Mean-field dynamics of infinite-dimensional particle systems: global shear versus random local forcing
MMean-field dynamics of infinite-dimensional particle systems:global shear versus random local forcing
Elisabeth Agoritsas Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
In infinite dimension, many-body systems of pairwise interacting particles provide exact analytical bench-marks for features of amorphous materials, such as the stress-strain curve of glasses under quasistatic shear.Here, instead of a global shear, we consider an alternative driving protocol as recently introduced in Ref. [1],which consists of randomly assigning a constant local displacement on each particle, with a finite spatial correla-tion length. We show that, in the infinite-dimension limit, the mean-field dynamics under such a random forcingis strictly equivalent to that under global shear, upon a simple rescaling of the accumulated strain. Moreover,the scaling factor is essentially given by the variance of the relative local displacements on interacting pairs ofparticles, which encodes the presence of a finite spatial correlation. In this framework, global shear is simplya special case of a much broader family of local forcing, that can be explored by tuning its spatial correlations.We discuss specifically the implications on the quasistatic driving of glasses –initially prepared at a replica-symmetric equilibrium– and how the corresponding ‘stress-strain’-like curves and the elastic moduli can berescaled onto their quasistatic-shear counterparts. These results hint at a unifying framework for establishingrigourous analogies, at the mean-field level, between different driven disordered systems.
I. INTRODUCTION
Theoretical descriptions of driven amorphous materials remain challenging, despite of decades of extensive analytical andcomputational studies [2–5]. The technical difficulties pertain to the interplay of competing sources of stochasticity – in particulartheir self-generated structural disorder – and to the resulting out-of-equilibrium nature of these systems. More generally, theseissues are common to driven complex systems in a broad sense, i.e. composed of many interacting degrees of freedom, andstretch to such dissimilar fields as active matter [6] or machine learning [7, 8]. Assessing both the similarities and discrepanciesin the statistical features of such systems upon different interactions and drivings, is thus key to allow for the transfer of knownresults between them. In that respect, the limit of infinite dimension provides a particularly valuable vantage point, as it canoften be exactly analytically tractable and thus allow for rigourous analogies already at a mean-field level.Dense many-body systems of pairwise interacting particles constitute a standard model for amorphous materials, allowingus to focus on the key role of their structural (positional) disorder. At sufficiently high packing fraction and low temperature,they behave as amorphous solids , meaning that they exhibit a rigidity which is sustained under external shear deformation,up to a sample-dependent maximum shear strain amplitude. Beyond this so-called ‘yielding point’, they might reach a steadystate with a finite built-in stress, and thus behave as yield stress fluids . Such a transition from arrested to flowing state is to beexpected in driven disordered systems, and has prompted the characterisation of corresponding phase diagrams as for instancein Refs [9, 10]. However, one important pending question is the following: are global versus local drivings fundamentallydifferent? This question should obviously be completed with the nature of the observables of interest, on the one hand mean-field quantities such as pressure or the average stress (taken as a proxy for the predicted macroscopic stress), and on the otherhand spatio-temporal correlations and their related features (such as possible transient or permanent shear bands). While thelatter are a priori highly sensitive to the built-in spatio-temporal structure of a specific driving, mean-field quantities are bettersuited – by their very definition – for establishing possible equivalences between different drivings.For sheared amorphous materials, the equivalence between global shear strain and random local displacements can be ad-dressed analytically in the limit of infinite spatial dimension, where their statics and dynamics become exactly mean-field. Thislimit has been extensively studied in the past years since it provides an exact benchmark for investigating the properties ofstructural glasses [11], such as the statistical features of their free-energy landscape [12], their equilibrium behavior [13–15] ortheir response to quasistatic drivings [16–20]. In fact, this framework can naturally be extended to the new driving protocol thathas been recently introduced in Ref. [1], namely the quasistatic driving of a glass through random local displacements, constanton each particle and spatially correlated. Under this new Athermal Quasistatic Random Displacements (AQRD) protocol, ithas been shown in Ref. [1] for two-dimensional numerical simulations (of Hertzian contact particles, under periodic boundaryconditions) that the stress-strain curves are qualitatively similar to those obtained under a standard Athermal Quasistatic Shear(AQS) protocol [21], as well as the distributions in the pre-yielding regime of (i) local elastic moduli along elastic branches, (ii) strain intervals between stress drops, and (iii) stress drop magnitudes. More importantly, it was shown that these mean-field-like metrics can be quantitatively collapsed one onto each other, in remarkably good agreement with the infinite-dimensionalpredictions. The aim of the present paper is to present the detailed derivation of the exact mean-field dynamics which led tothese predictions, and to discuss its implications for and beyond quasistatics. Note however that our formalism allows a finitetemperature, so our results will not be restricted to the strictly athermal case, although by convention we will keep thereafter theabbreviations AQRD and AQS to refer to the two types of quasistatic driving even at finite temperature. Conceptually speaking, a r X i v : . [ c ond - m a t . d i s - nn ] O c t our main statement will be the following: in the infinite-dimensional limit, traversing the potential energy landscape of suchmany-body systems is equivalent under a global shear or under a constant random local displacement field, in the sense that thestatistical sampling of the configurational phase space leads to the same mean-field metrics, up to a single rescaling factor.Thereafter we essentially adapt the derivation for the case of a global shear strain presented in Ref. [22], following thenotations and definitions of the recent extensive review on this topic [11]. We start in Sec. II by defining the random localdisplacements settings we consider, mirroring the global shear case, and we discuss in particular how we choose to encode theirspatial correlations. Secondly we sketch in Sec. III how we can go from the full many-body dynamics to an effective scalarstochastic process with such random local displacements. Thirdly we focus in Sec. IV on the quasistatic driving of a