FIG. 5: Motion of tagged particle in a box is shown for N = 1 , 10 , 70. Simulation results (squares) are taken from [25] (see text). At short time simulations exhibit normal dif- fusion 〈 ( x$_{T}$ ) $^{2}$〉 = 2 Dt as illustrated by the straight dashed line which is a guide to the eye. Our theory is expected to work well in the large N limit and when many collision events between tagged particle and surrounding Brownian particles took place. Hence agreement between theory (solid line) and simulation is reasonable at most only for N = 70 and not for too short times. FIG. 6: Scaled mean square displacement of the tagged par- ticle with Gaussian initial conditions of the packet of particles exhibits a transition between short time 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t 1 / 2 law to 〈 ( x$_{T}$ ) $^{2}$〉 ∝ t behavior. Dashed lines are short and long time asymptotic behavior Eq. (60), circles represent Eq. (59). g ( x, x$_{0}$, t ) = 1 √ 2 πDτ ( 1 − e − 2 t/τ ) exp [ − ( x − x$_{0}$e − $^{t/τ}$) 2 2 Dτ ( 1 − e − 2 t/τ ) ] , (61) 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 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 This solution is shown in Fig. 6 with its two limiting behaviors Inserting Eqs. (57,58) in Eq. (37) we find For short times √$_{Dt}$ ≪ ξ we have R ∼ 1 − √$_{2}$$_{Dt}$ πξ , namely most particles did not have time to cross the origin, while $^{in the opposite limit lim$_{t}$}$→∞ R = 1 / 2 due to the sym- metry of initial conditions. The calculation of r Eq. (34) using Eqs. (23,54) is straightforward where Erf(˜ x$_{0}$$_{/}$√$_{2)}$$_{/}$$_{2 =}$ ∫ ˜ x$_{0}$ 0 e − y $^{2}$/ $^{2}$d $_{y/}$√$_{2}$$_{π}$ is the error function [42]. Mathematica solves the integral in Eq. (56) and we find Changing variables according to y $^{2}$/ 2 = ( x − x$_{0}$ ) $^{2}$/ 4 Dt and using dimensionless parameter ˜ ξ = $_{ξ/}$√$_{2}$$_{Dt}$$_{, we find}$ 9 R = 1 2 + 1 ˜ ξ √ 2 π ∫ ∞ 0 d˜ x$_{0}$$_{e}$− (˜ x 0 ) 2 2 ˜ ξ 2 Erf ( ˜ x$_{0}$$^{$_{/}$√$_{2}$}$) 2 (56) R = 1 2 + 1 π arccot ( √$_{2}$$_{Dt}$ ξ ) . (57) r = 1 $^{$_{2}$√$_{πDt}$}$√1 + ξ $^{2}$/ 2 Dt , (58) 〈 ( x$_{T}$ ) $^{2}$〉 ∼ ξ 2 π N ( 1 + 2 Dt ξ 2 ) [ 1 4 − 1 π $_{2}$arccot 2 (√ 2 Dt ξ 2 )] , (59) 〈 ( x$_{T}$ ) $^{2}$〉 ∼ { $_{ξ}$√$_{2}$$_{Dt}$ N short times 2 Dt ≪ ξ 2 πD 2 N t long times 2 Dt ≫ ξ $^{2}$. (60)