Mode-coupling theory

David R Reichman¹ and Patrick Charbonneau

Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA 02138, USA

¹ Present address: Department of Chemistry, Columbia University, 3000 Broadway, NY 10027, USA
Email: reichman@chem.columbia.edu and pcharbon@fas.harvard.edu

Received 21 April 2005
Accepted 4 May 2005
Published 31 May 2005

Contents

• 1. Important phenomenology for mode-coupling theory
• 2. The mode-coupling theory of density fluctuations
  • 2.1. Memory functions
  • 2.2. Some properties of the solution(s) of the MCT equation
    • 2.2.1. Schematic MCT
    • 2.2.2. Solutions of full MCT equations
    • 2.2.3. Redux: an assessment of the successes and failures of MCT
  • 2.3. Field-theoretic description
• 3. Looking ahead: beyond `simple' MCT
  • 3.1. Coupling to currents
  • 3.2. New closures
  • 3.3. Four-point correlations and dynamical heterogeneities
• References

1. Important phenomenology for mode-coupling theory

Since our objective will be to sketch a derivation of what we will call mode-coupling theory (MCT), we will focus our attention on one observable in particular, namely density fluctuations. For this we will first define some of the concepts needed to do so.

We want to calculate a specific time correlation function. In general, such a function is expressed as follows:

It is an ensemble average of the evolution of the fluctuations of a variable in time, at equilibrium. As seen in figure 1, A(t) fluctuates around its average value in equilibrium, while C(t) measures the correlation of A at one time with the value of A at another time.

The density or particles in a liquid can be one example of A(t),

which we can Fourier transform,

In this case the correlation function will be labelled F(k, t), which is can be expressed as follows:

Note that we need to have (i.e. - k + k = 0!) to conserve momentum; otherwise the correlation function is equal to zero.

The variables labelled with k measure density fluctuations in reciprocal (`k = |k| ') space, which can be thought of as the inverse length. When k is small we are looking at long length scales, as we can see in figure 1. When it is large, we are probing very short scales.

The function F(k, t) is essentially what scattering experiments measure. At t = 0,

where S(k) is called the static structure factor of the liquid. Why that name? Consider the radial distribution function of a liquid g(r). The function g(r) is proportional to the probability that a particle is a distance r away from a particle at the origin. In a dense liquid, g(r) shows the structure of the solvation shells as depicted in figure 2. Also, it can be shown that [1-3]

where ρ = N/V is the density of the system, and thus S(k) is also indicating something about the liquid structure [1-3]. An example for a simple liquid is depicted in figure 2.

Figure 2. Left: radial distribution function g(r) for a simple liquid of size σ. Right: the corresponding structure factor S(k). A sample structure is also depicted where the solvation shells are indicated by the dotted lines. The exclusion radius can be seen in the absence of amplitude of g(r) for .

But how do we expect F(k, t) to behave? For high temperatures - above the melting point - F(k, t) will decay like a single-exponential function in time for k ≥ 2π/σ as plotted in figure 3. For supercooled liquids, the situation is different and a characteristic decay pattern can also be seen in figure 3. We observe a multi-step relaxation.

(1)	At short times decay is coming from free and collisional events that involve local particle motion. Consistent with a short-time expansion, in this regime [1-3]. This will be true at any temperature. We will not be concerned much with this part of the decay.
(2)	Intermediate times encompass a period during which particles appear trapped in cages formed by other particles. This regime is the β-relaxation regime. The decay to the plateau (IIa) may be fitted as f + At - a and the decay from the plateau (IIb) as f - Btb. Also, the exponents have a scaling consistent with the relationship
(3)	At long times, in the α-relaxation regime, the decay may be fitted to a stretched exponential law [4] with 0 < β < 1. Do not be confused by the notation. It is the β power that appears in the α-relaxation regime! In general β and τ will be k and temperature dependent.

In a later section, we will return to this kind of phenomenology. For now, we just make a few superficial remarks about things that will be covered in more depth by others in these lectures.

The constant τ that appears in the stretched exponential decay law is strongly temperature dependent. All transport coefficients - D (diffusion), η (viscosity), etc - are strongly temperature dependent as well. Over a rather wide range of temperatures, a fit to this temperature dependence may be [4]

Clearly, as T₀ is approached, relaxation times become so large that the system cannot stay in equilibrium. Other fitting forms, some that do not imply a divergence at finite temperatures, may be used to fit the data as well.