glassy state,starting from a replica-symmetric (RS) equilibrium configuration, and connect with the previously static results for quasistaticshear in infinite-dimension. The latter has first been discussed for hard spheres [23] and its further extensions and ramificationsare extensively reviewed in Ref. [11]. In Sec. V we focus furthermore on the implications for the quasistatic stress-strain curvesand the elastic modulus. Finally, in Sec. VI, we conclude and discuss some implications as possible perspectives to this work.This whole derivation will essentially allow us to show how our random local forcing and global shear turn out to be strictlyequivalent, in the infinite-dimensional limit, upon a simple rescaling of the accumulated strain: the scaling factor is then simplycontrolled by the variance of relative local displacements for a given pair of interacting particles, which encodes the finite spatialcorrelations of the local displacement field. This statement holds in particular for athermal quasistatic drivings, and the AQSprotocol can be interpreted, from that perspective, as a special case of the AQRD protocol. Note that this statement will beobtained for the replica-symmetric equilibrium case. Further work would be needed to extend the AQRD protocol to the full-replica-symmetry-breaking case (and thus all the way down to the yielding transition), as it has already been done for shear [16]. II. GLOBAL SHEAR
VERSUS
RANDOM LOCAL DISPLACEMENTS
We consider the same general settings as in Refs. [22, 24]: a system of N interacting particles in d dimension and of positions { x i ( t ) ∈ Ω ⊂ R d } i =1 ,...,N at time t . The region Ω has a volume | Ω | and thus a number density ρ = N/ | Ω | , and for simplicitywe assume Ω to be a cubic region with periodic boundary condition ( i.e. in the same spirit as the numerical settings for instancein Ref. [1]). Note that we always assume first the thermodynamic limit ( N → ∞ and | Ω | → ∞ at fixed ρ ), and secondly theinfinite-dimensional limit ( d → ∞ ). We consider the case of pairwise interactions between identical particles, with a genericradial potential v ( | r ij ( t ) | ) where r ij ( t ) = x i ( t ) − x j ( t ) fluctuates around a typical interaction length (cid:96) . This potential couldbe chosen to be a hard-sphere, soft-sphere or Lennard-Jones-like for instance, as long as it is thermodynamically stable in highdimension and has a well-defined infinite-dimensional limit upon rescaling: lim d →∞ v ( (cid:96) (1 + h/d )) = ¯ v ( h ) [11] . The rationalebehind this requirement is that, in the infinite-dimensional limit, the interparticle distances (for effectively interacting particles)have fluctuations of O (1 /d ) around (cid:96) , so that r ij ( t ) = (cid:96) (1 + h ij ( t ) /d ) with the gap h ij ( t ) ∼ O ( d ) , and the definition of therescaled potential ¯ v ( h ) allows to focus on this gap of order .A global shear strain of amplitude γ , in the plane { ˆ x , ˆ x } for instance, is defined by ˆ γ = γ · · · · · · ... ... ... ∈ R d × R d ⇒ x (cid:48) i = (cid:16) ˆ1 + ˆ γ (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) = ˆ S γ x i = x i, + γx i, x i, ... = x i + γ x i, ˆ x , (1)meaning that it assigns to each particle i a local displacement γ c i with c i = x i, ˆ x : this local displacement is always appliedalong the direction ˆ x and its amplitude depends on the configuration on which the shear strain step is applied, so it is continuallyupdated along the dynamics. The motion of particles in the laboratory frame can then be decomposed into the affine motion dueto the accumulated shear strain γ ( t ) , starting from a given initial configuration, and a ‘non-affine’ correction: x i ( t ) = x i (0) + γ ( t ) x i, (0) ˆ x + u i ( t ) ⇒ r ij ( t ) = r ij (0) + γ ( t ) r ij, (0) ˆ x (cid:124) (cid:123)(cid:122) (cid:125) = r (cid:48) ,ij ( t ) (for shear) + w ij ( t ) , (2)with u i ( t ) and w ij ( t ) the non-affine absolute and relative displacements, respectively. Note that, since the affine transformationis the same for both the absolute and relative positions (using the matrix ˆ S γ ) it makes sense to define a ‘co-shearing frame’where coordinates are directly given by the non-affine motion [21]. For random local displacements, we essentially release the We recall for instance from Sec. 2.1 of Ref. [14] that we have for soft harmonic spheres v ( r ) = (cid:15) d ( r/(cid:96) − θ ( (cid:96) − r ) = (cid:15) h θ ( − h ) = ¯ v ( h ) , for softspheres v ( r ) = (cid:15) ( (cid:96)/r ) αd ( d →∞ ) → (cid:15) e − αh = ¯ v ( h ) , and for Lennard-Jones potential v ( r ) = (cid:15) (cid:104) ( (cid:96)/r ) d − ( (cid:96)/r ) d (cid:105) ( d →∞ ) → (cid:15) (cid:104) e − h − e − h (cid:105) = ¯ v ( h ) . constraint for all these vectors { c i } i =1 ,...,N to be aligned to ˆ x , and we allow for the local displacement vector | c (cid:105) ≡ { c i } i =1 ,...,N to be a constant random vector in R Nd . We can then generalise the definition of non-affine motion from Eq. (2) as follows: x i ( t ) = x i (0) + γ ( t ) c i + u i ( t ) ⇒ r ij ( t ) = r ij (0) + γ ( t ) (cid:0) c i − c j (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) = r (cid:48) ,ij ( t ) (for random local displacements) + w ij ( t ) , (3)Contrary to global shear, we cannot define a ‘co-shearing frame’ per se , but we can still focus on the non-affine motion through u i ( t ) and w ij ( t ) .The definitions (2)-(3) will be key in the next section, when we focus on obtaining an effective dynamics for the non-affinemotion: the derivation will be very similar for shear and random local displacements, and their differences will mostly rely onreplacing r (cid:48) ,ij ( t ) by their corresponding explicit expressions. Moreover, regarding quasistatic driving of glasses and thus theconnection between AQS and AQRD, what will be relevant is specifically the statistical distribution of the affine motion at finiteshear strain. For AQRD, it is thus the distribution of the relative local displacement of pairs of particles c ij = c i − c j that willmatter, through the definition of the associated affine motion r (cid:48) ,ij ( t ) . An important point, though, is that we must impose thescaling in distribution | c ij | ∼ O (1 /d ) in order to mirror the scaling of local displacements under global shear: in the latter casewe have indeed | r ij, (0) ˆ x | = r ij, (0) ∼ O (1 /d ) since it involves one component of a d -dimensional vector r ij (0) whosenorm is of order . If we were considering a finite system, we should moreover take care of the finite-size and finite- N scalings,as discussed in Ref. [1]; here we recall that we consider on the contrary an infinite-size system and the thermodynamic limit, sothere is no such issue.In order to implement a local displacement vector |{ c i }(cid:105) ∈ R Nd as in the AQRD protocol introduced in Ref. [1], we firstgenerate a local displacement field C ( x ) continuously defined for x ∈ R d , with a Gaussian distribution given by C ( x ) = 0 , C ( x ) · C ( x (cid:48) ) = (cid:96) Ξ f ξ ( | x − x (cid:48) | ) /d , (4)where the overline denotes the statistical average over this ‘quenched random displacement field’, Ξ is a tunable amplitude whichhas the units of a length, and ξ > a finite correlation length. For