Dynamics of liquids and glasses in the large-dimensional limit

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A mean-field theoretical viewpoint: the Random First-Order Transition scenario

Here we present a theoretical scenario for the glass transition, the Random First-Order Transition (RFOT) scenario, which is strongly rooted in mean-field concepts. This thesis is a part of a research attempt to get an exact theory of liquids and structural glasses in the limit of large spatial dimension, which, as a very important corollary, gives a mean-field description of what is going on in the thermodynamics and dynamics of the system. As such it is relevant to compare it to the basic assumptions of RFOT, since it is predicted by its founders to hold exactly for high dimensions [225].
RFOT has emerged from the early insights of Adam-Gibbs-Di Marzio’s theory (AGDM) [154, 180,
181, 2] and Goldstein’s energy landscape interpretation [185], the dynamical input from the liquid com-munity with Mode-Coupling Theory [252, 33, 187] and some arguments from liquid theory (e.g. Density Functional Theory) [199], spin glass theory [279] and the mosaic scaling picture [224, 69]. It was formu-lated as a sort of patchwork [43] relating these ideas in a series of papers by Kirkpatrick, Thirumalai and Wolynes in the late 1980s [220, 225, 226, 221, 222, 223, 224].

Dynamics within Mode-Coupling Theory

First let us introduce a dynamic theory of the onset of the slowdown in the supercooled liquid phase, the Mode-Coupling Theory (MCT), a first-principle approach to get closed dynamical equations on cor-relation functions that has been successful in reproducing some of the above-mentioned aspects, which also compares well to experiments in the supercooled regime and serves as a dynamic justification for the whole RFOT approach. Work on this topic, in order to apply MCT to diﬀerent systems or regimes, or to find better extensions, has been impressive since the first results in the mid-1980s. Furthermore, once one makes the central (although somewhat crude and unjustified) approximation, the whole theory has enjoyed a rigorous treatment, followed by crucial numerical [232, 233, 295, 352, 234] and experimental tests [188, 116].
We quickly review here its outcomes, relevant for this thesis. We will not go into all details of the derivation since there are great reviews on this point, using the projection operator formalism [229, 199, Chap.9], or comparing it with diagrammatic techniques [326] first established in [71, 72]. MCT is fully detailed in G¨otze’s book [187] while the essence of it is reviewed in [186].

The MCT equation

MCT starts by projecting the Hamiltonian dynamics using the Mori-Zwanzig formalism [423, 292, 291] onto slow degrees of freedom of the system equilibrated at time t = 0. This is done by writing, for a relevant quantity we wish to study, its classical evolution with the Liouville operator [244] and performing a partial average over the slow degrees of freedom of the equilibrium initial conditions. The remaining fast modes’ contributions result in a fluctuating force. Integrating out the fast modes leads to a memory kernel in the dynamical equation ruling the slow (projected) part of the quantity under study, which is related to the autocorrelation of the fluctuating force. These slow modes may be the density in reciprocal space ρq(t) = Pi eiq•xi(t) and density currents, proportional to its time derivative. They are indeed slowly varying at small q (large lengthscales).
Another way is a diagrammatic approach [71, 72]. One starts from the microscopic dynamics of the system containing some disorder (such as a Brownian noise), e.g. for the particle density, Dean’s equation for Brownian dynamics [127] or the nonlinear fluctuating hydrodynamics of Das, Mazenko, Ramaswamy and Toner [120] for Newtonian dynamics. Then an expansion in the potential strength is performed, giving rise to a dynamical equation for the correlation and response of the considered field similar to the Schwinger-Dyson equation of quantum field theory [315, 422, 112] with a self-energy representing a memory kernel.
With this one gets exact equilibrium equations (TTI holds, see §2.5.3) for the coherent intermediate scattering function (1.8): φq(t) + Ωqφq(t) + dt q(t t )φq(t ) = 0 (1.16) with Ωq = q/ βmS(q). M is a memory kernel which encodes all the dynamics, and can be seen as the autocorrelation of some fluctuating force of the system in the Mori-Zwanzig viewpoint, or as the resummation of a certain class of diagrams in the diagrammatic approach.
Owing to the glassy phenomenology where a separation between fast transient liquid-like relaxation and slow rearrangement emerges, the memory kernel is splitted into two contributions, respectively:
Mq(t) = Mqreg(t) + Ωq2Mq(t) (1.17)
The regular part linked to fast modes is usually neglected close to the glass transition. For now this is just a sort of change of variables from describing the dynamics by φq(t) to Mq(t). Using a factorization ansatz for the kernel yields the Mode-Coupling approximation:
Mq(t) = Fq({φk(t)})
Fq({fk}) = Ve(q, k, p)fkfp
k+p=q (1.18)
V , , p) = ρS q S k S p ) {p • [kc(k) + pc(p)]}2
e(q k ( ) ( ) (
where c(k) is a direct correlation function related to the structure factor by the Ornstein-Zernicke equa-tion [199] similar to (1.7): S(k) = 1/[1 − ρc(k)].
At this stage one already sees that the whole dynamics is determined by the sole input of the structure factor, a static quantity. Besides, the memory kernel is local in time since it is a second order polynomial of φk for several wavevectors at the same time.

