Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics.

Abstract

We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Delta diffusing in a one-dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rhoT(yT,t|yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the N -particle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T , we arrive at an exact expression for rhoT(yT,t|yT,0) in terms of Jacobi polynomials or hypergeometric functions. Going beyond previous studies, we consider the asymptotic limit of large N , maintaining L finite, using a nonstandard asymptotic technique. We derive an exact expression for rhoT(yT,t|yT,0) for a tagged particle located roughly in the middle of the system, from which we find that there are three time regimes of interest for finite-sized systems: (A) for times much smaller than the collision time t<taucoll=1/(rho2D) , where rho=N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion; (B) for times much larger than the collision time t >taucoll but times smaller than the equilibrium time t<taueq=L2/D, we find a single-file regime where rhoT(yT,t|yT,0) is a Gaussian with a mean-square displacement scaling as t1/2; and (C) for times longer than the equilibrium time t>taue , rhoT(yT,t|yT,0) approaches a polynomial-type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems.

Full text

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arXiv:0907.0567v2 [cond-mat.stat-mech] 21 Dec 2009
Diffusion of Finite-Sized Hard-Core Interacting Particles In a One-Dimensional Box -
Tagged Particle Dynamics
L. Lizana∗
Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
T. Ambj¨ ornsson†
Department of Theoretical Physics, Lund University, S¨ olvegatan 14A, SE-223 62 Lund, Sweden, and
Department of Chemistry, Massachusetts Institute of Technology. 77 Massachusetts Avenue, Cambridge, MA 02139, USA.
(Dated: December 21, 2009)
We solve a non-equilibrium statistical mechanics problem exactly, namely, the single-file dynamics
of N hard-core interacting particles (the particles cannot pass each other) of size ∆ diffusing in a
one dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact
expression for the conditional probability density function ρT(yT,t|yT ,0) that a tagged particle T
(T = 1,...,N) is at position yT at time t given that it at time t = 0 was at position yT ,0. Using
a Bethe ansatz we obtain the N-particle probability density function, and by integrating out the
coordinates (and averaging over initial positions) of all particles but particle T , we arrive at an
exact expression for ρT(yT,t|yT ,0) in terms of Jacobi polynomials or hypergeometric functions.
Going beyond previous studies, we consider the asymptotic limit of large N, maintaining L finite,
using a non-standard asymptotic technique. We derive an exact expression for ρT(yT,t|yT ,0) for
a tagged particle located roughly in the middle of the system, from which we find that there are
three time regimes of interest for finite-sized systems: (A) For times much smaller than the collision
time t ≪ τcoll= 1/(̺2D), where ̺ = N/L is the particle concentration and D the diffusion constant
for each particle, the tagged particle undergoes normal diffusion; (B) for times much larger than
the collision time t ≫ τcoll but times smaller than the equilibrium time t ≪ τeq = L2/D we find a
single-file regime where ρT(yT,t|yT ,0) is a Gaussian with a mean square displacement scaling as t1/2;
(C) For times longer than the equilibrium time t ≫ τeq, ρT(yT,t|yT ,0) approaches a polynomial-
type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the
previously considered infinite systems.
I.INTRODUCTION
Recent development of single fluorophore tracking
techniques allow experimental studies of the motion of
particles in cellular environments with nanometer reso-
lution [1]. The cell interior represents a crowded envi-
ronment, in which the motion of an individual particle
is strongly effected by the presence of other particles.
Crowding affects, for instance, the folding of proteins,
diffusional motion [2, 3] as well as rates of biochemical
reactions [4, 5]. Crowding is also important during ri-
bosomal translation on mRNA [6], and binding protein
diffusion along DNA [7, 8], where bound proteins are hin-
dered from passing each other. Furthermore, advances
in nanofluidics allow studies of geometrically constrained
nano-sized particles [9, 10]. The system considered in
this paper, the diffusion of a tagged particle immersed
in a one-dimensional bath of hard-core interacting parti-
cles - in the literature referred to as single-file diffusion
(SFD) - represents one of the simplest systems governed
by crowding effects, but yet with possible applications for
obstructed one-dimensional protein diffusion along DNA
molecules and transport in nanofluidic systems.
∗Electronic address: lizana@nbi.dk
†Electronic address: tobias.ambjornsson@thep.lu.se
SFD phenomena emerge in quasi one-dimensional ge-
ometries. The particle order is under these circumstances
conserved over time t which results in interesting dynam-
ics for a tagged particle, quite different from what is pre-
dicted from classical diffusion (governed by Fick’s law).
Examples found in nature include ion or water trans-
port through pores in biological membranes [11], one-
dimensional hopping conductivity [12] and channeling in
zeolites [13]. SFD effects have also been studied in a num-
ber of experimental setups such as colloidal systems and
ring-like constructions [14, 15, 16, 17, 18]. One of the
most apparent characteristics of SFD is that the mean
square displacement (MSD) S(t) = ?(yT − yT ,0)2? of a
tagged particle is in the long time limit proportional to
t1/2in an infinite system with a fixed particle concentra-
tion (brackets denote an averageover initial positions and
noise, yT and yT ,0are tagged particle positions at time
t and t = 0, respectively). Also, the conditional prob-
ability density function for the tagged particle position
(tPDF) is Gaussian.
The first theoretical study showing the t1/2law of the
MSD and that the tPDF is Gaussian is Ref. [19]. Subse-
quent studies, proving the MSD law in alternative ways,
are found in [20, 21, 22, 23]. Simple arguments to its
origin are presented in [24, 25, 26], one of which [26]
uses a simple relationship between the displacement of a
single particle and particle density fluctuations, the lat-
ter known to be the same as for independent particles

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2
[19, 27]. The t1/2-law and Gaussian behavior have, in
the long time limit, been shown to be of general validity
for identical strongly over-damped particles interacting
via any short range potential in which mutual passage is
forbidden [28]. A generalized central limit theorem for
tagged particle motion has also been proven [29]. More
recent work include [30] where particles interacting via
a screened Coulomb potential (i.e. not perfectly hard-
core) was studied numerically (see also [31]), [32] deals
with SFD in an external potential, and [33, 34, 35] ad-
dress SFD dynamics with different diffusion constants.
A phenomenological Langevin formulation of SFD was
presented in [36].
Although much work has been dedicated to single-file
systems, to our knowledge, very few exact results for
finite systems with finite-sized particles have been ob-
tained.The one exception is [37] where the PDF for
N point-particles diffusing on a finite one-dimensional
line was derived. However, simplified expressions for the
tPDF was only considered in the thermodynamic limit
(N,L → ∞ where L denotes system length and the con-
centration ̺ = N/L kept fixed). In this paper, we go
beyond previous studies in the following ways: First,
finite-sized particles are considered and we show that the
N-particle probability density function (NPDF) can be
written as a Bethe ansatz solution. We obtain an exact
expression for the tPDF in terms of Jacobi polynomials
(or hypergeometric functions) which reduces to that in
[37] for the case of point particles. Second, we perform a
(non-standard) large N-analysis of the tPDF, keeping the
system size L finite. The expression for ρT(yT,t|yT ,0) in
the many-particle limit is new and is presented compactly
in terms of modified Bessel functions. An analysis of the
tPDF reveals the existence of three dynamical regimes for
a particle located roughly in the middle of the system:
(A) short times, t ≪ τcoll= 1/(̺2D) where τcolldenotes
the collision time and D is the diffusion constant. In
this limit the tagged particle undergoes standard Brow-
nian motion with a MSD S(t) ∼ t; (B) For intermediate
times, τcoll ≪ t ≪ τeq where τeq = L2/D is the equi-
librium time, we get a SFD regime where S(t) ∼ t1/2.
(C) for long times, t ≫ τeq, an equilibrium tPDF of
polynomial-type is found. Notably, only regimes (A) and
(B) exist in infinite systems.
This paper has the following organization: Sec. II con-
tains the formulation of the problem and a mapping onto
a point particle system. The tPDF is also formally stated
in terms of the NPDF to which governing dynamical
equations are introduced. In Sec. III, we provide the
solution to the equations of motion for the NPDF using
a coordinate Bethe ansatz. In Sec. IV the initial coor-
dinates as well as the coordinates for all particles except
the tagged one are integrated out in order to obtain an
exact expression for the tPDF. Also, asymptotic results
for large N for the tPDF are derived. In Sec. V the
asymptotic large N expression for the tPDF is expanded
for short and long times, and three different time regimes
(A) − (C) (see above) are identified. More technical de-
FIG. 1: (color online) Cartoon of the problem considered here:
N particles of linear size ∆ diffusing is a one-dimensional sys-
tem of length L. The particles have center-of-mass coordi-
nates yj and initial positions yj,0, j = 1,..,N, and are unable
to overtake. This implies that yj+1 ≥ yj+∆ (j = 1,..,N −1)
for all times. Also, the particles cannot diffuse out of the box,
i.e. y1 > −L/2 + ∆/2 and yN < L/2 − ∆/2.
tails are given in the appendices. A brief summary of
some of our results, corroborated with Gillespie simula-
tions (Monte Carlo-type), can be found in [38].
II.PROBLEM DEFINITION
In this paper we consider a system of N identical hard-
core interacting particles, each with diffusion a constant
D and a linear size ∆, diffusing in a finite one dimen-
sional system extending from −L/2 to L/2. A schematic
cartoon is depicted in Fig. 1. The particles each have
center-of-mass and initial coordinates ? y = (y1,...,yN)
and ? y0= (y1,0,...,yN,0), respectively. Due to the hard-
core interaction, the particles cannot pass each other and
retain their order at all times, i.e. yj+1 ≥ yj+ ∆, for
j = 1,...,N − 1. The end of the system are reflecting
(the particles cannot escape), i.e. y1> −(L − ∆)/2 and
yN< (L − ∆)/2.
The diffusion of finite-sized particles can be mapped
onto a point-particle problem. Introducing the rescaled
effective system length
ℓ = L − N∆,(1)
and making the coordinate transformation
xj = yj− j∆ +N + 1
xj,0 = yj,0− j∆ +N + 1
2
∆,
2
∆,(2)
leads to
R : − ℓ/2 < x1< x2... < xN< ℓ/2
where R denotes the phase space spanned by (3). The
phase space R0is also introduced for the initial coordi-
nates which satisfy −ℓ/2 < x1,0< x2,0... < xN,0< ℓ/2.
For convenience, we also introduce the short-hand nota-
tion ? x = (x1,...,xN) and ? x0= (x1,0,...,xN,0). Equa-
tions (1) and (2) maps exactly the problem of N finite-
sized hard-core particles in a box of length L onto a N
point-particle problem in a box of length ℓ.
(3)

