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Time-dependent active microrheology in dilute colloidal suspensions

In a microrheological set-up a single probe particle immersed in a complex fluid is exposed to a strong external force driving the system out of equilibrium. Here, we elaborate analytically the time-dependent response of a probe particle in a dilute suspension of Brownian particles to a large step-force, exact in first order of the density of the bath particles. The time-dependent drift velocity approaches its stationary state value exponentially fast for arbitrarily small driving in striking contrast to the power-law prediction of linear response encoded in the long-time tails of the velocity autocorrelation function. We show that the stationary-state behavior depends nonanalytically on the driving force and connect this behavior to the persistent correlations in the equilibrium state. We argue that this relation holds generically. Furthermore, we elaborate that the fluctuations in the direction of the force display transient superdiffusive behavior.


I. INTRODUCTION
While the static and dynamic properties of interacting many-particle systems in equilibrium encode the linear response via the fluctuation-dissipation theorem, corresponding principles generally applicable for systems driven far from equilibrium remain a grand challenge in statistical physics. Soft matter systems are ideally suited to study such nonequilibrium phenomena, since their defining characteristic is that they are strongly susceptible to forces. Conceptually, the simplest experimentally realizable system then consists of an interacting colloidal suspension that is driven out of equilibrium by a strong external force acting on a single probe particle. This setup constitutes the basic paradigm for active microrheology with the principle goal to infer material properties beyond the linear regime [1][2][3][4][5][6][7][8][9][10][11].
The nonlinear mobility in the stationary state µ(Pe) for a suspension of interacting Brownian particles has been derived to first order of the density of bath particles n in the seminal work by Squires and Brady [1] in terms of an asymptotic expansion where µ is the mobility of the probe particle at infinite dilution with diffusion coefficient D a , and the Péclet number Pe is a suitable dimensionless measure for the driving force (see below). The motion of the bath particles with diffusion coefficient D b enters in terms of the diffusion coefficient of the relative motion, D r = D a + D b . The leading term of the corresponding asymptotic expansion of the fluctuations around the average drift has also been achieved [4], revealing a long-time diffusion coefficient that can become arbitrarily large for strong driving. Active microrheology has found fruitful applications in dense colloidal systems in the vicinity of the glass transition, in particular, computer simulations have revealed (transient) superdiffusion and enhanced diffusivities [12,13]. Certain phenomena in the nonlinear regime have also been rationalized within a modecoupling approach [14][15][16][17][18][19][20], continuous-time random walks [21][22][23][24], Langevin equations [25,26], and kinetic theory [27].
Exact results beyond the stationary state have been accomplished for driven transport in lattice models, e.g., for a biased intruder in a dense crowded environment of mobile hardcore obstacles [28][29][30][31][32][33] and the complementary limit of a tracer in a dilute quenched array of obstacles [34][35][36]. Due to repeated encounters with the same obstacle one finds that the nonlinear force-dependent mobility in the stationary state becomes a nonanalytic function of the driving force.
For the case of colloids in continuum, one infers that the mobility also becomes a nonanalytic function in the Péclet number, signaled by |Pe| 3 in the asymptotic expansion [Eq. (1)], since the mobility has to be an even function of Pe.
These results immediately raise the question how the nonanalytic contributions in the nonlinear mobility emerge from a perturbative scheme and whether they are related to the persistent correlations in equilibrium? How fast is the nonequilibrium steady state approached from an initial equilibrium state in comparison to the predictions of linear response? What is the intermediate time-dependent behavior of the fluctuations connecting the short-time motion to the drastically enhanced long-time diffusion? These questions will be answered by solving the two-particle Smoluchowski equation for the time-dependent dynamics in the presence of a strong force for the first time. Thereby, we unify the previous approaches for the driven stationary state [1,4] and the time-dependent equilibrium dynamics [37][38][39][40] and reveal the interplay between persistent correlations and nonequilibrium driving. The solution enables us to fully address the time-dependent approach to the stationary state, in principle for all moments of the displacement along the force. The two lowest moments, the mobility and the fluctuations, will be elaborated and compared to computer simulations.
This work is organized as follows. In Sec. II the underlying model is introduced followed by the complete solution strategy. The main result of this section is the self-energy encoding the dynamics between probe and the bath particles. Readers who are primarily concerned about the results may skip this section and jump directly to Sec. III where we determine the time-dependent behavior in terms of the mobility and the fluctuations along the applied force. The obtained results from the analytic solution are compared to computer simulations and the phenomena involved in the transition from the initial equilibrium state to the new stationary state are discussed. In Sec. IV, the key results of this work are summarized followed by general conclusions.