Some systems, hard spheres for example, are not characterized by temperature, but by density of the packing fraction , where a is the particle radius. For such systems, one may fit with [4]

or with other forms.

2. The mode-coupling theory of density fluctuations

Our strategy will be to derive an exact equation of motion for F(k, t) and then to make approximations that allow us to solve them [5, 6]. The approximations are uncontrolled, and we will judge them by their success or failure.

2.1. Memory functions

Consider some classical function of phase space variables A(t), where the time dependence originates from that of the positions r_i and of the momenta p_i for an N-particle system. We know from Hamilton's equations that

where {,} is a classical Poisson bracket, which can be expressed as follows:

Also, for liquids of interest, is a classical Hamiltonian with pairwise interactions (r) between the particles,

We can thus identify the following:

It would be possible to integrate the differential equation to find , but this is not useful by itself.

We also need to define a scalar product of the variables as

Now, consider an operator called a projection operator  [1-3], [5, 6],

If A is a vector, (A, A) ^- 1 is thus the inverse of a matrix. Note also that . In geometrical terms, the projection operator finds the component of some variable B along the chosen direction A, as depicted in figure 4.

Figure 4. A two-dimensional version of the projection operator . The quantity B is projected unto the space A, which extracts the A component of B (indicated by a thick dashed line).

This is useful, as we can extract from an arbitrary B how much `character' of A it has. In particular, the operator A may be a slowly varying (quasi-hydrodynamic) variable. Consider the density as defined in equation (4),

and then

where j_k^L(t) is the longitudinal current. If k is small (large length scales), then is approximately small. This is what is meant by slow. In the limit k = 0, then , and the density is strictly conserved. As figure 1 indicates, if k is small, the area of density fluctuations is large, i.e. the rate at which the number of particles fluctuates is small.

We now want to find the exact equation of motion for a correlation function

where

Now, writing

can be obtained by differentiating both sides of the equation:

where the last equality follows from the fact that . As a result, we may write

where f(t) is called the fluctuating force:

What does this mean? The fluctuating force is obtained by taking the time derivative of A, using the complementary projection operator to remove the `A' character - perhaps the slow character - and is then propagated in the orthogonal - fast - space. To put it another way, if removes the slow character from a variable, then the fluctuating force is the remaining fast force. We will come back to this later.

It can be shown that (A, f(t)) = 0, by noticing that the definition of f(t) contains the factor. This means that f(t) is orthogonal to A, in accord with the discussion above. Noting that

the first term on the RHS of equation (22) allows the equation of motion to be rewritten as

where we define the memory function K(t):

This is a fundamental and exact equation for the time dependence of A(t).

Defining the correlation matrix

and using the equality (A, f(t)) = 0, we get

as an exact equation for the matrix of correlation functions C(t) that we will want to compute. The problem with computing C(t) is embodied in the difficulty of determining K(t).

Now, we want to focus this general framework on density fluctuations with respect to the bulk density ρ, which will allow us to get an expression for the intermediate scattering function. Consider [3]

where

and therefore

For the purpose of this demonstration, we will concentrate on the element in the lower left corner of the matrix, which is in this case (N/i q)(d²F(q, t)/dt²). At t = 0, the matrix reduces to

Also,

To obtain the last two equations, we used integration by parts, the property that the correlation of an observable and its derivative is always zero, and the statistical thermodynamics result that follows:

The random force f(0) is expressed as

We will now look at the equation of motion term by term. First,

Note that the lower left corner term equals (N/i q)(d²F(q, t)/dt²). Second,

Notice the lower left corner term is  - (qNk_BT/i mS(q))F(q, t) this time. Lastly, the memory matrix is

Concentrating on the lower left corner, using the equation of motion from equation (27), we find [1-3], [5]

This equation is exact, but impossible to solve. To make approximations, we will look at using some intuition. Recall that

and that

Also, note that in this last equation dp_i/dt is a force and therefore

where we made the momentum space transformation . We then see that hidden in the fluctuating force is a pair of densities. This illustrates an important point: at first, we may have suspected that the fluctuating force is a fast variable and that it decays on a short timescale because we removed the slow modes δρ_k from it - but we see that it contains, at leading order, a slow character (at least if δρ_k is slow) from the product of slow modes δρ_kδρ _- k ! Overall the time derivative of the current has the symmetry of δρ_kδρ_q - k, where the q factor comes from that multiplies the force in equation (41).

We will now approximate . As a convention, note that .