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The main quantity of interest in this study is the tPDF
ρT(xT,t|xT ,0), that is the probability density that a
tagged particle T (T = 1,...,N) is at position xT at time
t, given that it was at xT ,0at t = 0 [an ensemble aver-
age over over the initial (equilibrium) distribution of the
surrounding N −1 particles is implicit]. The equilibrium
tPDF is straightforwardly calculated from the ergodicity
principle: all points in the allowed phase space R are
equally probable. This leads the equilibrium NPDF
Peq(? x) =N!
ℓN
N−1
?
i=1
θ(xi+1− xi),(4)
where θ(z) is the Heaviside step function, θ(z) = 1 for
z > 0 and zero elsewhere. Using an extended phase space
integration technique (see Appendix C) it is easy to verify
that?
?
=
ℓN
NL!NR!
−ℓ/2
?ℓ/2
1
ℓN
NL!NR!
Rdx1···dxNPeq(? x) = 1. Integrating Eq. (4) over
all coordinates leaving out one, xT, gives the tPDF
ρeq
T(xT) =
R
dx′
1···dx′
1
Nδ(xT − x′
?xT
?ℓ/2
?ℓ
T)Peq(? x′)
?xT
N!
dx′
1···
−ℓ/2
dx′
T −1
×
xT
dx′
T +1···
xT
dx′
N
=
N!
2+ xT
?NL?ℓ
2− xT
?NR
, (5)
where δ(z) is the Dirac delta-function and NL (NR) is
the number of particles to the left (right) of the tagged
particle (N = NL+ NR+ 1). In the remaining part of
this section, we show how to calculate the complete time
evolution of the tPDF from the many-particle NPDF.
In order to obtain ρT(xT,t|xT ,0) one needs to first
introducetheN-particle
P(? x,t;? x0) which gives the probability density that the
system is in a state ? x and that it initially was in a
state ? x0. The joint probability density for the tagged
particle ρT(xT,t;xT ,0) is simply obtained from the joint
NPDF by integration over R and R0: ρT(xT,t;xT ,0) =
?
jointprobabilitydensities
[39],i.e.P(? x,t;? x0)
ρT(xT,t;xT ,0) = ρT(xT,t|xT ,0)ρeq(xT ,0), leads to
joint
probabilitydensity
Rdx′
x′
1···dx′
N
?
R0dx′
0).
1,0···dx′
Relating
N,0δ(xT
− x′
conditional
using
P(? x,t|? x0)Peq(? x0)
T)δ(xT ,0 −
T ,0)P(? x′,t;? x′
theand
rule
and
Bayes’
=
ρT(xT,t|xT ,0) =
1
ρeq(xT ,0)
?
×P(? x′,t|? x′
?
R
dx′
1···dx′
Nδ(xT − x′
T)
R0
dx′
1,0···dx′
0)Peq(? x′
N,0δ(xT ,0− x′
T ,0)
0),(6)
where ρeq
In order to get the tPDF using Eq. (6), we need to
calculate the NPDF P(? x,t|? x0). It is governed by the
T(xT ,0) is given in Eq. (5).
diffusion equation
∂P(? x,t|? x0)
∂t
= D
?∂2
∂x2
1
+
∂2
∂x2
2
+ ... +
∂2
∂x2
N
?
P(? x,t|? x0),
(7)
for ? x ∈ R (P(? x,t|? x0) ≡ 0 outside R). The equation cer-
tifying that neighboring particles cannot overtake reads
D
?
∂
∂xi+1
−
∂
∂xi
?
P(? x,t|? x0)
????
xi+1=xi
= 0. (8)
Also, reflecting boundaries are placed at the system ends
D∂P(? x,t|? x0)
∂x1
????
x1=−ℓ/2
= 0, (9)
D∂P(? x,t|? x0)
∂xN
????
xN=ℓ/2
= 0,(10)
making
[−ℓ/2,ℓ/2] at all times. Finally, the initial condition is
P(? x,0|? x0) = δ(x1− x1,0)···δ(xN− xN,0).
Summarizing this section, the problem of N hard-
core interacting particles of size ∆ diffusing in a one-
dimensional system of a finite length L was mapped onto
a point-particle problem using relationships (1) and (2).
The dynamics of the NPDF is governed by Eqs. (7)-(10).
Once they are solved (topic of Sec. III), the tPDF can
be calculated via Eq. (6) (as demonstrated in Sec. IV).
sure thatthe particlesare restricted to
(11)
III.
NPDF AS A COORDINATE BETHE
ANSATZ
In this section we obtain the NPDF for the diffusion
problem defined in previous section using a coordinate
Bethe ansatz. The Bethe ansatz has been proven useful
in solving a large variety of interacting particle problems
since its introduction by Hans Bethe in 1931 (see [40]
for a review). The Bethe ansatz solution for the present
problem reads
P(? x,t|? x0) =
?∞
−∞
dk1
2π
?∞
−∞
j=1φ(kj,xj,0)
dk2
2π···
?∞
−∞
dkN
2π
× e−E(k1,...,kN)tΠN
×
+ S21eik2x1+ik1x2+ik3x3+...+ikNxN
+ S32S31eik2x1+ik3x2+k1x3+...+ikNxN
+ all other permut. of {k1,k2,...,kN}?(12)
where E(k1,...,kN) is the dispersion relation, Sijare the
scattering coefficients, and φ(kj,xj,0) denotes a function
containing boundary and initial conditions. Each one of
these quantities are described below.
?eik1x1+ik2x2+ik3x3+...+ikNxN