II. MODEL AND SOLUTION STRATEGY
We solve for the time-dependent dynamics of a probe particle pulled by a force in the presence of other bath particles. In first order of the density of bath particles, the dynamics is completely encoded by the interactions of the probe particle with a single bath particle [1,48]. Discarding inertial effects it is sufficient to consider the two-particle Smoluchowski equation for probe and bath particle. The problem can be expressed via the independent motion of the center of diffusion and the relative distance between both particles. The dynamics of the relative distance is described in the frequency domain in terms of a self-energy which encodes all corrections to the dynamics of the probe particle due to interactions.

A. Two-particle Smoluchowski equation
We consider a single probe particle a interacting with a bath particle b with bare diffusion coefficients D a and D b , respectively. The particles interact by mutual hard-core exclusion with exclusion distance σ. At time t = 0 an external constant force F is switched on, pulling the probe particle and driving the system from its initial equilibrium state into a nonequilibrium stationary state. We describe the state of the system by the conditional probability density Ψ( r a , r b , t| r a , r b ) for probe particle a and bath particle b to be at positions r a , r b at time t provided they start at initial positions r a , r b at time t = 0. Since the system is initially in equilibrium, the corresponding initial condition reads Ψ( r a , r b , t = 0| r a , where the Heaviside-function ϑ(·) accounts for the mutual exclusion. The limit of large box sizes V → ∞ is anticipated throughout. At time t = 0, the force F is switched on and the probability density evolves according to the Smoluchowski equation with bare mobility µ = D a /k B T of the probe particle and the thermal scale k B T . The hard-core interaction between probe and bath particle is encoded in the no-flux boundary condition We introduce new coordinates for the center of diffusion R = (D b r a + D a r b )/(D a + D b ), and the relative distance r = r a − r b . After transformation, the Smoluchowski equation in these adapted coordinates reads with the diffusion coefficient for the relative motion, D r = D a + D b . Similarly, the no-flux boundary condition transforms to The Smoluchowski equation and the no-flux boundary condition in the new coordinates [Eq. (5) and (6)] reveal that the motion of the center of diffusion R and the dynamics of the relative motion r, are independent. Thus, the conditional probability factorizes into a simple Gaussian with diffusion coefficient D a D b /D r and drift µ F D b /D r for the center of diffusion R, and the conditional probability ψ( r, t| r ) to find relative distances r and r at times t and 0. It fulfills the reduced Smoluchowski equation and the initial condition is provided by the equilibrium state ψ( r, t = 0| r ) = ϑ(| r |−σ)δ( r− r ). Similarly, the no-flux boundary condition for the conditional probability ψ reads It is natural to measure the force in terms of the dimensionless Péclet number Pe = µF σ/D r , that already appeared in Eq. (1) [1,4], which can be written also as Pe = (F σ/k B T )D a /D r by using the Stokes-Einstein relation. This definition encompasses the case of equal sized colloids, Pe = F σ/2k B T , as well as the Lorentz model, Pe = F σ/k B T , where the bath particles are fixed in space (D b = 0).
Our main quantity of interest is the intermediate scattering function e −i q·∆ ra(t) for the displacement ∆ r a (t) = r a (t)− r a (0) of the probe particle, from which in principle all moments of the displacement can be extracted by derivatives with respect to the wave vector q. In new coordinates, ∆ r a (t) = ∆ R(t) + D a ∆ r(t)/D r , the intermediate scattering function can be expressed as where we used the independence of R and r [Eq. (5)]. Since the dynamics of the center of diffusion is Gaussian with diffusion coefficient D a D b /D r and drift µ F D b /D r , one finds immediately The remaining task is the calculation of the intermediate scattering function e −i(Da/Dr) q·∆ r(t) for the relative motion ∆ r(t) between probe and bath particle.

B. Dynamics of the relative motion
This part contains the detailed calculation of the relative motion and contains the heart of our analytic approach. The solution strategy is adapted from Felderhof's approach for the equilibrium dynamics [40].
