FIG. 4: For particles in a box we show 〈 ( x$_{T}$ ) $^{2}$〉 N/L 2 of the tagged particle, in equilibrium versus N . Exact expression obtained in [25] valid for all N (dots) converges in the limit of N → ∞ to the behavior predicted by our theory (dashed line) Eq. (52). For N = 70 clear deviations between our asymptotic theory and the exact result in [25] are found. ∑ ∞ n =1 , Odd 1 /n 2 = π $^{2}$/ 8 we have R = 1 − √$_{Dt}$ $_{L}$√$_{π}$ + · · · , and hence 〈 ( x$_{T}$ ) $^{2}$〉 ∼ 2 √$_{π}$ √$_{Dt}$ ρ . (50) This result was obtained in [4, 6], and it describes the dynamics of the tagged particle in a box, before particles have time to interact with the walls, namely when δ is small. In the opposite limit of long times and finite systems $^{we attain equilibrium, then lim$_{t}$}$→∞ R = 1 / 2 and hence Eq. (51) is a special case of the more general Eq. (21) since the single particle normalizing partition function for our example is Z = 2 L . As mentioned, in [25] the Bethe ansatz was used to solve the problem of tagged particle motion in a box. Among other things an exact expression for the long time limit of 〈 ( x$_{T}$ ) $^{2}$〉 , valid for all N was found [25] for x$_{0}$ > 0. As before we consider the tagged particle motion, which is initially at x$_{T}$ = 0, with N particles to its left and N to its right. With the free particle Green function g ( x, x$_{0}$, t ) Eq. (23) we proceed to find 〈 ( x$_{T}$ ) $^{2}$〉 . The reflection probability Eq. (31) is Consider particles without external forces V ( x ) = 0, in an infinite system. Initially particles are spread with a Gaussian packet with width ξ : In [25] numerical simulation of systems with N = 1 , 10 , 70 (maximum of 141 particles) were performed, and favorably compared with the Bethe ansatz solution. These simulations were made for finite size hard core par- ticles whose diameter is Δ. To make comparison between our theory and simulation we scale the size of the box according to 2 L → 2 L − (2 N )Δ. In Fig. 5 the scaled mean square displacement versus t/τ$_{eq}$ is shown, where τ$_{eq}$ = 4 L $^{2}$/D . Some general features of our theory can now be discussed. First, at short times simulations show normal behavior 〈 ( x$_{T}$ ) $^{2}$〉 = 2 Dt , which is a trivial ef- fect: particles did not have time to collide and hence the tagged particle diffuses normally as if it is free. After particles start to collide, deviations from normal diffusion are observed (for N = 70 but as expected not for N = 1). These are mainly due to collisions with other particles. Finally saturation due to the finite size of the system is found. While our theory shows the general trend of sim- ulations (say for N = 70 and for not too short times) it is clearly not in perfect agreement. We argue that this is due to the small number of particles N = 70, since as we showed in Fig. 4, at least for large times and large N our theory gives the exact result. It would be nice to have simulations with point particles, with larger N and since there are three phases of the motion: normal diffusion, single file diffusion without the influence of the walls and finally saturation to equilibrium. Separation of time scales is needed to demonstrate these behaviors clearly. Since our formalism is valid only in the large N limit, comparison of our solution to an exact result like Eq. (53) provides insight to the question of convergence. In Fig. $^{4 we plot lim$_{t}$}$→∞ 〈 ( x$_{T}$ ) $^{2}$〉 N/L 2 versus N us- ing Eq. (53) comparing it to our Eq. (52) which gives $^{lim$_{t}$}$→∞ 〈 ( x$_{T}$ ) $^{2}$〉 N/L 2 = 1 / 2. Not surprisingly we see that the two results yield asymptotically the same result. The figure illustrates that even for N = 100 deviations be- tween exact results and the present theory are observ- able. 8 B. Gaussian Packet f ( x$_{0}$ ) = √$_{2}$ √$_{πξ}$ exp [ − ( x$_{0}$ ) 2 2 ξ 2 ] (54) R = ∫ ∞ 0 d x$_{0}$ √ 2 ξ $^{2}$π $_{e}$− ( x 0 ) 2 2 ξ 2 ∫ ∞ 0 d x √$_{4}$$_{πDt}$ e − ( $^{x}$−x 0 ) 2 4 Dt . (55) lim t $_{→∞}$〈 ( x$_{T}$ ) $^{2}$〉 = L 2 ( 1 4 ) N +1 Γ ( 1 2 ) Γ [2 ( N + 1)] Γ ( N + 1) Γ ( N + 5 2 ) . (53) lim t $_{→∞}$〈 ( x$_{T}$ ) $^{2}$〉 = L $^{2}$/ 2 N. (52) P ( x$_{T}$ ) ∼ √$_{N}$ √$_{πL}$ e − N ( x$_{T}$ ) $^{2}$/L $^{2}$, (51)