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The dispersion relation has the form
E(k1,...,kn) = D(k2
1+ ... + k2
N),(13)
and relates ”energy” to the momenta k1,...,kN. Equa-
tion (13) is obtained by inserting the Bethe ansatz Eq.
(12) into the equation of motion, Eq. (7).
The scattering coefficients Sij describe pair-wise par-
ticle interactions and are in general functions of the mo-
mentum variables ki and kj, Sij = S(ki,kj). They are,
however, independent of the initial positions of the par-
ticles. In Appendix A it is demonstrated that the scat-
tering coefficients making sure that the particles cannot
pass each other [i.e. satisfying Eq. (8)] are given by
Sij= 1, (14)
which means that they are independent of momenta and
correspond to perfect reflection. For non-interacting par-
ticles Sij= 0, which reduces the Bethe ansatz to a stan-
dard Fourier transform.
The quantity φ(kj,xj,0) contains information about
the initial and boundary conditions of the problem, here
defined by Eqs. (9) - (11). The form of φ(kj,xj,0) satis-
fying these relationships is given by
φ(kj,xj,0) = 2cos[kj(xj,0+ ℓ/2)]
∞
?
m=−∞
eikj(2m+1/2)ℓ,
(15)
which is shown explicitly in Appendix A. Notably, for an
infinite system we have φ(kj,xj,0) = e−ikjxj,0[41, 42].
It is interesting to note that for SFD systems described
by Eq. (12)-(15) any macroscopic quantity that is in-
variant under interchange of any two particle positions,
xi ↔ xj, takes the same value as for a system of non-
interacting particles. This is in marked contrast to micro-
scopic quantities such as the tPDF which in general be-
have very differently for single-file and independent par-
ticle systems. In Appendix F we use the Bethe ansatz to
explicitly calculate two macroscopic quantities, the dy-
namic structure factor and the center of mass PDF, and
show that they agree with standard results for indepen-
dent particle systems.
Integration over momenta in Eq. (12) [using Eqs. (13)-
(15)] leads to the NPDF:
P(? x,t|? x0) = ψ(x1,x1,0;t)ψ(x2,x2,0;t)···ψ(xN,xN,0;t)
+ ψ(x1,x2,0;t)ψ(x2,x1,0;t)···ψ(xN,xN,0;t)
+ all other permut. of {x1,0,x2,0,...,xN,0}, (16)
where
ψ(xi,xj,0;t) =
1
(4πDt)1/2
∞
?
+ exp
×
m=−∞
?
exp
?
−(xi− xj,0+ 2mℓ)2
4Dt
?
?
−(xi+ xj,0+ (2m + 1)ℓ)2
4Dt
??
,(17)
is
(2π)−1?∞
in confined in a box of length ℓ.
The single-particle PDF given in Eq. (17) is, however,
not convenient for analyzing the long time limit t → ∞.
In order to get a more suitable expression we seek instead
the eigenmode expansion of ψ(xi,xj,0;t), which can be
done in a variety of ways. Here we use Bromwich in-
tegration. The Laplace transform of Eq. (17) is (see
Appendix B)
obtained
−∞dkjφ(kj,xj,0)e−Dk2
that ψ(xi,xj,0;t) is the single-particle PDF for a particle
fromtheinverse
jteikjxi. We
Fouriertransform
point out
ψ(xi,xj,0;s) =
?∞
√4Dssinh(ℓ?s/D)
?
+ cosh[(ℓ − |xj,0− xi|)
0
dte−stψ(xi,xj,0;t)
=
1
×cosh[(xi+ xj,0)
?
s/D]
?
s/D]
?
.(18)
The sought eigenvalue expansion is obtained as a sum of
residues of ψ(xi,xj,0;s) (see e.g. Ref. [43]), and reads
ψ(xi,xj,0;t) =1
ℓ
?
1 +
∞
?
m=1
Gm(xi,xj,0)Em(t)
?
(19)
where
Gm(xi,xj,0) = ν(+)
m cos
?mπxi
?mπxi
ℓ
?
cos
?mπxj,0
sin
ℓ
?
+ν(−)
m sin
ℓ
??mπxj,0
ℓ
?
, (20)
Em(t) = e−(mπ)2Dt/ℓ2
(21)
and
ν(±)
m
= 1 ± (−1)m.(22)
Elementary trigonometric identities [44] were used to
bring Gm(xi,xj,0) onto the form in Eq. (20). Equations
(19)-(22) agrees with well-known results [45]. The single-
particle PDF (19) is more convenient for obtaining the
long time limit as well as for numerical computations
compared to Eq. (17).
In summary, the many-particle NPDF for excluding
particles of size ∆ diffusing in a finite interval of length
L with reflecting boundaries is given by Eqs. (16), (19)
[or Eq. (17)] combined with the mapping equations (1)
and (2). For point-particles (∆ = 0) these results agree
with those presented in [37] where a different approach
was used [46]. Based on the explicit expression of our
NPDF, we will in the following section address the tPDF.
IV. tPDF - EXACT AND LARGE N RESULTS
In this section, we calculate the tPDF (6) by inte-
grating out the coordinates and initial positions of all

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non-tagged particles from the NPDF given in Eq. (16)
[47]. As is shown in detail in Appendix C we can, due
to the property that P(? x,t|? x0) is invariant under per-
mutations of xi ↔ xj, extend the integration from R
to the hypercubes xj ∈ [−ℓ/2,xT], j = 1,...,T − 1 and
xj ∈ [xT,ℓ/2], j = T + 1,...,N. A similar procedure
holds for integration over R0 (initial positions). Using
the extended phase-space technique, Eq. (6) becomes
ρT(xT,t|xT ,0) =fNL
LfNR
NL!NR!
?ℓ/2
?ℓ/2
R
?xT
?xT ,0
−ℓ/2
dx1···
?xT
−ℓ/2
dxT −1
×
?ℓ/2
?xT ,0
xT
dxT +1···
xT
dxN
−ℓ/2
dx1,0···(23)
×
−ℓ/2
dxT −1,0
xT ,0
dxT +1,0···
?ℓ/2
xT ,0
dxN,0P(? x,t|? x0),
where
fL= (ℓ/2 + xT ,0)−1,fR= (ℓ/2 − xT ,0)−1.(24)
Using a similar combinatorial analysis to the one in Ref.
[37], we arrive at Eq.(D1) (see Appendix D).
tPDF is, however, more conveniently expressed in terms
of Jacobi polynomials [44], P(α,β)
ties P(α,β)
n
(z) = (n + α)!(n + β)!/[n!(n + α + β)!][(z −
1)/2]−aP(−α,β)
n+α
(z) and P(α,β)
n
leads to
The
n
(z). Using the identi-
(−z) = (−1)nP(β,α)
n
(z) [48]
ρ(xT,t|xT ,0) =(NL+ NR− 1)!
NL!NR!
(ψL
L)NL(ψR
R)NR
×
?
+N2
(NL+ NR)ψΦ(0,0,0;ξ) + N2
L
ψLψL
ψL
L
Φ(1,0,0;ξ)
R
ψRψR
ψR
R
?ψRψL
Φ(0,1,0;ξ)
+NLNR
ψR
L
+ψLψR
ψL
R
?
Φ(0,0,1;ξ)
?
(25)
where
Φ(a,b,c;ξ) =
(NL− (a + c))!(NR− b)!
(NL+ NR− (a + b + c))!
×ξc(1 − ξ)NL−(a+c)P(c,NR−NL+a−b)
NL−(a+c)
?1 + ξ
1 − ξ
?
(26)
and
ξ =ψL
RψR
ψL
L
LψR
R
.(27)
were introduced. The quantities ψL, ψL
in Eq. (D3) and are integrals of the one particle propa-
gator ψ = ψ(xi,xj,0;t) [49]. For the general case we have
ξ ∈ [0,1] [50]. By using limiting results of ψL
and ψL
Rfrom Appendix E, one concludes that ξ → 0 for
short times (t → 0), and ξ → 1 in the long time limit
(t → ∞). It is also possible using standard relations for
Letc. are defined
L, ψR
R, ψR
L
the Jacobi polynomials [44] to express Φ(a,b,c;ξ) as a
Gauss hypergeometric function2F1(α,β,γ;z):
Φ(a,b,c;ξ) = 2F1(−NL+ a,−NR+ b,
−(NL+ NR) + a + b + c;1 − ξ) (28)
which is convenient for obtaining the long-time behavior
as will be seen in the next section.
Finally, we need to find explicit expressions for the
integrals of ψ(xi,xj,0;t) [Eq. (D3)]. Integrating Eq. (19)
[or integrating ψ(xi,xj,0;s) prior to Laplace inversion,
see Appendix E] yields (arguments left implicit)
ψL
L=
1
2+xT
ℓ
+
?1
?1
∞
?
2+xT ,0
2+xT ,0
ℓ
?−1
?−1
∞
?
∞
?
m=1
KmEm(t),
ψR
R=
1
2−xT
ℓ
+
2−xT ,0
ℓ
m=1
KmEm(t),
ψL=
1
2+xT
ℓ
+
m=1
Jm(xT,xT ,0)Em(t),
ψL =
1
ℓ
?
?
1 +
?1
?1
ℓ
?−1
?−1
L, ψL
∞
?
∞
?
m=1
Jm(xT ,0,xT)Em(t)
?
?
,
ψR =
1
ℓ
1 −
2−xT ,0
l
m=1
Jm(xT ,0,xT)Em(t),
ψR= 1 − ψL, ψR
with
L= 1 − ψL
R= 1 − ψR
R
(29)
Km =
1
(mπ)2
?
ν(+)
m sin
?mπxT
?
?mπz
?mπz
are given by Eqs. (21) and (22),
ℓ
?
sin
?mπxT ,0
??
?mπz′
?mπz′
ℓ
?
+ν(−)
m cos
?mπxT
m sin
ℓ
cos
?mπxT ,0
cos
ℓ
(30)
Jm(z,z′) =
1
mπ
?
ν(+)
ℓ
?
ℓ
??
?
,
−ν(−)
m cos
ℓ
?
sin
ℓ
(31)
where Em(t) and ν(±)
respectively. To summarize, the complete expression for
the tagged particle PDF is given by Eqs. (25)-(27) and
(29)-(30) together with Eqs. (1) and (2) for the case of
finite-sized particles. The expressions for ρT(xT,t|xT ,0)
can straightforwardly be computed numerically; our
MATLAB implementation is available upon request.
In the remaining part of this section we derive the
tagged particle PDF ρT(xT,t|xT ,0) valid for a large N
and (finite) system size ℓ. This large N expansion will
be used in the next section for identifying different time
regimes and to obtain ρT(xT,t|xT ,0) for short and inter-
mediate times. From Eq. (26) we note that the argu-
ment in the Jacobi polynomial, P(α,β)
val z ∈ [1,∞) (since ξ ∈ [0,1]) and that the number of
particles is related to the order n. A large N expansion
of Φ(a,b,c;ξ) therefore amounts to find a large order n
expansion valid for z ∈ [1,∞) (i.e. for all times) for the
m
n
(z), is in the inter-

