Correlations and forces in sheared fluids with or without quenching

Spatial correlations play an important role in characterizing material properties related to non-local effects. Inter alia, they can give rise to fluctuation-induced forces. Equilibrium correlations in fluids provide an extensively studied paradigmatic case, in which their range is typically bounded by the correlation length. Out of equilibrium, conservation laws have been found to extend correlations beyond this length, leading, instead, to algebraic decays. In this context, here we present a systematic study of the correlations and forces in fluids driven out of equilibrium simultaneously by quenching and shearing, both for non-conserved as well as for conserved Langevin-type dynamics. We identify which aspects of the correlations are due to shear, due to quenching, and due to simultaneously applying both, and how these properties depend on the correlation length of the system and its compressibility. Both shearing and quenching lead to long-ranged correlations, which, however, differ in their nature as well as in their prefactors, and which are mixed up by applying both perturbations. These correlations are employed to compute non-equilibrium fluctuation-induced forces in the presence of shear, with or without quenching, thereby generalizing the framework set out by Dean and Gopinathan. These forces can be stronger or weaker compared to their counterparts in unsheared systems. In general, they do not point along the axis connecting the centers of the small inclusions considered to be embedded in the fluctuating medium. Since quenches or shearing appear to be realizable in a variety of systems with conserved particle number, including active matter, we expect these findings to be relevant for experimental investigations.

The use of such models provides generic scenarios, which we expect to be relevant for physical systems which allow for shearing and/or quenching. Shear is directly experimentally accessible [48]. Quenches can also be realized, for instance by using effective interactions of particles which can be changed suddenly, e.g., by swelling particles [49] or through external fields [50]. Another type of quench concerns a sudden change of temperature, which is a perturbation often employed in order to obtain supercooled liquids [51]. Such quenches of temperature (or of noise strength) may be achieved experimentally also in active fluids [52][53][54], which in many respects can be described by the use of effective temperatures [27,55,56].
The manuscript is structured as follows. In Sec. II we begin with a detailed description of the system as well as of the model under consideration. Post-quench correlations in the absence of shear are briefly reviewed in Sec. III. In Sec. IV, this is followed by an analysis of the effect of conservation laws on steady state correlations of weakly sheared systems. The case of dissipative dynamics is discussed in Sec. IV A, while Sec. IV B deals with conserved density fluctuations. The dependence of the (equal-time) correlation function on space and time is computed for model B in Sec. V. Using the formal solution derived in Sec. V A, this quantity can be determined analytically in certain limits (Sec. V B). Correlations between points advected by the shear field are discussed in Secs. V B 2 and V C for various limiting cases. Section VI presents a formalism for computing non-equilibrium fluctuation-induced forces in the presence of quenching and shearing. This extends the framework of Ref. [35] to include shear. While this formalism holds for various geometries which do not couple to the shear flow, such as films formed by parallel plates, it is employed in Sec. VI B in order to compute forces between small inclusions embedded in steadily sheared systems, as well as for dynamic post-quench forces (PQFs) under shear (Sec. VI C). In Table I we provide a glossary of commonly used quantities.

A. Coarse-grained model: equilibrium properties
Aiming at the analysis of correlations in classical fluids, which extend far beyond microscopic length scales, we employ classical field theory based on the Landau-Ginzburg theory for a scalar order parameter field φ [1,2]. With a one-component fluid in mind, φ describes density fluctuations φ(x, t) = ρ(x, t) − ρ(x, t) , where ρ(x, t) is the (snapshot) number density distribution. In the case of binary liquid mixtures (in the mixed state), ρ(x, t) corresponds to a local concentration of the particles. The vector x is a d-dimensional position vector, and t denotes time. Thermodynamically far from phase transitions, a Gaussian Hamiltonian H is expected to capture the leading influence of the fluctuations. Thus we consider which induces a (bulk) correlation length ξ = κ/m. In thermal equilibrium and for d > 2, the Hamiltonian in Eq. (1) gives rise to the following two-point correlation function [2]: where S d = 2π (d+1)/2 /Γ( d+1 2 ) is the surface area of a ddimensional unit sphere. T is the temperature and k B is the Boltzmann constant. For systems far away from phase transitions, ξ is small so that the lower line in Eq. (2), i.e., |x| ξ applies. In this regime, the correlation function decays exponentially as a function of x/ξ. (Here we do not consider the presence of long-ranged forces such as van der Waals forces, which asymptotically give rise to an algebraic decay even for |x| ξ.) Within the Gaussian approximation, m in Eq. (1) can be expressed in terms of the isothermal compressibility [57][58][59] (V is the system volume): where ρ 0 is the mean bulk density.