practical purposes, whenever we need an explicit expressionwe will assume a Gaussian function f ξ ( x ) = e − x / (2 ξ ) / (cid:112) πξ . Secondly we associate to each particle a local displacementfixed by the value of the displacement field at its initial position, so that the local displacements are also Gaussian distributed: c i = C ( x i (0) ⇒ c i = 0 , c i · c j = (cid:96) Ξ f ξ ( | r ij (0) | ) /d . (5)Note that the explicit scaling in d is chosen precisely such as to match the scaling of local displacements in AQS, as discussedabove, and that we implicitly assume a statistical isotropy of the local displacements with a radial function for the correlator.From there, we can directly characterise the distribution of the relative local displacements c ij = c i − c j : it is also Gaussianof zero mean, but its variance considerably simplifies in the infinite-dimensional limit, using that r = (cid:96) (1 + h/d ) with thegap h ∼ O (1) and thus f ξ (cid:0) r ij (0) (cid:1) = f ξ ( (cid:96) ) + f (cid:48) ξ ( (cid:96) ) (cid:96)d h ij (0) + O (1 /d ) . This implies on the one hand that, for different pairs ( i, j ) (cid:54) = ( i (cid:48) , j (cid:48) ) , correlations are subdominant in high dimension and eventually vanish in the limit d → ∞ : d c ij · c i (cid:48) j (cid:48) = (cid:96) Ξ (cid:104) f ξ (cid:0) r ii (cid:48) (0) (cid:1) + f ξ (cid:0) r jj (cid:48) (0) (cid:1) − f ξ (cid:0) r ij (cid:48) (0) (cid:1) − f ξ (cid:0) r i (cid:48) j (0) (cid:1)(cid:105) = (cid:96) Ξ f (cid:48) ξ ( (cid:96) ) (cid:96)d (cid:2) h ii (cid:48) (0) + h jj (cid:48) (0) − h ij (cid:48) (0) − h i (cid:48) j (0) (cid:3) ( d →∞ ) → (6)whereas on the other hand the variance of a given pair remains finite: d c ij = 2 (cid:96) Ξ (cid:104) f ξ (0) − f ξ (cid:0) r ij (0) (cid:1)(cid:105) ( d →∞ ) → (cid:96) Ξ (cid:2) f ξ (0) − f ξ ( (cid:96) ) (cid:3) ≡ F (Ξ , (cid:96), ξ ) , (7)Another way to phrase these results is that spatial correlations between the local displacements c i = C ( x i (0)) only affect thevariance of a given pair relative displacements c ij , since different pairs of particles effectively do not interact in infinite dimen-sion (or rather their contribution becomes irrelevant, in the limit d → ∞ , to path-integral statistical averages). We can chooseto decompose these relative displacements as √ d c ij ≡ ˜ c ij ˆ c ij , where ˆ c ij is a unitary vector with a uniform distribution by sta-tistical isotropy; the factor F defined in Eq. (7) should then be understood as the variance of the amplitude ˜ c ij , restricted to thepairs whose interactions are relevant for the dynamics ( i.e. r ij (0) ≈ (cid:96) ). Beware that ˜ c ij is not the norm, since we allow it to takenegative values for a given choice of unitary vector, so that its probability distribution is simply the Gaussian function: ¯ P (cid:0) ˜ c ij (cid:1) = e − ˜ c ij / (2 F ) / (cid:112) π F ⇒ ˜ c ij = 0 , ˜ c ij = F . (8)Note that we choose a slightly different convention than in Ref. [1]: here the vector | c (cid:105) is a displacement field and has thus theunits of a length, and it differs only from a factor (cid:96) compared to the unitless strain field in Ref. [1]; otherwise other quantitiesare defined with the same units, and in particular √ F has the units of a length.This dependence of the variance F on the spatial correlations of local displacements is in fact physically expected and can besimply understood in the following two opposite limits: for an infinite correlation length ξ , all the { ˜ c i } would be the same, andthere would be no relative displacements, so for ξ → ∞ their distribution should consistently tend to P ( c ij ) ∝ δ ( | c ij | ) ; in theopposite limit, the variance F diverges, so we actually need to have a finite correlation ξ > to keep it regular, as it is physicallythe case when we consider discrete interacting particles instead of a true continuum of local displacements. If we assume f ξ ( x ) to be a normalised Gaussian function, we can compute explicitly F : f ξ ( x ) = e − x / (2 ξ ) (cid:112) πξ ⇒ F (Ξ , (cid:96), ξ ) (7) = 2 √ π (cid:96) Ξ (cid:96)ξ (cid:16) − e − ( (cid:96)/ξ ) / (cid:17) ( (cid:96)/ξ (cid:29) = √ π (cid:96) Ξ (cid:96)ξ ( (cid:96)/ξ (cid:28) = √ π (cid:96) Ξ (cid:0) (cid:96)/ξ (cid:1) , (9)thus predicting a crossover of the ξ -dependence depending on the ratio (cid:96)/ξ , with F ∼ /ξ at (cid:96)/ξ (cid:29) and F ∼ /ξ at (cid:96)/ξ (cid:28) .Technically, these specific scalings rely on the functional form of the even correlator f ξ ( x ) = ξ − f ( x/ξ ) , valid in particularfor f ξ being a Gaussian function. Nevertheless, the decrease of F with an increasing correlation length ξ must be qualitativelyrobust with respect to alternative (physical) choices for f ξ ( x ) .As a concluding remark, we note that such a correlation length exists under global shear (thus for the AQS protocol as well),and indeed it was the motivation for the creation of the AQRD protocol in Ref. [1]. Indeed, when we shear a finite-size systemwith periodic boundary conditions, the individual affine motions are by construction correlated on a finite portion of the wholesystem, with a periodicity of twice the system size, so the value of ξ is set once and for all. Note that this statement is only truein the laboratory frame, since in the co-shearing frame the affine motions are by definition set to zero, so the associated localdisplacements have to be defined in the laboratory frame in order to be able to establish a direct connection between the globalshear and random-local-displacements protocols. An intermediate case is to have regularly patterned local displacements, chosensuch as to be compatible with the periodic boundary conditions), where we can tune the correlation length ξ . Such settings hasalso been examined, for two-dimensional numerical simulations, in Ref. [1]. Here we considered the case of Gaussian randomlocal displacements because it directly relates to the shear case in infinite dimension, as we will see in the next section. However,these different alternatives will only affect the quantitative distribution of affine motions r (cid:48) ij ( t ) but not the effective dynamicsfor the non-affine motions u i ( t ) and w ij ( t ) , so we could a priori generalise our derivation to the patterned local displacementsas well (but this is kept for future work). III. FROM MANY-BODY TO EFFECTIVE SCALAR DYNAMICS
