Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond

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8.3 Ergodicity-restoring mechanisms 411

simply assumed that crystallization does not occur. Note that at T

the barrier to nucleation

is inﬁnite. However, in theoretical analysis for the deeply supercooled state an appropriate

model for the dynamics should include the competition with the crystallization process and

a logical resolution of the Kauzmann paradox.

8.3 Ergodicity-restoring mechanisms

So far we have considered the simple form of the mode-coupling model in which the

ergodic–nonergodic (ENE) transition occurs at a critical density or temperature. The static

or thermodynamic properties of the system do not undergo any drastic change on pass-

ing through the transition. These structural properties merely determine the strength of the

mode-coupling effects and hence the critical values of the control parameters for the tran-

sition point. The ENE transition is therefore essentially a dynamic transition. However,

as discussed above, evidence of such an ENE transition in experiments on glass-forming

liquids has been obtained only through indirect means. At best one can only conclude that

there is a crossover in the dynamic behavior of the liquid around the mode-coupling T

A very pertinent question to ask then is that of whether the sharp ENE transition described

in different models described above ﬁnally survives in a more careful analysis of the the-

ory. From a theoretical point of view, the occurrence of this sharp dynamic transition in

a real liquid having a ﬁnite number of participating degrees of freedom is unexpected.

This criticism of the ENE transition also holds even for the wave-vector-dependent model.

On the other hand, the ENE transition in the self-consistent model is not an artifact of

some approximate solution of the dynamics, although this may be the case in certain

speciﬁc models, e.g., in the case of a nonlinear oscillator (Siggia, 1985). Here the non-

linear equations for the dynamics are linked to an effective potential energy, which is

non-Gaussian, and the corresponding ENE transition predicted from the perturbative solu-

tions of the dynamic model is destroyed due to the nature of the equilibrium states. An

important point to note here is that the ENE-transition model discussed in the earlier section

is obtained with a completely Gaussian interaction free energy for which the possibility

of such disordering agents is absent. This is clear from the special form of the nonlin-

earities in the reversible part of the dynamics of the generalized Navier–Stokes equation

(Das et al., 1985a).

The crucial issue which needs to be addressed here is that of whether the MCT model

really preserves this simple form (with the prediction of an ideal transition) under proper

consideration of the equations governing the dynamics of the dense ﬂuid. In the following,

we address the question: do the equations of ﬂuctuating hydrodynamics applied to a com-

pressible liquid allow the ENE transition to occur? A careful analysis (Das and Mazenko,

1986) of the nonlinearities in the equations governing the dynamics of compressible liquids

shows that ergodicity holds at all densities. This is the case even with the liquid dynam-

ics being controlled by an effective Gaussian potential energy with a single minimum and

without any thermally activated hopping.

412 The ergodic–nonergodic transition

8.3.1 Ergodic behavior in the NFH model

In order to facilitate the discussion, we note the equivalence of the following asymptotic

behaviors of the density correlation function and its Laplace transform:

lim

t→∞

ρρ

(t) = 0,

ρρ

(z) ∼ 1/z,

ρρ

(ω) ∼ δ(ω). (8.3.1)

Let us now consider the model equations for the compressible liquid introduced in Section

3.3.1 to study the nature of the asymptotic dynamics. In the nonlinear ﬂuctuating-

hydrodynamics formulation discussed above, renormalization of the viscosity is obtained

from the self-energy 

ˆg ˆg

and the one-loop contribution to this self-energy involves the

product of the density correlation functions. As we have seen in earlier chapters, the

integral relation between L(ω) and G

ρρ

(t) (see eqn. (7.3.63)) gives rise to the nonlin-

ear feedback mechanism which forms the very basis of the self-consistent mode-coupling

approach to the glass physics. Therefore an ENE transition is characterized by a persistent

time dependence of the density correlation function and this implies a diverging viscosity.

This is equivalent to saying that the self-energy 

ˆg ˆg

blows up at small frequencies:



ˆg ˆg

=−Aδ(ω) +RT, (8.3.2)

where RT represents terms that are regular in the ω → 0 limit and A is the amplitude

of the singular term. In writing the above expression we are not ignoring the wave-vector

dependence but suppressing it to keep the notation simple.