The intermediate scattering function, viz. the characteristic function of the relative displacement ∆ r(t) = r(t) − r(0), allows extracting the moments by series expansion in the wave vector q. To make analytic progress, we only consider the axially symmetric case, where the wave vector and force are aligned q F . This enables us to determine the motion in the direction of the force, the time-dependent motion perpendicular to the force is left for future analysis. We perform a temporal Fourier-Laplace transform and a spatial Fourier transform, The transformed quantityψ q ( r, ω) is connected to the propagator G(q, ω) via a spatial Fourier transform G(q, ω) := d 3 r e −i q· rψ q ( r, ω)/ √ V . From Eq. (7) and the initial condition one derives the equation of motion [40] In particular,ψ q ( r, ω) = 0 for | r| < σ, reflects the hard-core exclusion. Without interaction, we denote the respective conditional probability byψ 0 q which evolves according to The free motion allows for a plane-wave solution of the form which is valid for all r ∈ V and the respective free propagator can be read off as In particular, the plane-wave solution can be written asψ 0 q ( r, ω) = G 0 (q, ω)e i q· r / √ V . To make further progress, we observe that the difference of Eqs. (13) and (14), vanishes for terminal distance | r| larger than the exclusion distance σ. We are interested in the forward-scattering amplitude which we separate into contributions outside and inside of the region of overlap of probe and bath particle: In the second term we observe that the conditional probabilityψ q vanishes for terminal positions inside the obstacle. Hence, there we can immediately compute the integral To calculate the first term in Eq. (19), it is advantageous to convert the volume integral into a surface integral relying on the equations of motion. We use the auxiliary variable χ( r) =ψ q ( r, ω) −ψ 0 q ( r, ω) and investigate the identity which follows from Eq. (17). Then, for the drift, we obtain | r|>σ where we introduced the unit vectorˆ r = r/| r| and the surface element dS r of the sphere of radius σ. Similarly, for the diffusive contribution, we derive Together, this yields The first term on the right-hand side simplifies by the no-flux boundary condition [Eq. (8)] and the plane wave solutionψ 0 q [Eq. (15)]: where in the last line the flux integral of a constant vector vanishes. Collecting results, we obtain an expression for the forward scattering amplitude in terms of a surface integral of the difference of the conditional probabilities: In order to determine the conditional probabilities, we return to Eq. (17). For the solution of the homogeneous part of Eq. (17), we use the imaginary "gauge transformation"X( r, ω) = [ψ q ( r, ω) −ψ 0 q ( r, ω)]e −µ F · r/2Dr such that the drift term is absorbed in the Laplacian by completing the square, ∇ → ∇ − µ F /2D r [1,49]. This step will have crucial implications for the results at finite forces. Then, the new quantityX fulfills the 3d source-free Helmholtz equation with complex wavenumber κ 2 σ 2 = −iωτ + (Pe/2) 2 and diffusive time scale τ = σ 2 /D r . We write the general axially symmetric solution in the form for | r| > σ, with modified spherical Bessel functions of the second kind k (·) and Legendre polynomials P (·). The expansion coefficients a for the different angular channels have to be determined via the no-flux boundary condition [Eq. (8)]: With the explicit solutions [Eqs. (15) and (28)], we find that the expansion coefficients a have to fulfill where we abbreviated η = cos(ϑ) and used the Péclet number Pe = µF σ/D r . Performing the derivatives the relation can be expressed as where we used the recursion formula ( + 1)P +1 (η) = (2 + 1)ηP (η) − P −1 (η). Using the orthogonality relation of the Legendre polynomials, 1 −1 dη P (η)P (η) = 2δ /(2 + 1), this becomes a tridiagonal matrix equation: The inhomogeneity b is defined as The remaining integral can be be solved using the Rayleigh identity e −iz cos ϑ = ∞ =0 (−i) (2 + 1)j (z)P (cos ϑ), with spherical Bessel function j (·) and relying again on the orthogonality relation of the Legendre polynomials: where the prime indicates a derivative. For the special case of vanishing wavenumber q = 0 and the stationary state ω = 0, we recover the tridiagonal matrix derived in Ref. [1]. For the forward scattering amplitude [Eq. (26)] we calculate the integral over the spherical surface with radius r = σ: The integral on the right-hand side can be evaluated again by inserting the Rayleigh identity.