The dynamical transition

Then MCT proceeds with mathematical implications of the above equations (1.16),(1.17),(1.18). The long-time limit fq = lim φq(t) ∈ [0, 1], called Debye-Waller factor or non-ergodic parameter in the liquid t→∞ literature [199, 187], or Edwards-Anderson parameter in the spin glass literature [141, 142], changes discontinuously from 0 to fqc > 0, the critical form factor, at the dynamical transition temperature Td (or density if one follows a isothermal protocol). Indeed, at long times, equation (1.16) gives (see e.g. §2.5.5) fq = Fq({fk}) (1.19) where a saddle-node bifurcation of solutions occurs at Td, which does not depend upon any wavevector. Charaterizing the distance to the singularity by = (Td − T )/Td or = (ρ − ρd)/ρd, one has the usual square-root approach to the bifurcation fq = fqc + hqr (1.20) 1−λ +O( ) where C is a constant and λ ∈ , 1 is the so-called MCT parameter. This bifurcation means that at T the plateau becomes infinite, resulting in a breaking of ergodicity. In the pictorial view of the cage, the particles do not escape anymore and the structure does not relax, even at long times. This can be interpreted as a spurious eﬀect of the Mode-Coupling approximation since in reality the real system must relax and avoid the ergodicity breaking, whatever time it takes, e.g. by an activation mechanism as emphasized earlier. The MCT transition may be seen as an idealized version of the glassy crossover. Indeed, as an example, it can be rigorously proven [300] that the self-diﬀusion coeﬃcient cannot go continuously to zero at thermal equilibrium and finite temperature and pressure.
Above the dynamical temperature, the intermediate scattering function develops a plateau (or be-comes exponentially damped at higher temperatures), observed in simulations and experiments, as in fig-ure 1.5(a). This points out that MCT may contain some of the right ingredients to describe the glass crossover, in spite of a spurious sharp transition.

MCT scaling laws

From studying the previous MCT equations, a number of scaling laws in several well-identified relaxation regime are recovered [186, 187]. They are valid close to the dynamical transition (small > 0), where the plateau is well formed and the α-relaxation still occurs at long times.
We denote by t0 a transient time scale. Upon approaching the plateau close to the dynamic transition, i.e. in the -relaxation window, a scaling law holds with ˆ = 0: φq(ttˆ0) = fqc + hqtˆ−a + O(tˆ−2a) (1.21)
This is valid for times t0 t τβ with τβ = t0/(C| |)1/2a. The latter timescale represents the β-relaxation timescale, which diverges at the transition.
Similarly, upon leaving the plateau (for small negative ), the so-called von Schweidler’s scaling law
is obtained for β with ˜ = α: t > τ t t/τ ˜ c ˜b ˜2b ) (1.22) φq(tτα) = fq − hqt + O(t where τα can be defined by φq(τα) = fqc/2 up to an -independent factor. We shall call it in the same way as the α-relaxation time since it can be viewed as a practical definition of it. The scaling with τα is reminiscent of the time-temperature superposition of §1.1.7.
One can prove that both MCT exponents a ∈ 0, 12 , b ∈]0, 1] are given by the MCT parameter and thus related by: λ = (1 − a)2 = (1 + b)2 (1.23) (1 − 2a) (1 + 2b)
Finally for larger times τq = (fq/hq)1/bτα, Fuchs [165] has shown that during the α-relaxation process one retrieves Kohlraush-Williams-Watts’ stretched exponential law [235, 403] for large wavevectors, as in (1.10) with the exponent β0 = b: q→∞ q q q − t∗b lim φ (t∗τ ) = fc exp (1.24)
For a generic wavevector, one can try such an ansatz but then the exponent depends upon the wavevector. It has been put forward to explain why spectra generically do not follow Kohlraush-Williams-Watts’ law precisely.
For a generic observable A coupled to density fluctuations, the same laws hold for its correlation replacing fq and hq by similar quantities fA and hA [186, 187]. λ (hence the exponents), t0 and τα remain the same. As a result, (1.21) and von Schweidler’s law (1.22) emphasize a remarkable property of MCT (from which they are actually derived), the so-called factorization property, which states that close to the plateau (in the β-relaxation window), the fluctuations of correlations from the plateau value (e.g. φq(t) − fqc) factorize into a wavevector-dependent (or space, by inverting the Fourier transform) function only (e.g. hq) times a time-dependent function only. This is a stringent test of MCT in simulations [232, 233].
The diﬀusion coeﬃcient can be computed from the MCT approach. Note that, as an example, the MSD D(t, t0) = [x(t) − x(t0)]2 is provided by the incoherent or self-intermediate scattering function for which a similar MCT equation can be obtained. One gets, due to isotropy,
q( )=N* N + q→0 1 − 2d D( )+() (1.25)
φs t, t0 1 eiq•[xi(t)−xi(t0)] = q2 t, t0 O q4
The second timescale τα verifies τα ∝ t0 ∼ −γ, γ = 1 + 1 (1.26) (C )γ 2a 2b
From (1.16) and (1.25), when the plateau diverges one has D(t) = 2dDt with D ∝ τα−1 ∼ γ (1.27)
The relaxation time diverges and the diﬀusion vanishes as a power law at the transition, whose exponent is determined by the MCT parameter λ.