Page 6

6
Jacobi polynomial. One such expansion was derived in
[51] (see also [52, 53]), and applying it to Eq. (26) yields
Φ(a,b,c;ξ) ≈ [N − (a + b + c)]1/2ξ(2c−1)/4
×(2πζ)1/2
×Ic[(N − (a + b + c))ζ]?1 + A(ζ)?
?1 − ξ
4
?[N−(a+b+c)]/2
(32)
where [54]
ζ =1
2ln
?1 +√ξ
1 −√ξ
?
,(33)
and Iα(z) is the modified Bessel function of the first kind
of order α. Stirling’s formula [44] was also used to ap-
proximate factorials involving NLand NR. The correc-
tion term appearing in Eq. (32) is
A(ζ) =
B0(ζ)
N − (a + b + c)
1
2
Ic+1[(N − (a + b + c))ζ]
Ic[(N − (a + b + c))ζ]
ζ− cothζ
B0(ζ) =
??c2− 1/4??1
−[(δN+ a − b)2− 1/4]tanh(ζ)
?
?
,(34)
where δN = NR− NL was introduced. The expression
for A(ζ) was found by explicitly evaluating an integral in
Ref. [51], and it is a straightforward matter to show that
the correction term A(ζ) is indeed always small for all
ξ ∈ [0,1] provided that 1/N,|δN|/N ≪ 1. Inserting Eq.
(32) in Eq. (25) and using the same approximations as
above, we obtain our final large N result for the tPDF:
ρT(xT,t|xT ,0) = (ψL
L)(N−δN−1)/2(ψR
?1/2?
+ψRψR
ψR
R
?ψRψL
R)(N+δN−1)/2
×(1 − ξ)(N−1)/2
+N
2
?
ζ
√ξ
(1 − ξ)1/2ψI0[Nζ]
?ψLψL
?
ψL
L
?
I0[Nζ]
+N
2
ξ
ψR
L
+ψLψR
ψL
R
?
I1[Nζ]
?
.(35)
We point out that this expression, in contrast to pre-
vious asymptotic expressions [20, 22, 37], is valid for a
finite box of size ℓ, assuming only that the number of
particles is large N ≫ 1 and that the tagged particle is
approximately in the center of the system: |δN|/N ≪ 1.
V. THREE DIFFERENT TIME REGIMES
In this section we show that for large N, the finite SFD
system considered here has three different time regimes to
which expressions for ρT(xT,t|xT ,0) are derived. Math-
ematically, the different cases appear due to the magni-
tude of Nζ [found in the argument of the Bessel functions
in Eq. (35)], and if ζ is small or large. Utilizing Eqs.
(E14) and (33), these cases can be turned into different
time regimes if introducing the collision time
τcoll=
1
̺2D,
(36)
where ̺ = N/ℓ is the concentration of particles, and the
equilibrium time
τeq=ℓ2
D.
(37)
For a particle located roughly in the middle, |δN|/N ≪ 1
(i.e. Eq. (35) applies), the three cases are given by
A. short times, Nζ ≪ 1, i.e. t ≪ τcoll,τeq,
B. intermediate times, ζ ≪ 1 and Nζ ≫ 1, corre-
sponding to τcoll≪ t ≪ τeq,
C. long times, ζ ≫ 1, i.e. t ≫ τcoll,τeq.
Time regimes (A)-(C) are analyzed in detail below.
A.Short times, t ≪ τcoll,τeq
For short times we have Nζ ≪ 1, and may therefore
use the approximations Iα(z)|z≪1≈ (z/2)α/Γ(α+1) [44],
ζ ≈√ξ and ξ ≪ 1. In this limit, one finds that the first
term in Eq. (35) dominates which in combination with
Eq. (2) leads to
ρT(yT,t|yT ,0) = (4πDt)−1/2exp
?
−(yT − yT ,0)2
4Dt
?
,
(38)
for which the MSD is
S(t) = 2Dt.(39)
In the short time regime, almost no collisions with the
neighboring particles (nor the box walls) have occurred
and the tPDF is therefore a Gaussian with width 2Dt as
for a free particle in an infinite one-dimensional system.
B. Intermediate times, τcoll≪ t ≪ τeq
In the intermediate time regime the tagged particle
has collided many times with its neighbors but not yet
reached its equilibrium tPDF. For this regime, where
ζ ≪ 1 but Nζ ≫ 1, we get the tPDF as follows. First,
the first term in Eq. (35) is neglected (this is checked
at the end of the calculation). Second, the Bessel func-
tion is approximated with Iα(z)|z≫1 ≈ ez/√2πz.
straightforward expansion of Eq. (35) for ψR
(i.e.√ξ ≪ 1), together with Stirling’s formula, gives:
A
L,ψL
R≪ 1
ρT(yT,t|yT ,0) ≈1
2e−δN(ψL
R−ψR
L)
?
N
2π
?ψL
RψR
L
?−1/4

Page 7

7
×e−N
2
?√
ψL
R−√
ψR
L
?2?
ψLψL+ ψRψR
+ψRψL
?ψL
R
ψR
L
?1/2
+ ψLψR
?ψR
L
ψL
R
?1/2?
. (40)
If we furthermore assume that the average of the absolute
value of η = (xT−xT ,0)/√4Dt is small (which is checked
after the calculation), Eq. (E15) may be used which in
combination with Eqs. (40), (1) and (2), keeping only
lowest order terms in η, leads to the SFD result:
ρT(yT,t|yT ,0) =
1
√2π
?
1
4Dt
π(1−̺∆