Equations of motion
The dynamical description employed here is based on the Hamiltonian given in Eq. (1) and on Langevin equations for the field φ within model A and model B [32]. These models consider non-conserved and conserved dynamics, respectively. As far as shear is concerned, we consider a simple shear velocity profile v, so that any feedback effects of the field φ onto the velocity profile, as well as fluctuations of v, are neglected (in contrast to model H [1], which includes these couplings). Using v =γye x with shear rateγ ≡ ∂v x /∂y [48], the Langevin equation reads where the white noise obeys the spatio-temporal correlations The mobility operatorμ encodes whether φ is conserved or not:μ We note that the coefficients µ A/B carry different dimensions. Defining the Fourier transform F as f (k) = vector between two fixed or two co-moving points in the shear flow, respectively Sec. V B 2 a (u) for any vector u = (ux, vector connecting two stationary (L = L0) or co-moving (L = L(t)) inclusions in shear flow Sec. VĨ λ,ξ λ, ξ rescaled by |x| (correlation functions) or |L0| (forces) Eq. (24), Sec. VI steady-state force (in model A/B) between two inclusions in shear flow Sec. VI B Fs(L0, t) post-quench force (PQF) between two stationary inclusions separated by L0 Sec. VI C 2 Fc-m,γ(L(t), t) PQF between two co-moving inclusions in a sheared fluid Sec. VI C 3 Fc-m,0(L(t), t) PQF between inclusions following the co-moving trajectory L(t), but fluid is unsheared Sec. VI C 3 Table I. Glossary of the quantities most frequently used in the present study. (4) and (5) can be expressed in Fourier space: where we have introduced the operator and representedμ(x) from Eq. (6) in terms of its spectrum The quantity of our interest is the time-dependent structure factor C(k, t). It is defined as C(k, t) depends on time because the system is out of equilibrium. It is evaluated at equal times, and evolves according to which has the general solution where C(k, t = 0) is the structure factor at t = 0. Importantly,Ô(k, {∂ ki }) comprises powers of k i and ∂ ki with i = 1 . . . d, and therefore the exponents must be expanded using the Zassenhaus formula [60]. Expressions such as in Eq. (12) have been discussed in the literature -see, e.g., Refs. [41][42][43][44][45][46]. (We note that there are discrepancies of a factor of 2 among the latter references regarding the coefficients of the terms in Eq. (11). According to our derivation, Eq. (11), which follows directly from the Langevin Eq. (7), fixes these constants via Eq. (8).) However, our aim is to obtain explicit expressions in position space: using Eqs. (10) and (12), the time-dependent equal-time correlation function can be found by Fourier inversion. Here |x| is the distance in the bulk between two points the correlation of which is being considered.

Quenching at t = 0
We consider the dynamics defined by Eqs. (4) and (5) subject to a quench at time t = 0; this amounts to a sudden change of one or more of the parameters in these equations. For instance, this parameter can be the temperature T . Such a description in terms of instantaneous changes of parameters is based on the assumption that processes at small length scales relax on short time scales, so that the mesoscopic parameters rapidly attain their new values. Thus, in the general solution given by Eq. (13), the first term on the rhs depends on the parameter values before the quench (in the following denoted with subscript I), while the second term on the rhs depends on the parameter values after the quench (for which no subscript is used).
Physically, the parameters in Eqs. (4) and (5) are in general not independent; for example, a change in temperature may also change the coefficent m via Eq. (3). However, we treat these quantities as being independent, thereby allowing for a wide variety of quenching scenarios.
Regarding Eq. (2), we note that, in the absence of shear, the correlation function depends on the ratio k B T /m and on the correlation length ξ. Therefore a quench induces a non-equilibrium, transient dynamics if one of these parameters is changed.

Important length scales
It is useful to introduce the collective diffusion coefficient D [61], which follows from Eq. (4): We recall that, in model A, D vanishes in the limit ξ → 0, because in the absence of correlations, the relaxation mechanism of model A is local and not diffusive. D gives rise to two length scales: Here λ is the length scale on which shear and diffusion have comparable effects, i.e., regions with |x| λ are diffusion-dominated, while regions with |x| λ are shear-dominated. The quantity t is the typical distance covered by diffusion within the time t.
Thus the Langevin equation (4) depends on the length scales |x|, t , λ, and ξ, where |x| is the observation scale of a given observable. Regarding notation, we shall employ vectors x when referring to points in the bulk, and vectors L when denoting vectors connecting external objects immersed in the fluid (e.g., for computing forces between certain objects in Sec. VI).

III. QUENCH IN THE ABSENCE OF SHEAR
Here, we briefly recall the main findings of Refs. [26,27], in which quenches in the absence of shear were studied.
The explicit evaluation of Eqs. (12) and (13) forγ = 0 yields, within model B and to leading order in ξ (recall that indices I denote parameters before the quench), The dimensionless quantity is obtained by rescaling time by the diffusive time scale across the distance |x| [see Eq. (15)]. Equation (16) shows that a quench gives rise to non-equilibrium LRCs, which, by virtue of their algebraic spatial decay, extend beyond the correlation length ξ. These LRCs are, to leading order in ξ, independent of ξ. The rescaled time [compare Eqs. (16) and (2)] illustrates that t = √ Dt plays the role of a time-dependent correlation length. For long times, the correlation function in Eq. (16) decays algebraically in time, as the system approaches the new equilibrium state. The result within model A is qualitatively different in that the range of the correlations is restricted by ξ, so that for |x| ξ, de facto no correlations are present. Therefore the conservation law associated with model B is the key ingredient which explicitly gives rise to the aforementioned non-equilibrium LRCs.

