Hydrodynamic Limit and MSD

The dynamics of a tagged particle provides the theoretical basis to derive an equation of mo-tion for the MSD of tracer particles in host environments. A fundamental relamo-tion between the SISF and the MSD is seen by considering the limit of low-q in equation (3.3.3), which reveals

By extractingS_0,0^s (q, t)from equation (3.3.12) and inserting this relation yields an equivalent set of coupled equations forδr²(t)of the tagged particle, reading

∂_tδr²(t) + lim The structure of this equation exceeds the complexity of its passive version through additional couplings to low-q Sl,0(þq, t) correlation functions. It is pointed out that the equation for the MSD is invariant under the rotation of the wavevector þq according to the transformation law given by equation (3.1.7), as it is expected since it results from the isotropic (0,0) component.

Nevertheless, it still needs to be checked that allq→0terms are well-defined by inspecting the low-qbehaviour of all quantities that are involved. One first step to do so is to note thatS_l,l^α,β′ (þq, t) is of orderq^|l−l′|. This follows by observing that there only arise non-vanishing contribution from the time evolution operator that result from a|l−l^′|-fold action ofδΩ^†onρ^β_l′(þq)which generates terms that are at least of order q^|l−l′|.

This is sufficient to perform a first consistency check of the equation of motion for the MSD by validating that it indeed reproduces the well-known analytic expression of the MSD for a free ABP from equation (2.2.10). To do so it is convenient to introduce theq-independent isotropic correlation function For a free particle, the memory-integrals in equation (3.4.2) can be dropped which leads to the following equation of motion for the MSD

∂tδr² = 4D^s_t−^Ø Checking the individual terms in the second line reveals that the first three are of order O(q) while the last one is of orderO(q³). Thus by multiplying both sides with(1/q)e^±iϕq and taking q→0 yields

∂tφˆ^s_±1,0(t) = iv^s₀

2 −D_r^sφˆ^s_±1,0(t), ⇒ φˆ^s_±1,0(t) = iv^s₀

2D^s_r(1−e^−Dsrt), (3.4.6)

where the initial condition S_±^s_1,0(0) = 0was used. Inserting this solution back into (3.4.4) and performing an integration while using the initial conditionδr²(0) = 0finally results in equation (2.2.10).

To solve the equation of motion for the MSD in the general case of interacting particles requires to determine the hydrodynamic limits of the matrix elements of the memory-kernel and the inverse of the translational part of the frequency matrix. The latter results for the trivial case v₀^s = 0 of a passive tracer in ω^s_T⁻¹

l,l′(þq) = 1/q²δ_l,l^′. For v₀^s Ó= 0 a Taylor-expansion of (3.3.17) shows the following expression in leading order of q:

ω^s_T⁻_l,l′¹(þq) =e^{−i(l−l′)ϕq}i^|l−l′| C 1

v^s₀q −|l−l^′|D_t^s (v₀^s)²

D

+O(q). (3.4.7)

This means that the limits þq→0 and v₀^s→0 do not commute as the low-q asymptote changes discontinuously fromO(q⁻²)toO(q⁻¹)when switching on the activity of the tracer particle. This means that any small activity is detectable in the system if only the length scale that is observed is chosen sufficiently large. It will also lead to entirely different equations of motion for the MSD in the respective cases of passive and active tracers as seen below.

The low-q expansion of the tracer memory-kernel is presented in the following by starting with an expansion of the vertex functions V_eq(þq, þk) and δV(þq, þk) given by the equations in (3.3.14).

Setting þk = þq−þp and writing (þq·þk)k = (q² −þq·þp)k, one can exploit the Taylor-expansion for a function of a scalar variable f(k) = f(p) −f^′(p)þq·þp/p+O(q²). Making further use e^±iαk =−e^±iαp+O(þq·þeϕp) the Taylor expansion of the both equilibrium and non-equilibrium in equation (3.3.14) read in leading order ofq

!V^eq"s,γ_l,l1¹_,l^,γ₂²(þq, þk) = (D^s_t)²c^γ1,s(p)c^γ2,s(p) (þq·þp)²δ_l,l2δ_l1,0+O(q³), (3.4.8)

δVl,l^s,γ1¹,l^,γ2²(þq, þk) =iD_t^s

2 e^{−i(l−l1−l2)ϕp}δ_{|l−l1−l2|} I1

(þq·þp)p−(þq·þp)²

p −q²p²

×^Ø

ǫÓ=s

c^ǫ,s(p)c^γ2,s(p)¹v₀^γ1S^ǫ,γ1(p) x_γ1

δ_l,l2 −v₀^sc^γ1,s(p)δ_l1,0δγ1,ǫ

2

−(þq·þp)²∂_p^{è Ø}

ǫÓ=s

c^ǫ,s(p)c^γ2,s(p)¹v^γ₀¹S^ǫ,γ1(p) xγ1

δ_l,l2−v^s₀c^γ1,s(p)δ_l1,0δ_γ1,ǫ^2é

+O¹(þq·þp)(þq·þe_ϕp)² J

+O(q³).