̺
)2
?1/4
×exp

−
(yT − yT ,0)2
?
2
4Dt
π(1−̺∆
̺
)2
(41)
where the MSD is
S(t) =1 − ̺∆
̺
?
4Dt
π
. (42)
Equation (42) justifies the assumption that the expec-
tation value of |η| is a small number. Also, comparing
the magnitude of the first term in Eq. (35) with respect
to the second and the third, shows indeed that our first
assumption above was correct [55]. For point-particles
∆ = 0, Eq. (41) agrees with standard results [20, 37], in
which N,L → ∞ while keeping the concentration ̺ fixed.
The result above shows that SFD behavior appears also
in a finite system with reflecting ends, as an interme-
diate regime (for a particle roughly in the middle). In
addition, note that the simple rescaling ̺ → ̺/(1 − ̺∆)
takes us from previous point-particle results to those of
finite-sized particles.
C.Long times, t ≫ τeq
In the long-time limit we have ζ ≫ 1, for which the
tPDF can be obtained exactly for arbitrary N. Using
the exact expression for ρT(xT,t|xT ,0) found in Eqs. (25)
and (28), together with2F1(α,β,γ;z = 0) = 1 [44] and
Eq. (E7) [also using Eqs. (1) and (2)] gives
ρeq
T(yT) =
1
(L − N∆)N
?L
?L
(NL+ NR+ 1)!
NL!NR!
×
2+ yT − ∆(1/2 + NL)
?NL
?NR
×
2− yT − ∆(1/2 + NR). (43)
where ρeq
agrees with the equilibrium tPDF given in Eq. (5) (for
∆ = 0), as it should [56]. The results above shows the
consistency of our analysis, and illustrates the ergodicity
T(yT) = ρT(yT,t → ∞|yT ,0). This equation
of the finite SFD system. Calculating the second moment
of the equilibrium tPDF, S(t → ∞) = Seq, gives
?NR+1?L − N∆
Seq=
?1
42
?2
Γ(1/2)Γ(2(NR+ 1))
Γ(NR+ 1)Γ(NR+ 5/2),
(44)
for NL= NRwhere Γ(z) is the the gamma function. For
N ≫ 1 we can simplify the expression for the equilibrium
tPDF as well as Seq. If a symmetric file is assumed, i.e.
NL= NRand T = (N + 1)/2, an asymptotic expansion
of ρeq
T(yT) (using Stirling’s approximation [44]) gives the
Gaussian PDF
ρeq
T(yT) ≈
?
2N
π
1
(L − N∆)2
?
×exp−2N
?
yT
L − N∆
?2?
, (45)
from which we read off that
Seq≈(L − N∆)2
4N
. (46)
Equation (46) can also be found directly from a large
N expansion of Eq. (44) using Stirling’s formula and
assuming N ≈ 2NR. We point out that ρeq
N,L → ∞ even if the concentration ̺ = N/L is kept
fixed. This is consistent with the long time limit of Eq.
(41) which indeed goes to zero for large times. Finally,
the analysis in this subsection also gives an estimate on
the time required to reach equilibrium, namely t ≫ τeq.
T(yT) → 0 for
VI.CONCLUSIONS AND OUTLOOK
In this study we have solved exactly a non-equilibrium
statistical mechanics problem: diffusion of N hard-core
interacting particles of size ∆ which are unable to pass
each other in a one-dimensional system of length L
with reflecting boundaries. In particular, we obtained
an exact expression for the probability density function
ρT(yT,t|yT ,0) (denoted tPDF) that a tagged particle
particle T is at position yT at time t given that it at
time t = 0 was at position yT ,0. We derived the tPDF by
first finding the N-particle probability density function
(NPDF) via the Bethe ansatz, and then integrating out
the coordinates and taking the average over the initial
positions of all particles except one. The exact expres-
sion for ρT(yT,t|yT ,0) is found in Eqs, (1), (2), (25) and
(26), and constitutes the main result of the paper. For
a large number of particles and for a tagged particle lo-
cated roughly in the middle of the system, an asymptotic
expansion of the tPDF was derived [see Eq. (35)]. Based
on this equation, we found three time regimes of interest:
(A) For short times, i.e. times much smaller than the col-
lision time t ≪ τcoll= 1/(̺2D), where ̺ = N/L is the
particle number concentration, the tPDF coincides with

Page 8

8
the Gaussian probability density function which charac-
terizes a free particle [Eq.
ate times, t ≫ τcoll but much smaller than the equilib-
rium time t ≪ τeq = L2/D, a sub-diffusive single-file
regime was found in which the tPDF is a Gaussian with
an associated MSD proportional to t1/2[Eq. (41)]. (C)
For times exceeding the equilibrium time t ≫ τeq, the
tPDF approaches a probability density of polynomial-
type [Eq. (43)].
We point out that the sub-diffusive behavior for a
tagged particle in time regime (B) is of fractional brow-
nian motion type [33, 57, 58] rather than that of
continuous-time random walks (CTRWs) characterized
by heavy-tailed waiting time densities [59, 60, 61]. For
such CTRW processes the probability density function is
not a Gaussian as for the current system. Further com-
parisons between sub-diffusion in single-file systems and
that occurring in CTRW theory was pursued numerically
in [62].
The Bethe ansatz is often employed in quantum me-
chanics when many-body systems are studied (e.g. quan-
tum spin chains) [40, 63], and also for stochastic many-
particle lattice problems [64, 65]. We hope that the theo-
retical analysis based on the Bethe ansatz presented here
will stimulate further progress in the field of single-file
diffusion and that of interacting random walkers. For
instance, it would be interesting to see whether our anal-
ysis could be extended to derive exact results also for
particles interacting through potentials other than hard-
core type [31], and for particles having different diffusion
constants [34].
From the applied point of view, our exact expres-
sion for the tPDF covers all time regimes and is
straightforward to implement for numerical computa-
tions. Therefore, we believe that our explicit formula
for ρT(yT,t|yT ,0), as well as the approximate results for
regimes (A)−(C), will be useful for experimentalists (see
for instance [18]) seeking to extract system parameters
such as particle size ∆, system size L, the particle’s dif-
fusion constants (D), and the number (NL and NR) of
particles to the left and right of the tagged particle.
We finally note that the use of fluorescently labeled
(tagged) particles is of much use for studying biological
systems. The understanding how the motion of such par-
ticles correlate with its environment is therefore the key
for grasping the behavior of such systems in a quantita-
tive fashion.
(38)].(B) For intermedi-
VII.ACKNOWLEDGMENTS
We are grateful to Bob Silbey, Mehran Kardar, Eli
Barkai, Ophir Flomenbom and Michael Lomholt for valu-
able discussions. L. L acknowledges support from the
Danish National Research Foundation, and T. A. from
the Knut and Alice Wallenberg Foundation.
APPENDIX A: THE BETHE ANSATZ
In this section we show that the Bethe ansatz, Eq.
(12), is a solution to the problem defined by Eqs. (7)-
(11). First, it is demonstrated that Eq. (12) satisfies the
boundary conditions at the ends of the box. Second, we
show that the requirement that the particles are unable
to pass each other is satisfied by setting the scattering
coefficients to unity. Finally, it is demonstrated that Eq.
(12) also satisfies the initial condition.
1.Boundary conditions at the ends of the box
In this subsection it is proven that Eq. (12) satisfies
the the reflecting boundary conditions Eq. (9) and (10)
at ±ℓ/2 with an appropriate choice for φ(kj,xj,0) [Eq.
(15)]. The scattering coefficients are set to Sij= 1 which
is proven to be correct in the following subsection. First,
we define the function
λ(k,z) = φ(k,z)e−ikℓ/2= 2cos[k(z+ℓ/2)]
∞
?
m=−∞
e−2ikmℓ
(A1)
which has the symmetry relation
λ(k,z) = λ(−k,z). (A2)
By taking the derivative of Eq. (12) with respect to x1,
and evaluating at the left boundary x1= −ℓ/2 gives
∂P(? x,t|? x0)
∂x1
x1=−ℓ/2
?∞
×
...
+ikieikix1?eik2x2+...+ik1xi+...+ikNxN
+ perm. of {k2,...,ki−1,ki+1,...,kN})
...
+ ikNeikNx1?eik2x2+...+ik1xN
+ perm. of {k1,...,kN−1})
????
=
−∞
?
dk1
2π···
?∞
−∞
dkN
2πe−D(k2
1+...+k2
N)tΠN
j=1φ(kj,xj,0)
ik1eik1x1?eik2x2+...+ikNxN+ perm. of {k2,...,kN}?
?
x1=−ℓ/2
=
?∞
?∞
−∞
dk2
2π···
dk1
2π(ik1)e−Dk2
?∞
−∞
dkN
2πe−D(k2
2+...+k2
N)tΠN
j=2φ(kj,xj,0)
×
−∞
1tλ(k1,x1,0) +
...
+
?∞
−∞
dk1
2π···
?∞
i−1+k2
−∞
dki−1
2π
?∞
N)tΠN
−∞
dki+1
2π
···
?∞
−∞
dkN
2π
×e−D(k2
1+...+k2
i+1+...+k2
j=1,j?=iφ(kj,xj,0)