IV. PERTURBATIVE ANALYSIS OF WEAK SHEAR IN STEADY STATE
Having reviewed the quenching process without shear in Sec. III, we proceed by analyzing the case of shear without quenching, i.e., the case of a steadily sheared system.

A. Non-conserved density fluctuations: model A
Since solving Eq. (11) for arbitraryγ is challenging, we treat shearing perturbatively, i.e., we expand the correlation function according to with C (0) and C (1) evaluated atγ = 0, i.e., without shearing. Such an expansion is valid if the length scale λ in Eq. (15) is the largest one to be considered, i.e., Thus, the above expansion in terms of powers of the shear rate is valid for small observation scales |x|, short times t, and small correlation lengths ξ. Since here we consider steady states (i.e., times long after the quench), t in Eq. (19) is replaced by τ , where τ is the time scale for the relaxation of density fluctuations in the system.
The structure factor C(k, t) obeys the differential equation (11) with µ k = µ A , and the steady state can be obtained from the limit t → ∞. C (0) is found as [2] Fourier inversion of Eq. (20) yields Eq. (2) above.

B. Conserved density fluctuations: model B
In the case of model B dynamics, C(k, t) obeys Eq. (11) with µ k = k 2 mµ B . The expression for zero shear is identical to Eq. (20), reflecting the fact that the choice of the dynamic model has no influence on the equilibrium (Boltzmann) distribution. The term linear in the shear rate follows from the perturbative expansion of Eq. (18), which in this case gives As above, this expression can be Fourier inverted analytically for d = 3 [62], yieldinġ We recall from Eq. (2) that equilibrium correlations decay exponentially for |x| ξ. In stark contrast, the correlations in Eq. (26) extend beyond ξ, decaying algebraically. Accordingly, shear is a qualitative correction, in contrast to the above findings for model A. In order to illustrate this further, we consider the limit of small ξ = ξ/|x|, i.e.,ξ 1: The latter result exhibits again the aforementioned difference to Eq. (22) in that it is scale-free with respect to |x|. However, it is also structurally distinct from Eq. (16) in that it carries ξ as a prefactor.
The expression in Eq. (26) diverges for ξ → ∞, which in this limit points to a non-analytic dependence oṅ γ [61]. Indeed, Eq. (19) requires ever smaller values ofγ in order for Eq. (26) to remain the leading term. This is in contrast to model A, for which λ increases with ξ.

V. QUENCH AND SHEAR
In Secs. III and IV the effects of quenching and shearing were discussed separately. Here, we shall analyze their combined effects. Since in model A no post-quench LCRs are found, we restrict our studies to model B throughout. As before, quantities before the quench are denoted with subscript I, and parameters after the quench carry no subscript.

A. Formal solution
The formal solution given in Eq. (12) yields where we have introduced the advected wave-vector and the function The first term on the rhs of Eq. (28) represents the relaxation of the initial equilibrium correlations, with In the absence of a closed solution of the Fourier inversion of Eq. (28), we shall analyze this expression in various limits. For |x|/ξ = ∞, Eq. (28) can be inverted analytically, as discussed in Sec. V B. Subsequently, we shall provide perturbative expressions for finite but large values of |x|/ξ in Sec. V C.

Correlations between spatially fixed points
In the limit of a large observation length scale relative to ξ, i.e., |x|/ξ → ∞, the terms k 2 κ in the time integral in Eq. (28) [see also Eq. (30)] can be dropped, and the integral can be carried out explicitly. We find (with This expression can be Fourier-inverted, yielding a result which is valid in any order of the shear rate (for |x|/ξ → ∞, we drop the first term in Eq. (31), which amounts to a local contribution ∝ δ (d) (x)): Here x ⊥ is the component of x perpendicular to the flow direction e x . We note that C has precisely the functional form of the probability density of a particle diffusing in shear flow [compare, e.g., Ref. [61]]. Equation (32) differs, however, in that here the time is twice as large. This is because C follows Eq. (11), which carries an extra factor of two compared to the diffusion equation. (This is a generic observation when comparing dynamics of a stochastic variable and its correlation function.) The prefactor in Eq. (32) shows that this long-ranged contribution is absent without a quench. C(x, t) in Eq. (32) Here, in the final step, we have expanded the expression for λ { t , |x|} (i.e., linear response inγ) and λ { t , |x|} (strong shear), and used Ω 2 At zero shear (γ = 0), the correlations of Eq. (16) are recovered.
In steady state, C(x, t → ∞) vanishes. Equation (33) shows that the correlation between two points after a quench depends on the orientation of the vector connecting them (relative to the shear velocity v). Equation (33) is illustrated in Fig. 1 for certain choices of parameters, where the functional form in the second line of Eq. (33) was used. The vertical axis is truncated, because the limit t → 0 is (infinitely) sharply peaked.