MCT and the p-spin model

Standard MCT, as briefly described above, gives predictions about the dynamics of a liquid if we input the static structure factor S(q). Since these equations are quite cumbersome in the general case, simplified models have been first studied, which reproduces all behaviours shown above. This is the schematic MCT, and the first such equation studied by Leutheusser [252] and Bengtzelius-G¨otze-Sj¨olander [33] consisted in keeping only the dominant first peak q0 of the structure factor, thus getting rid of all wavevector dependence, the memory kernel becoming a simple quadratic function of φ(t) = φq0 (t). The schematic MCT equation can be written as φ(t) + νφ(t) + Ω φ(t) + Ψ dt φ (t t )φ(t ) = 0 (1.28)
This is the same equation, dropping the inertial term irrelevant for large-time dynamics, as the one ruling the exact dynamics of the p-spin spherical model [107, 81, 417] for p = 3, a spherical model of spin glasses first introduced for Ising spins by Derrida [129, 130, 192, 109] analyzed in chapter 2, see (2.164). Spin-glass theory was developped starting from the 1970s, initially to describe the strange behaviour of disordered magnetic alloys [56]. This compelling parallelism was noticed by Kirkpatrick and Thirumalai [220], and is one of the reasons that make this mean-field spin glass model a paradigm for the structural glass transition. This also hints at MCT being some kind of mean-field theory for the dynamics of supercooled liquids, as was argued by Andreanov, Biroli and Bouchaud [11]. The fact that Td is in experiments greater than Tg may be seen as another (slight) evidence, since mean-field eﬀects are known to increase free energy barriers (see next sections concerning this interpretation). In experiments, adjustments of Td and λ must then be made, which is tricky especially when dealing with the diﬃculty of obtaining accurate measurements at very long times. Some experiments have claimed that the MCT prediction for the relaxation time was not fullfilled close to the crossover in colloidal HS [74, 272]. The power-law prediction is indeed very diﬀerent from VFT-like fits shown in §3.7.4. Another serious concern is the work of Berthier and Tarjus [50, 51, 52], where they used a model of glass former where the attractive part of the interaction between particles can be switched on and oﬀ in such a way that the static structure factor is left unchanged, and is determined accurately numerically. The resulting dynamics is seen to change radically between the two systems (the one with attraction being much slower than the one without), even in the weakly supercooled regime, undermining the MCT basic tenet that static pair correlations determine the dynamics. Currently lots of works are aimed at understanding the MCT foundations and improving it [73], see the short review in §1.3.

