], and as 1/x 2 in integrable models [31,32]. We explain and quantify this phenomenon by developing a universal theory, which we dub ballistic macroscopic fluctuation theory (BMFT). The BMFT is a hydrodynamic large-deviation theory, solely based on the emergent Euler hydrodynamic data of the model, that characterises fluctuations and correlations at the ballistic hydrodynamic scale. In order to illustrate the general phenomenon and the BMFT predictions, we compare with 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 an excellent agreement.
Ballistic long-range correlations.-We first make more precise the meaning of ballistic long-range correlations and the corresponding physical interpretation. We show that correlations in the large wavelength initial state (2), between observablesô 1 ( x 1 , t) andô 2 ( x 2 , t) that are macroscopically far from each other, generically do not decay exponentially out of equilibrium, but instead decay as −1 . These long-range correlations imply that the connected correlation function ô 1ô2 c := ô 1ô2 − ô 1 ô 2 has a nonzero Euler-scaling limit [33] : when x 1 = x 2 . Note that, under the usual assumption of independent fluid cells, this limiting two point function should be delta correlated as in the initial state. In the following, we will demonstrate that there is an additional nonzero contribution if three conditions are satisfied: (a) the initial state is inhomogeneous, in such a way that there is fluid motion; (b) the system is interacting, in the precise sense that its hydrodynamic flux Jacobian A j i depends on the state; and (c) the model has at least two hydrodynamic modes with different fluid velocities. The latter two conditions are generic in many-body systems that admit ballistic transport. This includes translation invariant Hamiltonian systems such as the anharmonic chain [34], integrable quantum spin chains [8], and integrable gases such as the one-dimensional interacting Bose gas [35] and soliton gases [16]. The result (4) means that correlations decay algebraically at large times and distances. If the initial state has short-range correlation, which is the case for (2), then as time passes, the length over which correlations occur grows linearly from zero, and within it, correlations decay algebraically. As this is on macroscopic distances and times, it means that macroscopically separated points have correlations that decay more slowly than exponentially -thus fluid cells develop in time mutual correlations. In particular, total charges on macroscopic regions, separated by macroscopic distances, have macroscopic correlations. Due to a nonlinear hydrodynamic response of the system, there exist "coherent" fluid modes -emitted from fluid cells in pairs (or larger groups) with correlations -which move from their initial positions due to inhomogeneity. The long-range correlations are the product of such coherent fluid modes of different velocities and scattering amongst each other. Nonlinear response is possible only if interaction is present, and long-range correlations can only occur, after the emission event, if fluid modes propagate at different velocities; this explains the three conditions above. This is a new mechanism, inherently nonlinear and distinct from the well-known mechanism for timecorrelations coming from standard hydrodynamic linear response [19,20,23,36], whereby Euler-scale correlations occur when a mode emitted byô 1 at time t 1 is probed byô 2 at t 2 > t 1 .
In order to prove the result in Eq. (4) and quantify the emergent long-range correlations, we introduce the ballistic macroscopic fluctuation theory (BMFT). The BMFT is similar in spirit to the macroscopic fluctuation theory (MFT) [37][38][39][40][41][42][43][44], which, however, pertains to the diffusive hydrodynamic scale. For the full development of the BMFT, see the companion manuscript [45]. The result (4), implying nonzero correlations between fluid cells, calls for a new understanding of the principles of Euler hydrodynamics to replace independent local entropy maximisation of fluid cells. We believe the BMFT provides the first steps towards this new understanding.