Let us now enquire whether the above assumption (8.3.2) is compatible with the set of

Schwinger–Dyson equations corresponding to the MSR action (7.2.8). On putting

eqn. (8.3.2) back into eqn. (7.2.31) we obtain a δ(ω) peak in G

ρρ

as long as the response

function G

ρ ˆg

is not zero in the ω → 0 limit. This result follows simply on setting both α

and β equal to ρ in eqn. (7.2.31). It is straightforward to obtain that the singular contribu-

tion of G

ρρ

comes from 

ˆg ˆg

in the form

ρρ

= G

ρ ˆg



ˆg ˆg

ˆgρ

+ RT. (8.3.3)

For an ENE transition to occur, it is necessary that the response function G

ρ ˆg

not vanish as

ω → 0. The response functions G

are calculated from the relation (7.2.34), where the

are as given in Table 7.2. The response function G

ρ ˆg

therefore has the form

ρ ˆg

, (8.3.4)

where D is given by eqn. (7.2.35). This requires that ρ

goes to a nonzero value in the

zero-frequency limit and that the determinant D does not blow up as ω → 0. We assume,

8.3 Ergodicity-restoring mechanisms 413

with no reason to expect otherwise, that the ω → 0 limits of ρ

, γ , c

, and L are nonzero.

As a result D(ω → 0) is not inﬁnite

and hence G

ρ ˆg

= 0 in the low-frequency limit.

From the relation (7.2.31) it also follows that the correlation functions G

ρv

and G

have a delta-function contribution coming from 

ˆg ˆg

, provided that G

ˆgv

is nonzero in the

small-frequency limit. This is the case if the self-energy contribution γ(ω=0) = 0. To

demonstrate this, we consider the case in which α and β in eqn. (7.2.31) are both equal to

the current v, to obtain

= G

v ˆg



ˆg ˆg

ˆgv

+ RT. (8.3.5)

Similarly, G

ρv

is obtained by setting α and β equal to ρ and v, respectively,

ρv

= G

ρ ˆg



ˆg ˆg

ˆgv

+ RT. (8.3.6)

Using the expressions for N

provided in Table 7.2, it follows from eqns. (8.3.5) and

(8.3.6), respectively, that the correlation functions G

and G

ρv

both have a diverging con-

tribution in the ω → 0 limit if the quantity γ (or equivalently 

ˆvρ

) is nonvanishing in the

same limit. To summarize, from eqn. (7.2.31) it follows that all three correlation functions

ρρ

, G

ρv

, and G

exhibit a δ(ω) component, provided that γ is nonzero. On the other

hand, the correlation functions involving a momentum index g do not have a delta-function

peak at zero frequency. To demonstrate this, we note that if either index α or β on the LHS

of (7.2.31) is the momentum density g then the singular contribution due to 

ˆg ˆg

is coupled

to the response function G

g ˆg

. However, from Table 7.2 it follows that

g ˆg

D(ω)

(8.3.7)

vanishes as ω → 0 as long as D(ω =0) = 0. Therefore, the correlation functions involving

a momentum index g do not have a delta-function peak at zero frequency.

Next we consider the implications of the FDT (7.3.29) on the above results. Since G

vρ

and G

blow up, it follows that the imaginary parts of the response functions G

ˆgρ

and

ˆgv

, respectively, blow up. Considering the explicit form of these response functions from

Table 7.2, we obtain

vρ

=−2β

−1

Im(ρ

∗

)

∗

, (8.3.8)

=−2β

−1

Im{ρ

(ω +iq

γ)D

∗

}

∗

. (8.3.9)

This implies that we require simultaneously that D

∗

D is bounded and that the imaginary

parts of both ρ

∗

and (ω + iq

γ)D

∗

diverge. But, since both D



and D



, denoting

the real and imaginary parts of D, respectively, are bounded, ρ

and γ must diverge in

order to make this possible. However, in that case it follows from (7.2.35) that D must

also blow up and we have a contradiction. This leads to the obvious conclusion that the

