Tagged Particle Dynamics

The ABP-MCT for mixtures includes the specific case in which a single tracer particle is im-mersed in a surrounding bath. This scenario is of particular interest in the context of experiments and simulations where the statistics of single-particle quantities is far easier accessible rather than those of collective quantities. It further allows to study different interesting scenarios, like the motion of an active tracer particle in a host of passive hard-disks or the response of a passive particle to an active bath. Last but not least, studying the tagged particle dynam-ics in the limit of low wavenumbers gives rises to equations of motion for the MSD as seen later.

The tagged particle shall be charaterized by the coordinates Γ_s= (þr_s, θ_s)and ABP parameters (D_t^s, D^s_r, v₀^s)furtheron. This means that the evolution of the probability density of the combined phase space Γ = Γs×Γ_b of bath- and tracer particles is determined by the total Smoluchowski operator given by Ω=Ωs+Ωb with

Ωs=∇þs

1∇þs−β þF_s²+D^s_r∂_θ²_s−v₀^s∇ ·þ þo_s. (3.3.1) In the following, it will be convenient to introduce the concentration resolved fluctuating density which results in a concentration rescaled ISF as follows:

˜

ρ_l^α(þq) := 1

√x_αρ^α_l(þq), S˜^α,β(þq, t) := 1

√x_αx_βS^α,β(þq, t). (3.3.2) The quantity of interest that characterizes the dynamics of the tracer particle in the bath is the self-intermediate scattering function (SISF) defined as

S˜^s,s_l,l′(þq, t) :=S_l,l^s′(þq, t) =^eρ˜_l^s(þq)^∗e^Ω†tρ˜_l^s′(þq)^f, (3.3.3) where the introduction of a rescaled density has ensuredS^s(þq, t)∼ O(1)as desired. An equation of motion forS^s(þq, t) can be derived from equation (3.1.41) by considering a(r+ 1) component mixture in the limit of a dilute tracer density x_s→0. The time evolution of the concentration resolved ISF reads

∂_tS˜(þq, t) +ω(þq˜ ) ˜S⁻¹(q) ˜S(þq, t) + ˆ t

0

dt^′ m(þq, t˜ −t^′) ˜ω⁻_T¹(þq)^!∂_tS(þq, t˜ ^′) + ˜ω_RS˜⁻¹(q) ˜S(þq, t^′)^"= 0, (3.3.4) where the tilted quantities are defined as in (3.3.2). It is further instructive to consider a schematic representation, where tagged- and collective contributions are split into different

blocks. Following equation (3.1.6) one finds in leading order ofx_sthat S˜⁻¹(þq) =

A

2ΛL+1 −ρc^b,s√x_s

−ρc^s,b√xs ( ˜S⁻¹)^b,b B

, (3.3.5)

where the first row and first column accounts for tagged particle variables. An inversion of this matrix can be performed by exploiting a matrix-inversion formula for block matrixes. LetK= C^{(2ΛL+1)×(2ΛL+1)} and letA∈K^1×1,D∈K^r×r be square matrices withAinvertible, B∈K^1×r andC ∈K^r×1matrices withD−CA⁻¹Binvertible then there holds

AA B C D

B₋1

=

AA⁻¹+A⁻¹B(D−CA⁻¹B)⁻¹CA −A⁻¹B(D−CA⁻¹B)⁻¹

−(D−CA⁻¹B)⁻¹CA⁻¹ (D−CA⁻¹B)⁻¹ B

. (3.3.6) Applying this formula to (3.3.5) reveals

S(q) =˜ A

2ΛL+1 −ρS˜^b,b·c^b,s√x_s

−ρc^s,b·S˜^b,b√x_s S˜^b,b

B

+O(x_s). (3.3.7)

This implies that a schematic representation of the frequency-matrix reads

˜ ω(þq) =

A ω^s(þq) O(√x_s) O(√x_s) ω˜^b,b

B

+O(x_s), (3.3.8)

with the tagged particle frequency matrixω^s(þq) :=−éρ˜^s_l(þq)^∗Ω^†ρ^s_l′(þq)ê=ω_T^s(þq) +ω_Rwhere the matrix elements are