As a result, we find Collecting results, one finds The first term on the right-hand side merely reflects that the free propagator G 0 (q, ω) = (−iω + D r q 2 + iµF q) −1 allows for particles starting also inside of the obstacle, thereby renormalizing the residue of the perturbed propagator. We may safely drop this term. Conventionally, the result is expressed in terms of a self-energy Σ(q, ω) via the Dyson equation G = G 0 + G 0 ΣG. To first order in the density, the self-energy is merely proportional to the number density n = N/V of the bath particles, and we obtain the self-energy as our main result of the analytic calculations The self-energy encodes the density-induced corrections of all moments of the relative motion along the force. The leading factor q reflects the particle-conservation law. Without external driving, Pe = 0, the tridiagonal matrix for the coefficients a [Eq. (32)] becomes diagonal yielding Thus, we recover the known result for the self-energy for equilibrium [37][38][39][40]: In the case of driving, only terms of order = O(Pe) significantly contribute for small wave numbers and the matrix [Eq. (32)] may be safely truncated for numerical evaluation. The matrix inversion does not generate nonanalytic behavior and one infers that the coefficients a ≡ a (qσ, κσ, Pe) should be analytic functions in the arguments. In particular, the matrix is suited for a perturbative approach for small Péclet numbers, in particular, the linear response results can be derived.
The matrix in Eq. (32) determines the coefficients a and thereby the complete solution of the self-energy [Eq. (38)]. Up to matrix inversion and a temporal Fourier back-transform, which have to be implemented numerically, we have elaborated a complete time-dependent analytic solution for the relative motion of the probe particle in the presence of bath particles for the dilute case.

III. CUMULANTS OF THE DISPLACEMENT
The intermediate scattering function generates the moments of the displacement along the field by taking derivatives with respect to the wavenumber. In this section we elaborate explicitly how the mean displacement, respectively the time-dependent nonlinear mobility, and the fluctuations along the field can be calculated from the self-energy of the relative motion.
The starting point are the intermediate scattering functions for the displacement [Eqs. (9) and (10)] where we put the wave vector along the field, q F , which is chosen as the z direction. The corresponding cumulant generating function is obtained by taking the logarithm: This formula establishes the connection between the relative motion z(t) and the displacement of the probe particle z a (t).

A. Average motion
First, we discuss the time-dependent nonlinear mobility defined by characterizing the mean motion of the probe particle, upon switching on the force in the z direction. From the cumulant generating function of the displacement ∆z a (t) [Eq. (41)], the mean-displacement ∆z a (t) is obtained as The contribution from the relative motion can be obtained from the intermediate scattering function [Eq. (11)] as the first derivative ∂/∂q| q=0 . The dynamics of the relative distance ∆z(t) between probe and bath particle is contained in the propagator where, in the second line, we used the relation D r /σ = µF/Pe and expressed the spherical Bessel functions j (·) in terms of (−i)i (Pe/2) = (−i) j (iPe/2) with modified spherical Bessel function i (·). The coefficients a = a (qσ, κσ, Pe) are to be evaluated for vanishing wavenumber q = 0. In the above equation, the first term corresponds to the free motion and the second encodes the density-induced response. Then, the mean displacement of the probe particle along the field, ∆z a (t) , is determined via Eq. (43) leading to For the time-dependent case, we perform a numerical Fourier inversion of Eq. (47) and compare to event-driven Brownian dynamics simulations at low density [ Fig. 1]. Initially, the probe particle experiences only the drag of the pure solvent, µ(t → 0, Pe) = µ. Then the dynamics slows down due to the interaction with the bath particles and approaches its stationary mobility µ(Pe) := µ(t → ∞, Pe). Its dependence on the Péclet number is discussed in Sec. III C below.
