3. Mode-Coupling Equations 13
3.2. Asymptotic Equations
The glass transition is formally characterized by a bifurcation scenario of the non-ergodicity parameterf(þq). Usually the determination off(þq)is not carried out by explicitly calculating the ISF on very large time scales, but by using a time-independent iterative algebraic equation, which is derived from the Laplace-transformation of the Mori-Zwanzig equation under the assumption of vanishing derivatives of the correlation functions close to the plateau value. Following this assumption an equation forf(þq)reads
f(þq) +S(q)ω−1(þq)m[f(þq),f(þq)]ω−T1(þq)!f¯(þq)−S(q)"= 0, (3.2.1) wheref¯(þq) :=f(þq)+ωR´∞
0 dtS−1(q)S(þq, t). Here the memory-kernel was expressed in terms of the mode-coupling functional as a bilinear form of two correlation functions withm[f(þq),f(þq)]
denoting an evaluation of the memory-kernel where the involved ISFs have been replaced by f(þq). The appearance of the term f¯(þq) is due to the hopping term and leads to the fact that equation (3.2.1) is no longer self-consistent becausef¯(þq) is required to be explicitly eval-uated in the time-domain by integration of the ISF at least on time-scales of the persistence time, because ωRS−1(q)S(þq, t) will always reveal a decay at least after τr. This observation is manifested in a crossover behaviour of the ISF that has not been observed in previous MCTs and is exemplified for the one-component theory in [47]: At times t ≈ τr, there emerges a transition between two different pleateau values. This transition marks the crossover of a glass with an infinite persistence time τr, where f¯(þq) = f(þq) and a glass with finite τr with f¯(þq)Ó=f(þq).
Expressing the non-ergodicity parameter in terms of the mode-coupling functional constitutes the starting point for the glass transition asymptotics for the ISF. However, the requirement of equation (3.2.1) to evaluate dynamical quantities bears some problems in this case because a reliable asymptotic expansion requires a very precise determination of the glass transition point, which is not possible due to the necessity of very many numerical evaluations of the hopping integral close to the bifurcation. It is therefore instructive first to investigate the case Dr = 0 in more detail. In this case, the hopping term vanishes and equation (3.2.1) results in the
usual algebraic equation known from passive MCT which implies the ISF to reveal the known asymptotic regimes depicted in figure 3.0.1. Even though the limitsDr →0andt→ ∞are not necessarily expected to commute and different scaling laws are expected to emerge for Dr Ó= 0, studying the case Dr= 0can serve as an interesting first starting point to investigate the non-trivial influence of activity on the details of the glass transition asymptotics. One might further expect some similarities between the cases Dr = 0 and Dr Ó= 0 for systems below the glass transition whose structural relaxation time is still exceeded by the persistence timeτr, meaning that the crossover behaviour for t≫τr is cancelled. Moreover, the case Dr = 0 can serve as a simplified model to study active systems with arrested rotational degrees of freedom, be it dense suspensions of self-propelled elongated particles or spherical active particles with rough surfaces that experience strong contact friction forces at high densities.
Asymptotic expansions of MCT equations have been worked out in great detail for passive mix-tures [38] and first results have already been presented for the monodisperse ABP-MCT with Dr= 0[96]. In the latter case, the calculations can be conducted exactly as same as for the the-ory of passive mixtures and do not provide any new insights. The following steps therefore only aim to sketch the readily known standard MCT-asymptotics and are focused to summarize the necessary quantities to predict the divergence behaviour of the α-relaxation process. This is in particular interest in the context of this work as this allows to characterize transport coefficients close to dynamical arrest. For a more rigorous derivation of the MCT-asymptotics with addi-tional higher-order expansions, the reader is referred to [38, 78].