It is useful to note that for the z → 0 limit the quantity L(z +iγ q

) in D does not diverge even when L ∼ 1/z becomes large,

since

Lγ q

remains ﬁnite in the nonhydrodynamic regime ω ∼ q

414 The ergodic–nonergodic transition

original assumption of a nonergodic phase is not supported in the model. The key self-

energy contribution is therefore 

ˆvρ

, or equivalently γ . If for some reason this quantity

vanishes at zero frequency, then G

ρv

and G

vanish as ω goes to zero. Then G

ρv

and G

do not have a δ(ω) component and one does not have the constraints on ρ

, γ , and D.In

this case one may have an ENE transition. The presence of the nonzero self-energy 

ρ ˆv

therefore crucial and ensures the absence of the ENE transition. This self-energy element

is a direct consequence of the presence of 1/ρ nonlinearities in the NFH equations. The

nonlinear constraint g =ρv takes care of this nonlinearity and allows construction of the

MSR action functional to all orders in perturbation theory. This avoids any truncation of a

nonpolynomial action in the perturbation action (Yeo, 2010). It is important to note that the

cutoff mechanism results from the nonperturbative treatment of the density nonlinearities

and does not violate any time-reversal symmetry.

The ergodicity-restoring mechanism which removes the sharp ENE transition has often

been referred to in the literature as one signifying hopping processes. Such a description

can (somewhat vaguely) be justiﬁed as follows. The sharp ENE transition, which is a result

of the nonlinear interaction of the density ﬂuctuations, signiﬁes the cage effect for the

single-particle motion in the dense system. At higher densities the persistence of the cages

grows in time until at a critical density a dynamic transition of the liquid to a nonergodic

glassy phase occurs. This is similar to other models for the glass transition, such as the free-

volume model (Grest and Cohen, 1981) and the trapping-diffusion model (Odagaki, 1995;

Odagaki and Hiwatari, 1990). The process of trapping of a particle in the cage formed

by its neighbors can be interpreted as being produced by a static random potential. The

ergodicity-restoring mechanism is then viewed as a relaxation via jumps over the almost

static potential barriers due to cage formation. However, calling this a “hopping process”

is clearly a matter of interpretation and could even be misleading.

8.3.2 The hydrodynamic limit

For small q and ω we can further analyze the ergodicity-restoring mechanism described

above. The density correlation function, which is the central quantity in the MCT, is

computed from the corresponding response function in this case,

ρρ

(q,ω) = 2β

−1

ρρ

(q)Im G

ˆρρ

(q,ω), (8.3.10)

where χ

ρρ

(q) is the Fourier transform of the equal-time correlation function. The Laplace

transform of the density–density correlation function normalized w.r.t. to its equal-time

value is obtained by inverting the G

−1

matrix as

ρ ˆρ

(q, z) =

zρ

+iL

(ω

−q

) + iL(z +iq

γ)

. (8.3.11)

Note that this relation holds only in the hydrodynamic limit and hence the one-loop expression for the cutoff function

presented in the following is valid only for small

q and ω.

8.3 Ergodicity-restoring mechanisms 415

The renormalization of the longitudinal viscosity L(q, z) is obtained from the longitudinal

part 

ˆgv

of the corresponding self-energy matrix 

ˆg

of the isotropic liquid,

L(q, z) = L



ˆgv

(q, z). (8.3.12)

As already discussed in the previous section, the absence of the ENE transition is primar-

ily a consequence of the nonzero self-energy 

ˆvρ

. If we ignore the self-energy 

ˆvρ

,the

expression (8.3.11) is identical to the conventional expression (8.1.31) for the density cor-

relation function. This crucial self-energy element 

ˆvρ

appears in the renormalized theory

as a consequence of the nonlinear constraint (6.2.22) which is introduced in order to deal

with the 1/ρ nonlinearity in the hydrodynamic equations.