Correlations in the frame co-moving with shear
It is instructive to consider two points which are advected by shear flow, i.e., separated by the vector x(t) = x 0 +vt = x 0 +y 0γ te x , because this is the natural trajectory of a particle in flow. The corresponding postquench correlations in the co-moving frame (as indicated by subscript "c-m") can be inferred from Eq. (32); for d = 3 we find Here x ⊥ 0 = (y 0 , z 0 ) labels the components of x 0 perpendicular to the shear direction e x , with a ≡ |x ⊥ 0 |/|x 0 |. In the last line, t * is an abbreviation for t * |x0| = Dt/|x 0 | 2 [see Eq. (17)], i.e., we rescale time by the time scale of diffusion across the initial separation |x 0 | = 0.  16), i.e., for the system without shear and evaluated at a distance |x 0 | between the two points, as given by the curve with λ = ∞. We note that, especially at early times, the correlations of Eq. (34) can be larger than the ones of the corresponding quiescent system, even though the correlations are taken between points at larger distances. The maximum of the curve can be tuned by choosing 1]. However, at late times shearing speeds up the decay of correlations. While equilibrium correlations decay as (t * ) −3/2 , the curve corresponding to Eq. (34) decays as (t * ) −5/2 [see Fig. 2 The correlation function of an unsheared fluid after a quench [see Eq. (16)] evaluated at the distance |x(t)|. This result, which still depends onγ (and thus λ) via x(t), illustrates the effect of moving the observation points along a "shear trajectory", in contrast to shearing the medium itself. These correlations are in general weaker than those resulting from Eq. (34), and exhibit a qualitatively different behavior in that they decay exponentially at late times. The latter occurs because, for any finite a, the two points move apart faster than the diffusion of the correlations. For a → 0, the correlations between co-moving points in the unsheared system collapse onto the curve corresponding toλ = ∞, because in this limit x(t) → x 0 ∀t, i.e., the stationary case is recovered. Generically, correlations are maximal for a = 0 both in sheared and unsheared systems.

C. Perturbative solution for non-zero |x|/ξ with weak shear
In this subsection we evaluate C(k, t) from Eq. (28) in order to include effects of a nonzero correlation length ξ, especially regarding the change of ξ during a quench. The expression in Eq. (28) can be determined analytically in the limit of small shear and large |x|/ξ. For d = 3 we find 16 √ 2(πt * ) 3/2 . All solid curves forλ = ∞ decay asymptotically as ∝ (t * ) −5/2 ; differences at early times are shown in the inset. For further details see main text after Eq. (34). where Above, t * = t * |x| [see Eq. (17)], ∆ξ = ξ − ξ I reflects any change in correlation length during the quench, and ∆ξ n ≡ ξ n − ξ n I . Further, we usel = l/|x| for l ∈ {ξ, λ, ∆ξ}.
Equations (35) and (36) reveal explicitly the origin of the various contributions due to quenching and shearing. The first line of Eq. (35) is generated by a change of the ratio k B T /m during the quench (denoted by the superscript "Q"), expanded in terms of small shear and a large length ratio |x|/ξ. This contribution mostly aligns with the discussion in Sec. V B, extended by including finite values of |x|/ξ.
The first term in the second line is the only one with a nonzero long-time value. This term describes a system with shear (denoted by a superscript "S") starting at t = 0 in the absence of a quench. It thus relaxes to the result of Eq. (27), with lim t * →∞ h Ṡ γ,ξ 2 (t * ) = 3 2π . The second term in the second line represents relaxation (denoted by a superscript "R") after a quench of the correlation length ξ itself, as captured by ∆ξ. Concerning the nomenclature for the functions h, subscripts denote the order of perturbation in shear and correlation length, respectively. Figure 3 shows the time dependence of the various amplitudes. Panel (a) provides linear scales, while panel (b) displays the long-time behavior on logarithmic scales. We see that the steady-state contribution h Ṡ γ,ξ 2 (t * ) is approached algebraically at late times. All shear-dependent contributions exhibit a dependence on the orientation of x.
Finally we point out the difference between quenching the ratio k B T /m and quenching the correlation length ξ (under the proviso that in an experiment these quantities can be quenched independently): at leading order (in t * ), a quench of ξ only renders correlations which decay more rapidly in time than the corresponding correlations due to quenching the ratio k B T /m. (This is easily inferred from the exponents of the algebraic longtime tails in Fig. 3.) The spatial algebraic prefactors of the various contributions in Eq. (35) also differ, so that (for fixed ξ and λ), the contributions ∝ h Q 0,ξ 2 and h Q/Ṙ γ,0 are shorter-ranged than those ∝ h Q 0,0 and h

VI. NON-EQUILIBRIUM FORCES BETWEEN TWO SMALL INCLUSIONS IN A SHEAR FIELD
Long-ranged correlations give rise to fluctuationinduced forces between objects which confine the fluctuations [4]. Such forces occurring after a quench have been analyzed, for instance, in Refs. [26,27,[33][34][35]. In the spirit of the above discussions, we now investigate post-quench forces between inclusions in shear. We start by generalizing the formalism of Ref. [35] for the computation of time-dependent non-equilibrium forces after a quench with shear, and then apply these results to the case of small inclusions, using the correlation functions computed above.