Page 9

9
×
?∞
−∞
dki
2π(iki)e−Dk2
itλ(ki,xi,0) +
...
+
?∞
−∞
dk1
2π···
?∞
−∞
dkN−1
2π
?∞
e−D(k2
1+...k2
N−1)t
×ΠN−1
= 0,
j=1φ(kj,xj,0)
−∞
dkN
2π(ikN)e−Dk2
Ntλ(kN,xN,0)
(A3)
where
see Eq.
this calculation does not rely on any specific form of
φ(kj,xj,0).It is only required that the symmetry re-
lation (A2) holds. Furthermore, the dispersion relation
E(?k) = D(k2
N) was also used in the above
derivation. However, it would work equally well for any
?∞
−∞dkikie−Dk2
(A2)] was used in the last step.
itλ(ki,xi,0) = 0 [odd integrand,
Note that
1+ ... + k2
dispersion relation as long as E(?k) =
E(ki) = E(−ki) is valid.
A similar analysis as just presented shows that the
Bethe ansatz solution also satisfies the reflecting condi-
tion at +ℓ/2 [Eq. (10)]. In fact, since the Bethe ansatz is
invariant under the coordinate transformation xi↔ xj,
gives [∂P(? x,t|? x0)/∂xj]xj=±ℓ/2= 0 for all xj.
?
iE(ki) with
2.Single-file condition: particles are unable to
overtake
In this subsection it is shown that the condition that
the particles are unable to pass each other, Eq. (8), is
satisfied for scattering coefficients given by Sij = 1 in
the Bethe ansatz solution Eq.
expressing P(? x,t|? x0) in two alternative ways:
(12). We start off by
P(? x,t|? x0) =
?∞
×
+eik2xj?eik1x1+ikjx2+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikNxN+ all other permut. of {k1,k3,...,kN}?
...
+eikNxj?eik1x1+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikjxN+ all other permut. of {k1,...,kN−1}??
?∞
×
+eik2xj+1?eik1x1+ikj+1x2+...+ikj−1xj−1+ikjxj+ikj+2xj+2+...+ikNxN+ all other permut. of {k1,k3,...,kN}?
...
+eikNxj+1?eik1x1+...+ikj−1xj−1+ikjxj+ikj+2xj+2+...+ikj+1xN+ all other permut. of {k1,...,kN−1}??
Using the above equations one finds
?????
×
−eikjx1+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikNxN+ all other permut. of {k2,...,kN}?
+ik2eik2xj?eik1x1+ikj+1x2+...+ikj−1xj−1+ikjxj+1+ikj+2xj+2+...+ikNxN
−eik1x1+ikjx2+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikNxN+ all other permut. of {k1,k3,...,kN}?
...
+ikNeikNxj(eik1x1+...+ikj−1xj−1+ikjxj+1+ikj+2xj+2+...+ikj+1xN
−eik1x1+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikjxN+ all other permut. of {k1,...,kN−1}??= 0
−∞
?
dk1
2π
?∞
−∞
dk2
2π···
?∞
−∞
dkN
2π
e−D(k2
1+...+k2
N)tΠN
j=1φ(kj,xj,0)
eik1xj?eikjx1+...+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+...+ikNxN+ all other permut. of {k2,...,kN}?
(A4)
and
P(? x,t|? x0) =
−∞
?
dk1
2π
?∞
−∞
dk2
2π···
?∞
−∞
dkN
2πe−D(k2
1+...+k2
N)tΠN
j=1φ(kj,xj,0)
eik1xj+1?eikj+1x1+...+ikj−1xj−1+ikjxj+ikj+2xj+2+...+ikNxN+ all other permut. of {k2,...,kN}?
. (A5)
?∂P(? x,t|? x0)
?
∂xj+1
−∂P(? x,t|? x0)
∂xj
xj+1=xj
=
?∞
−∞
dk1
2π
?∞
−∞
dk2
2π···
?∞
−∞
dkN
2πe−D(k2
1,...,k2
N)tΠN
j=1φ(kj,xj,0)
ik1eik1xj?eikj+1x1+...+ikj−1xj−1+ikjxj+1+ikj+2xj+2+...+ikNxN
(A6)
where it was used that each parenthesis is identically zero
due to the cancellation of the 2(N−1)! terms after permu-
tation over all allowed momenta. Note that this deriva-

Page 10

10
tion is independent of the choice of φ(kj,xj,0) and E(?k).
3.Initial condition
In this subsection we show that the Bethe ansatz Eq.
(12) agrees with the initial condition Eq.
t → 0. By defining
?∞
Eq. (12) reads
(11) when
Ω(x,y) =
−∞
dkj
2πeikjxφ(kj,y), (A7)
P(? x,t → 0|? x0) = Ω(x1,x1,0)Ω(x2,x2,0)···Ω(xN,xN,0)
+ Ω(x1,x2,0)Ω(x2,x1,0)···Ω(xN,xN,0)
+ all other permut. of {x1,0,...,xN,0}.(A8)
Using the explicit expression for φ(k,y) found in Eq. (15)
and that δ(x − z) = (2π)−1?∞
Ω(x,y) =
?
−∞dk eik(x−z)leads to
∞
m=−∞
δ(x − y + 2mℓ) + δ(x + y + (2m + 1)ℓ).
(A9)
For all m ?= 0 the δ-functions are non-zero for coordi-
nates lying outside of R and R0 where, per definition,
P(? x,t|? x0) ≡ 0 (see discussion in section Sect. II). For
m = 0, it is only the first δ-function in the sum, δ(x−y),
that contributes to the NPDF. Furthermore, since all
terms except the first one in Eq. (A8) are zero due to
the fact that P(? x,t|? x0) = 0 outside R, one obtains
P(? x,t → 0|? x0) = δ(x − x0)···δ(xN− xN,0)
as t → 0 which is the desired result.
(A10)
APPENDIX B: LAPLACE TRANSFORM OF
ψ(xi,xj,0;t)
In this section it is shown explicitly how one can go
from the one-particle PDF ψ(xi,xj,0;t) for a particle in
a box expressed in terms of Gaussians [Eq. (17)], to the
eigenmode expansion [Eqs. (19)-(22)] used in this paper.
First, the summation in Eq. (17) is divided schemat-
ically into
?∞
s−1/2e−a√s(a > 0) [44] one finds
ψ(xi,xj,0;s) = Q(s) +e−(xi+xj,0+ℓ)√
?e−(xi−xj,0)√
+e−(xi+xj,0+ℓ)√
√4Ds
∞
?
m=−∞fm = fm=0+?∞
m=1[fm+ f−m].
Then, using the Laplace transform L[(πt)−1/2e−a/(4t)] =
s/D
√4Ds
+
s/D+ e(xi−xj,0)√
√4Ds
s/D+ e(xi+xj,0+ℓ)√
s/D]
s/D
?
×
m=1
e−2mℓ√
s/D
(B1)
1 x
x2
x12 x
=
x
1 x > x2
x1<
2
FIG. 2: Integration area (the darker upper area, x1 < x2)
for the integral defined in Eq. (C2). For any function which
is symmetric under x1 ↔ x2 the integration over the lower
triangle (x1 > x2) yields the same result as integration over
the same function over the upper triangle (x1 < x2). One
may therefore extend the integration area to the full rectangle
above provided one divides the corresponding result by 2.
where
Q(s) =
?
(4Ds)−1/2e−(xi−xj,0)√
(4Ds)−1/2e−(xj,0−xi)√
s/D, xi≥ xj,0
s/D, xi≤ xj,0.
(B2)
Considering the cases xi> xj,0and xi< xj,0separately,
and that?∞
ψ(xi,xj,0;s) =
√sDsinh[ℓ?s/D]



which after elementary trigonometric manipulations re-
sult in Eq. (18).
m=1e−2mℓ√
s/D= (e2ℓ√
s/D−1)−1, leads to
1
×








cosh[(ℓ/2 + xi)?s/D]
×cosh[(ℓ/2 − xj,0)?s/D], xi≤ xj0
cosh[(ℓ/2 − xi)?s/D]
×cosh[(ℓ/2 + xj,0)?s/D], xi≥ xj,0
(B3)
APPENDIX C: EXTENDED PHASE SPACE
INTEGRATION
When the tPDF is integrated out from the NPDF we
need to resolve the following type of integral:
I(xT) =
?
R\xT
dx′
1···dx′
T −1dx′
T +1···dx′
NP(? x′,t|? x0)
(C1)
over the region R [Eq. (3)] with the tagged particle co-
ordinate xT left out. As pointed previously in the pa-
per, the Bethe ansatz solution [Eq. (12)] is symmetric