For computing the effect of the ergodicity-restoring processes on the ﬁnal decay of the

density correlation function, we start from (8.3.11) in the following form:

φ(q, z) =



z + i δ(q, z) −



z + iL(q, z)



−1

, (8.3.13)

where we have used the deﬁnition

δ(q, z) = γ(q, z)



iL(q, z)

z + iL(q, z)



. (8.3.14)

In the asymptotic limit, for long enough times or correspondingly for small enough z, when

the longitudinal viscosity is large due to the feedback effect from the coupling of density

ﬂuctuations we have z  L(q, z). The density–density correlation function φ(q, z) given

by eqn. (8.3.13) reduces to the form

φ(q, z) =

z + i

(

−1

+˜γ q

)

. (8.3.15)

We have used above the leading-order result in q, L(q, z) ∼ q

L and γ(q, z) ∼ q

˜γ .

If the term involving ˜γ in the denominator of the RHS of eqn. (8.3.15) is ignored, the

longest time scale of relaxation for the density correlation function will be given by

−1

dynamic transition occurs in this model, with φ(q, z) ∼ 1/z corresponding to the situation

in which

L tends to diverge beyond a critical density. The enhancement of

L occurs due to

the feedback effects from slowly decaying density ﬂuctuations. For example, we have seen

in Section 8.1.2 that such a situation occurs at a packing fraction ϕ

=0.525 if the model

equations are considered with a hard-sphere structure factor. Owing to the presence of the

term γ the 1/z pole in φ is avoided even if

L becomes arbitrarily large due to feedback

effects. In this case, ˜γ q

determines the pole of the density correlation function. However,

this is not a new hydrodynamic mode in the system and there is no extra conservation

law involved. Since the viscosity remains ﬁnite, to leading order the ﬁnal relaxation of

the density correlation is controlled self-consistently by

−1

(Latz and Schmitz, 1996).

416 The ergodic–nonergodic transition

Similar ergodicity-restoring mechanisms removing the sharp ENE transition of the MCT

have been discussed by other authors (Götze and Sjögren, 1987; Schmitz et al., 1993; Das,

1996; Chong, 2008).

So far our discussion of the cutoff function is qualitative and treats the self-energy 

ˆvρ

in a nonperturbative manner. For estimating the extent of slowing down due to the den-

sity feedback mechanism from the theoretical model, a quantitative approach is needed.

This is done by evaluating the relevant self-energies at one-loop order. The evaluation of

the contribution to ˜γ at one-loop level is outlined in Appendix A8.3. In doing this we

make use of the relation (7.3.55), which follows from the FDT (7.3.29) in the hydrody-

namic limit. Thus γ is computed from the self-energy 

ˆv ˆv

.Theone-loop expression that

we use for the cutoff function is valid only in the hydrodynamic limit in which the rela-

tion (7.3.55) holds. The asymptotic behavior of the density correlation functions φ(q, t)

is obtained from the solution of eqn. (8.3.11) retaining the cutoff function γ(q, z).Both

these equations are combined in time and space in the form of a coupled set of integro-

differential equations (Das, 1987, 1990). With this extended model a qualitative change in

the dynamics is observed around the ideal transition point in calculations. A similar behav-

ior in experimental data obtained from the study of a large number of liquids classiﬁed as

fragile liquids was reported (Taborek et al., 1986).

Finally, the above discussion on the ergodicity-restoring mechanism applies only to

models involving both mass and momentum densities. What about the DDFT models,

which are formulated in terms of density ﬂuctuations only? The analysis presented in

Section 8.1.6 demonstrates that the DDFT model at one-loop level gives rise to the same

dynamic transition as that of the ﬂuctuating-hydrodynamics model. It is important to stress

here that the Kawasaki rearrangement (see Section A8.2) applied for this analysis is ad

hoc and can give rise to contradictory results if one extends it to two-loop order (Mazenko,