(1)	Make the replacement , where we define the new projection operator and where Ak1, k2 = δρk1δρk2. The operator simply projects Rq onto its dominant slow product mode. We neglect the in the exponent, simply because it is hard to compute anything keeping it there. However, to , it may be shown that this neglect has no consequences.
(2)	Factorize four-point density terms into products of two-point ones.

Using this algorithm, we get

where

The denominator has a product of four density variables that will be factorized into products of two structure factors. The numerator has terms like

where the result was obtained by integration by parts, and

Let us calculate one of the terms in equation (46),

where we used the result of equation (34), to complete the calculation. The other term similarly gives

The term in equation (47) is hard to compute directly, but within the convolution approximation, it can be reduced as follows [1, 5]:

After treating all static density fluctuations within the Gaussian (and convolution) approximations, we find that the vertex V_q(k₁, k₂) can be expressed as a function of only two wavevectors. As a consequence of translational invariance, we would be left only with terms involving the difference of wavevectors , which allows us to write V_q(k₁, k₂) = V_k, q - k. Also note that the summation is now only over k. Combining all terms gives

where we have rewritten the result using the direct correlation function . So, piecing this together,

where we used Wick's factorization and where

We need to convert the discrete sum to the continuous integral, , and multiply by the m/Nk_BT prefactor as obtained in equation (39), to get the final MCT equation:

with

2.2. Some properties of the solution(s) of the MCT equation

2.2.1. Schematic MCT. Via the approximation discussed by Bengtzelius et al we can reduce our MCT equation to a schematic form,

where Φ(t) ~ F(k, t), before we neglect the coupling wavevectors. The solutions of equation (56) have been discussed by Leutheusser [7] and Bengtzelius et al [8]. The most striking feature of this equation, and of the full MCT equation from which we `derived' it, is that there is a transition to a completely non-ergodic phase for particular Ω₀² and λ (or T and ρ for the real MCT equation). The two cases are depicted in figure 5.

The transition from ergodic to non-ergodic at a sharp, well-defined set of parameters may be interpreted as the transition from a liquid to a solid. The fact that correlations do not decay as is indicative of this. However, no information of an ordered state was used or imposed. Thus, the solid could only be a disordered one, i.e. a glass.

2.2.2. Solutions of full MCT equations. If we denote by T_c the temperature where MCT predicts a glass transition, the relaxation time τ scales as [5-8]

which means that when , τ diverges as a power law with a universal exponent γ. This form may fit data, but only over a limited temperature range.

Also, the decay in the β-relaxation regime is indeed given by [5-8]

with

at least for T very close to T_c, i.e. when . This is a great triumph of the MCT equations and is fully consistent with simulations and experiments [5, 6]. Furthermore, the power γ is related to the exponents a and b as

For the α-relaxation regime, an approximate solution of the full MCT equations is indeed approximately given by

and this stretched exponential function describes well experiments and simulations. The schematic equation, however, only exhibits exponential decay.

More generally, MCT predicts that for a correlator at temperature T a time-temperature superposition holds [5, 6]:

where C(t, T) is a correlation function, is some master function and τ(T) is the α-relaxation time. This is also generally consistent with experiments and simulations.

2.2.3. Redux: an assessment of the successes and failures of MCT. Even a scientist who is opposed to the spirit and approximations that go into MCT ought to be impressed by its success, where it succeeds. Furthermore, it is essentially the only first-principles theory of glassy liquids. That is, from the structure of the liquid alone (S(k), the structure factor) a detailed set of dynamical predictions emerge. We will now spell out where MCT works and where (we think) it does not.

(1)	The best-known failure of MCT is that it predicts a sharp glass transition at a temperature Tc, but Tc >  Tg. This means that MCT predicts kinetic arrest to a non-ergodic phase at temperatures where the system is still ergodic and liquid.
(2)	MCT also predicts power-law divergence of transport coefficients and relaxation times as in equation (57), but this is less accurate over a wide range of temperatures than the temperature dependence of transport coefficients given in equation (9) [4]. It is a reasonable fitting form over several decades of relaxation time for mildly supercooled liquids. In addition, the parameters β, a, b, ... are predicted to be constant in MCT, at least for temperatures for . But in actuality, they are mildly temperature dependent. One should be careful, however, not to take asymptotic predictions of MCT and apply them to cases where (T - Tc)/Tc is not small [5].
(3)	Another failure of MCT is in the prediction of certain indicators of collective relaxation. In general, timescales and length scales of such heterogeneous motion can be probed by multi-point correlations. A simple, non-multi-point function that seems to correlate crudely with the timescale of such motion is the non-Gaussian parameter where for a tagged particle and similarly for the other term. Usually, the behaviour of is similar to what is depicted in figure 6. However, MCT predictions of α2(t) are quite inaccurate. In general, MCT fails to accurately depict properties such as the non-Gaussian parameter and the breakdown of the Stokes-Einstein relation.