General derivation
In the context of model A and model B dynamics, Dean and Gopinathan derived a formalism for computing non-equilibrium forces which emerge between immersed objects after a quench in systems described by bilinear Hamiltonians of the form [35] The force F (t) = − ∇ L H[φ(t)] (averaged over noise realizations) is computed from the instantaneous configuration of φ, and L is the relevant vector separating the external objects (e.g., two plates or two finite-sized inclusions). In Appendix A, we extend this formalism in order to include shear flow. The main result is that the Laplace transform of the time-dependent (nonequilibrium) force following a quench can be computed from an effective equilibrium theory: where f (s) = L[f (t)](s) = ∞ 0 dt e −ts f (t) denotes the Laplace transform of f (t). Equation (38) states that the non-equilibrium force is given by the equilibrium force corresponding to an s-dependent Hamiltonian with ∆ (γ) corresponds to the advection term in the Langevin equation (4). This s-dependent Hamiltonian leads to the following s-dependent partition sum: Accordingly, the force is obtained by taking the gradient with respect to the separation L, as stated in Eq. (38).
We emphasize that Eq. (38) rests on the assumption that the external objects do not alter the shear flow. This is expected to be valid in the case of plates oriented parallel to shear, or for the small inclusions which will be investigated below.
Remarkably, at times long after the quench, the force in Eq. (38) adopts exactly the equilibrium form, but with a shear-dependent pseudo-partition sum. In Eq. (40) and in what follows, we suppress the implied average for brevity [compare Eq. (38)].

Two inclusions of finite size
We now apply this result to determine nonequilibrium forces between two inclusions with volumes V 1 and V 2 , respectively, separated by a vector L pointing from the first to the second inclusion, in the limit of large separation (L V 1/d i ). We model these inclusions in terms of local, quadratic contributions to the Hamiltonian: with the first (bulk) term given in Eq. (1), and where  For inclusions held at a fixed relative position, one has L = L0. For inclusions following a co-moving (advected) trajectory in shear flow, L = L(t) ≡ L0 +γtL0,yex. The separation of the inclusions is taken to be much larger than their radii. In this limit, L becomes independent of the choice of reference points within Vi to be connected by L.
represents the inclusions in terms of coupling constants c 1 and c 2 . Accordingly, the above formalism can be applied. Thus the inclusions are modelled by a contrast of the mass inside the inclusions relative to the bulk value m: For simplicity, we consider k B T I /m I = 0, i.e., there are no fluctuations before the quench (corresponding, e.g., to a low initial temperature). The quench gives rise to a non-equilibrium force which can be expressed in terms of a pseudo-potential V, derived from a cumulant expansion of H tot (see Appendix A): In Eq. (45), the displacement vector L can depend explicitly also on time, for instance if one considers the force between moving objects, as discussed below. This case will be indicated by L(t), while L 0 refers to spatially fixed inclusions.
In the long time limit, one has so that the force in the steady state with shear can be inferred easily from the equal-time two-point correlation function in the bulk, adopting the same form in terms of correlation functions as in equilibrium.

B. Forces in steady state under shear
Using Eq. (46) and the results of Sec. IV, one can directly provide the non-equilibrium forces in the steady state under shear. For model A, the correlations in Eq. (22) yield the following force vector: Here L 0 denotes the (stationary) vector joining the inclusions [compare L(t) in Subsec. VI C 3), the orienta-tion of which is captured by Ω α ≡ Ω (L0) α ; see Eq. (23)]. F A (t → ∞) decays exponentially forξ → 0.
For model B, we use Eq. (27) in order to obtain the steady state force between the inclusions to leading order in shear: Unlike F A , F B (ξ → 0) vanishes according to a power law. Thus, in model B, shear gives rise to a qualitatively relevant steady state force between the inclusions even for small correlation lengths. Strikingly, and in stark contrast to equilibrium steady states, the conservation laws of the underlying dynamics strongly influence the phenomena observed in shear-induced steady states. We also note that, in general, neither F A nor F B are parallel to L 0 ; this also differs from the equilibrium case where, by symmetry, the forces are necessarily along the separation vector [26,35].

C. Forces after quenches
In this subsection we compute time-dependent forces between the two inclusions after a quench. We focus on the limit of vanishing ξ, in which no post-quench correlations are observed within model A. Accordingly, the remaining analysis proceeds in terms of model B dynamics; henceforth, the corresponding subscript "B" for the force will be dropped.