We consider the following many-body Langevin dynamics, adapted from the standard shear case [22] for a finite ‘shear rate’ ˙ γ ( t ) (such that the AQRD case corresponds to the limit ˙ γ ( t ) → ): ζ (cid:2) ˙ x i ( t ) − ˙ γ ( t ) c i (cid:3) = F i ( t ) + ξ i ( t ) , with F i ( t ) = − (cid:88) j ( (cid:54) = i ) ∇ v (cid:0) | x i ( t ) − x j ( t ) | (cid:1) (10)and a microscopic Gaussian noise { ξ i ( t ) } i =1 ,...,N , of mean and variance respectively given by (cid:10) ξ i,µ ( t ) (cid:11) ξ = 0 , (cid:10) ξ i,µ ( t ) ξ j,ν ( t (cid:48) ) (cid:11) ξ = δ ij δ µν [2 T ζδ ( t − t (cid:48) ) + Γ C ( t, t (cid:48) )] , (11)where the brackets (cid:104)· · ·(cid:105) ξ denote the statistical average over realisations of the noise ξ . ζ is the (local) friction coefficient and T = β − the temperature of a thermal bath (with the Boltzmann constant k B = 1 setting the units). The generic noise kernel Γ C ( t, s ) can be chosen such as to describe a wide variety of physical situations, as discussed extensively in Sec. 2 of Ref. [24]. Byessentially tuning its ‘persistence time’ τ , we can consider anything from an isotropic active matter with Γ C ( t, s ) = f e −| t − s | /τ ,to a pure thermal bath ( τ = 0 ) with Γ C ( t, s ) = 0 , and constant random forces (‘ τ = ∞ ’) with Γ C ( t, s ) = f . In addition, weneed to specify the distribution of the initial configurations at time t = 0 ; for explicit computations we can focus on the particularcase of an equilibrium (possibly supercooled) liquid phase at a temperature T = β − , where positions are sampled from aGibbs-Boltzmann distribution ∝ e − (cid:80) i (cid:54) = j v ( | r ij (0) | ) .These settings are very similar to the standard case under shear strain, that we have examined in Ref. [22]: here we essentiallyreplaced for each particle i the time-dependent local displacements γ ( t ) x i, ( t )ˆ x under a global shear (in the plane { ˆ x , ˆ x } )by the random local displacements γ ( t ) c i . In Ref. [22] we focused on the non-affine motion by using the change of variables ofEq. (2), which is equivalent to working in the co-shearing frame. We can similarly use Eq. (10) to obtain the dynamics for thenon-affine motion under random local displacements: ζ ˙ u i ( t ) = F i ( t ) + ξ i ( t ) , with F i ( t ) = − (cid:88) j ( (cid:54) = i ) ∇ v (cid:16) | r (cid:48) ,ij ( t ) + u i ( t ) − u j ( t )) | (cid:17) (12)with the uniform initial condition u i (0) = ∀ i . Compared to global shear, we do not have a term ζ ˆ˙ γ ( t ) u ( t ) so the dynamics is de facto isotropic, and the affine motion is given by r (cid:48) ,ij ( t ) ≡ r ,ij + γ ( t ) c ij instead of r (cid:48) ,ij ( t ) = ˆ S γ ( t ) r ,ij (see Eqs. (2)-(3)).Note that if we start from a statistically isotropic initial condition, such as equilibrium, with such an isotropic dynamics we canalways assume statistical isotropy to hold.This many-body dynamics becomes exactly mean-field in infinite dimension, as shown in previous works for the isotropiccase [24] or in presence of global shear [22]. In a nutshell, this can be understood as a consequence of two key physicalassumptions: particles always stay ‘close’ to their affine motion with respect to their initial position – in the sense that theirnon-affine displacements are of O (1 /d ) – and each particle has numerous uncorrelated neighbours. Intuition can be gained fromthe simpler case of an isotropic random walk: each particle has so many directions towards which it can move, that it effectivelyexplores a volume whose typical radius shrinks with an increasing dimensionality; moreover, in a dense system each particlehas O ( d ) neighbours, and once it interacts with a given neighbour it is very unlikely that it would interact with it again, makingthem effectively uncorrelated. The recipe for obtaining the mean-field dynamics is thus as follows: we start from the many-bodydynamics (10), focus on the non-affine displacement in Eq. (12), and Taylor-expand the interaction force F i ( t ) to leading orderin the infinite-dimension limit. Technically, the difficulty is to correctly identify the scaling in d of the different contributions tothe dynamics and the leading-order terms. Physically that amounts to neglecting the contributions due to collective feedbacksinvolving more than two particles, which are subdominant in the infinite-dimension limit when computing statistical averages ofobservables.Starting from Eq. (12), we can directly adapt the mean-field dynamics previously obtained for global shear (specif-ically Eqs. (23)-(24) of Ref. [22]). The interactions with other particles are fully described by three tensorial kernels { ˆ k ( t ) , ˆ M R ( t, s ) , ˆ M C ( t, s ) } , defined as averages involving the force ∇ v ( r ij ( t )) = ∇ v ( r (cid:48) ,ij ( t ) + w ij ( t )) as a result of theTaylor-expansion of F i ( t ) . From this point, we can drop on the indices ( i, j ) only keeping in mind that c stands for the rela-tive local displacements c ij . r (cid:48) ( t ) is set by the distributions of the initial condition and of the local strains, and the non-affinedisplacements follow an exact mean-field dynamics given by the following vectorial stochastic processes: ζ ˙ u ( t ) = − ˆ k ( t ) u ( t ) + (cid:90) t d s ˆ M R ( t, s ) u ( s ) + √ Ξ ( t ) ,ζ w ( t ) = −
12 ˆ k ( t ) + 12 (cid:90) t d s ˆ M R ( t, s ) w ( s ) − ∇ v (cid:0) r (cid:48) ( t ) + w ( t ) (cid:1) + Ξ ( t ) , u (0) = 0 , w (0) = 0 , (cid:10) Ξ µ ( t ) (cid:11) Ξ = 0 , (cid:10) Ξ µ ( t )Ξ ν ( t (cid:48) ) (cid:11) Ξ = δ µν (cid:20) T ζδ ( t − t (cid:48) ) + 12 Γ C ( t, t (cid:48) ) (cid:21) + 12 M µνC ( t, t (cid:48) ) , (13)where the noise has a zero mean by statistical isotropy, and the following self-consistent equations for the tensorial kernels with r (cid:48) ( t ) = r + γ ( t ) c : k µν ( t ) = ρ (cid:90) d r g in ( r ) (cid:90) d c ¯ P ( c ) (cid:10) ∇ µ ∇ ν v ( r (cid:48) ( t ) + w ( t )) (cid:11) w ,M µνC ( t, t (cid:48) ) = ρ (cid:90) d r g in ( r ) (cid:90) d c ¯ P ( c ) (cid:10) ∇ µ v ( r (cid:48) ( t ) + w ( t )) ∇ ν v ( r (cid:48) ( t (cid:48) ) + w ( t (cid:48) )) (cid:11) w ,M µνR ( t, s ) = ρ (cid:90) d r g in ( r ) (cid:90) d c ¯ P ( c ) δ (cid:10) ∇ µ v ( r (cid:48) ( t ) + w ( t )) (cid:11) w , P δP ν ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = . (14)Here ρ is the density, g in ( r ) the initial distribution of inter-particle distances in the laboratory frame at t = 0 [24, 25], and ¯ P ( c ) the distribution of relative local displacements. The brackets (cid:104)•(cid:105) w denote the dynamical average over the stochastic process w ( t ) (thus self-consistently defined), starting from a given set { r , c } . P ( s ) is to be understood as a perturbation field added intothe dynamics, inside the interaction potential as ∇ v ( r (cid:48) ( t ) + w ( t ) − P ( t )) . If we start from a replica-symmetric equilibrium atinverse temperature β , we would have g in ( r ) = g eq ( r ) = e − β v ( r ) . Finally, as discussed extensively in the previous sectionaround Eq. (7), we consider specifically vectors c ≡ ˜ c ˆ c / √ d with ˆ c a unitary vector uniformly distributed (by statistical isotropy)and the norm ˜ c ∼ O (1) with the Gaussian distribution of zero mean and variance F given in Eq. (8). Physically, we emphasisethat ˆ k ( t ) characterises the averaged divergence of forces at time t , ˆ M C ( t, t (cid:48) ) is nothing but the two-time force correlator, and ˆ M R ( t, s ) the averaged response on the force at time t after a perturbation at an anterior time s .We can in fact reduce further these high-dimensional vectorial stochastic processes, using the fact that the pairwise interactionpotential is a radial function of the distance r ( t ) ≡ | r ( t ) | = | r (cid:48) ( t ) + w ( t ) | ≈ (cid:96) (1 + h ( t ) /d ) of typical interaction length (cid:96) andthat the definition of the gap h ( t ) ∼ O (1) allows us to focus on fluctuations of O (1 /d ) around r ( t ) ≈ (cid:96) . Exactly as for shear (inSec. IV.B of Ref. [22]), we can decompose this inter-particle distance in three contributions as follows: | r ( t ) | ≡ | r (cid:48) ( t ) + w ( t ) | ≈ r (cid:48) ( t ) + ˆ r (cid:48) ( t ) · w ( t ) + w ( t ) r (cid:48) ( t ) with ˆ r (cid:48) ( t ) · w ( t ) ≡ (cid:96)d y ( t ) , w ( t ) r (cid:48) ( t ) ≈ (cid:104) w ( t ) (cid:105) (cid:96) ≈ (cid:96)d ∆ r ( t ) , (15)where we used that | w ( t ) | (cid:28) r (cid:48) ( t ) in high dimension, defined y ( t ) as the projection of w ( t ) on the affine relative displacement ˆ r (cid:48) ( t ) , and introduced the rescaled mean-square-displacement (MSD) function ∆ r ( t ) = d(cid:96) (cid:10) | u ( t ) | (cid:11) . We recall that for a globalshear we would have instead r (cid:48) ( t ) = r + γ ( t ) r , ˆ x , with relative displacements c fully prescribed by the initial conditionas c = r , ˆ x , or –equivalently– the special case where ¯ P ( c ) = δ ( c − r , ˆ x ) . The implications of the more generic defini-tion r (cid:48) ( t ) = r + γ ( t ) c are better reflected by on the gap associated to the third contribution in Eq. (15), namely the affinecontribution r (cid:48) ( t ) : r (cid:48) ( t ) = (cid:12)(cid:12)(cid:12)(cid:12) r + γ ( t ) ˜ c √ d ˆ c (cid:12)(cid:12)(cid:12)(cid:12) ( d →∞ ) ≈ r (cid:32) h d + γ ( t ) d ˜ cr √ d ˆ r · ˆ c + γ ( t ) d ˜ c r (cid:33) , (16)where h is the initial gap, r ≈ (cid:96) and we truncated higher orders in /d . Note that the scalar product of the two random unitvectors ˆ r · ˆ c scales in distribution as / √ d (despite the fact that it is a sum of d terms each of O (1 /d ) , because of the randomnessof both vectors). We have thus g c ≡ √ d ˆ r · ˆ c ∼ O (1) , normal distributed with zero mean and unit variance. Further, we canrewrite Eqs. (15)-(16) for the total gap as | r ( t ) | = | r (cid:48) ( t ) + w ( t ) | ≈ (cid:96) (1 + h ( t ) /d ) with h ( t ) = h (cid:48) ( t ) + y ( t ) + ∆ r ( t ) ,h (cid:48) ( t ) = h + γ ( t ) g c (˜ c/(cid:96) ) + γ ( t ) (cid:0) ˜ c/(cid:96) (cid:1) . (17)This allows us to rewrite the high-dimensional vectorial stochastic processes Eqs. (13)-(14) into a scalar stochastic process, withits three associated kernels (generalizing Eqs. (46) and (48) from Ref. [22]): (cid:98) ζ ˙ y ( t ) = − κ iso ( t ) y ( t ) + (cid:90) t d s M iso R ( t, s ) y ( s ) − ¯ v (cid:48) ( h ( t )) + Ξ( t ) ,h ( t ) = h + γ ( t ) g c ˘ c + γ ( t ) c + y ( t ) + ∆ r ( t ) , Initial condition: y (0) = 0 , γ (0) = 0 , ∆ r (0) = 0 , random (cid:8) h , ˘ c = ˜ c/(cid:96), g c (cid:9) ∼ O (1) , Gaussian noise: (cid:10) Ξ( t ) (cid:11) Ξ = 0 , (cid:10) Ξ( t )Ξ( s ) (cid:11) Ξ = 2 T ˆ ζδ ( t − s ) + G C ( t, s ) + M iso C ( t, s ) , (18)with the rescaled friction coefficient ˆ ζ = (cid:96) d ζ and noise kernel G C ( t, t (cid:48) ) = (cid:96) d Γ C ( t, t (cid:48) ) , and the three rescaled kernels: κ iso ( t ) = (cid:98) ϕ (cid:90) ∞−∞ d h e h g in ( h ) (cid:90) d˜ c (cid:90) d g c ¯ P ( g c , ˜ c ) (cid:10) ¯ v (cid:48)(cid:48) ( h ( t )) + ¯ v (cid:48) ( h ( t )) (cid:11) h | h , ˜ c,g c , M iso C ( t, t (cid:48) ) = (cid:98) ϕ (cid:90) ∞−∞ d h e h g in ( h ) (cid:90) d˜ c (cid:90) d g c ¯ P ( g c , ˜ c ) (cid:10) ¯ v (cid:48) ( h ( t ))¯ v (cid:48) ( h ( t (cid:48) )) (cid:11) h | h , ˜ c,g c , M iso R ( t, s ) = (cid:98) ϕ (cid:90) ∞−∞ d h e h g in ( h ) (cid:90) d˜ c (cid:90) d g c ¯ P ( g c , ˜ c ) δ (cid:10) ¯ v (cid:48) ( h ( t )) (cid:11) h | h , ˜ c,g c , P δ P ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P =0 . (19)We recall from Refs. [24]-[22] that instead of the density ρ we have in those definitions a dependence on the rescaled packingfraction (cid:98) ϕ = ρV d (cid:96) d /d with V d = π d/ / Γ( d/ the volume of the unit sphere in d dimensions. If we start from equilibriumat temperature T = β − , we simply have g in ( h ) = g eq ( h ) = e − β ¯ v ( h ) , and the statistical average over the relative localdisplacements takes the explicit form (cid:90) d˜ c (cid:90) d g c ¯ P ( g c , ˜ c ) ( . . . ) (cid:55)→ (cid:90) R d g c e − g c / √ π (cid:124) (cid:123)(cid:122) (cid:125) ≡D g c (cid:90) R d˜ c e − ˜ c / (2 F ) √ π F ( . . . ) . (20)Finally, the dynamical equations for the correlation and response functions (and in particular of the MSD function) are strictly thesame as for the case of global shear, see Ref. [22] (specifically Eq. (41) and the definitions Eqs. (31)-(33) and (40)). If we wishto add an acceleration term with a particle mass m and/or a retarded friction kernel Γ R ( t, s ) , in the many-body dynamics (10)that we took as a starting point, we can similarly adapt the dynamical equations given in Ref. [24] (see specifically its summarysection VII).We emphasise that the case of global shear can be seen as the special case of Eqs. (16)-(17), with r (cid:48) ( t ) = r + γ ( t ) r , ˆ x ⇒ c = r , ˆ x ⇔ (cid:40) rescaled amplitude: ˜ c/r ≡ √ d c/r = √ d ˆ r , ≡ g direction: ˆ c = ˆ x ⇒ g c = √ d ˆ r , ≡ g (21)where we used the definition for the unit vector coordinates ˆ r ,µ ≡ g µ / √ d with g µ ∼ O (1) [19, 22], and the average over relativedisplacements (20) simplifies into (cid:82) D g (cid:82) D g ( . . . ) . In other words, global shear has the same scalar mean-field dynamics asrandom local displacements with F /(cid:96) ≡ . The other way around, we can tune the variance F , for instance by playing with theamplitude Ξ or the spatial correlation length ξ in Eq. (7), allowing for a more general family of such random local driving. Inthe expression (18) of the gap h (cid:48) ( t ) , if we define γ eff ( t ) = γ ( t ) √ F /(cid:96) , then ˘ c can be taken Gaussian distributed of zero meanand unit variance, exactly as the random variable g appearing in the global shear expressions. This means that we can go onestep further than Eq. (20): (cid:90) d˜ c (cid:90) d g c ¯ P ( g c , ˜ c ) (cid:16) . . . (cid:8) γ ( t ) , ˜ c, g c (cid:9) . . . (cid:17) (cid:55)→ (cid:90) D g c (cid:90) D ˘ c (cid:18) . . . (cid:110) γ eff ( t ) = γ ( t ) (cid:112) F /(cid:96), (cid:96) ˘ c, g c (cid:111) . . . (cid:19) . (22)Consequently, and this is our main result, in the limit of infinite dimension, the many-body dynamics under a global shearor spatially correlated random local displacements can be exactly reduced on scalar mean-field dynamics which are strictlyequivalent, upon the above rescaling of the accumulated strain. The dependence on the spatial correlation length ξ persiststhrough the variance F of relative local displacements, with a functional form which is for instance given by Eq. (9) for Gaussiandistributed local displacements.Technically, this simple connection with global shear relies ultimately on the separation of the affine and non-affine motions r (cid:48) ( t ) and w ( t ) , respectively. In infinite dimension, the explicit dependence on the ‘accumulated shear strain’ γ ( t ) only appearswithin the affine motion which keeps a memory of the initial condition r (cid:48) ( t ) and its associated gap h (cid:48) ( t ) . As for the non-affinemotion, the feedback of γ ( t ) is treated in an exact mean-field way through the three kernels (19). Our new protocol essentiallyallows for a more general distribution of relative displacement amplitude ¯ P (˜ c ) , admittedly Gaussian with zero mean and variance F , but it is the exact mean-field reduction of the dynamics at d → ∞ which allows us to do so without any loss of generality (seethe discussion surrounding Eqs. (6)-(7) in the previous