Let therefore beDαr = 0for allαin the following. The starting point of an asymptotic description close to the glass transition point is to note, that the ISF evolves very close to the non-ergodicity parameter where ∂tS(þq, t) = 0. This leads to a preliminary assumption of a power series at the critical point where the difference to the plateau value is regarded as a small quantity, i.e.,
S(þq, t)−fc(þq)∼h(þq)(t/t0)−a+O(t/t0)−2a), t≫t0, (3.2.2) where t0 denotes a typical time scale of the short-time relexation and a denotes an unknown exponent for now. Exploiting the identity∂t´t
0dt′f(t−t′)g(t′) =´t
0dt′f(t−t′)∂t′g(t′)+f(t)g(0), the Mori-Zwanzig equation can written as
S(þq, t) =S(q)M(þq, t)S(q)−S(q)∂t ˆ t
0
dt′M(þq, t−t′)S(þq, t′), (3.2.3) where the abbrevation M(þq, t) := ω−1(þq)m(þq, t)ω−1(þq) has been introduced. Inserting the expansion forS(þq, t) in the EOM and exploiting the identity
∂t
ˆ t 0
dt′(t−t′)−xt′−y= Γ(1−x)Γ(1−y)
Γ(1−x−y) t−x−y (3.2.4) allows counting powers in t which yields self-consistent equations for fc(þq) and the so-called critical amplitude h(þq). Counting terms of ordert0 and dropping theþq-dependence for brevity reveals
fc+ScM[fc,fc](fc−Sc) = 0, (3.2.5)
with the superscriptc denoting critical quantities at the transition point. This is nothing but equation (3.2.1) evaluated at the critical point for vanishing rotational diffusion coefficients.
Collecting terms in t−a shows
h= 2(Sc−fc)M[fc,h](Sc−fc) :=C[h], (3.2.6) which has been expressed in terms of an eigenvalue equation for the left eigenvectorh induced by the linear map C[h]. Correspondingly there exists a right eigenvector hˆ of C which fulfils hCˆ = ˆh. Both left and right eigenvectors are uniquely determined up to two normalization coefficients, where it will prove to be convenient to fix the following conditions to simplify later occuring terms:
hˆ†:h= 1, (3.2.7)
hˆ†:h(Sc−fc)−1h= 1. (3.2.8) Here the contraction operator:was defined as
A:B:= Ø
þ q, l1,l2
γ1,γ2
Aγl11,l,γ22(þq)Blγ22,l,γ11(þq). (3.2.9)
The introduced quantities are sufficient to calculate the exponent a which describes the initial decay towards the plateau value. This exponent is linked to the so-called exponent parameter λ for which the following relationship can be derived by extending the expansion to the next higher order [38]:
λ:= Γ(1−a)2
Γ(1−2a) = ˆh†: (Sc−fc)M[h,h](Sc−fc). (3.2.10) For times that exceed typical microscopic time scales of the system, MCT states that all cor-relation functions will evolve close to the plateau value according to the same power-law with exponent a on a certain time window tσ. It can be shown that this time scale grows like tσ ∼ |σ|−1/2a, where σ is in first-order linear in the separation parameter ǫ= (φ−φc)/φc that defines the distance to the glass transition point. Ifǫ < 0 another time-windowt′σ emerges for t≫ tσ that describes the transition towards the finalα-relaxation regime. A nontrivial state-ment of MCT ist that the scaling laws for the ISF towards theα-regime can be obtained by the replacement a → −b in the previous expansion, which reveals the well-known von-Schweidler law with an exponent bthat fulfils
λ= Γ(1 +b)2
Γ(1 + 2b). (3.2.11)
A further statement that can be derived within the MCT expansion is that the implying time-window of the α-relaxation process diverges like t′σ ∼ |σ|−γ, with the non-universal ex-ponent
γ := 1 2a + 1
2b. (3.2.12)
The resulting divergence behaviour simultaneously holds for the ISFs of all wavenumbers, which is one of the central cornerstones of MCT. This is of particular interest in the context of this work since transport coefficients predicted from MCT are nothing but functionals of the
ISF, meaning that they are well-specified with the knowledge of the α-relaxation time. Con-sequently, transport coefficients close to the critical point predicted from MCT follow the same power-law with exponent γ that can be derived by calculating the exponent parameter λ.