Prerequisites
In order to compute the time-dependent force after quenching, the Laplace transform of the correlation function is required. For two points in the bulk at large separations compared to the correlation length, i.e., |x|/ξ 1, the Laplace transform of Eq. (33) reads, in terms of an expansion for small shear rates: where s * = |x| 2 s/D is the rescaled, dimensionless Laplace variable, and Ω α ≡ Ω  (49), the force is expanded in terms of powers n of the shear rate: where the vector L connects the first inclusion to the second one. Equation (50) is obtained by evaluating the relevant correlation functions at x = L. First we consider the force between two inclusions which are placed at fixed positions (L = L 0 ). In a second step, we provide the force between two inclusions advected by the shear flow (L = L(t)).

Post-quench force between inclusions at fixed positions
We now consider the dynamics of the force given in Eq. (48). In accordance with Eq. (50), at order n, the force between the two (stationary) inclusions is The subscript "s" has been introduced in order to distinguish the present case of stationary inclusions from the co-moving inclusions considered in Subsec. VI C 3.
Regarding notation, we note that here (and in the following subsections) the vector L 0 sets the length scale relative to which we defineλ = λ/|L 0 | and the diffusive time t * |L0| = Dt/|L 0 | 2 [see Eq. (17)]; henceforth this dependence will not be indicated explicitly. The components f   few orders of Eq. (50) can be computed explicitly: In the following, Ω α ≡ Ω  At late times after the quench, the components of the force decay algebraically in time: x/y (t * 1) = From Eqs. (51) and (52) we can construct the magnitude and the unit vector of each order contributing to the force. Introducing N (n) = α 1 α 2 k B T /(|L 0 | 7−2n λ 2n ), we find While the zeroth-and first-order shear corrections decay as (t * ) −5/2 at late times after the quench, the second-order correction decays more slowly as (t * ) −1/2 . This is a remnant of Eq. (33), which shows that higher orders in shear are important at late times. Shear thus appears to dominate forces at late times. However, this regime is not accessible within the present approach. Indeed, Eq. (33) (evaluated at x = L 0 ) provides the condition for the crossover between the regimes of weak shear at short times and strong shear at late times, which must be satisfied for the expansion in Eq. (51) to hold.
In Fig. 5 we show the time-dependent components of the force at zeroth order in shear. This limit corresponds to the result in Ref. [26] for the post-quench force in a homogeneous system. Due to diffusing correlation fronts passing the inclusions [26] (also see Fig. 1), the force changes sign at the reduced time In the absence of shear, the force is parallel to the separation vector L 0 . With shear, the force is modified. Figure 6 shows the magnitude of the first shear correction. This correction depends sensitively on the orientation of L 0 , and is maximal when L 0 lies in the x-y plane, i.e., for ϑ = π/2. Additionally, a larger separation along the y-axis implies a corresponding larger difference in shear velocity. The sign of this correction depends, inter alia, on the orientation of L 0 , i.e., on ϑ, ϕ, and t * . Accordingly, the inclusions may experience an increase or decrease of the force due to shear, depending on their orientation relative to the shear plane.
The angle between L 0 and F s (L 0 , t) is determined by the unit separation and force vectors: Using Eq. (52), Eq. (56) is expanded in terms of powers of the shear rateγ = D/λ 2 : The lowest order represents the change of sign as it occurs in the absence of shear [see Eq. (55)]. Due to symmetry, δ s does not carry a first-order shear correction ∝λ −2 . The second-order shear correction in δ s displays a singularity at t * = t * sgn , associated with the divergence of the normalization of the unit vectors in Eq. (56), so that the weak-shear expansion becomes invalid. The angle between the force and the vector connecting the inclusions depends on the orientation of L 0 as well as on time, because the post-quench correlations are distorted by shear and the magnitude of this distortion depends on the position of the inclusions in the shear field. Formally, one can compute the angle at long times from Eq. (56) by taking lim t * →∞Fs ·L 0 ; this renders [63] lim t * →∞ cos δ s = Ω x sgn(Ω y ).
Thus the final angle of the force formed with L 0 appears to be independent of many details. However, the detailed study of this regime of late times requires investigative tools which go beyond those employed here.
In summary, depending on the orientation of L 0 , the distortion of the post-quench correlations by shear can either increase or decrease the strength of the postquench force between the inclusions. In addition, the angle between the separation vector of the inclusions and the force gains a dependence on time (due to the evolution of sheared correlations) and on the orientation of the inclusions.

