Mode-coupling equations

The scheme of the projection-operator formalism above is very general and is also used e.g. to compute velocity correlation functions or transverse collective modes of shear waves in liquids far below the glass transition [50]. Everything depends on the choice of the dynamic variables 𝐴_𝑘(⃗𝑟^𝑁, ⃗𝑝^𝑁). Considering MCT at the glass transition, the choice of slow variables are the particle densities𝑛 and the particle current𝑗⃗that depends on the particle velocities𝑣⃗_𝑖. It turns out to be useful to write them in Fourier space as The last equation is known as the continuity equation telling us that temporal changes in density are proportional to the longitudinal modes of the current𝑗_𝐿(𝑞, 𝑡) =⃗ 𝑞⃗⋅𝑗⃗(𝑞, 𝑡)∕⃗ |𝑞⃗|. Since our Hamiltonian is isotropic, there is no correlation between the velocities in orthogonal directions, so that𝑣_𝑥, 𝑣_𝑦and𝑣_𝑧are uncorrelated. Therefore longitudinal modes and transverse modes of the current are uncorrelated as well and consequently the particle densities𝑛do not couple to the transverse modes of the current so that the choice of dynamical variables can be restricted to𝐀= {𝑛(𝑞, 𝑡), 𝑗⃗ _𝐿(𝑞, 𝑡)}⃗ . Our particular interest is in the normalized coherent density correlator (see equs.1.10and1.13)

𝜙(𝑞, 𝑡) =𝐹(𝑞, 𝑡)∕𝑆(𝑞, 𝑡) =(

𝑛(𝑞, 𝑡)⃗ |𝑛(𝑞⃗))

∕(

𝑛(𝑞⃗)|𝑛(𝑞)⃗)

(1.37) As we assume the structure of our liquid or glass to be isotropic𝜙does not depend on the direction of𝑞⃗. With a more involved calculation one can solve the set of Equations1.36for𝜙[12] to yield

𝜕²_𝑡𝜙(𝑞, 𝑡) + Ω²(𝑞)𝜙(𝑞, 𝑡) +

The squared frequency matrixΩ(𝑞)²can be determined from the limit 𝑡 → 0(omitting the integral) to be Ω² = 𝑞²𝑣²_𝑇 where one assumes the particles velocities to be on average𝑣_𝑇 = √

𝑘_𝐵𝑇∕𝑚, which is the thermal velocity. The so-calledregular part of the memory function𝑀^reg governs the short time dependence of𝜙(𝑞, 𝑡)(particle collisions) and is often simplified by using a Gaussian Ansatz or a delta function [45, 50]. In this form the MCT equation is the equation for a damped harmonic oscillator with a time-dependent damping described by the memory kernel𝑀(𝑞, 𝑡). In colloidal systems, that are addressed in this work, the system is in the overdamped situation and one has to switch from the Liouville to the Smoluchowski operator that describes Brownian dynamics. This removes the complications of 𝑀^regand simplifies the equation to:

𝜏(𝑞)𝜕_𝑡𝜙(𝑞, 𝑡) +𝜙(𝑞, 𝑡) +

𝑡

∫

0

𝑀(𝑞, 𝑡−𝑡^′)𝜕_𝑡𝜙(𝑞, 𝑡^′)d𝑡^′ = 0 (1.39) The microscopic relaxation time𝜏(𝑞)is now the left-over of the frequency matrixΩ(𝑞). It is also deduced from the the limit𝑡→0:

𝜕_𝑡𝜙(𝑞, 𝑡) = −𝜏⁻¹𝜙(𝑞, 𝑡) ⇒ 𝜙(𝑞, 𝑡) = 1 −𝜏⁻¹𝑡+(𝑡²)

A comparison with Equation1.25gives𝜏 =𝑆(𝑞)∕𝑞²𝐷, so that the short time dynamics is simply particle diffusion, as expected in an overdamped system. The interesting physics is hidden in the memory func-tion 𝑀(𝑞, 𝑡) that governs the long-time dynamics. Up to now, no approximations have been made but they become necessary in constructing𝑀(𝑞, 𝑡). The mode coupling approximation is conducted by pro-jecting the fluctuating forces𝐟(𝑡)(see Equ.1.34) onto pair products of the density𝑛(𝑞, 𝑡)𝑛(⃗ 𝑝, 𝑡)⃗ (two-point correlation functions). The rationale is that the density is the main slow variable and its pair products are the simplest functions orthogonal to it. In a further step static three-point correlation functions are approximated with the two-point correlation function𝑆(𝑞). One ends up with:

𝑀(𝑞, 𝑡) = 1

(2𝜋)³∫d⃗𝑘 𝑉(𝑞, ⃗⃗ 𝑘, ⃗𝑝)𝜙(𝑞, 𝑡)𝜙(𝑝, 𝑡) (1.40) and the vertex𝑉(𝑞, ⃗⃗ 𝑘, ⃗𝑝)connecting the two density correlators given by

𝑉(𝑞, 𝑘, 𝑝) = 𝑛 2𝑞²𝛿(

⃗

𝑝− [𝑞⃗−⃗𝑘])

𝑆(𝑞)𝑆(𝑘)𝑆(𝑝) [𝑞⃗

𝑞 ⋅

(⃗𝑘 𝑐(𝑘) +𝑝 𝑐(𝑝)⃗ )]²

(1.41) the variable𝑝⃗is introduced as a short cut,𝑉(𝑞, ⃗⃗ 𝑘, ⃗𝑝)is only non-zero if𝑝⃗=𝑞⃗−𝑘⃗. The direct correlation function𝑐(𝑞)is linked to the structure factor by the Fourier transformed Ornstein-Zernike relation𝑐(𝑞) = 𝑛⁻¹(

1 −𝑆⁻¹(𝑞)) .

