content_image/431918.jpg content_image/431919.jpg content_image/431932.jpg content_image/431935.jpg content_image/431936.jpg content_image/431922.jpg content_image/431933.jpg content_image/431934.jpg
<img src='content_image/431918.jpg'>

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Changing variables according to y $^{2}$/ 2 = ( x − x$_{0}$ ) $^{2}$/ 4 Dt and using dimensionless parameter ˜ ξ = $_{ξ/}$√$_{2}$$_{Dt}$$_{, we find}$ 

<img src='content_image/431932.jpg'>

where Erf(˜ x$_{0}$$_{/}$√$_{2)}$$_{/}$$_{2 =}$ ∫ ˜ x$_{0}$ 0 e − y $^{2}$/ $^{2}$d $_{y/}$√$_{2}$$_{π}$ is the error function [42]. Mathematica solves the integral in Eq. (56) and we find 

For short times √$_{Dt}$ ≪ ξ we have R ∼ 1 − √$_{2}$$_{Dt}$ πξ , namely most particles did not have time to cross the origin, while $^{in the opposite limit lim$_{t}$}$→∞ R = 1 / 2 due to the sym- metry of initial conditions. The calculation of r Eq. (34) using Eqs. (23,54) is straightforward 

Inserting Eqs. (57,58) in Eq. (37) we find 

<img src='content_image/431935.jpg'>

This solution is shown in Fig. 6 with its two limiting behaviors 

<img src='content_image/431936.jpg'>

For short times the particles do not have time to disperse; hence, the motion of the tagged particle is slower than normal, increasing as t 1 / 2 which is similar to the single file diffusion with a uniform density Eq. (50). Roughly speaking, for short times the tagged particle sees a uni- form density of particles with ρ = N/ξ . For long times we recover the behavior in Eq. (27) since the scale of diffusion is much larger than ξ . Hence if we start with a Gaussian or delta function packet we get in the long time limit similar behavior, as we showed. 

## C. Particles in Harmonic Oscillator 

Consider particles in a harmonic potential V ( x ) = mω $^{2}$x$^{2}$/ 2 where ω > 0 is the harmonic frequency. The single particle undergoes an Ornstein Uhlenbeck process [38] and the corresponding single particle Green function is 

<img src='content_image/431922.jpg'>

<img src='content_image/431933.jpg'>

<img src='content_image/431934.jpg'>