section on that point).Finally, we recall the dynamical definition of the stress under a global shear strain (see Eq. (49) in Ref. [22]): ˆ σ shear ( t ) ≡ βσ shear ( t ) d ρ = (cid:98) ϕ (cid:90) ∞−∞ d h (cid:90) D g D g e h g in ( h , g , g ) g g (cid:10) ¯ v (cid:48) ( h ( t )) (cid:11) h | h ,g ,g ,γ ( t ) , (23)which consists in a statistical average over the initial condition ( i.e. the inital gap h and rescaled components of ˆ r in the shearplane, g µ = √ d ˆ r ,µ for µ = 1 , ) of the average force amplitude (cid:10) ¯ v (cid:48) ( h ( t )) (cid:11) under an accumulated strain γ ( t ) . This definitioncan be generalised to our new protocol, using the translation stated in Eq. (21) and the decomposition of the gap h ( t ) of Eq. (17),as follows: ˆ σ ( t ) ≡ βσ ( t ) d ρ = (cid:98) ϕ (cid:90) ∞−∞ d h e h g in ( h ) (cid:90) D g c (cid:90) R d˜ c e − ˜ c / (2 F ) √ π F ˜ cr g c (cid:10) ¯ v (cid:48) ( h ( t )) (cid:11) h | h ,g c , ˜ c/ r ,γ ( t ) = (cid:98) ϕ (cid:90) ∞−∞ d h e h g in ( h ) (cid:90) D g c (cid:90) D ˘ c ˘ c g c (cid:10) ¯ v (cid:48) ( h ( t )) (cid:11) h | h ,g c , ˘ c,γ eff ( t ) . (24)The last equality emphasises once again the equivalence with global shear, upon the rescaling of the accumulated strain, sincewe can simply replace g by ˘ c = ˜ c/(cid:96) , g by g c , and g in ( h , g , g ) by g in ( h ) (the latter by statistical isotropy). In addition, itprovides a quite intuitive definition of the stress in the system: it is the statistical average over the scalar product, for a giveninteracting pair of particles, of the interparticle force at time t and its assigned relative local displacement, i.e. ˜ cr g c (cid:10) ¯ v ( h ( t )) (cid:11) = 1 r √ d ˜ c ˆ c (cid:124) (cid:123)(cid:122) (cid:125) ≡ c · ˆ r (cid:10) v (cid:48) ( r ( t )) (cid:11) (cid:96)d ≈ d (cid:68) c ij · ∇ v (cid:0) | r ij ( t ) | (cid:1)(cid:69) , (25)where we have reinstated the pair indices just to emphasise that it is computed over pairs, and the brackets denote here thedynamical average at fixed initial conditions. This definition is in agreement with Eq. (4) in Ref. [1], where the stress iscomputed as the scalar product in the N d -dimensional phase space of the local relative displacement vector |{ c ij }(cid:105) with theforce vector |{ F ij ( t ) }(cid:105) prescribed by the configuration at time t . It enforces the notion that the stress is defined with respect toa given driving direction in configuration space , as further discussed in Ref. [1]. IV. QUASISTATIC DRIVING OF GLASSY STATES
We now focus on the quasistatic driving of glassy states, meaning that we start from an equilibrium initial condition belowthe dynamical transition temperature [11]. We aim in particular at comparing the AQRD and AQS protocols, as discussed inRef. [1]. Such quasistatic driving can be investigated directly with static approaches [16–20, 23], and in Refs. [22, 24] weprovided a dynamical derivation of such ‘state-following protocols’. Because of the equivalence with the case of global shear,that we have established in the previous section at the level the dynamics , we can straightforwardly adapt the results presented inRef. [22] (specifically in Sec. V), using the notations and definitions of the review book [11]. However, we emphasise that ourformalism allows us to include more generally the possibility of a finite temperature, directly in the many-body dynamics givenby Eqs. (10)-(11), so we are not restricted to the pure athermal case even though we refer by convention to the AQRD and AQSprotocols.In the ‘solid’ glassy phase we have at short timescales equilibrium-like fluctuations ( i.e. satisfying fluctuation-dissipationrelations), within the local metabasins of a disordered energy landscape, whose statistics depend on the preparation protocol ofthe system (we assume it to be replica-symmetric, for now). By definition, quasistatic driving amounts to slowly modifying thislandscape while always enforcing equilibrium at short timescales. In practice, we substitute such a timescale separation ansatz inour dynamical mean-field equations (DMFE) and we assume that the mean-square-displacements (MSD) functions saturate to afinite plateau at long times, which can in turn be interpreted as the typical size of the local metabasins. The set of self-consistentequations for this plateau value has a solution as long as the system behaves as a ‘solid’, which can eventually be broken byeither increasing the temperature, lowering the packing fraction, shearing too much, or increasing the variance of random forces.Note that we focus here on the RS phase, but the system might undergo a transition to a full replica-symmetric phase [16, 23]before actually yielding ‘for good’.We consider the case of a ‘strain’ γ smoothly applied over a finite timescale τ , such that at long times the affine motion simplybecomes r (cid:48) ( t ) → r + γ c ≡ ˆ S γ c r . We assume the same ansatz as in Eq. (51) [22], which thus leads to the same restrictedequilibrium distributions as in Eq. (55) [22], replacing only ˆ S γ in AQS by ˆ S γ c in AQRD. In addition to integrating over theinitial condition (cid:82) d r g in ( r ) , we have to average over the local random displacements (cid:82) d c ¯ P ( c ) ( . . . ) . We complement theansatz by assuming that we have at long times the MSD plateaus D r and D , with the following definitions (recalled fromEq. (56) [22]): D r = 1 d lim t →∞ (cid:68) | x ( t ) − x (0) | (cid:69) , D = 1 d lim t →∞ lim | t − s |→∞ (cid:68) | x ( t ) − x ( s ) | (cid:69) , A ≡ D r − D . (26)We assume that we start from an RS equilibrium at inverse temperature β , that we are in contact with a thermal bath atinverse temperature β , and for completeness we allow for having constant random forces with Γ C ( t, t (cid:48) ) = f in Eq. (11). Thesequantities have to satisfy the following set of equations, given first in their high-dimensional vectorial form: D − AD = − β f − ρd (cid:90) d r e − β v ( r ) (cid:90) d c ¯ P ( c ) e A ∇ ∇ e D ∇ e − βv ( ˆ S γ c r ) e D ∇ e − βv ( ˆ S γ c r ) , D = − ρd (cid:90) d r e − β v ( r ) (cid:90) d c ¯ P ( c ) e A ∇ ∇ (cid:104) e D ∇ e − βv ( ˆ S γ c r ) (cid:105) . (27)where we used the compact notation e ˜∆2 ∇ f ( r ) = (cid:82) d x e − x