Example: the hard-rod gas.-Before developing the BMFT in full generality, we illustrate the result on the many-body interacting hard-rod model. This model has a well-established Euler hydrodynamics description [18,[46][47][48]. The rods with unit-length a = 1 have ordered positions x i (t) ∈ R that move in time t freely at velocities v i , except at collisions x i+1 (t)−x i (t) = 1 where velocities are exchanged. We consider the rod densitŷ and evaluate the following object: Of course, in real systems, the result (6) is applied for large but finite values of ; in our numerical analysis we find that / micro ≈ 100 is sufficient for a 10% relative accuracy. We choose the initial state · · · as a random distribution of two types of rods, uniformly with velocities v i ∈ {±1} under the constraint that rods do not overlap, and with the "bump" density profile, The use of a discrete velocity distribution allows to make long-range correlations more neatly detectable. In this case, indeed, scattering events, which induce correlations, necessarily occur between normal modes with opposite velocities v and −v (in the hard-rod model, normal modes are the velocity tracers). In the case of a continuum velocity distribution, instead, long-range correlations become smaller as they spread among the whole continuum of available normal modes. In the numerical simulations we observe that correlations decay as 1/ , and thus a finite result is obtained in the limit → ∞ by multiplying by . This is precisely the definition in Eq. (4) of the Euler scaling limit for connected, two-point correlation functions. The result predicted by BMFT, compared with computer simulation of the hard-rod dynamics, is displayed in Fig. 1. Striking are both the fact that the result is nonzero, thus long-range correlations are present, and that the BMFT correctly predicts it. In the following, we quantify this effect for interacting many-body systems with an Euler hydrodynamics description by developing the BMFT.
BMFT.-Euler-scale hydrodynamics is justified through local relaxation of averages in long-wavelength states, such as the state (2). This implies that localobservables averages are function of the densities q only: lim →∞ ô( x, t) = ô β(x,t) := o[q(x, t)] [6,25]. Applying this assumption to the current averages j i [q(x, t)], the Euler equation (3) follows. The fundamental principle of the BMFT can be obtained by extending the above assumption. Namely, we first assume that at the Euler scale mesoscopic charge densities can be identified with classical fluctuating variablesq( x, t) := q(x, t). Local relaxation of fluctuation then implies that, at the Euler scale, not only do averages relax locally, but also meso- The initial state is the density bump in Eq. (7). The collapse of the numerical data in terms of is evident and it thereby numerically confirms the existence Euler-scale long-range correlations of order 10 −2 . Fundamentally, these correlations are captured by the BMFT (red points) with an excellent agreement. The numerical data, corresponding to three different macroscopic scales = 250, 500 and 1000, are reported with the associated uncertainty bars. The rod length is a = 1, while velocities can take only two values v = ±1 with equal probability. The numerical data are obtained by simulating the deterministic hard-rod dynamics and by averaging over a large number of independent realizations of the rods initial positions and velocities. The data corresponding to the scales = 250 and 1000 are shifted horizontally by ±0.04 to make the collapse of the numerical data clearly visible.
where dµ[q(·, ·)] is the flat measure for functions on the space-time region S := R × [0, T ]. This equation implies that fluctuations of mesoscopic densities stems from those of the initial condition and are then deterministically propagated in space-time according to Euler equation. Initial fluctuations of densities q(·, 0) are encoded into the factor e − F[q(·,0)] , where F[q(·, 0)] is the large deviation function, see, e.g., Refs. [40,41,44], representing the state (2) at the Euler scale: with s[q] and f [β] the entropy and free energy density, respectively. With this the BMFT average is given by which we identify with the local equilibrium average • . Formally the delta function is understood as the Laplace where H(x, t) is an auxiliary field. As we are only interested in the large-limit, all results can be obtained by saddle-point calculations -thus we have reduced the problem of Euler-scale correlations to a functional minimisation problem. From Eq. (8) all the BMFT predictions follow.
The BMFT is similar in spirit and inspired by the wellestablished MFT [38], in the sense that they are both large-deviation theories based on an action formalism and saddle-point analysis describing space-time configurations of densities and currents. It is, however, crucial to emphasize that the BMFT and the MFT pertain to different hydrodynamic scales, ballistic and diffusive, respectively, and as such they cannot be derived from one another. In the companion manuscript [45], we also introduce a multi-scale hydrodynamic fluctuation theory describing both the ballistic and the diffusive scale, and we discuss its relation to the BMFT, the MFT and nonlinear fluctuating hydrodynamics.