ω_T,l,l^s ′(þq) =D_t^sq²−iv^s₀q

2 e^{−i(l−l′)ϕq}δ_|l−l′|,1, ω_R,l,l^′ =D_r^sl²δ_l,l^′. (3.3.9) The inversion ofω˜_T(þq)directly follows from (3.3.6) and reads schematically

˜

ω_T⁻¹(þq) =

A ω^s_T⁻¹ O(√x_s) O(√x_s) (ω˜^b,b_T )⁻¹

B

+O(x_s). (3.3.10)

By knowing the scaling behaviour of the structure factor with the tracer density, a tedious inspection of all terms in equation (3.1.47) that is not elaborated here shows thatm˜^α,β takes a similar form and can be written as

m(þq, t) =˜

A m^s O(√x_s) O(√x_s) m˜^b,b

B

+O(xs), (3.3.11)

with the tracer memory-kernel m^s(þq, t) := ˜m^s,s(þq, t) ∼ O(1). One can also show that all cou-plings to tracer variables vanish in the limitx_s→0inm˜^b,b. This expected since a vanishing small concentration of tracer particles will not be detectable in the dynamics of the bath in the ther-modynamic limit. Extraction of the(s, s)component from equation (3.3.4) in leading order ofx_s

subsequently allows to derive an equation of motion for the SISF:

∂_tS^s(þq, t) =−ω^s(þq)S^s(þq, t)− ˆ t

0

dt^′m^s(þq, t−t^′)ω_T^s−1(þq)^è∂_t^′S^s(þq, t^′) +ω^s_RS^s(þq, t^′)^é. (3.3.12) One further makes the same observation as in passive MCT that m^s(þq, t) does not couple to mixed correlation functions between tracer and bath variables in leading order ofxs. As derived in detail in appendix A.2 the contributing parts of the tracer memory-kernel can then be written as

m^s_l,l′(þq, t)≈ρ

ˆ d²p (2π)²

Ø

l1,l2

γ1Ó=s,γ2Ó=s

V_l,l^s,γ1¹,l^,γ2²(þq, þq−p)þ S_l^s_2,l′(þp, t) S_l^γ₁¹_,0^,γ2(þq−þp, t), (3.3.13)

with a static vertex function V^s(þq, þk) =V_eq(þq, þk) +δV(þq, þk) that combines an equilibrium part exactly given as from passive MCT and an active contribution explicity entered by the activities of both bath and tracer particles. The vertex functions are

!Veq

"s,γ1,γ2

l,l1,l2 (þq, þk) = (D^s_t)²c^γ1,s(k)c^γ2,s(k)(þq·þk)²δ_l,l2δ_l1,0, δVl,l^s,γ1¹,l^,γ2²(þq, þk) =iD^s_tþq·þk

2 ke^{−i(l−l1−l2)ϕkØ}

ǫÓ=s

c^ǫ,s(k)c^γ2,s(k)¹v^γ₀¹S^ǫ,γ1(k)

x_γ1 δ_l,l2−v^s₀δ_γ1,ǫδ_l1,0²δ_{|l−l1−l2|,1}, (3.3.14) where the interaction between tracer and bath is entered through the tracer-bath direct corre-lation functions c^α,s(k). An alternate derivation of these equations in a monodisperse system has already been carried out in [84] by following the strategy to use a mixed density projection operator given by

P^s₂ := ^Ø

1,2,3,4

--ρ^s₁ρ₂^,g_1,2,3,4^{s +}ρ^s₃^∗ρ^∗₄^-- (3.3.15) to derive the mode-coupling approximation for the tracer memory-kernel. This ansatz incor-porates the interaction between bath and tracer particles through product states of respective densities. The result is fully consistent with that through the theory of mixtures presented here which offers a generalization to arbitrary tracer environments.

In analogy to equation (3.2.5) there follows an algebraic equation for non-ergodicity parameter f^s(þq) of the tracer particle in the caseD_r^s= 0 given by

f^s(þq) +ω^s−1(þq)m^s[f,f^s]ω^s−1(þq)^!f^s(þq)− ^"= 0, (3.3.16) wherem^s[f,f^s]denotes an evaluation ofm^swhere the correlation functions have been replaced by f(þq) and f^s(þq).

