content_image/414376.jpg content_image/414378.jpg content_image/414380.jpg content_image/414381.jpg content_image/414383.jpg content_image/414377.jpg content_image/414388.jpg content_image/414384.jpg content_image/414382.jpg content_image/414386.jpg content_image/414385.jpg
efficient of exp( iφ ) is the probability that Δ α$_{j}$ is equal one (i.e. P$_{LL}$P$_{RL}$ ), the coefficient of exp( iφ ) = 1 (i.e. φ = 0) yields the probability Δ α$_{j}$ = 0, and similarly for exp( − iφ ). Since the summands in Eq. (3) are indepen- dent, Fourier analysis gives: 

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We average Eq. (6) with respect to the initial conditions x$_{0}$ which are assumed independent identically distributed random variable and we find 

where from Eq. (5) 

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Here 〈 P$_{ij}$ 〉 is the probability of starting in i = L, R (rel- ative to the the Jepsen line) and ending in j = L, R and 〈· · ·〉 denotes an average over initial condition. The prob- ability 〈 P$_{LR}$ 〉 is given in terms of the Green function of the non interacting particle g ( x, x$_{0}$, t ) and the initial den- sity of particles situated initially to the left of the Jepsen line f$_{L}$ ( x$_{0}$ ) 

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We see that for 〈 P$_{LR}$ 〉 we average over initial conditions in the domain ( − L, 0) weighted by f$_{L}$ ( x$_{0}$ ) (since the starting point is to the left of the Jepsen line) and also integrate over x in the domain ( vt, L ) with the weight g ( x, x$_{0}$, t ) (since the end point is to the right of the line). Similarly 

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As mentioned f$_{L}$ ( x$_{0}$ ) [ f$_{R}$ ( x$_{0}$ )] is the PDF of initial po- sitions of particles that initially are at x$_{0}$ < 0 ( x$_{0}$ > 0) respectively, hence f$_{L}$ ( x$_{0}$ ) = 0 when x$_{0}$ > 0 and ∫ 0 − $_{L}$f$_{L}$ ( x$_{0}$ )d x$_{0}$ = 1, while f$_{R}$ ( x$_{0}$ ) is normalized and non zero only in (0 , L ). 〈 P$_{LL}$ 〉 and 〈 P$_{RR}$ 〉 are defined simi- larly, and they satisfy 〈 P$_{LL}$ 〉 = 1 − 〈 P$_{LR}$ 〉 , and 〈 P$_{RL}$ 〉 = 1 − 〈 P$_{RR}$ 〉 . 

## III. DYNAMICS OF THE TAGGED PARTICLE 

We now apply the central limit theorem to analyze the sum Eq. (3) using Eq. (7) when N → ∞ . The small φ expansion of the structure function Eq. (8) is 

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where 

. as expected, and the variance σ 2 Δ α = 〈 (Δ α ) $^{2}$〉 − 〈 Δ α 〉 2 is 2 

Using the central limit theorem we have the probability of zero total crossing of the Jepsen line, namely α = 0 in the limit N → ∞ 2 

<img src='content_image/414384.jpg'>

This simple result, valid for a large class of Green func- tions and initial conditions, is suited for the investigation of a large number of single file problems. 

The probability P$_{N}$ ( α = 0) is according to our earlier discussion equal to the probability that the tagged par- ticle is in the interval vt < x$_{T}$ < vt + δx , where δx is the distance between the Jepsen line and the position of the second particle to the right of the Jepsen line. We as- sume that δx is small relative to the typical length scale the tagged particle traveled. For example if the den- sity of interacting particles ρ is a constant, we assume 1 /ρ << √ 〈 ( x$_{T}$ ) $^{2}$〉 . Hence we may replace vt → x$_{T}$ to transform P$_{N}$ ( α = 0) to the PDF of the tagged particle P ( x$_{T}$ ) [41]. 

Making the replacement vt → x$_{T}$ , using Eq. (14) we have in the limit of large N 

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where C is a normalization constant. In Eq. (15) and what follows 〈 P$_{ij}$ ( x$_{T}$ ) 〉 is given by Eqs. (9,10) with vt → x$_{T}$ . Eq. (15) yields the PDF of the center par- ticle P ( x$_{T}$ ) in terms of 〈 P$_{ij}$ ( x$_{T}$ ) 〉 which according to Eq. (10) depends on the free particle green function and the initial conditions. Thus the information contained in the non-interacting Green function is sufficient for the deter- mination of the single file diffusion of the tagged particle. 

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