In linear response, Pe = 0, we recover the equilibrium solution [Eq. (49)] as anticipated by the fluctuation-dissipation theorem. In particular, for the equilibrium case, the approach to the stationary state is completely monotone and the time dependence can be worked out explicitly by Laplace inversion of Eq. (49) [38], leading to with Fresnel integrals S(x) = x 0 du sin(πu 2 /2) and C(x) = x 0 du cos(πu 2 /2) [50,51]. Inserting the asymptotic expansions for the Fresnel integrals for large arguments, we recover the algebraic approach of ∼ t −3/2 to the stationary mobility which is due to repeated collisions of the probe particle with the same bath particle: The persistent memory in the system also emerges as a long-time tail of the form ∼ t −5/2 in the velocity-autocorrelation function, k B T dµ(t)/dt. For finite driving, expansion of Eq. (47) yields which is analytic in the complex wavenumber κ and the Péclet number. Moreover only even terms in Pe appear. The nonanalytic dependence on the frequency and the Péclet number arises via the square root κσ = −iωτ + (Pe/2) 2 . For vanishing forces κσ = √ −iωτ and the term O(κ) 3 generates the long-time tail in Eq. (49). For small but finite forces, the singularity is shifted in the complex plane and this term yields the long-time behavior in the time domain Therefore the initial decay becomes more rapid and the long-time tail is followed only for small forces up to some driving-dependent crossover time τ F := τ /Pe 2 , where the tail is decorated by a decaying exponential. Yet, there is a second non-analytic contribution at finite driving O(Pe 2 κ) (see also Ref. [36] for the corresponding lattice case) which yields in the temporal domain Comparing expressions in Eqs. (51), (52) reveals that for times t τ F = τ /Pe 2 the latter decays more slowly and is the relevant one. Therefore, the divergent time scale τ F in the problem separates two different regimes. For t τ F the linear response prediction remains qualitatively correct, while for t τ F the non-equilibirium driving dominates. The second regime also explains the counterintuitive sign change in Fig. 1 such that in the terminal regime the velocity of the probe particle speeds up again for small and intermediate Péclet numbers. For large Péclet number the approach to the stationary state becomes monotonic again.
The simulations quantitatively confirm the theory for a low density of bath particles (see Appendix A for simulation details). Note that the simulations were performed at a finite density of nσ 3 = 0.01 [ Fig. 1]. Nevertheless, the agreement for the approach to the stationary behavior of the mobility extends to relative order 10 −3 . In particular, for equilibrium, we observe the persistent power-law correlations due to repeated interactions over roughly one decade in time.

B. Fluctuations along the force
The next interesting quantity for the motion of the probe particle are the time-dependent fluctuations around the drift motion, which is the second cumulant of the fluctuating probe displacement ∆z a (t). From the cumulant generating function [Eq. (41)], it is obtained via where, in the second line, we already inserted the explicit expression for the second cumulant of the relative displacement ∆z(t). First, we calculate the second moment along the field in the frequency-domain: where we introduce auxiliary functions In particular, with the auxiliary function for the mobility, ∆μ(ω, Pe), the frequencydependent nonlinear mobility [Eq. (47)] can be written as In the time-domain, the mean-square displacement then follows as where we introduced the inverse Laplace transform L −1 {·}(t). To calculate the square of the mean displacement ∆z(t) , we first express the frequency-dependent mean displacement given in Eq. (45) in the time-domain leading to Then, the square of the mean-displacement to first order in the density is given by Collecting results, the time-dependent diffusion coefficient along the field, D z (t, Pe) := (1/2)dVar z (t)/dt, can be written as where we used the relation For small times the probe particle does not interact with the bath particles and the diffusion coefficient is given by the bare one D z (t → 0, Pe) = D a . Only in the equilibrium case, the diffusion coefficient decreases monotonically [see Fig. 2] to its stationary-state value D eq z /D = 1 − (D a /D r )2πnσ 3 /3.  Let us rationalize the superdiffusive behavior in terms of an asymptotic model for the limit of strong driving. Therefore we adapt an earlier asymptotic model valid for the driven lattice Lorentz model [35]. Here, the probe particle's motion is dominated by the drift with constant velocity µF along the force until it hits a bath particle for the first time and then slowly slides along its surface. In this time regime, the motion becomes essentially onedimensional such that the probability distribution for the relative motion, P(∆z, t), consists of freely moving particles with fixed velocity or particles that are transiently blocked by bath particles. The free path lengths are exponentially distributed, since at low density the positions of the bath particles are independent, and the probability distribution can be estimated directly to P(∆z, t) = δ(∆z − vt)e −∆z/l * + (1/l * )e −∆z/l * ϑ(vt − ∆z), where l * = 1/nΣ * denotes a mean-free path length. The first term corresponds to the freely moving probe (with v = µF ), while the second term accounts for the blocked probe particles. From the distribution of the displacements [Eq. (64)], it is straight forward to calculate the mean and the mean-square displacement. Then, the growth of the fluctuations for the probe particle are obtained from Eq. (54), where we discarded the diffusive contribution since we are only interested in the drift motion. As a result, the distribution of the displacements implies a strong initial growth of the variance with which is also confirmed by Brownian dynamcis simulation [ Fig. 3]. Empirically we find, that the relevant scattering cross section Σ * ≈ σ 2 is smaller than the geometric cross section πσ 2 , leading to a mean-free path length of l * ≈ 1/nσ 2 . The smaller scattering cross section can be readily interpreted since only head-on collisions effectively stop the directed motion [in the lattice variant no empirical correction was necessary, the scattering cross section coincides with the geometrical one [35]]. Matching the superdiffusion with the short-time asymptote 2D a t yields as crossover time ∼ Pe −3/2 . Similarly the terminal time of superdiffusion is set by the time the probe particle needs to pass a bath particle σ/µF ∼ 1/Pe leading by crossover matching to the scaling prediction D z (Pe) ∼ n Pe for the diffusion coefficient consistent with Ref. [4]. Thus, the window of superdiffusion expands as Pe 1/2 as the force is increased.