Page 11

11
under the transformation xi ↔ xj when all scattering
coefficients are given by Sij = 1. This allows the in-
tegration of P(? x,t|? x0) in Eq. (C1) to be extended to
the whole hyperspace xj ∈ [−ℓ/2,xT],j = 1,...,T − 1
and xj∈ [xT,ℓ/2],j = T + 1,...,N. This is most easily
demonstrated in an example which then is extended to
the general situation. Consider the case of three particles
where particle three is tagged T = 3:
?
I(x3) =
−ℓ/2<x1<x2<x3<ℓ/2
dx1dx2P(x1,x2,x3,t|? x0).
(C2)
The integration area in the (x1,x2)-plane is sketched in
Fig. 2 (upper dark triangle). Since P(? x,t|? x0) is invariant
under x1↔ x2, integration over the lower triangle gives
the same result, i.e.:
I(x3) =
?
−ℓ/2<x2<x1<x3<ℓ/2
dx1dx2P(x1,x2,x3,t|? x0).
(C3)
If Eqs. (C2) and (C3) are added, I(x3) can be expressed
as an integral over the full rectangle
I(x3) =1
2
?xT
−ℓ/2
dx1
?xT
−ℓ/2
dx2P(x1,x2,x3,t|? x0) (C4)
For the general case, the integration can be extended for
any particle number to the left of the tagged particle.
This means that it is possible to go from integration over
the phase space −ℓ/2 < x1< ... < xT −1< xT to −ℓ/2 <
xj< xT for j = 1,...,T − 1 provided that we divide by
NL!. This holds also for NR particles to the right and
Eq. (C1) can in general be written as
I(xT) =
1
NL!NR!
?xT
−ℓ/2
dx1
?xT
−ℓ/2
?ℓ/2
dx2···(C5)
×
?xT
−ℓ/2
dxT −1
?ℓ/2
xT
dxT +1···
xT
dxNP(? x,t|? x0).
Since P(? x,t|? x0) is also invariant under xi,0 ↔ xj,0, a
similar extended phase-space technique is valid for inte-
grations over the initial particle positions R0.
APPENDIX D: FROM THE NPDF TO THE tPDF
In this section the tPDF is obtained from the integral
(6) using the Bethe ansatz NPDF (16) explicitly. A sim-
ilar combinatorial analysis to that found in [37] gives
ρ(xT,t|xT ,0) =
1
NL!NR!
?
min{NL,NR}
?
min{NL−1,NR}
?
q=0
H0(q)(ψL
L)NL−q(ψL
R)qψ(ψR
L)q(ψR
R)NR−q
+
q=0
HL
L(q)(ψL
L)NL−q−1(ψL
R)qψLψL(ψR
L)q(ψR
R)NR−q
+
min{NL,NR−1}
?
min{NL−1,NR−1}
?
q=0
HR
R(q)(ψL
L)NL−q(ψL
R)qψRψR(ψR
L)q(ψR
R)NR−q−1
+
q=0
HR
L(q)(ψL
L)NL−q−1(ψL
R)q+1ψRψL
×(ψR
L)q(ψR
R)NR−q−1
+
min{NL−1,NR−1}
?
q=0
HL
R(q)(ψL
L)NL−q−1(ψL
R)qψLψR
×(ψR
L)q+1(ψR
R)NR−q−1?
(D1)
with the combinatorial factors
H0(q) =
?NL
?NL− 1
?NL
?NL− 1
?NL
q
??NR
q
?
NL!NR!,
HL
L(q) =
q
??NR− 1
??NR
??NR− 1
??NR
q
?
?
NL!NR!NL,
HR
R(q) =
q
q
NL!NR!NR,
HR
L(q) =
q
q + 1
?
?
NL!NR!NL,
HL
R(q) =
q + 1
q
NL!NR!NR, (D2)
and integrals (leaving arguments xT ,0and xT implicit)
?xT
?ℓ/2
?ℓ/2
?xT
ψL(t) =
−ℓ/2
?ℓ/2
ψL(t) = fL
−ℓ/2
?ℓ/2
ψL
L(t) = fL
−ℓ/2
dxi
?xT ,0
?ℓ/2
?xT ,0
?ℓ/2
−ℓ/2
dxj,0ψ(xi,xj,0;t)
ψR
R(t) = fR
xT
dxi
xT ,0
dxj,0ψ(xi,xj,0;t)
ψR
L(t) = fL
xT
dxi
−ℓ/2
dxj,0ψ(xi,xj,0;t)
ψL
R(t) = fR
−ℓ/2
dxi
xT ,0
dxj,0ψ(xi,xj,0;t)
?xT
dxiψ(xi,xj,0;t)
ψR(t) =
xT
dxiψ(xi,xj,0,t)
?xT ,0
dxj,0ψ(xj,xj,0;t)
ψR(t) = fR
xT ,0
dxj,0ψ(xj,xj,0;t)(D3)
The prefactors fL and fR are found in Eq. (24), and
correspond to uniform distributions to the left and right
of the tagged particle according to which the surrounding
particles are initially placed. They appear when integrals
over initial coordinates are performed. Also, it is easy to
see from normalization that ψL(t) + ψR(t) = 1, ψL
ψR
R(t) = 1.
L(t) +
L(t) = 1 and ψR
R(t) + ψL

Page 12

12
If considering a single particle in a box of length ℓ,
the integrals defined in Eq. (D3) are easily interpreted
as follows. First, ψL
R) is the probability that single
particle is to the left (right) of xT at time t given that it
started, with an equal probability, anywhere to the left
(right) of xT ,0. Similar interpretations hold also for ψR
and ψL
R. The quantity ψL (ψR) is the probability that
a single particle is at position xT given that the particle
started somewhere to the left (right) of xT ,0. Finally,
ψL(ψR) is the probability that a single particle is to
the left (right) of xT at time t given that it started at
position xT ,0.
L(ψR
L
APPENDIX E: INTEGRALS OF ψ(xi,xj,0;s) IN
LAPLACE-SPACE
In this section, exact expressions as well as limiting
forms for the integrals appearing in Eq. (D3) are given
in the Laplace-domain. Using the Laplace-transformed
one-particle PDF ψ(xi,xj,0;s) found in Eq. (B3), the
integrals defined in Eq. (D3) (with time t replaced by
Laplace variable s, and xT and xT ,0left out) are given by
ψL(s) =
1
ssinh[ℓ?s/D]
sinh[(ℓ/2 + xT)?s/D]
×











×cosh[(ℓ/2 − xT ,0)?s/D], xT ≤ xT ,0
sinh[ℓ?s/D] − sinh[(ℓ/2 − xT)?s/D]
×cosh[(ℓ/2 + xT ,0)?s/D], xT ≥ xT ,0
(E1)
ψL(s) =
fL
ssinh[ℓ?s/D]
sinh[ℓ?s/D] − cosh[(ℓ/2 + xT)?s/D]
×











×sinh[(ℓ/2 − xT ,0)?s/D], xT ≤ xT ,0
cosh[(ℓ/2 − xT)?s/D]
×sinh[(ℓ/2 + xT ,0)?s/D], xT ≥ xT ,0
(E2)
ψR(s) =
fR
ssinh[ℓ?s/D]
cosh[(ℓ/2 + xT)?s/D]
×











×sinh[(ℓ/2 − xT ,0)?s/D], xT ≤ xT ,0
sinh[ℓ?s/D] − cosh[(ℓ/2 − xT)?s/D]
×sinh[(ℓ/2 + xT ,0)?s/D], xT ≥ xT ,0
(E3)
ψL
L(s) =
1
ssinh[ℓ?s/D]
×























(1 − (xT ,0− xT)fL)sinh[ℓ?s/D]
−fL
×sinh[(ℓ/2 − xT ,0)?s/D], xT ≤ xT ,0
sinh[ℓ?s/D]
−fL
×sinh[(ℓ/2 + xT ,0)?s/D], xT ≥ xT ,0
1
ssinh[ℓ?s/D]
sinh[ℓ?s/D]
−fR
×sinh[(ℓ/2 − xT ,0)?s/D], xT ≤ xT ,0
(1 − (xT − xT ,0)fR)sinh[ℓ?s/D]
−fR
×sinh[(ℓ/2 + xT ,0)?s/D], xT ≥ xT ,0.
The remaining three integrals follow from the normaliza-
tion conditions (arguments left implicit):
?
D
ssinh[(ℓ/2 + xT)?s/D]
?
D
ssinh[(ℓ/2 − xT)?s/D]
(E4)
ψR
R(s) =
×