One may take from this last result that MCT is not capable of saying anything about dynamically heterogeneous motion in supercooled liquids [12], but perhaps this statement is too strong. We will explore this further in the next section.

2.3. Field-theoretic description

Before finishing this section, we provide a sketch of the field-theoretic approach to schematic MCT [10, 11]. In some sense, the memory function approach can be thought of as arising from coupled Langevin equations for the modes δρ_q and j_q^L. For example,

As a toy model for this (forgetting vector labels and wavevectors), we get

with . We also define

and

where G₀ is the bare response function, or propagator. It must be zero if t < t '. The solution for (t), with (0) = 0 is, in graphical terms,

where an arrow represents G₀, a × represents the noise, and a factor of g/2 is associated with each branching point. These terms are simply obtained as a solution from integrating equation (66). In a more compact notation,

where `·' is a simple product.

We define two kinds of functions,

and

where the last equality is true for Gaussian noise. Again, C(t, t ') can be defined on the entire (t, t ') plane but G(t, t ') only for t > t '. Let us construct a series for the two functions. The zeroth-order contribution to C(t, t ') is given as follows:

The bracket average implies connecting the ×-vertices to form diagrams with none of these left. Diagrams that do not pair up all noise vertices (i.e. those with an odd number of ×-vertices) average to zero.

Going beyond zeroth order, we get the following:

where to obtain the last line we defined

What about G(t, t ') ?

where the lone × has to be attached to a × on the tree and then the remaining diagram is closed. Thus,

where we similarly defined

This is an exact, formal representation of the perturbations series. In fact, in some sense it is simply a definition of the kernels D and Σ. We can appeal to the structure of the perturbation series to justify this.

Also, the lower limit of the second integration ensures that t₂ > t '. In fact, let us take a closer look at the limits of integration for a sample diagram, the second term in equation (77), which is reproduced on the left-hand side of figure 7 with additional labels. The incoming branch imposes t > t₁, the central loop t₁ > t₂, and the outgoing branch t₂ > t ', for an overall t > t₁ > t₂ > t '. The resulting integration limits are thus .

If you have followed so far, you might be asking yourself what happened to the diagrams like the one appearing on the right in figure 7. Such closed loops are called tadpoles. They do not contribute to the time dependence of C or G, and we assume that their contribution is absorbed into μ(t). The next lowest order terms are then the diagrammatic forms we last drew in equations (74) and (77).

To make a self-consistent approximation we replace G₀ and C₀ in our second-order approximations for D and Σ by G and C. We will call this the mode-coupling approximation, a name that will be clear in meaning at the end [10, 11]. This is equations (74) and (77) with

These two equations can be further manipulated by noting that

and so we can multiply both sides of equations (74) and (77) by G₀ ^- 1:

where I is the identity operator. In other notation,

This still does not look like the MCT equations derived before. We will manipulate the RHS of the second equation to achieve this. Taking the first term, we make the substitution

We can replace because of the restriction on time arguments in G(t ', t ' ') and we can substitute G for C as written by using the fluctuation-dissipation theorem (FDT) and assuming that the system is at equilibrium. The result can now be integrated by parts,

Note that we can neglect the δ-function part of D here, since it provides no contribution. So, we denote the regular part of D as D '.

Again, using the FDT, we can make the substitution

yielding an integral which can be combined with the other term of equation (85) to obtain

Now, using the FDT in the reverse direction and integrating by parts, we get

Combining all terms on the right-hand side of equation (85), we get

Using the fact that, in equilibrium, these functions are time-translation invariant,

and making the transformation,

we find

where

This is just the schematic model with

The only difference is that appears instead of . This actually makes no difference as far as the glassy properties are concerned. In fact, for models of overdamped systems, such as Brownian colloidal spheres, is what appears naturally in the reduction of the full MCT equations of the schematic model. This completes the relationship between the memory function/projection operator MCT derivation and a field-theoretic approach.

3. Looking ahead: beyond `simple' MCT

In this section we will outline some thoughts on attempts to do better than the MCT derived thus far. This is not an exhaustive discussion, but is meant to give an idea about what can be done.