For small driving and small frequency, we can determine the coefficients a of the auxiliary function ∆μ(ω, Pe) [Eq.
Similarly, for the stationary diffusion coefficient D z (Pe), we perform the limit of small frequencies in Eq. (62) and obtain With our solution, we recover the leading correction O(Pe 2 ) to the equilibrium case, which has been calculated earlier [4]. Furthermore, our calculation extends this result to arbitrary order and reveals the emergence of nonanalytic contributions similar to the mobility in the stationary state. As can be inferred from the stationary mobility [

D. Time-dependent pair-distribution function
Our solution provides also the time-dependent pair-distribution function g( r, t) for the relative distance r of the tracer and the bath particles by integrating the conditional probability distribution ψ( r, t| r ) over all r . In the frequency domain,ĝ( r, ω) =ψ q=0 ( r, ω) √ V and with Eq. (28), we find for | r| > σ − iωĝ( r, ω) = 1 + e µ F · r/2Dr ∞ =0 a k (κr) k (κσ) P (cos ϑ), where the coefficients a ≡ a (qσ, κσ, Pe) are to be evaluated at zero wavenumber q = 0. In particular, in equilibrium all coefficients vanish a = 0 and the pair-distribution function reduces to the stationary step function. Simulation results for the time-evolution of the pair-distribution function are displayed in Fig. 6. Shortly after switching on the force, the pair-distribution function is still almost spherically symmetric, there has been no time to propagate the information of the strong perturbing force to the surroundings. Quickly the tracer's motion is obstructed by the bath particles such that probability accumulates in front of the tracer particle. For longer times, a trailing wake evolves where probability is depleted behind the tracer particle. The numerical results for the pair-distribution function in the frequency domain shown in Fig. 6 corroborate this picture. For small frequencies we recover the stationary distribution calculated in Refs. [1,2]. For high Péclet numbers the stationary two-particle distribution function becomes strongly asymmetric as has been shown first Ref. [1] and more systematically in Ref. [2]. Probability piles up in a narrow boundary layer in front of the pulled particle particle and leaves behind a wake extending to larger and larger distances O(Pe) as the driving force is increased.

IV. SUMMARY AND CONCLUSION
We have derived an analytic solution for the full time-dependent response of a probe particle driven out of equilibrium by a step force in first order of the density of bath particles (discarding inertial and hydrodynamic effects). The response is completely encoded in the self-energy from which, in principle, all moments along the force can be generated. In comparison to the known self-energy in equilibrium, the force results in a complex shift of the frequency −iωτ → −iωτ + (Pe/2) 2 = κ 2 σ 2 encoded in the complex wavenumber κσ, an explicit regular variation of the expansion coefficients a via the boundary condition, and a shift in the wavenumber qσ → qσ + i Pe/2. The shift in the frequency reveals that the nonanalytic frequency behavior in equilibrium, as manifested in the long-time tails, and the Top: Simulation results for the time-dependent pair-distribution function g( r, t) for Péclet number Pe = 32 and density nσ 3 = 0.01. As time progresses t · D a /σ 2 = 4 · 10 −4 , 4 · 10 −2 , 4 (left to right) the tracer piles up probability in front and leaves a trail of depleted probability. Bottom: Analytic result for the real part of the pair-distribution function in the frequency domain Re[−iωĝ( r, ω)] at the same driving Pe = 32 for frequencies ω · σ 2 /2πD a = 1/(4 · 10 −4 ), 1/(4 · 10 −2 ), 1/4 (left to right). nonanalytic dependence on the driving, are merely two sides of the same coin. In particular, this explains why at any finite driving the approach of the nonlinear mobility to its stationary value becomes exponentially fast. Furthermore, this reveals that for finite times, all response functions are analytic functions in the driving, however, this does not hold for infinite times since the limits do not commute.