Post-quench force between two inclusions advected by shear flow
In a typical experimental setup, the inclusion may be advected by the shear flow. In the following we shall study this scenario for two cases. In addition to the force F c-m,γ (L(t), t) ≡ F s (L 0 → L(t), t) between advected inclusions in the sheared fluid, we shall also compute the force F c-m,0 (L(t), t) ≡ F (0) s (L 0 → L(t), t) between two inclusions following the same advected trajectories, but in a system in which the correlated fluid is not sheared. (In both cases the subscript "c-m" refers to co-moving inclusions.) This allows us to disentangle the effects of motion of the inclusions and those of shearing the post-quench correlations in the fluid. In both cases, the displacement vector is L(t) = L 0 +γL 0,y te x . As before, one hasλ = λ/|L 0 | and t * = Dt/|L 0 | 2 . For the orders in shear, we obtain (see Fig. 7) and The components g   In those expressions, the quantity Ω α ≡ L 0,α /|L 0 | and thus the angles are given with respect to the initial separation vector. At zeroth order inγ, both forces in Eqs. (59) and (60) naturally reduce to the force between stationary inclusions in an unsheared medium.
At the various orders of the shear expansion, the forces in Eqs. (59) and (60) can also be decomposed into magnitudes and unit vectors. The results of this procedure are shown in Fig. 7 for the first-and second-order corrections to |F c-m,0 | and |F c-m,γ |. These corrections clearly display a dependence on the initial orientation of L(t) (i.e., the orientation of L 0 described in terms of ϑ and ϕ). The corrections are always maximal for ϑ = π/2, i.e., if L 0 lies in the x − y plane. We find that, at late times, |F (1) c-m,0 | and |F (1) c-m,γ | approach the same asymptotes [see Eq. (61)]. Indeed, the shear corrections of the forces acting on the co-moving inclusions relax more slowly at long times than the shear-free contribu- We conclude that at first order in shear, the force at late times is affected more strongly by the motion of the co-moving particles than by the shearing of the post-quench correlations. However, in the intermediate regimes, the two contributions differ with respect to their time-dependence. In turn, at second order, the correction to |F c-m (t * 1)| differs for the unsheared and the sheared system. This indicates that the combined effect of shearing and co-motion is visible at second order in shear at late times. Furthermore, shear corrections for the co-moving particles (for both the unsheared and the sheared system) relax more slowly at late times than those of the stationary inclusions [compare Eqs. (54) and (61)]. For both the unsheared and sheared system, successive orders in the shear corrections are longer-lived, decaying with ever increasing powers of t * . As mentioned before, this indicates that the shear expansion is invalid at late times [also see Eq. (33)]. Consequently we also expect the angle between the force and L(t) to be determined by the shear corrections at late times, provided that λ is large enough for the expansion to be valid in that regime.
Therefore we consider the angular dependence of these forces explicitly. The time-dependent unit vector connecting the two co-moving inclusions isL(t) = t * Ωỹ Thus the angle δ c-m,0/γ between the force and the inclusions is determined by cos δ c-m,0/γ =F c-m,0/γ ·L(t).
Therefore in the unsheared system, F c-m,0 is always parallel to the vector L(t) connecting the inclusions, which is a welcome cross check of our computations. In turn, at the expansion orders provided [compare Eq. (57)], the angle between L(t) and F c-m,γ is the same as the one between L 0 and F s .