Thouless-Anderson-Palmer free energy

To develop a static mean-field interpretation, let us analyze first the case of the mean-field Ising model in zero external field. In its Curie-Weiss or Bragg-Williams treatment, one gets the free energy as a function of the order parameter, as in Landau theory of phase transitions [247, 248], the global magnetization m. For T < Tc the up-down (Z2) symmetry is spontaneously broken and the system becomes non-ergodic: there are two minima with either positive or negative magnetization ±m∗(T ), and since the barrier between the two basins of the free energy diverges with the system size, the system in the thermodynamic limit is trapped in one of these two pure states [279, 308, 81, 339]. These pure states are disjoint sets of configurations having either positive or negative global magnetization m. They can be selected by imposing a vanishing external field with either sign on the system, which plays a similar role in choosing the state of the system to boundary conditions in physical systems. Both pure states are related by the Z2 symmetry. The partition function can be decomposed into a sum of partition functions restricted to either pure state. When the system falls in one of the basins, then only the corresponding restricted partition function is relevant to compute its thermodynamic properties: it will explore ergodically all configurations inside this state as time goes by, but will never encounter one of the configurations of the other pure state.
For disordered magnetic systems, a similar procedure has been followed by Thouless, Anderson and Palmer (TAP) in 1977 [384]: they considered a mean-field (fully connected) disordered model of a magnet, the Sherrington-Kirkpatrick (SK) model [358], where the coupling between pairs of spins is a random variable, giving rise to amorphous configurations of spins at low temperature (a spin glass). They com-puted its free energy as a function of local magnetizations mi. Minimizing the free energy with respect to the local magnetization gives the TAP states, which are an operational definition of pure states in the disordered case. Each TAP state is identified by its set of local magnetizations {mi} and gathers all spin configurations that have these magnetizations. The free energy landscape is obtained by scanning the values of the free energy over all possible sets of local magnetizations. As usual in mean-field, the free energy barrier between these states are infinite and the system can be trapped metastable states (local but not absolute minima of the free energy landscape) as well as it may in equilibrium (lowest-lying) states.
In the case of liquids and glasses, in principle this procedure may be repeated. One can think of it as a coarse-graining [85] of the system, or by using a lattice gas model [43]. Consider a thermodynamic potential of a lattice gas defined by occupation number7 ni > 0 on site i (number of particles whose center falls into the volume occupied by site i), Hamiltonian H and local chemical potentials µi (on each site i, acting as external fields): Ω({µi}) = −β ln exp −βH({ni}) + β µini (1.29) where the average is generated by the partition function e−βΩ.
We may put ourselves in a situation where one fixes the average particle number of site i rather than imposing local chemical potentials. We may achieve this by performing the Legendre transform µi ↔ ρi: F ({ρi}) = Ω({µi∗}) + i µi∗ρi { µi=µi∗
From these definitions, using the sum rule, one has ∂F = µ∗ (1.32) ∂ρi i
Note that for the hard core case ni ∈ {0, 1} with homogeneous chemical potential there is a mapping to the Ising ferromagnet [412, 249], which can be obtained by a coarse-graining process. so that when no external field is present the profiles {ρi} are determined as stationary points of the free energy, as for the TAP computation. In the next sections, we will generalize this thanks to liquid theory to a continuum [199], expressing the free energy as a functional of the local particle density ρ(x). Defining the fluctuation δρ(x) = ρ(x)−ρ where ρ = N/V is the global particle density of the liquid, we see that the situation is comparable to the TAP case. In the homogeneous liquid phase, δρ = 0 which is analogous to the high temperature paramagnet. At low T , the amorphous glassy phases can be described by a space-dependent δρ(x) as for a spin glass phase. Crystalline states would not be homogeneous but would have a periodic δρ(x), similarly to the anti-ferromagnet [323], even if the analogy does not go much further: the mean-field paramagnetic/antiferromagnetic transition is very diﬀerent from the freezing transition.