The emergence of non-analytic behavior and a divergent time scale for Pe → 0 calls for an explanation in terms of physics. The stationary Smoluchowski equation is non-uniform at small Péclet number [the same physics has been discussed for the advection-diffusion equation in the seminal contribution by Acrivos and Taylor [52]]: it displays an outer region at distances r σ/Pe where advection dominates, and an inner one where diffusion is the dominant contribution. For the transport coefficient already to order O(Pe 2 ) both regions need to be calculated and matching the solutions makes the non-analytic contributions evident [2]. For non-zero frequencies or finite times the advected Smoluchowski equation is regular and solutions decay exponentially fast on the 'Skin penetration' depth D r /ω ∼ √ D r t. Correspondingly, if the Skin penetration depth is smaller than the inner region, i.e. for times √ D r t, σ/Pe, i.e. t τ F = τ /Pe 2 , one can safely ignore the presence of the outer region and the system behaves as in linear response. In contrast, for large enough times, the Skin penetration depth covers the outer region and the nonlinearities becomes important. The interplay of the divergent boundary layer and the Skin penetration depth is the origin of non-commuting limits. Mathematically related, but not quite identical, is the problem of the time-dependent motion at small but finite Reynolds number [53].
The full solution provides the first direct access to the time-dependent fluctuations along the force. Here we explicitly characterized the transient superdiffusion which connects the short-time bare diffusion and the long-time enhanced diffusion [4]. The emergence of superdiffusion has been rationalized by considering the distribution of the free path lengths.
Let us comment also on hydrodynamic interactions. Progress has been made for certain limiting cases: In equilibrium, the stationary diffusion coefficients have been estimated by including instantaneous hydrodynamic interactions at the level of the Oseen tensor, as well as including near-field corrections. Results for the density-induced suppression depend somewhat on the approximation of the hydrodynamic interactions but are close to the diffusion coefficient neglecting hydrodynamics [37,[54][55][56]. For the driven case, hydrodynamic interactions have been accounted for in the stationary state both for the structure deformation as well as the mobility for all Péclet numbers [41]. The force-induced corrections to the stationary diffusion coefficient have been elaborated only recently, and it has been shown that hydrodynamic interactions do not lead to qualitatively new behavior, although the numerical values change. In particular, the low-force corrections are still O(Pe 2 ), and in the regime of strong forces they still scale as O(Pe), however with a slow convergence to the asymptotic result [56].
A second effect due to hydrodynamics arises due to frequency-dependent hydrodynamic interactions implying hydrodynamic memory. Here, the slow vortex diffusion of transverse momentum in the fluid leads to a characteristic algebraic decay of the form Bt −3/2 , B > 0 for the velocity-autocorrelation function of the particle [57,58]. Hence, one may ask the question if the long-time tail due to hydrodynamics dominates the long-time tail −At −5/2 , A > 0 due to repeated collisions of the probe particle with the bath particles. Taking physical values from experiments [59,60] and comparing both tails shows that, in principle, a window of time opens where the algebraic decay due to the collision of the probe particle with bath particles dominates before hydrodynamics becomes relevant at larger times.
Our predictions for the time-dependent response of a driven colloid can be tested, in principle, in laboratory experiments on a colloidal suspension via particle tracking. The general scenario persists also for soft spheres and is not restricted to dilute systems. To first order in the density, the different diffusivities of probe and bath particles can be trivially accounted for, e.g., by a rescaling of time. In particular, the case of a dilute and quenched array of obstacles (Lorentz model) is also included. Simulation of the Lorentz model, where the bath particles are pinned, were added to the comparison of the mobility and diffusivity with theory in Figs. 4 and 5.
Our analysis of the driven colloid shows that the nonequilibrium stationary state is inherently a nonanalytic function of the driving such that the transport coefficients can not be expanded in a Taylor series beyond linear response. Although this has been derived to first order in the densities only, this behavior is anticipated to be generic and valid for arbitrary densities. Arguably, such relations should hold universally in a general nonlinear response framework. This view is supported by recent predictions for a two-dimensional or three-dimensional driven lattice Lorentz gas [34][35][36], where qualitatively the same scenario applies.
infrastructure LEO of the University of Innsbruck.