(3.4.9) It is convenient to introduces isotropized correlation functions asSˆ_l,l^α,β′ (p, t) :=e^{i(l−l′)ϕp}S_l,l^α,β′ (þp, t) which allows to express the correlation functions that appear in the þp integration of equation

(3.3.13) in terms of

S_l^s_2,l′(þp, t) =e^{−i(l2−l′)ϕp}Sˆ_l^s_2,l′(p, t), S_l^γ₁¹_,0^,γ2(þk, t) =e^{−il1(ϕp+π)1}1−þq·þp

p ∂_p+O(þq·þe_ϕp)²Sˆ_l^γ₁¹_,0^,γ2(p, t) +O(q²). (3.4.10) After inserting the expansions (3.4.8), (3.4.9) and (3.4.10) into equation (3.3.13), the þp inte-gration is performed in polar coordinates, where the ϕ_p integration can be done analytically.

The resulting matrix elements of the tracer memory-kernel that will be relevant for the latter calculation of the MSD are given in leading order inqas follows:

!meq"s

Having determined the low-q behaviour of all quantities, the resulting equations of motion for δr²(t)after performing the limits in equation (3.4.2) are given in the following by distinguishing the scenarios of active and passive tracer particles. This is necessary because there arise different equation of motion as one notes that the components of the memory-kernel that couple in leading order to δr²(t) change in the respective scenarios. As derived in detail in appendix A.3.1 the

equation of motion for a passive tracer reads

This is the same equation that is already known from passive MCT, but with an extra memory-kernel δm^s_0,0(þq, t). The activity of the host particles is being entered in two different ways. On the one hand, there arises an implicit contribution through a faster relaxation of the S0,0(þq, t) correlation function within(meq)^s_0,0(þq, t) when increasing activity. This can be seen as a renor-malization of the density as activity shifts the glass transition density. On the other hand, there emerges an explicit activity-induced termδm^s_0,0(þq, t)that translates typical features of ac-tive motion like superdiffusive behaviour to the passive particle through couplings toS_±^α,β_1,0(þq, t) correlation functions of the active bath.

The equation of motion for δr²(t) in the case of an active tracer particle becomes more com-plicated because additional couplings to low-q Sl,0(q, t) correlation functions are involved, that are governed by a separate equation of motion. Using the expansions for the memory-kernels and equation (3.4.7), the same structure as for the free particle is observed that only φˆ^s_±1,0(t) correlators surive in leading order of q. As derived in detail in appendix A.3.2 the equation of motion for δr²(t) is given by Where the isotropized andq-independent memory-kernels are defined as

ˆ If the system experiences dynamical arrest, the tracer particle remains in a localized state for infinite times, meaning thatlimt→∞δr²(t)approaches a constant, while becoming unbounded in the fluid state. In the first case, the plateau value of the MSD indicates the localization length of the tracer via the relation 4l²_c = lim_t→∞δr²(t) which can be calculated from the long-time behaviour of the memory-kernels. This is achieved by dropping ∂_tδr²(t) at long times wherein the case of a passive tracer, both equations in (3.4.15) reveal

l_c = 1/^ñlim

t→∞mˆ^s(t). (3.4.19)

It is therefore possible to immediately calculate the localization length from the non-ergodicity parameters f(þq) and f^s(þq) by replacing the correlation functions in mˆ^s(t) with them. In the specific case ofD_r = 0, this constitutes an efficient way to calculatel_cby using the self-consistent iterative equations forf(þq)and f^s(þq).

In the case of an active tracer, the equation for the localization length changes entirely and can be obtained forD^s_r= 0 by

A transport coefficient that is specific for ABPs is given by the effective swimming velocity which underlying definition stems from the projection of the velocity vector in the direction of orientation. Both vectors are in general not expected to be equally aligned in an interacting system as for a free self-propelled particle. The effective swimming velocity, therefore, accounts for the reduction of motility through interactions with the host environment. An expression for v^α results from the overdamped Langevin-equation for þr˙_i^α by projecting on þo_i^α and taking the transient ensemble average. This yields

After applying the ITT formula (2.3.7) the effective swimming velocity is expressed as a Green-Kubo integral by usingδΩpeq =−β^q_(j,γ)v₀^γFþ_j^γ·þo_j^γpeq. This results in where the integral has been expressed in terms of the correlation function C^γ,α(z) in Laplace space that reads This is a straightforward generalization of the Green-Kubo expression that has already been derived and approximated in the framework of the linear-response theory [76] as well as in the MCT-ITT approach [84] for a single-component active system. Before this MCT-ITT approach can be extended to the theory of mixtures, the derived ITT expression demands some further