Goldstein’s energy landscape picture

An interpretation of the dynamical facts presented above in terms of static (free) energy landscape has been formulated in a very influential paper by Goldstein in 1969 [185]. The argument is reviewed in detail in [83, Sec. V.]. He imagined what the trajectory of the system would be in the potential energy landscape, its position given by the coordinates of all particles and the value of the whole potential energy at this phase space point. This multi-dimensional landscape can be seen as a very rugged one, like hills separated by narrow valleys [112, 323], see figure 1.10. Local minima of the potential energy landscape are called inherent structures in the literature [369, 83]. They enjoy the property of being mathematically well defined, but are not necessarily thermodynamically relevant. Therefore, it is more correct to talk about the free energy landscape in this way, whose shape changes according to temperature and density, and denote the basins as pure TAP states rather than inherent structures; the main drawback of the former compared to the latter is that pure (metastable) states are not well defined in non-mean field models, though attempts building on the eﬀective separation of timescales have been suggested by e.g. Biroli and Kurchan [63], based on the works of Gaveau and Schulman [175, 176, 177, 174], defining metastable states as eigenvectors of the ground states of a Fokker-Planck operator of the system.
In some fully connected models such as SK or the p-spin TAP states and inherent structures are equivalent [108] as well as for other analytical approaches of liquids in the RFOT spirit [284, 283, 103], which adds to the confusion [60].
The deepest minima, which are very few compared to the total number of local minima, represent crystalline configurations. They are very narrow and thus hard to find but separated by very high energy barriers to the rest of the valleys so that nucleation of liquid-like configuration is heavily suppressed for < Tm. At low T , such that energy barriers are greater than T , the system is stuck in some local minimum and the passage from a valley to another can be done only through an activated event [198]. These events represent local rearrangement of the structure, like hopping from a cage. The time spent in a basin, visiting configurations within this basin may give an interpretation of the β-relaxation process while the activated jumps allow for broader relaxations allowing in fine to visit the whole phase space, recovering ergodicity, as in the α-process. When the temperature is very high, there is no need to cross barriers since the activated energy scale T is greater: the relaxation proceeds by following simple paths from a point to another, climbing valleys and descending saddles which have unstable directions. The nature of the relaxation is very diﬀerent. These two mechanisms separate two temperature regions, and one may think that this occurs at some definite temperature Tx. By analogy with MCT one may identify the two temperatures Tx ‘ Td (1.33) giving a connection between statics and dynamics. Simulations [349] and experiments [364] have shown it to be consistent.
The topology of the potential energy landscape of mean-field spin glass models, such as the p-spin, has been explored since they are amenable to an analytic treatment [81, 82, 237]. It was shown that for E < Eth that the extrema of the Hamiltonian are dominated by minima, which means that the eigenvalues of the Hessian at these points are strictly positive, while at E = Eth marginal directions (with zero eigenvalues) appear, and for E < Eth the extrema are dominated by saddles having a nonzero fraction of the eigenvalues negative [84]. Besides, the energy E(T ) of the typical minima trapping the system for T < Td can be computed and grows with temperature, meaning that the system is trapped by higher energy minima the higher the temperature. In addition, one finds E(Td) = Eth (1.34)
Figure 1.10: A schematic illustration of the potential energy landscape. The longitudinal axis represents configurations of all dN coordinates. (a) High temperature liquid, the typical barrier height is less than the thermal energy, and all configurations can be accessed as indicated in blue. (b) Low temperature glass. The barrier height between basins is now much higher than T . [Reprinted from [338]]
confirming the guess (1.33). The p-spin model realizes exactly Goldstein’s scenario. Pure states appearing at this crossover from saddles to minima are called threshold states. The fact that for T > Td the plateau is already well formed before the dynamical transition can be interpreted in light of these results: the system first relaxes exponentially along the many stable directions that constitutes the vast majority of the spectrum of the Hessian of a typical extremum of the potential energy landscape, resulting in the -relaxation and the emergence of the plateau. Later, the system finds its way through the few almost marginal unstable directions, and leaves the saddle. This triggers the α-relaxation which is here of a non-activated nature. Nevertheless, for T < Td, since only minima with stable directions are present, the α-relaxation must proceed through an activated event, while the β-relaxation is a similar process of visiting stable directions. The p-spin is rather peculiar and one may question the applicability of the concepts derived from it to real glass former where analytical computations are much harder, but numerical studies seem to confirm this picture (see e.g. [13, 77, 191]).

Low temperature thermodynamics of glasses

Configurational entropy and Kauzmann’s paradox

The above static interpretation of the two-step relaxation process has been turned into an operative scheme to analyze the thermodynamics of the system. The picture of a temporary relaxation inside metastable states followed by cooperative rearrangements leading to an ergodic sampling of the phase space is translated at the level of the entropy of the system. The β-relaxation process leads to a restricted equilibrium defined by the time window τβ t τα, to which we can associate an entropy, the vibrational entropy, counting logarithmically the number of configurations visited in a typical basin, corresponding to vibrational motion inside cages. The α-relaxation aims at visiting disconnected basins and we count the number of these in the so-called configurational entropy. We may write the total entropy of the (supercooled) liquid as Sliq(T ) = Svib(T ) + Sc(T ) (1.35)
This equation assumes a sharp definition of metastable states and independence of the two relaxational processes. The basins trapping the dynamics are somewhat considered as equivalently populated in terms of configurations (or their distribution has a meaningful typical value, i.e. is not fat-tailed), which is only approximate in real glass formers, and better holds the lower the temperature, similarly to Goldstein’s scenario. A way to compute the configurational entropy is to regard the vibrational motion and its associated set of configurations as similar to the one observed in crystals, where harmonic modes perturb the lattice equilibrium positions for small non-zero temperature. These are not exactly the same as in crystals but it seems a reasonable approximation. Then we identify the vibrational entropy and the crystal entropy at the same temperature Svib(T ) ‘ Scry(T ), and using the definition of the specific heat C(T ) = T ∂S/∂T N,V, one obtains
The configurational entropy for some substances is shown in figure 1.11 from experimental measure-ments. Upon cooling down the configurational entropy tends to decrease up to the glass transition point where the system falls out of equilibrium and the structural relaxation is frozen, as in figure 1.2. Then one measures only a sort of oﬀ-equilibrium vibrational entropy. If one extrapolates the data to lower temperatures, one finds a point TK at which the configurational entropy vanishes. This would mean that if somehow we were able to equilibrate the system below the experimental glass transition temperature we would observe that at some point the entropy of the supercooled liquid becomes lower to the one of the crystal. This is the Kauzmann paradox [215]; this seems rather weird since we are used to think of a liquid as very disordered and thus expect its entropy to be larger than the ordered crystal. Kauzmann suggested that above TK there should be a kinetic spinodal, meaning that the relaxation time becomes larger than the time to nucleate the crystal and thus such extrapolations have no meaning, since the supercooled liquid cannot exist anymore near TK. Then, AGDM proposed that TK must be associated with a phase transition to an ideal glass state, where ergodicity is broken, the system being trapped in the lowest-lying states that are a few (subexponential) since Sc = 0 [83, 154, 180, 181, 2], inducing a downward jump of the specific heat. This was actually verified in some mean-field spin glass models, such as the spherical p-spin [109, 81, 417, 279], see §2.3. However, some spin-glass models do not ex-hibit such transitions, such as finite-dimensional spin plaquette models [205]. The experimental evidence (see figure 1.11) that the extrapolated Kauzmann transition point and the VFT divergence point seem to coincide, TK ‘ T0, gives an interpretation of the dynamic slowdown as a critical slowing down linked to a true thermodynamic phase transition towards the ideal glass, hence the search of a diverging correlation length. These temperature regions being out-of-reach experimentally and numerically, at least for now, only such speculations relying on theoretical hypothesis can be put forward.