3.3.1 Exact Inversion

Due to the simplified structure of the tagged particle frequency matrix, it is possible to obtain an exact inversion formula for its translational part ω^s_T

l,l′(þq) for arbitrary values of the rotational

cutoff-number Λ_L. This can be achieved by closing a recurrence relation for the inverse of tridiagonal matrices [97] in the special case of Toeplitz matrices as it isω^s_T(þq). A cumbersome calculation that is skipped here reveals in the limitΛ_L→ ∞

ω_T^s−_l,l′¹(þq) =e^{−i(l−l′)ϕq}

!iv^s₀q^"|l−l′|

∆^!D_t^sq²+ ∆^"|l−l′|, ∆ :=^ñ(D^s_tq²)²+ (v₀^sq)². (3.3.17) This expression is straightforwardly verified by showing that ω_T^s(þq)·ω^s_T⁻¹(þq) = is fulfilled for arbitraryΛ_L. It displays a peculiar behaviour for varyingv₀^s, that provides some insightful interpretations. In figure 3.3.1, ω_T^s−1_0,0(q) is plotted against q in a double logarithmic represen-tation, withv₀^s varying by several orders of magnitude. There emerges a crossover from a 1/q² behaviour, equally found for a passive tracer, to a1/q behaviour at q≈v₀^s/D^s_t. This crossover can be rationalized by the motion of a free ABP, as the ballistic regime is only resolved on length scalesl≫l_ν = 2D_t^s/v₀^sthat is accounted for byω^s_T⁻_0,0¹(q)if2π/q≫l_ν while the regime2π/q ≪l_ν displays the features of passive Brownian diffusion. A further curious behaviour is seen when

10

⁻³

10

⁻²

10

⁻¹

10

⁰

10

¹

10

²

q

10

⁻⁴

10

⁻²

10

⁰

10

²

10

⁴

10

⁶

ω

s−1 T0,0

( q, t )

v0^s= 1 v0^s= 10

v0^s= 100

v0^s= 1000 1/q² 1/q

Figure 3.3.1.:Analytical solution forω^s_T⁻_0,0¹ according to equation(3.3.17)for differentv^s₀. Black lines are 1/q and 1/q² asymptotes as indicated.

comparing numerical solutions for ω^s_T⁻_0,0¹(q) at finite cutoffs Λ_L with the predictions from the analytic expression, as presented in figure 3.3.2 (a)-(d). The solutions for finiteΛL reveal clear deviations from the presented analytical expression below a certain crossover wavenumber, that marks the emergence of different low-q asymptotes, depending on whether Λ_L is chosen even or odd. These discrepancies arise because for fixedΛL the numerical inversion of the low-q asymp-tote is considering the regime of smallq·Λ_L, whereas the true low-q asymptote that is relevant for physical application is only correct in the limit q·Λ_L → ∞ and this is always fulfilled for the presented analytical expression. One further notes that to correctly resolve the behaviour

of ω_T^s−1

0,0(q), the cutoff Λ_L must be chosen larger with increasing v₀^s. To correctly resolve the behaviour in a regime of relevant wavenumbers for the integration of the memory-kernel, the requiredΛ_Lquickly approaches regimes that are far beyond the MCT numerics, which complex-ity rapidly increases with Λ_L. This observation provides a hint for the instabilities that occur at high self-propulsion velocities in the numerical routines that will be used later to solve the ABP-MCT dynamics since the non-commutative behaviour of inversion and cutoff formation presumably also occurs for the ISF in the numerical scheme and becomes most striking at large v₀.

Figure 3.3.2.: (a)-(d) Comparison between numerical solutions of ω^s_T⁻¹_0,0 at finite Λ_L (dashed coloured lines) with predictions from equation (3.3.17)(black solid lines) for differentv₀^sandΛ_L.

Im Dokument Transport Coefficients in Dense Active Brownian Particle Systems (Seite 46-50)