22 ˜∆ (2 π ˜∆) d/ f ( r + x ) . The corresponding glassy free energy f g , which isthe quantity that would be computed and studied using replicæ in statics approaches [11], is − d β f g = (cid:20) πD ) + AD (cid:21) + β f D ρd (cid:90) d r e β v ( r ) (cid:90) d c ¯ P ( c ) e A ∇ log (cid:104) e D ∇ e − βv ( ˆ S γ c ) (cid:105) , (28)and one can check that we can recover in particular Eqs. (27) from the extremalization conditions ∂ ∆ f g = 0 and ∂ A f g = 0 , as itshould be.For explicit computations of observables, such as the stress-strain curves, we need to work with the scalar formulation ofthe free energy, which is much more user-friendly and essentially involves scalar Gaussian convolutions. It can be obtainedeither starting from Eq. (28), or from the scalar mean-field dynamics with the scalar counterpart of the quasistatic ansatz.Thereafter we directly use the expressions from the review book [11], starting from the definitions of the rescaled MSD functions ∆ r = d (cid:96) D r , ∆ = d (cid:96) D , and A = 2∆ r − ∆ , as well as the rescaled random forces variance ˆ f = (cid:96) d f (same rescaling as for G C ( t, s ) = (cid:96) d Γ C ( t, s ) , given after Eq. (18)). The glass free energy takes the now standard form [11]: − d β f g = (cid:20) (cid:16) π(cid:96) ∆ /d (cid:17) + 2∆ r − ∆∆ (cid:21) + β ˆ f ∆ + (cid:98) ϕ (cid:90) R d h e h q γ (2∆ r − ∆ , β ; h ) (cid:124) (cid:123)(cid:122) (cid:125) only dependence on γ log q (∆ , β ; h ) , (29)with (cid:98) ϕ the rescaled packing fraction as defined after Eq. (19), and the following definitions: q (∆ , β ; h ) ≡ (cid:90) R d h e − β ¯ v ( h ) e − ( h − h − ∆ / √ π ∆ , (30) q AQS γ (cid:16) (cid:101) ∆ , β ; h (cid:17) ≡ (cid:90) d g e − g / √ π q (cid:16) (cid:101) ∆ + γ g , β ; h (cid:17) ≡ (cid:90) D g q (cid:16) (cid:101) ∆ + γ g , β ; h (cid:17) , (31) q AQRD γ (cid:16) (cid:101) ∆ , β ; h (cid:17) ≡ (cid:90) d˜ c e − ˜ c / (2 F ) √ π F q (cid:32) (cid:101) ∆ + γ ˜ c (cid:96) , β ; h (cid:33) = (cid:90) d˘ c e − ˘ c / √ π q (cid:101) ∆ + γ F (cid:96) (cid:124) (cid:123)(cid:122) (cid:125) γ eff ˘ c , β ; h . (32)In that case, we see that the free energy (from which all static observables can be derived) is strictly equivalent for AQS andAQRD provided that we rescale accordingly the accumulated strain , specifically by replacing γ by γ eff = γ √ F /(cid:96) . This is adirect consequence of the equivalence between global shear and constant random local displacements – that we have shown inSec. III to be valid for the whole dynamics – which had to hold in particular for quasistatic drivings.All those quantities can be computed at least numerically, for a given interaction potential v ( r ) = ¯ v ( h ) . In practice, the set ofequations for { ∆ , A = 2∆ r − ∆ } is obtained by extremalization of the free energy (29), with ∂ ∆ f g = 0 and ∂ ∆ r f g = 0 , yieldingthe following equations: − A ∆ = − β ˆ f − (cid:98) ϕ (cid:90) R d h e h q γ ( A , β ; h ) ∂ ∆ q (∆ , β ; h ) q (∆ , β ; h ) ,
1∆ = − (cid:98) ϕ (cid:90) R d h e h ∂ A q γ ( A , β ; h ) log q (∆ , β ; h ) . (33)Once we have self-consistently determined the values of { ∆ , A} or { ∆ , ∆ r } satisfying these equations (if such a solutionexists, we are in the solid phase or at least it can be sustainable), we can compute observables such as the shear stress or thepressure [11]. The principal interest of including the case with constant random forces is that Eq. (33) can then be used moregenerally to study not only ‘strain’-controlled protocols (controlling γ ( t ) at ˆ f = 0 ), but also ‘stress’-controlled counterparts(controlling f at γ ( t ) = 0 ), but this will be further discussed in future work. In the next section, we focus on the former case,which was the case under study in Ref. [1], for which the corresponding stress is computed as σ = ∂ γ f g (see Eq. (10.15) of [11]). V. AQRD
VERSUS
AQS STRESS-STRAIN CURVES AND ELASTIC MODULUS
We have established in Sec. III the equivalence between global shear and constant random local displacements at the level ofthe dynamics, which includes the quasistatic case as explicitly shown in Sec. IV. The implications on the quasistatic stress-straincurves and their associated elastic moduli, in the comparison between the AQRD or AQS protocols, are quite straightforwardto obtain, as we discuss below. We recall that our formalism allows a finite temperature, so our predictions are not restricted tothe athermal case, although we use thereafter the AQS/AQRD abbreviation by convention and in direct reference to the resultsannounced in Ref. [1].For a quasistatic global shear in the pre-yielding regime, starting from a RS equilibrium glass, the stress can be computed bytaking the derivative of the glass free energy given by Eqs. (29)-(32) with respect to the ‘true’ strain, σ AQS ( γ ) = ∂ γ f AQS g ( γ ) [11].This definition has been used for instance to compute the quasistatic mean-field stress-strain curves given in Refs. [16, 18, 20, 23],for sheared hard-sphere systems. Consequently, their AQRD counterparts are simply obtained by the following rescaling: γ eff = γ (cid:112) F /(cid:96) ⇒ σ AQRD ( γ ) ≡ ∂ γ f AQRD g ( γ ) = ∂ γ f AQS g ( γ eff ) = √ F (cid:96) ∂ f AQS g ( γ eff ) ∂γ eff ≡ √ F (cid:96) σ AQS ( γ eff ) . (34)As for the elastic modulus at zero strain, using the definition given by Eq. (10.18) in [11], it is simply given by: µ AQS ≡ d σ AQS ( γ )d γ (cid:12)(cid:12)(cid:12) γ =0 ⇒ µ AQRD ≡ d σ AQRD ( γ )d γ (cid:12)(cid:12)(cid:12) γ =0 = F (cid:96) d σ AQS ( γ eff )d γ eff (cid:12)(cid:12)(cid:12) γ eff =0 = F (cid:96) µ AQS , (35)where µ AQS is fixed by the rescaling packing fraction (cid:98) ϕ , the initial and final inverse temperatures { β , β } , and of course thespecific interaction potential v ( r ) = ¯ v ( h ) with its typical interaction length (cid:96) fixing the units of length. Eq. (35) is particularlyinteresting since it predicts the ratio κ of the initial elastic moduli to be directly given by the unitless variance F /(cid:96) . These pre-dictions can be rewritten as follows, denoting by ˜ γ the ‘random strain’ controlled in AQRD as opposed to γ shear the corresponding0AQS strain, in order to match the notations used in Ref. [1] (see its Eq. (7)): κ ≡ µ AQRD µ AQS = F (cid:96) , γ shear = ˜ γ √ κ , σ AQS = σ AQRD (˜ γ ) / √ κ . (36)These relations prescribe how one should rescale the AQRD mean-field stress-strain data to make them collapse on their AQScounterpart, and the other way around how to generate AQRD curves from an AQS one. These predictions, obtained in theinfinite-dimensional limit, might extend to lower dimensions as far as mean-field-like quantities are considered. They have beentested in Ref. [1] on two-dimensional numerical simulations, and the quantitative agreement was remarkably good for the stress-strain curves, the distributions of elastic moduli, of strain-step between stress drops ( i.e. of the distance between two successiblesaddle points of the potential energy landscape), and of stress drops themselves.According to these predictions, µ AQRD ∝ F µ shear . If we assume in particular f ξ ( x ) to be a normalised Gaussian function, asin Eq. (9), we then expect the elastic modulus to have a crossover depending on the ratio (cid:96)/ξ , with F ∼ /ξ at (cid:96)/ξ (cid:29) and F ∼ /ξ at (cid:96)/ξ (cid:28) , the latter case corresponding to a global shear, where ξ is of the order of the system size [1]. In bothcases, this implies that the elastic modulus decreases with increasing ξ , as we numerically observe and physically expect: it iseasier to deform a glass with local displacements which are more coordinated, i.e. with a larger correlation length. Less obviousis that a larger correlation length between individual displacements implies a smaller variance of the relative displacements, i.e. F must decrease with increasing ξ . This crossover is illustrated in Fig. 1, on the left for F ∝ κ ≡ µ AQRD /µ AQS and on theright for a set of stress-strain curves, obtained by rescaling σ AQS ( γ ) computed for hard spheres in Ref. [23] (see its Fig. 2, for apacking fraction (cid:98) ϕ = 6 ). Because of the power-law dependence of F on ξ , these curves can easily be shifted by several order ofmagnitude (including their ending point, which is related to the yielding point).