The replica method

The number of stationary points (minima for low temperatures) of the free energy, i.e. the number of TAP states N , is given by the configurational entropy, also called complexity Σ = Sc in the spin glass context. In mean-field models they are shown to scale exponentially with the system size [81, 417]: N (f, T, N) ∼ eNΣ(f,T ) (1.37) with the free energy per particle f = F/N. The fact that the number of metastable states is exponential in the system size leads to a unusual thermodynamics and makes this entropic contribution compete with the free energy restricted to these states. When computing the partition function one gets, assuming a clear-cut separation in states indexed by α Z(T, V, N) = e−βH(C) ∼ e−βH(C) = e−βNfα = N (fα, T, N)e−βNfα C α C∈α α fα
Shifting from a discrete view of the free energy levels to a continuous one in the thermodynamic limit, we may use a saddle-point approximation:
fmax Z T, V , N ) ∼ d f eN[Σ(f,T )−βf] ( fmin (1.39) F (T, V, N) = −T ln Z ∼ f∗(T ) − T Σ(f∗(T ), T ) where f∗(T ) is a solution of the following saddle-point equation 1 = ∂Σ(f, T )
At the dynamical point Td the solution of this equation is f∗(Td) = fmax(Td) while when the temperature is lowered the solution is between fmin(T ) and fmax(T ). At some point TK the solution to this equation becomes f∗(TK) = fmin(TK). The thermodynamics is then dominated by the states that have zero complexity, i.e. states that are in subexponential number compared to the size of the system. This is nothing but the Kauzmann temperature at which the ideal glass transition takes place. For T < TK the thermodynamics is dominated by the same states that have zero complexity.
Figure 1.12: The complexity curve in the mean-field treatment of structural glasses. [Reprinted from [389]]
We observe that in order to characterize statically the system we must find a way to compute the complexity. This is provided by the real replica method, developed by Monasson, Mézard and Parisi [288, 277, 283, 284, 276, 285, 278]. The idea is to introduce m replicas (or clones) of the system, and introduce an attraction coupling of vanishing strength . This coupling, here put by hand, can be seen as emerging from imposing an external disordered field on the system, that will pin it into an amorphous configuration we wish to study [288]. As is usual with explicit disorder, this introduction can be analyzed through the use of replicas, see §2.3 for a detailed example. The Mari-Kurchan model [270] studied in chapter 4 can be seen as a practical realization. Another way is to take a clone of the system, let it equilibrate and couple the system to it: the clone then provides the pinning to an amorphous configuration, similarly to the spirit of the point-to-set pinning in §1.1.5. This is a second equivalent procedure, the Franz-Parisi potential method [156]. These are answers to the diﬃcult problem of pinning the system in a given pure state when there is no obvious symmetry breaking occuring and no explicit description of this pure state. Indeed, in the simple example of the ferromagnetic Ising model where the Z2 symmetry allows to compute easily restricted equilibrium measures in one of the two ferromagnetic pure states at T < Tc.
When the strength of the coupling → 0, for T > Td the coupling has no eﬀect, the m systems are independent ergodic liquids, since states do not have yet an influence on the thermodynamics of the system. For T < Td the coupling is eﬀective, there is a first-order transition to a phase where the m systems have the same free energy: they are trapped by the same metastable state [277, 283, 284, 276, 285, 278]. The m clones of a particle composing the system condense in the same cage, forming a kind of molecule of m atoms.