Long-range correlations from the BMFT.-Euler-scale correlation functions Sq 1,q2 (x 1 , t; x 2 , t) of mesoscopic densities (4) are computed by identifying them with correlation functions of the associated classical random variables q i1,2 , described by the BMFT measure (8), To compute this object, we define Λ = λ 1 q i1 (x 1 , t 1 ) + λ 2 q i2 (x 2 , t 2 ) for the generating function of Euler-scale correlations, for which we have where F Λ [q(·, ·)] = F[q(·, 0)] − Λ. The result follows by saddle-point analysis as → ∞ with q * minimizing the BMFT action The correlation function can be then obtained simply as Using the fact that the action is stationary for q = q * , i.e., δS Λ [q * , H] = 0, we can further recast (15) into where we dropped the asterisk and λ := λ 1 . In terms of the Lagrange multipliers β, the BMFT equations satisfied by the optimal β and H read (using AC = CA T ) with the boundary condition β i (x → ±∞, t) = β i ini (x → ±∞) and H i (x → ±∞, t) = 0. The value of T is of no importance so long as t 1 < T . The formal manipulations of the BMFT equations generically imply that the equal-time correlator Sq i 1 ,qi 2 (x 1 , t 1 ; x 2 , t 1 ) shows longrange correlations where U λ (t, t ) is the time-evolution operator associated to Eq. (17c) that time-evolves (nonlinearly) β(t ) as β(t) = U λ (t, t )β(t ). The first term in (18) is the linear response contribution [19,20,23,36], which is present already under the assumption of independent fluid modes which embodies the assumption of independent fluid cells at t 1 > 0, and it is supported only when x 1 = x 2 . The second term, instead, is generically supported on x 1 = x 2 (the operator U λ from Eq. (18) depends on x 1 ) and accounts for long-range correlations. It describes nonlinear correlated normal modes production and scattering through all the times prior to t 1 . It vanishes when t 1 = 0 (U λ (0, 0) = 1), since in this case the initial state (2) by definition has no correlations; and also when the state is homogeneous (U λ (t 1 , 0)β ini = β ini ) and the model is noninteracting (U λ does not depend on λ). In such cases linear response theory applies. This also justifies the conditions (a) and (b) (see after Eq. (4)) concerning inhomogeneity of the state and interactions, respectively. The condition (c), the presence of more than one fluid velocity, can in fact be directly verified from the Euler equation for Sq i 1 ,qi 2 (x 1 , t 1 ; x 2 , t 1 ) obtained from hydrodynamic projections [6,18,19,[49][50][51][52], see [45] ( hydrodynamic projection formulae also follow from the BMFT formalism). In general, it is challenging to bring the second term on the right hand side of Eq. (18) to an explicit and calculable form. In fact, solving the BMFT equations (17) is generically tricky as shock solutions, where entropy is not conserved, may appear [53,54]. In integrable models, however, this problem does not arise. The Euler hydrodynamics of these systems is described by GHD, which is known to display no shock solutions. Moreover, the specialization of Eqs. (17) to integrable systems can be exactly solved using the method of characteristics [55] leading to the calculable expression Sq i 1 ,qi 2 (x 1 , t 1 ; x 2 , t 1 ) = C i1i2 (x 1 , t 1 )δ(x 1 − x 2 ) + E i1i2 (x 1 , x 2 ; t 1 ) for (18). The function E i1i2 (x 1 , x 2 ; t 1 ) is obtained by solving an integral equation, see [45]. In the case of the hard-rod model, this leads to the result reported in Fig. 1.
Conclusions.-We have shown that long-range correlations (4) generically develop over time in many-body systems at the Euler scale. Such correlations build in time under three rather generic conditions: inhomogeneity in initial conditions, interaction, and the presence of more than one fluid velocity. This result invalidates the fundamental tenet of Euler hydrodynamics that at each time fluid cells undergo entropy maximization independently of each other. In this work (and its companion manuscript [45]), we have provided, to our knowledge, the first observation and description of this universal phenomenon. To achieve this task, we introduced the BMFT, which is a hydrodynamic fluctuation theory describing correlations and fluctuations at the Euler scale. We have evaluated the ensuing expression for these correlations in the hard-rod model observing an excellent agreement with numerical simulations. We expect that these long-range correlations can be experimentally observed using quantum gas microscopes [56].