3.1. Coupling to currents

Götze and Sjögren [13] as well as Das and Mazenko [14] have developed theories that remove the sharp transition at T = T₀. In both cases, it is the couplings to certain current modes that are ignored in the expressions we have just derived that restore ergodicity below T_c.

In the theory of Götze and Sjögren, the Laplace transform of our exact equation of motion for F(k, t) is given as

where Ω(k) = k²k_BT/mS(k) and M(k, z) is the Laplace transform of the memory function

and, similarly,

Essentially, within the `extended' MCT of Götze and Sjögren,

where K(k, z) is the ordinary MCT memory function given in equation (39). The expression for δ_h(k, z) is complicated, but to see what it does, note that:

(1)	If δh = 0, we recover exactly the ordinary MCT that we have derived before. This can be checked by applying a Laplace transform the old expressions.
(2)	If δh(k, z) has no singularities as , the strict transition at Tc in the MCT that we have previously derived is removed since relaxation is governed by M ~ 1/|δh| at long times and not K(k, z), which yields a pole singularity in z space.

In the theory of Das and Mazenko, a hydrodynamic approach is used. The kinetic energy of the free energy functional in terms of current j and density ρ has the form

This is like the usual p²/2m kinetic energy. However, the 1/ρ part coupled to the current rounds off the strict singularity at T_c, like in the Götze and Sjögren theory.

Note that in both theories, we need currents to restore ergodicity. For some systems, like (simulated) colloidal hard spheres undergoing Brownian motions, these currents do not exist! Thus, the Götze and Sjögren and the Das and Mazenko theories cannot be used to improve ordinary MCT there.

3.2. New closures

As mentioned above, the extended MCT of Götze and Sjögren and of Das and Mazenko cannot tell us anything (beyond ordinary MCT) for Brownian hard sphere systems. An interesting proposal was recently put forward by Szamel [15]. The main idea is not to factorize the memory function expression leading to the ordinary MCT given in equation (39), but to write an exact equation of motion for it, and then factorize the memory function for the new equation. Here is a sketch of the idea.

Recall our old approach to MCT,

where essentially is a four-point function of density variables. In the old approach, the closure involved

and this allowed us to solve for F(k, t).

Instead of factorizing the four-point memory kernel, let us write an exact equation of motion for it, following the same lines of reasoning as before:

The wavevector indices are suppressed to simplify the notation in order to clarify the idea behind the manipulations. This has the same form as before, but with new frequencies Γ and a new memory function R(t - τ).

Schematically, is a six-point function! We can close the equation for K and F, by making the approximation R ≈ K·F, the product of a four-point and a two-point function. This yields two coupled sets of integro-differential equations that may be solved self-consistently yielding a converged F(k, t).

This approach has not been considered for the full dynamics of F(k, t), but yields a better estimate for T_c (i.e. the T_c that is extracted is closer to the measured glass transition).

3.3. Four-point correlations and dynamical heterogeneities

It was mentioned in section 2 that MCT does not describe well

and that α₂(t) seems to correlate well with the timescale of maximal dynamical heterogeneity [16, 17]. It turns out that this timescale is in the late β-regime. This highlights the fact α₂(t) yields information on transiently mobile particles that jump due to the destruction of cages. It should be noted that there is no length scale dependence in α₂(t).

To gain some information about a growing (dynamical) length scale, a multi-point dynamical generalization of the static structure factor may be studied [18-21]. The limit of this structure factor, as Sharon Glotzer discusses in her lectures, is the susceptibility

where the function θ_a(|r₁ - r₂|) equals one when |r₁ - r₂| ≤ a, and zero otherwise [18]. The timescale at which χ₄(t) peaks is generally in the α-regime. The growing length scale associated with dynamic heterogeneity is associated with slow moving, transiently caged particles.

Given the superficial similarity with α₂(t), as shown in figure 8, one might conclude that MCT cannot compute objects like χ₄, but this is not the case. Recent work by Biroli and Bouchaud [19], motivated by the earlier insight of Franz and Parisi [20] and Kirkpatrick and Thirumalai [21], shows that MCT may make quantitative statements about the scales of length and time associated with dynamical heterogeneity. So far, absolute length scales have not been computed, but dynamical exponents z relating timescales τ and length scales ξ have:

where z = 2γ and γ is given in equation (59).