Very similar equations to1.39-1.41can be derived for the tagged particle correlator𝜙_𝑠involving products 𝜙_𝑠(𝑞, 𝑡)𝜙(𝑝, 𝑡)in the memory function. The MSD can be derived from𝜙_𝑠 by a Taylor expansion of the definition of𝜙_𝑠=𝐹_𝑠(see Equ.1.11) for𝑞→0which yields:

⟨𝑟²(𝑡)⟩= 2𝑑lim

𝑞→0

1 −𝜙_𝑠(𝑞, 𝑡)

𝑞² (1.42)

MCT predicts a bifurcation of the long time dynamics with a complete vanishing of the correlations 𝜙(𝑞, 𝑡→∞) = 0in the liquid/ergodic state and a finite value𝜙(𝑞, 𝑡→∞)>0in the glassy/nonergodic state. The non-ergodicity parameter𝑓(𝑞) = 𝜙(𝑞, 𝑡 → ∞)≥ 0can therefore be interpreted as the order parameter for a kinetic transition from the liquid to the glassy state, where𝑓(𝑞)jumps from 0 to a finite

1.3 Mode coupling theory value. For a determination of the transition line the computation of𝑓(𝑞) is sufficient, it is not neces-sary to compute the complete time evolution of𝜙(𝑞, 𝑡). To this end, the basic Equation1.38is Laplace transformed in the time domain, where we use the definition

𝑔(𝑠) =̃

∞

∫

0

𝑔(𝑡)𝑒^−𝑠𝑡d𝑠 (1.43)

Applying the rules for convolution and differentiation in the time domain together with the initial value 𝜙(𝑞, 𝑡= 0) = 1we can rewrite Equation1.38as

𝜙(𝑞, 𝑠)̃

1 −𝑠 ̃𝜙(𝑞, 𝑠) =𝑀̃(𝑞, 𝑠) +𝜏 (1.44) The use of the final value theoremlim

𝑡→∞𝑔(𝑡) = lim

𝑠→0𝑠 ̃𝑔(𝑠)yields 𝑓(𝑞)

1 −𝑓(𝑞) =𝑀(𝑞, 𝑡= ∞) =𝑀{𝑓(𝑞)} = 1

(2𝜋)³∫d⃗𝑘 𝑉(𝑞, ⃗⃗ 𝑘, ⃗𝑝)𝑓(𝑞)𝑓(𝑝), (1.45) an equation that can even be solved iteratively. One starts with𝑓⁽⁰⁾(𝑞) = 1and then goes further with iteration steps𝑓^(𝑗+1) = 1∕(1∕𝑀{𝑓^(𝑗)} − 1). In the limit𝑗 →∞the iteration converges to the solution 𝑓(𝑞)which is a fixed point of Equation1.45[19].

Physical picture

As we have seen above, the only input to the mode coupling equations is indeed the static structure factor 𝑆(𝑞)together with the average particle density𝑛. This is the outstanding feature of the theory, that only the structural information is needed to compute the dynamics. Additionally, the diffusion constant𝐷can be used to give the resulting correlators the appropriate time scaling, which is of course not important for the determination of the transition lines via the non-ergodicity parameter𝑓(𝑞).

The approximation for the memory functions to be a coupling of density modes 𝑛(𝑞, 𝑡) has given the name “mode coupling theory”. The idea is that the fluctuating forces governing the density fluctuations themselves have their origin in products of density fluctuations. The physical picture is the so-called cage effect: In a supercooled liquid each particle is trapped in a cage of its nearest neighbours and it is allowed to escape only if one of the caging particles moves. But in order to allow this, other neighbours of that particle also need to move and so on and so forth. Small density fluctuations are not enough to bring in large particle movements, only cooperative motions of many particles which are the “coupled” density fluctuations can explain the breaking of cages. They eventually give rise to the vanishing correlation function for𝑡→∞in supercooled liquids.

One should mention that the equations above describe the idealized MCT where there is a clear singular transition from an ergodic to an ideal non-ergodic glass state. In an extension of MCT that does not only include density-fluctuation pairs in the memory function but also the coupling to currents, the transition from ergodic to non-ergodic vanishes in a way that all correlation functions relax to zero [19,51]. The current modes can be interpreted as phonons and they contribute to the relaxation by bringing in an additional way to break the cages.