Table des matières

0 Liquides surfondus et transition vitreuse
0.1 Phénoménologie de base et diagramme des phases
0.1.1 Fondus de liquides !
0.1.2 Un peu de relaxation
0.1.3 Sonder la structure locale
0.1.4 L’effet de cage : relaxation en deux temps
0.1.5 Hétérogénéités dynamiques
0.1.6 Relation de Stokes-Einstein
0.1.7 Vieillissement
0.2 Sphères dures amorphes en dimension infinie
0.3 Aper¸cu et questions traitées dans cette thèse
1 Supercooled liquids and the glass transition
1.1 Basic phenomenology and phase diagram
1.1.1 Becoming supercool(ed)
1.1.2 Relaxation matters
1.1.3 Probing the local structure
1.1.4 The caging effect: two-step relaxation
1.1.5 Heterogeneous dynamics
1.1.6 Stokes-Einstein relation
1.1.7 Aging
1.2 A mean-field theoretical viewpoint: the Random First-Order Transition scenario
1.2.1 Dynamics within Mode-Coupling Theory
1.2.2 Thouless-Anderson-Palmer free energy
1.2.3 Goldstein’s energy landscape picture
1.2.4 Low temperature thermodynamics of glasses
1.2.5 Out-of-equilibrium dynamics
1.2.6 Scaling arguments beyond mean-field
1.2.7 Alternative theories
1.3 Dynamical theories of structural liquids and glasses
1.3.1 In the low-density liquid phase
1.3.2 Liquids to supercooled liquids
1.3.3 Theory of the glass crossover
1.4 Invariant curves in the phase diagram of liquids
1.4.1 Static and dynamic scalings
1.4.2 Isomorph theory
1.4.3 Exploiting the invariance
1.5 Amorphous Hard Spheres in high dimension
1.6 Outline and questions addressed in this thesis
2 Formalism of many-body disordered systems
2.1 The virial expansion in liquid theory
2.1.1 The grand potential
2.1.2 Legendre transform
2.2 The virial expansion of Hard-Sphere liquids in high dimension
2.3 Statics and the replica method: example of the spherical p-spin glass model
2.3.1 Replicated partition function
2.3.2 Replica-symmetric solution
2.3.3 One-step replica-symmetry-breaking solution
2.3.4 Full replica-symmetry breaking
2.4 Dynamics: the supersymmetric formalism
2.4.1 Introduction
2.4.2 Superfields and the superspace notation
2.4.3 Matricial representation: analogy with 2 × 2 block matrices
2.4.4 Derivation of a scalar field with respect to a superfield
2.4.5 Other useful identities
2.5 Analogy with static replica computations: application in the p-spin spherical model
2.5.1 The Lagrange multiplier
2.5.2 Field-theoretical formulation of the dynamics in SUSY notation
2.5.3 (Super)symmetries and equilibrium relations
2.5.4 The Mode-Coupling equation
2.5.5 Dynamical transition
3 Dynamics of liquids and glasses in the large-dimensional limit
3.1 Introduction
3.1.1 Definition of the model
3.1.2 Crystal cleared
3.1.3 The convenience of the spherical model
3.1.4 Outline of the derivation
3.2 Formulation of the dynamics
3.2.1 The dynamical action
3.2.2 Derivation of the generating functional using a virial expansion
3.2.3 Spherical setup
3.2.4 Translation of the dynamics into superfield language
3.3 Translational and rotational invariances
3.3.1 Functional spherical coordinates: invariances using the mean-squared displacement
3.3.2 Scalings in the infinite d limit
3.3.3 Ideal gas term
3.3.4 Interaction term
3.3.5 Final result in the limit d
3.4 Saddle-point equation
3.4.1 Explicit form of the kinetic term
3.4.2 Saddle-point equation for the dynamic correlations
3.4.3 Simplification of the saddle-point equation
3.5 Equilibrium hypothesis
3.6 Free dynamics
3.6.1 Saddle-point equation
3.6.2 Brownian diffusion on the sphere Sd(R)
3.6.3 Newtonian dynamics
3.7 Equation for the equilibrium dynamic correlations
3.7.1 Mode-coupling form of the saddle-point equation and the effective stiffness
3.7.2 Effective Langevin process
3.7.3 Memory kernel in equilibrium: resumming trajectories from the remote past
3.7.4 Relaxation at long times in the liquid phase
3.7.5 Getting rid of the sphere Sd(R): infinite radius limit
3.7.6 The Lagrange multiplier
3.7.7 Choice of the dynamics
3.8 Physical consequences of the equilibrium dynamical equations