References

[1]

Balucani U and Zoppi M, 1994 Dynamics of the Liquid State (New York: Oxford University Press) (General reference on liquid state theory and on memory functions) 

[2]

Boon J-P and Yip S, 1980 Molecular Hydrodynamics (New York: McGraw-Hill) (General reference on liquid state theory and on memory functions) 

[3]

Hansen J-P and McDonald I R, 1986 Theory of Simple Liquids (London: Academic) (General reference on liquid state theory and on memory functions) 

[4]

Ediger M D, Angell C A and Nagel S R, Supercooled liquids and glasses, 1996 J. Phys. Chem. 100 13200 (A good review of glass-forming liquids) 
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Kob W, Supercooled liquids, the glass transition, and computer simulations, 2002 Lecture Notes for Les Houches 2002 Summer School - Session LXXVII: Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter p 47 [cond-mat/0212344] (A good review of MCT and tests of MCT with simulation) 
Preprint

[6]

Götze W, Recent tests of the mode-coupling theory for glassy dynamics, 1999 J. Phys.: Condens. Matter 11 A1 (A good review of MCT and tests of MCT with simulation) 
IOPscience

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Leutheusser E, Dynamical model of the liquid-glass transition, 1984 Phys. Rev. A 29 2765 (A discussion of the schematic MCT) 
CrossRef

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Bengtzelius U, Götze W and Sjölander A, Dynamics of supercooled liquids and the glass-transition, 1984 J. Phys. C: Solid State Phys. 17 5915 (A discussion of the schematic MCT) 
IOPscience

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Dawson K, Foffi G, Fuchs M, Götze W, Sciortino F, Sperl M, Tartaglia P, Voigtmann T and Zaccarelli E, Higher-order glass-transition singularities in colloidal systems with attractive interactions, 2001 Phys. Rev. E 63 011401 (Predictions for behaviour in attractive colloidal suspensions) 
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Bouchaud J P, Cugliandolo L, Kurchan J and Mezard M, Mode-coupling approximations, glass theory and disordered systems, 1996 Physica A 226 243 (A discussion of the field-theoretic approach to schematic MCT) 
CrossRef

[11]

Bouchaud J P, Cugliandolo L, Kurchan J and Mezard M, Out of equilibrium dynamics in spin-glasses and other glassy systems, 1997 Preprint cond-mat/9702070(A discussion of the field-theoretic approach to schematic MCT) 
This was also published in Young A P (ed) 1998 Spin Glasses and Random Fields (Directions in Condensed Matter Physics) (Singapore: World Scientific) 
It should also be noted that in the above two references there are some typos that lead to an MCT equation with an incorrect sign. This was corrected in this version of the derivation. 

[12]

Donati C, Douglas J F, Kob W, Plimpton S J, Poole P H and Glotzer S C, Stringlike cooperative motion in a supercooled liquid, 1998 Phys. Rev. Lett. 80 2338 (A discussion of the non-Gaussian parameter and dynamic heterogeneity) 
CrossRef

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Götze W and Sjögren L, The glass-transition singularity, 1987 Z. Phys. B 65 415 (Extended MCT via coupling to currents) 
CrossRef

[14]

Das S and Mazenko G, Fluctuating nonlinear hydrodynamics and the liquid-glass transition, 1986 Phys. Rev. A 34 2265 (Extended MCT via coupling to currents) 
CrossRef

[15]

Szamel G, Colloidal glass transition: beyond mode-coupling theory, 2003 Phys. Rev. Lett. 90 228301 (New closure for MCT) 
CrossRefPubMed

[16]

Donati C, Glotzer S C, Poole P H, Kob W and Plimpton S J, Spatial correlations of mobility and immobility in a glass-forming Lennard-Jones liquid, 1999 Phys. Rev. E 60 3107 (A discussion of the non-Gaussian parameter and dynamic heterogeneity) 
CrossRef

[17]

Kob W, Donati C, Plimpton S J, Poole P H and Glotzer S C, Dynamical heterogeneities in a supercooled Lennard-Jones liquid, 1997 Phys. Rev. Lett. 79 2827 (A discussion of the non-Gaussian parameter and dynamic heterogeneity) 
CrossRef

[18]

Glotzer S C, Novikov V N and Schroder T B, Time-dependent, four-point density correlation function description of dynamical heterogeneity and decoupling in supercooled liquids, 2000 J. Chem. Phys. 112 509 (Simulations and definition of the χ₄(t) function) 
CrossRef

[19]

Biroli G and Bouchaud J-P, Diverging lengthscale and upper critical dimension in the mode-coupling theory of the glass transition, 2004 Europhys. Lett. 67 21 (MCT theory of χ₄(t)) 
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