2008). In the NFH description the inclusion of the 1/ρ nonlinearities or the current gives

rise to a theory that is generally inconsistent with an ENE transition. The 1/ρ nonlinearity

is absent in the DDFT model. As we have seen above, by integrating out the momentum

variable from the NFH equations and applying what is termed the adiabatic approximation,

the ﬂuctuating equation for the single density variable can be obtained. The 1/ρ nonlin-

earities play a very nontrivial role in reaching the ﬁnal form of the stochastic equation for

ρ in DDFT. As the momentum variable is integrated out, the noise, which is of additive

nature in the NFH case, becomes multiplicative in the stochastic nonlinear equation for the

density ρ. The full nature of the dynamics which follows from the DDFT equation with

multiplicative noise and, in particular, the issue of restoring the ergodic behavior in this

case still remain unclear. Even the very feasibility of such a scheme is not evident, since in

the DDFT model the dynamics is formulated in terms of density ﬂuctuations only and the

momentum density has been integrated out of the problem at an early stage. It would also

be useful to note in this regard that, in computer simulations with Newtonian dynamics or

Brownian dynamics of the particles, an ENE transition to the nonergodic phase has never

been observed. It is therefore natural to expect that ergodic behavior will be restored in the

DDFT model ﬁnally.

8.3 Ergodicity-restoring mechanisms 417

Let us summarize the discussion above. The standard MCT with a sharp dynamic

transition is obtained when the cutoff function is ignored. In the full theory or the

so-called extended MCT the ENE transition is removed. We have shown here that

this result is nonperturbative and is reached without using the hydrodynamic limit.

However, the explicit one-loop expression for the cutoff function (of the ergodicity-

restoring mechanism) in terms of correlation functions has been reached only in the

hydrodynamic limit. The origin of this ergodic behavior has been shown to be the 1/ρ

nonlinearities appearing in the equations of motion. An important point in this respect is

the validity of the basic equations for describing compressible liquids. We have discussed

in the previous chapters how these equations, which are taken as plausible generalizations

of the hydrodynamics, are obtained. This involves the extension of the NFH description

to large wave vector or short wavelengths and these equations constitute the primary tools

available for studying the dynamics of a many-particle system. It is important to note in

this respect that, be it through the so-called memory-function formalism or kinetic-theory

approaches, a renormalized theory for self-consistent MCT for all wave vectors remains

elusive.

Phenomenological extensions of the mode-coupling theory have been made by extend-

ing the set of slow modes by including transverse modes and defect densities (Kim, 1992;

Das and Schilling, 1994; Yeo and Mazenko, 1995) or order parameters for structural relax-

ation (Liu and Oppenheim, 1997). A viable mechanism for understanding the boson peak

has also been constructed (Das and Schilling, 1999; Götze and Mayr, 2000; Cang et al.,

2005) using mode-coupling models involving transverse sound modes in the presence of

defects in a disordered solid (see also Kojima and Novikov (1996)). Evidence for the

role of such vibrational modes has indeed been found in recent computer experiments

on glass-forming liquids (Shintani and Tanaka, 2008).

8.3.3 Numerical solution of NFH equations

The equations of ﬂuctuating nonlinear hydrodynamics we have discussed in detail in the

earlier chapters give rise to the mode-coupling model. However, the standard MCT model

is a result obtained by considering the renormalized theory for the nonlinear dynamics

only up to one-loop order. Thus one is considering only to ﬁrst order in perturbation

theory an effect that is becoming large as the dynamic transition is approached. Hence,

while the conclusions reached can be qualitatively correct, quantitatively deﬁnite results

are unlikely to be obtained. The MCT for the structural relaxation has not been extended to

consider renormalization to higher order in a controlled manner. Indeed, this goes back to

the usual problem of dense-liquid-state theory where no suitable small parameter exists for

constructing a quantitatively accurate theory. The NFH equations have also been solved

numerically (Lust et al., 1993; Sen Gupta et al., 2008) to obtain results from a nonper-

turbative approach. Stable numerical solutions for the density ﬁeld on a lattice of ﬁnite

size have been obtained with a local coarse graining of the ﬂuctuating density ﬁeld. The

418 The ergodic–nonergodic transition

normalized dynamic correlation function of density ﬂuctuations δρ at times t

and t + t

is deﬁned as

C(t + t

, t

) =

δn(t + t

)δn(t

)

δn(t

)δn(t

)

. (8.3.16)

For large waiting time t

, the system equilibrates and time translational invariance holds.