VII. CONCLUSIONS AND PERSPECTIVES
We have presented a systematic Gaussian study of spatial correlation functions as they occur after a quench in a sheared fluid. The quantity undergoing a quench could be either k B T /m, i.e., the temperature and/or the compressibility of the fluid, or the correlation length ξ, or a combination of both. We have studied the sheared post-quench dynamics in the limit of small ξ, and as a function of the shear-induced length scale λ = D/γ. The presence (model B) or absence (model A) of the conservation of density fluctuations φ strongly influences correlations and forces. Our findings can be summarized as follows: 1. In a steady state, correlations under weak shear with dissipative dynamics decay as e −1/ξ for ξ → 0, as it is the case in equilibrium. In contrast, for conserved dynamics, the steady state correlation function displays long-ranged correlations which vanish algebraically for ξ → 0. Thus shear produces quantitative and qualitative corrections to correlation functions in systems with conserved dynamics.  Fig. 1), so that points can be more strongly or more weakly correlated than in an unsheared medium, depending on their displacement relative to the shear flow. (ii) Correlations between two points following an advected trajectory depend strongly on the initial displacement between the points (see Fig. 2). (iii) For non-zero values of ξ, the different contributions to the post-quench correlation function due to quenching k B T /m or ξ, as well as their dependence on weak shear, have been identified (see Fig. 3). At leading order, terms stemming from quenching k B T /m decay more slowly than those arising from a quench of ξ.
3. We have extended the formalism of Ref. [35] for computing post-quench fluctuation-induced forces, in order to include shear. This description applies to both the time-dependent and the steady state forces following a quench under shear, and can be used for a variety of geometries (e.g., parallel plates), thereby opening perspectives for numerous future research projects. Here, the formalism was applied to the force between finite-sized inclusions (as sketched in Fig. 4), rendering a far-field force with properties resembling those of the aforementioned correlations. 4. In contrast to a homogeneous system, transient as well as steady state post-quench forces in a sheared medium are not parallel to the vector connecting the inclusions. Indeed, the forces depend strongly (both in magnitude and direction) on the (initial) relative orientation of the inclusions. 5. In a steady state with weak shear, forces decay exponentially as ξ → 0 for model A, but algebraically for model B. In both cases, the orientation of the inclusions relative to the flow affects the magnitude as well as the direction of the force. The conservation law of the underlying dynamics therefore influences the observed non-equilibrium steady states; this differs strongly from equilibrium phenomena which are independent of the type of dynamics.
6. Transient post-quench forces have been studied for the following cases: (i) static inclusions in a sheared medium, (ii) inclusions advected with the shear flow, and (iii) inclusions following advected trajectories in an unsheared medium. In the absence of shear, all cases reduce to the known result for a homogeneous system (see Fig. 5). All forces are long-ranged, decaying algebraically with the (initial) vector L 0 connecting the inclusions, as L −7+2n 0 λ −2n at the nth order in the shear rateγ. Figures 6 and 7 summarize the angular and temporal dependence of the forces.
We conclude that conservation laws play an important role in determining the character of nonequilibrium correlations.
For conserved dynamics, quenches give rise to long-ranged effects, both in the transient and in the steady state regimes, even in the limit of small correlation lengths. If in addition the medium is sheared, strong spatial and orientational variations of fluctuation phenomena are observed. Based on this knowledge, correlations (and the associated fluctuation-induced forces) can be selectively enhanced or diminished.
The phenomena studied here are expected to have a large variety of experimental realizations, either for passive fluids under shear without a quench, or for active matter for which quenches can easily be introduced in addition. Indeed, non-equilibrium rheology is being explored experimentally and theoretically [64]. In particular, our findings are an important step toward harnessing the combination of fluctuation effects and shear in order to engineer interactions, e.g., between colloidal particles in correlated, quenched fluids. As far as physical realizations of quenches are concerned, suspensions of colloidal particles with tunable interactions [49] are promising candidates.
Future studies may address the role of momentum conservation (corresponding to the so-called model H [32]). This would facilitate a connection to Refs. [18,29,30] which deal with fluctuation phenomena in hydrodynamic systems subject to shear. The above formalism can also be applied to forces in other geometries (e.g., thin films), so that other experimentally relevant setups (such as fluctuating wetting films) can be explored in the future. Extending the above formalism to time-dependent (e.g., oscillatory) shear would provide a further avenue for theoretical exploration and would potentially allow one to make contact with experiments [64].

ACKNOWLEDGMENTS
We thank M. Gross for valuable discussions. During the preparation of this study, M. Krüger was supported by Deutsche Forschungsgemeinschaft (DFG) under the grants numbers KR 3844/2-1 and KR 3844/2-2.

Appendix A: Extension of the formalism in
Ref. [35]

to include shear
In general, Gaussian Hamiltonians can be cast into the form so that, e.g., the Hamiltonian with inclusions [Eq. (41)] corresponds to the kernel where m(x) is given in Eq. (43) and L is the separation vector of the inclusions. More generally, ∆ may also incorporate boundary conditions for the surfaces of immersed objects [35]. The framework presented in Ref. [35] can be used to compute forces between objects which can be cast in terms of ∆. Then, as in Eq. (A2), ∆(x, x , L) gains a dependence on the separation vector L of the objects. In thermal equilibrium, the force between the external objects can be computed from the partition function with Turning to dynamics, the Langevin equation with shear [see Eq. (4)] can be written as upon introducing the operator notation AB(x, x ) ≡ d d x A(x, x )B(x , x ) (integration over repeated coordinates is implied). Here R encodes dissipative or conserved dynamics for φ (i.e., model A or B), and can be mapped onto Eq. (6) according to R(x, x ) = µ A/B (x)δ(x − x ). Due to the fluctuation-dissipation theorem, R also appears in the noise correlator: η(x, t)η(x , t ) = 2k B T δ(t − t )R(x, x ). (A5) The operator S(x, x ) =γy∂ x δ(x − x ) represents the advection term for simple shear, and R (γ) = R + S∆ −1 .
For a given configuration of φ(x), the mean force can also be computed directly from the Hamiltonian: x, x , L). Inspired by the analysis in Sec. II A in Ref. [35], the equivalence of Eqs. (A3) and (A6) can be exploited for instantaneous configurations of the field φ(x, t). First, we note that, for a quench from T I = 0 to T , the (temporal) Laplace transform of the correlation function C(x, t) from Eq. (13) can be written as This implies that, according to Eq. (38), the timedependent (non-equilibrium) forces emerging after a quench can be computed from an effective equilibrium theory. Forγ = 0, our results reduce exactly to those of Ref. [35]. However, the analysis holds also forγ = 0, because Eq. (A4) is still a linear Langevin equation and S has a local kernel ∝ δ(x − x ) for simple shear, i.e., γ = const. From ∆ (γ) s [see below Eq. (38)] one infers that the source of LRCs can be either the inherent correlations manifest in the Hamiltonian (via ∆), or the presence of a conservation law (via R).