Equation (2.1.1) is a stochastic differential equation which describes the evolution of trajectories under single realisations of the white noise terms. Following the theory of Ito-calculus [70]
it can be translated into an equivalent equation of motion for the noise-averaged conditional probability distribution p(Γ, t|Γ0, t0) of the combined N-particle phase space Γ = (Γþr,Γθ) at time t under the condition that the system was prepared in the phase space configuration Γ0 = (Γþr0,Γθ0) at t = t0. The time evolution of p(Γ, t|Γ0, t0) is governed by the so called
Smoluchowski equation
∂tp(Γ, t|Γ0, t0) =Ω(Γ)p(Γ, t|Γ0, t0), (2.2.1)
Ω(Γ) = ØN i=1
Dt∇þi1∇þi−β þFi
2+Dr∂θ2i−v0∇þi·þoi, (2.2.2)
with the initial condition p(Γ, t0|Γ0, t0) =δ(Γ−Γ0) and the Smoluchowski operator Ω, which consists of an equilibrium part describing Brownian diffusion and the particle-particle inter-actions, and a non-equilibrium part that accounts for activity, thus it is convenient to write Ω=Ωeq+δΩwith δΩ=−qiv0∇þi·þoi. Defining the translational and rotational probability currentsþjt,i :=Dt1∇þi−β þFi2−v0þoi and jr,i=Dr∂θi, the differential equation forp(Γ, t|Γ0, t0) can be expressed equivalently as a continuity equation:
∂tp(Γ, t|Γ0, t0) =Ø
i
!∇þiþjt,i+∂θijr,i"p(Γ, t|Γ0, t0). (2.2.3) Integrating out the translational degrees of freedom and dropping the surface terms of the translational probability current, the conditional distribution of the orientationsp(Γθ, t|Γθ0, t0) =
´ dΓþrdΓþr0p(Γ, t|Γ0, t0)fulfills the differential equation
∂tp(Γθ, t|Γθ0, t0) =DrØ
i
∂θ2ip(Γθ, t|Γθ0, t0). (2.2.4) This is nothing but a diffusion equation which can be factorized into the independent solution for the rotational degrees of freedom of each particle, meaning that the solution can be developed on a single-particle level. The resulting probability distribution of the orientation θ of single particle is given by the well-known solution of a Wiener process
p(θ, t|θ0, t0) = 1
ð4πDr(t−t0)exp A
− (θ−θ0)2 4Dr(t−t0)
B
. (2.2.5)
One further defines the joint-probability distributionp(θ, t, θ0, t0) = (2π)−1p(θ, t|θ0, t0)of having the orientation angle evolved from θ0 at t0 to θ at t by following the assumption of equally distributed inital orientations. This allows to perform an exact calculation of the autocorrelation-function of the orientation vector of a spherical ABP defined as
+þo(θ(t))·þo(θ0(t0)),:=
ˆ dθ
ˆ
dθ0p(θ, t, θ0, t0)þo(θ(t))·þo(θ0(t0)) =e−Dr∆t. (2.2.6) As one expects from a Markovian-process, the dependence is only on the time difference ∆t= t−t0 and the result yields a characteristic correlation time, often denoted as the so-called persistence time τr := Dr−1, which indicates a typical time scale it takes to randomize the orientation vector from an initial configuration. It translates into an associated length scale, the persistence length lp := v0τr, that indicates the distance which the particle covers balistically during this time scale on average.
2.2.1 Free Particle
For non-interacting systems, the noise- and ensemble averaged motion of a single ABP can be characterized even more precisely. For brevity let t0 = 0 and þri(0) = 0 in the following. The calculation of the mean displacements proceeds by integrating the equation of motion forþri(t) (2.1.1) in time and exploiting+d þWi,= 0 which yields which is an expected result since there is no favorable orientation of the ABP. In a similar fashion, the mean-squared displacement can be written as [71]
+þr2(t),= 2Dt allows to derive the following expression after carrying out the integration
+þr2(t),:=δr2(t) = 4Dtt
where the Péclet number was introduced as P e := v02/(2DrDt). When considering this exact solution for δr2(t) in the different temporal regimes t≪ τr and t ≫ τr, the different states of motion of the free ABP can be analyzed more precisely by using a Taylor expansion up to the second-order. This yields crossover times for the characteristic stages of motion that are given by where the crossover times have been associated to corresponding length scales. Figure 2.2.1 depicts a schematic representation of δr2(t) as well as the derived crossover length- and time scales. For t ≪ τν, the MSD of the free ABP shows the same Brownian short-time diffusion as seen for a passive particle until the crossover length scale lν before switching to a ballistic regime for τν ≪ t ≪ τl on length scales lν ≪ l ≪ ll. Finally the MSD is characterized by an enhanced long-time diffusive behaviour at times t ≫ τl with an effective diffusion coefficient Deff = Dt(1 +P e). Knowing these transition points between the different states of motion is
10
−510
−310
−110
110
310
5t/t 0
10
−410
−210
010
210
410
610
8δr 2 ( t ) /σ 2
4D
tt
∼ v
20t
2∼ 4D
t(1 + P e)t
Figure 2.2.1.: Schematic sketch of the mean-squared displacement δr2(t) of a free ABP. The dashed lines represent the crossover time scales τν and τl.
of fundamental importance to understand active motion in the case when additional competing length scales emerge. If these length scales are large compared to ll, it constitutes a promising strategy to map the ABP to a passive Brownian particle with an effective diffusion constant. The reliability of this approach can be verified experimentally in low-density systems of ABPs, e.g. in systems with sedimenting active Janus particles [72]. If the sedimentation length of the particles becomes much larger then their persistence length, the height distribution is well described by a Boltzmann distribution ρ(h) ∼ e−mg h/kbTeff with an effective temperature kbTeff = Deff/µ.
On the other hand, such a distribution profile is not observed if the persistence length exceeds the sedimentation length. When describing active transport phenomena in combination with volume exclusion effects, an additional length in the form of the cageing length emerges, which is easily exceeded by typical persistence lengths of microswimmers. This makes the simple-minded approach of an effective diffusion highly unreliable at high densities as will be seen later.
Despite its simplicity, the model of non-interacting ABPs still remains subject to current pub-lications. Very recently interesting connections between equilibrium polymer models and the ABP model have been shown. Notably, the probability distribution for the end-to-end distribu-tion of the worm-like-chain model of semi-flexible polymers, which has been investigated back in 1952 [73] long before the ABP model, obeys the same Smoluchowski equation as the free ABP under the absence of thermal noise. Shee et al. have demonstrated in [74] that it is possible to construct a polymer model that yields an exact mapping to an ABP with thermal noise and have exploited that mapping to derive exact expressions of all moments of the ABP in arbitrary dimensions.