3.8.1 Plateau and dynamical transition
3.8.2 Relation with rigorous lower bounds for sphere packings in high dimensions
3.8.3 Diffusion at long times
3.8.4 Connections with the microscopic model
3.8.5 Stokes-Einstein relation
3.9 Relation to the standard density formulation of Mode-Coupling Theory
3.9.1 MCT exponents
3.9.2 Intermediate scattering functions
3.9.3 The self part in infinite dimension
3.9.4 The factorization property
3.9.5 Comparison with MCT
3.10 Out-of-equilibrium dynamics
3.10.1 From the SUSY equations to the dynamical equations in an off-equilibrium regime
3.10.2 Equilibrium phase
3.10.3 Non-stationary temperature drive protocol at finite times
3.10.4 Density-driven dynamics: inflating spheres
4 Thermodynamics of the liquid and glass phases
4.1 Introduction
4.2 Setting of the problem
4.2.1 Definition of the model
4.2.2 Replicated partition function
4.2.3 Liquid phase entropy and distinguishability issues
4.2.4 Pair correlation function
4.2.5 The role of random rotations
4.2.6 Summary of the results
4.3 Rotational invariance and large-dimensional limit
4.4 Hierarchical matrices and replica symmetry breaking
4.4.1 Liquid (replica symmetric) phase
4.4.2 The 1-RSB glass phase
4.4.3 The full-RSB glass phase
4.4.4 Relation with previous works
4.5 Saddle-point equation for the order parameter
4.5.1 Derivation of the saddle-point equation
4.5.2 A microscopic expression of the memory kernel: force-force, stress-stress correlations, and the shear modulus
4.5.3 Replica symmetric solution
4.5.4 1-RSB solution
4.6 Connection between statics and dynamics: the formal analogy
4.7 Conclusion
5 Scale invariance in the phase diagram of particle systems
5.1 Introduction
5.2 Mayer integral contributions
5.2.1 Liquid free energy
5.2.2 The different regimes of the Mayer function
5.2.3 The effective diameter and the gap fluctuation scaling
5.2.4 Some examples
5.3 A second-order pseudo-transition
5.4 Isomorphs and effective potential
5.5 Dynamics and reduced units
5.5.1 With the supersymmetric analogy
5.5.2 Without the supersymmetric analogy
5.5.3 Reduced units
5.6 Glassy phases of the system
5.7 Virial-energy correlations
5.7.1 Virial truncation
5.7.2 The case of the exponential potentials
5.7.3 The slope of the isomorphs
5.8 Other types of potentials
5.9 Discussion
5.9.1 Simulation results from the Roskilde group
5.9.2 Summary
6 Conclusions and outlook
6.1 Main results
6.2 An overview of perspectives
A Recap of notations
A.1 Definition of basic quantities of the model
A.1.1 Basic definitions
A.1.2 Static quantities
A.1.3 Dynamic quantities
A.2 Replica coordinates
A.3 SUSY
A.4 Gaussian integrals and special functions
A.5 Averages
B Algebra of hierarchical matrices
B.1 RS matrices
B.2 1-RSB matrices
B.3 Full-RSB matrices
C A pedestrian presentation of the Martin-Siggia-Rose-De Dominicis-Janssen generating functional
C.1 The idea
C.2 Pedestrian way: discretization
C.2.1 Setting
C.2.2 Time scaling of the noise term
C.2.3 -discretization
C.2.4 The action in d dimensions
C.3 Additional remarks
C.4 `A la Feynman
D Derivation of the dynamic generating functional through the Mari-Kurchan model
E Dynamics from an equilibrium initial condition
F Derivation of the static free energy
F.1 Integrals for rotationally invariant functions
F.2 One-particle integrals: normalization of the density and ideal gas term
F.3 Two-particle integrals: the interaction term
F.3.1 Change of variables
F.3.2 Scaling of the mean square displacement
F.3.3 Mayer function
F.4 Final result
G Equivalence with previous static computations
G.1 The ideal gas term
G.2 The cofactor
G.3 Cayler-Menger determinant: rotation and translation invariances
H Equivalence between the Mari-Kurchan model and Hard Spheres without random shifts
H.1 With the random shifts
H.2 Without the random shifts
H.3 The Mayer function and scalings in d ! 1
I Group property of FDT superfields
I.1 Product
I.2 Inverse
Bibliography