In this case C(t + t

, t

) ≡ C(t). The decay of C(q

, t) is in agreement with the corre-

sponding MD-simulation results of Ullo and Yip (1985) for a one-component ﬂuid inter-

acting through the purely repulsive cut Lennard-Jones potential. The input c(r) used in

the numerical solution of the ﬂuctuating-hydrodynamics equations is taken from the stan-

dard integral-equation solutions for the structure of a uniform liquid interacting through

the same cut Lennard-Jones potential. The bare transport coefﬁcients which determine the

correlations of Gaussian noise in the NFH equations are such that the dynamics agrees

with computer-simulation results at short times. The agreement between the correlation

functions obtained from the MD simulations (Ullo and Yip, 1985) and those from numer-

ical solution of the NFH equations improves considerably at higher density. This reﬂects

the fact that at higher densities the mean free path of the ﬂuid particles gets smaller and

approaches the atomic length scale. As a result, the validity of generalized hydrodynamic

equations at short length scales (corresponding to wave vector q ∼ q

) improves with

increasing density.

The four-point correlation function χ

, t) at the structure-factor peak is deﬁned as

, t) =



F(k

, t)

F(−k

, t)

|

F(k

, 0)|

, (8.3.17)

where

F(k

, t) =δρ (k

, t)δρ(k

, 0) shows a peak near the α-relaxation time scale (see

Fig. 8.4). The peak height and position grow with supercooling roughly in same the manner

as that in which the α-relaxation grows. This four-point function in fact acts as a signature

of the departure from Gaussian nature of collective density ﬂuctuations. Note that a similar

behavior is seen at the single-particle level from the study of the so-called non-Gaussian

parameter (see Figs. 4.12(a) and 8.4). Therefore the NFH equations (which give rise to the

MCT) do indeed conform to the behavior reﬂecting the dynamical heterogeneities.

The numerical solution of the NFH equations for a one-component Lennard-Jones sys-

tem does not display many features of the two-step relaxation process. The various relax-

ation regimes predicted in the simple MCT model are distinct only when the different

characteristic time scales of the dynamics are separated and do not overlap. Indeed, as is

clear from our discussion above, the asymptotic behavior of the density correlation func-

tion strongly depends on how small the cutoff function γ is. The smaller γ is, the longer is

the time scale over which the feedback mechanism is effective in enhancing the transport

coefﬁcient. For small enough γ , the time scale of ﬁnal decay is widely separated from the

two-step relaxation of simple MCT described in Sections 8.1.1 and 8.1.2. On the other

hand, if γ does not become small enough self-consistently, a lot of the relaxation scenarios

8.4 Spin-glass models 419

that are predicted in the simple MCT will be screened out (Liu and Oppenheim, 1997),

since the system does not get close enough to the dynamic transition point.

8.4 Spin-glass models

We introduce in this section models of supercooled liquids formulated in terms of mod-

els developed for disordered systems referred to as spin glasses (Binder and Young, 1986;

Mézard et al., 1987; Fischer and Hertz, 1991; Binder and Kob, 2005). In particular, we

focus our discussion on the aspects of the spin-glass dynamics which are related to the feed-

back mechanism of the mode-coupling theory and the associated ENE transition discussed

earlier in this chapter. It is useful to note at the outset a few points about the two types

of disordered system, namely the structural and spin glasses, between which the anal-

ogy is being made here. Indeed, both represent systems without any long-range order. In

each case the difference between the glassy state and the corresponding high-temperature

disordered state, i.e., the normal liquid and the paramagnet, respectively, is in terms of

the time-dependent behavior. However, there are obvious differences between these sys-

tems as well. In the structural glass the most stable state having the lowest free energy is

the crystalline state with long-range order, whereas the supercooled liquid is metastable