Left:
Variance of relative displacements F as a function of the correlation length ξ (both rescaled by the typical interparticledistance (cid:96) ), assuming a Gaussian correlation as in Eq. (9) with an overall amplitude Ξ = 1 . This amplitude can be tuned to shift the crossoverin the scaling of ξ , with respect to the value F /(cid:96) = 1 which indicates the collapse onto the AQS case. Right:
Rescaled AQRD stress ˆ σ AQRD (as defined in Eq. (24)) as a function of the ‘random’ strain ˜ γ , generated by rescaling of the AQS stress-strain curve in dotted black(reproduced from Fig. 2 in Ref. [23], for hard spheres at a packing fraction (cid:98) ϕ = 6 ) according to Eq. (36). The three continuous curvescorrespond to ξ/(cid:96) ∈ { . , . , } (respectively purple, blue and red), as indicated by the black arrow for increasing ξ . They correspond to F /(cid:96) ∈ { . , . , . } for the amplitude fixed at Ξ = 1 . Note that the overall amplitude Ξ is also a control parameter of our protocol,whereas in Ref. [1] its value is fixed by the imposed normalisation (cid:104) c | c (cid:105) = 1 . VI. CONCLUSION
We have established the exact equivalence between global shear and a local forcing – in the form of imposed constant randomlocal displacements – for infinite-dimensional particle systems with pairwise interactions. By adapting the derivation for globalshear detailed in Ref. [22], we have shown that this equivalence holds at the level of the full mean-field dynamics upon a simplerescaling of the accumulated effective strain (see Sec. III). This statement holds in particular for quasistatic drivings (in Sec. IV),and culminates in the prediction that the AQS and AQRD stress-strain curves and their initial elastic modulus can be collapsedon each other via Eq. (36) (see Sec. V).The key parameter of this equivalence is the unitless variance F /(cid:96) of the relative local displacements – restricted on interactingpairs of particles – which keeps an explicit dependence on the spatial correlation of the local forcing. An increasing correlationlength implies a decreasing variance: such a coordinated forcing deforms an amorphous material less efficiently than a com-pletely uncorrelated random forcing, in the sense that it corresponds to a smaller effective accumulated strain γ eff = γ √ F /(cid:96) . In1our derivation, we first assume a Gaussian distribution for the local deformation field. In addition, for explicit computations weconsider a Gaussian two-point correlator f ξ (in Eq. (7)) for which we predict a crossover from F ( ξ ) ∼ /ξ at ξ/(cid:96) (cid:28) to ∼ /ξ in the opposite limit, the latter being relevant to compare to the global shear case (see Sec. II). Remarkably, except for thesescalings, our construction is not specific to these assumptions. It relies indeed on the conjunction of two features which are exactin the infinite-dimensional limit: (i) different pairs of particles effectively do not interact, in the sense that their contributionbecomes irrelevant in path-integral statistical averages (see for instance Ref. [24]), so the effective mean-field dynamics is con-trolled by single-pair statistics; (ii) the statistics of relative displacements tends to a Gaussian distribution, uncorrelated betweendifferent pairs and fully characterised by the variance on a given pair. We could consequently consider alternative deformationfields, as for instance wave-like patterned field as in Ref. [1], and their sole relevant feature for the mean-field dynamics ininfinite dimension will always be their variance F (albeit with a different functional dependence on ξ ).A pending issue regarding such infinite-dimensional results is their relevance regarding systems in low dimensions. Asemphasised in the introduction, this limit has the advantage that exact mean-field predictions might be within reach, while onthe other hand important physics due to spatial correlations might be completely washed out. Since this work has been done inparallel to the numerical study presented in Ref. [1], we were able to directly address this issue by comparing our mean-fieldpredictions (36) to 2D numerics under either AQRD or AQS protocols. Beyond qualitatively similar behaviours, we found aremarkably good quantitative agreement regarding the stress-strain curves and the avalanches statistics. First, we could collapsethe different stress-strain curves and the distribution of stress drops and strain steps via a rescaling controlled by the ratio of initialelastic moduli κ = µ AQRD ( ξ ) /µ AQS . Secondly we could measure the unitless variance F ( ξ ) /(cid:96) and show that it coincides with κ (as predicted in infinite dimension) for wave-like patterned displacement field, and also for the Gaussian random displacementsthat we considered here. For the latter, quantitative discrepancies nevertheless appear when the correlation length ξ decreases,hinting at the increasing role of spatial correlations in these low-dimensional systems.Although both Ref. [1] and our Sec. IV-V focus on quasistatic driving, our statement about the exact equivalence betweenglobal shear and local forcing holds for the whole dynamics of infinite-dimensional particle systems with pairwise interactions.It holds in particular for a finite, possibly time-dependent, shear rate ˙ γ ( t ) . We keep the investigation of the resulting rheologicalbehaviours for future work, but for now we only emphasise that in that case, the local forcing on each particle is strictly speakinga local force , equal to ζ ˙ γ ( t ) c i in Eq. (10). It thus corresponds to having a local force field correlated in space and with anamplitude proportional to the shear rate. In low dimensions, the similarity between the rheology at constant shear rate of 2Dparticle systems, either under global shear or with self-propulsion, has been addressed in Ref. [26], but assuming a fixed normof the local forces: a missing ingredient to quantitatively connect these two types of rheology might be to allow for a spatialcorrelation of the local forcing. Remarkably, the idea of applying such a local forcing to mimic the rheology at a constant shearrate has already been considered in Ref. [27], by adding a non-conservative force on the spherical p -spin model and invoking a so-called ‘schematic’ modelling of mode-coupling relations. In that respect, our results allow us to bypass the possible objectionsregarding either the analogy between the driven p -spin model and the rheology of dense interacting-particle systems, or thelimitations of the mode-coupling approximation: again, in our infinite-dimensional framework, dynamics under global shear orrandom local forcing are strictly equivalent.From a technical point of view, we adapted the previous derivation presented in Ref. [22], where we had used a so-called‘dynamical cavity’ approach in order to obtain the dynamical mean-field equations under a global shear protocol. This approachconsists, in a nutshell, in performing a self-consistent Taylor expansion at leading order in the infinite-dimensional limit. Acomplementary but technically more involved derivation was earlier presented in the companion paper [24] –focused on theisotropic case– based on a super-symmetric path-integral formulation and its infinite-dimensional saddle point. Here our newderivation focuses on the implementation of spatial correlations in the local forcing, either on local displacements (key for thequasistatic limit of vanishing shear rate) or on local forces (at any finite shear rate ˙ γ ( t ) in Eq. (10)). The concrete translation ofsuch spatial correlations in a path-integral formulation would further support our dynamical cavity derivation, properly justifyingthe underlying assumptions as exact features of the d → ∞ path-integral saddle-point. Moreover, it would allow for directconnections to other models with spatially-correlated disorder, such as elastic random manifolds for which disorder correlator isthe key quantity to follow upon functional renormalisation approaches [28].These results support the physical picture that there is indeed a proper equivalence between global shear and local forcing,regarding the statistical sampling of the configurational phase space, at least as long as we focus on mean-field metrics. Apromising perspective to this work would be to challenge this picture in other driven disordered systems, and to systematicallydisentangle such mean-field behaviour from spatial-correlation effects, depending on the specific nature of the driving. Acknowledgments
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