with respect to the crystal. No such state exists corresponding to the disordered spin-glass

phase. Furthermore, in the spin glass the disorder is “quenched,” depicting the situation in

which the exchange-interaction coupling constants of the spins are random variables and

are time-independent on all experimental time scales. On the other hand, in the structural

glass the randomness is self-generated. It is important to note in this regard that spin-

glass-like behavior has been seen also in systems without any intrinsic disorder (Bouchaud

and Mézard, 1994; Marinari et al., 1994a, 1994b; Chandra et al., 1995). The presence of

frustration in such systems is crucial in order to produce the spin-glass state rather than

quenched randomness (Bouchaud et al., 1997; Cugliandolo, 2003). Indeed, systems with

and without quenched disorder have similar behavior as long as a large number of uncor-

related metastable states are associated. Another point of difference between the two types

of disordered systems lies in the nature of the transition from the high-temperature state to

the corresponding glassy state in the two cases. In the case of a spin glass a second-order

phase transition (in zero magnetic ﬁeld) characterized by a diverging relaxation time and

nonlinear magnetic susceptibility has been seen in experiments (Mydosh, 1993; Binder and

Young, 1986). The existence of an ideal thermodynamic structural glass transition, on the

other hand, remains speculative. This issue of a thermodynamic phase transition will be

discussed further in Chapter 10.

We ﬁrst discuss the scenario of the dynamic feedback mechanism of the self-consistent

MCT in the mean-ﬁeld spin models. Here we will consider the dynamics around the equi-

librium state similarly to the MCT for the supercooled liquids, which is almost exclusively

done for ﬂuctuations around the equilibrium state. Time translation invariance (TTI) holds

in such a situation, and the two-point correlation function is dependent only on the time

difference between the points at which the ﬂuctuations are considered. In recent years

420 The ergodic–nonergodic transition

good progress has been made in our understanding of the nonequilibrium problem through

spin models that reduce to a similar form (at least in terms of the mathematical structure)

to the corresponding MCT for supercooled liquids. These nonequilibrium aspects will be

considered in Chapter 9.

8.4.1 The p-spin interaction model

The connection between structural and spin glasses is proposed (Kirkpatrick and Thiru-

malai, 1987a, 1987b) through the p-spin (p > 2) interaction Hamiltonian. These theo-

ries show that the dynamical equations for the spin ﬂuctuations in the soft version of

the mean-ﬁeld p-spin interaction model are similar to the self-consistent mode-coupling

equations for the liquids. Unlike the Ising-type spin glasses, p-spin models lack reﬂec-

tion symmetry, and this is somewhat reminiscent of the case of real liquids. The spin–

spin interaction Hamiltonian with quenched disorder and involving interaction of p spins

(p > 2) is of the form

=−



···<i

···i

···σ

. (8.4.1)

The above form of the Hamiltonian needs to be regularized with a constraint to include

all possible values of the random variable J

···i

. The regularization used in the soft-spin

version (8.4.2) is done by including the function

H(σ ) with a σ

term. The Hamiltonian

of the mean-ﬁeld p-spin spin-glass model takes the following soft-spin version:

H = H



i=1



H(σ

) − h

, (8.4.2)

where the part

H of the Hamiltonian is given by

H(σ ) =



¯r

+ uσ

. (8.4.3)

The couplings {J

···i

} are independent random interactions whose probability distribution

function is given by

P(J

···i

) =



p−1

π p!



1/2

exp



−J

···i

p−1



. (8.4.4)

···i

therefore has zero mean and variance (J

···i

)

= p!/(2N

p−1

). The dependence

of the distribution on N in (8.4.4) is chosen such that the relative ﬂuctuation in energy

in the limit N →∞is vanishing, making the thermodynamic limit well behaved. The

length of the soft spin σ

is allowed to vary continuously from −∞ to +∞.Thisisa

mean-ﬁeld model since all spins interact with all other spins. The strength of the couplings

does not decay with distance and hence there is no role of an underlying lattice. These

are termed fully connected or inﬁnite-dimensional mean-ﬁeld models. Such models with