?
D
ssinh[(ℓ/2 + xT)?s/D]
?
D
ssinh[(ℓ/2 − xT)?s/D]
(E5)
ψL+ ψR=1
s, ψL
L+ ψR
L=1
s, ψR
R+ ψL
R=1
s
(E6)
Laplace inversion of the above relationships, using e.g.
residue calculus [43], gives Eq. (29). In the following
subsections we give asymptotic results in the (1) long
and (2) short time limit for the expressions above.
1. Long-time behavior
The long-time behavior of Eqs. (E1)-(E16) is obtained
from a series expansion for ℓ?s/D ≪ 1, and reads (ar-
1
sℓ, ψL= ψL
?1
The inverse transforms are found from L−1(s−1) = 1 [44].
guments left implicit)
ψ = ψL= ψR=
L= ψL
R=1
s
?1
2+xT
ℓ
?
,
ψR= ψR
R= ψR
L=1
s2−xT
ℓ
?
,(E7)
2. Short-time behavior
Short times is defined here as (ℓ ± xT)?s/D,(ℓ ±
to diffuse across the entire box. The short time behavior
of Eqs. (E1)-(E16) is given by
2se−(xT ,0−xT)√
1
s
1 −1
xT ,0)?s/D ≫ 1, i.e. times shorter than the time it takes
ψL(s) ≈
?
1s/D,xT ≤ xT ,0
, xT ≥ xT ,0
?
2e−(xT−xT ,0)√
s/D?
(E8)

Page 13

13
ψL(s) ≈
?
fL
s
fL
?
1 −1
2e−(xT ,0−xT)√
s/D?
,xT ≤ xT ,0
xT ≥ xT ,0
2se−(xT−xT ,0)√
s/D,
(E9)
ψR(s) ≈
?
fR
2se−(xT ,0−xT)√
fR
s
1 −1
s/D,xT ≤ xT ,0
, xT ≥ xT ,0
?
2e−(xT−xT ,0)√
s/D?
(E10)
ψL
L(s) ≈












1
s
?
−fL
?
1 − (xT ,0− xT)fL
?
1 −fL
2
D
se−(xT ,0−xT)√
?
se−(xT ,0−xT)√
s/D?
,xT ≤ xT ,0
, xT ≥ xT ,0
1
s2
D
se−(xT−xT ,0)√
s/D?
(E11)
ψR
R(s) ≈






1
s
?
?
−fR
1 −fR
1 − (xT − xT ,0)fR
?
2
?
se−(xT−xT ,0)√
Ds/D?
, xT ≤ xT ,0
1
s
2
D s/D?
,xT ≥ xT ,0
(E12)
The remaining integrals follow from Eq. (E16).
Equations (E9)-(E12) can be inverted exactly into time
domain. Using standard formulas [44] and introducing
η =xT − xT ,0
√4Dt
, (E13)
leads to
ψL(t) =fL
2(1 − erfη), ψR(t) =fR
ψL(t) =1
2(1 + erfη)
2(1 + erfη), ψR(t) =1
?
π
R(t) = 1 −fR
2
2(1 − erfη)
ψL
L(t) = 1 −fL
2
4Dt
?
?
e−η2+√πη(erfη − 1)
e−η2+√πη(erfη + 1)
?
?
ψR
?
4Dt
π
(E14)
where erfz is the error-function [44]. The expressions
above are valid for times such that 4Dt/(ℓ/2±xT ,0)2≪ 1
and 4Dt/(ℓ/2 ± xT)2≪ 1. Expanding the result in Eq.
(E14) for small η and using the normalization conditions
Eq. (E16), yields
ψL(t) =fL
2
?
1 +2η
1 −2η
√π
?
, ψR(t) =1
, ψR(t) =fR
2
?
?
1 +2η
√π
?
?
ψL(t) =1
2
?
√π
?
2
1 −2η
√π
ψR
L(t) =fL
2
?
?
4Dt
π
4Dt
π
?1 −√πη?,
?1 +√πη?.ψL
R(t) =fR
2
(E15)
and (arguments left implicit)
ψL+ ψR= 1, ψL
L+ ψR
L= 1, ψR
R+ ψL
R= 1. (E16)
The limit where t → 0 limit (s → ∞) is most conveniently
found from Eqs. (E1)-(E16) which, in combination with
L−1(s−1) = 1, reads
?0, xT ≤ xT ,0
ψL(t → 0) =
1, xT ≥ xT ,0
(E17)
ψL(t → 0) =
?fL, xT ≤ xT ,0
0,xT ≥ xT ,0
(E18)
ψR(t → 0) =
?0,xT ≤ xT ,0
fR, xT ≥ xT ,0
(E19)
ψL
L(t → 0) =
?1 − (xT ,0− xT)fL, xT ≤ xT ,0
1,xT ≥ xT ,0
(E20)
ψR
R(t → 0) =
?1,xT ≤ xT ,0
1 − (xT − xT ,0)fR, xT ≥ xT ,0
(E21)
Again, the remaining integrals follow from the normal-
ization condition (E16).
APPENDIX F: MACROSCOPIC DYNAMICS -
THE DYNAMIC STRUCTURE FACTOR AND
CENTER-OF-MASS MOTION
Macroscopic quantities for a single-file system is the
same as for a system consisting of non-interacting par-
ticles [19, 27]. By “macroscopic” we here refer to any
quantity which is invariant under xi→ xj, i.e. any one
which do not ”notice” if two particle are interchanged in
the system. In this Appendix we explicitly evaluate two
such macroscopic quantities: (1) the dynamic structure
factor and (2) the PDF for the center-of-mass coordinate.
We demonstrate that indeed they agree with known re-
sults for non-interacting systems.
1.Dynamic structure factor
The dynamic structure factor is tractable via scat-
tering experiments (see e.g. Ref. [66]), and is widely
used in condensed matter physics and crystallography.
Considering a set of noise-driven stochastic trajectories
X1(t),...,XN(t), the dynamic structure factor is [67]
S(Q,t) =
1
N
?
i,j
?eiQ(Xi(t)−Xj,0)?, (F1)
where the brackets denote an average over different re-
alizations of the noise. In terms of the NPDF and the
equilibrium NPDF (see Sec. II), it is given by
S(Q,t) =
1
N
?
?
R
dx1···dxN
?
R0
dx1,0···dxN,0
×
i,j
eiQ(xi−xj,0)P(? x,t|? x0)Peq(? x0) (F2)

Page 14

14
where we averaged over the (equilibrium) initial posi-
tions.Since the integrand above is invariant under
xi ↔ xj we can apply the technique explained in Ap-
pendix C to extend the integrations over coordinates as
well as initial positions to [−ℓ/2,ℓ/2], which leads to
?ℓ/2
?ℓ/2
×
i,j
S(Q,t) =
1
N
1
N!
1
ℓN
−ℓ/2
dx1
?ℓ/2
−ℓ/2
dx2···
?ℓ/2
−ℓ/2
dxN
×
−ℓ/2
?
dx1,0
?ℓ/2
−ℓ/2
dx2,0···
?ℓ/2
−ℓ/2
dxN,0
eiQ(xi−xj,0)P(? x,t|? x0), (F3)
where also Eq. (4) was used. Inserting explicitly the
NPDF from Eq. (16) yields
S(Q,t) =
1
N
1
N!
1
ℓN
?ℓ/2
−ℓ/2
dx1
?ℓ/2
−ℓ/2
dx2···
?ℓ/2
−ℓ/2
dxN
×
?ℓ/2


×ψ(x1,x1,0;t)ψ(x2,x2,0;t)···ψ(xN,xN,0;t)
+ remaining N! − 1 terms
−ℓ/2
N
?
dx1,0
?ℓ/2
−ℓ/2
dx2,0···
?ℓ/2
−ℓ/2
dxN,0


i=1
eiQ(xi−xi,0)+
?
i,j,i?=j
eiQ(xi−xj,0)


?
.(F4)
Defining
C(Q) =
1
ℓ
?ℓ/2
?ℓ/2
×ψ(xi,xj,0;t)ψ(xk,xl,0;t)eiQ(xi−xl,0)
and noticing that all the N! terms corresponding to per-
mutations of the initial particle positions in Eq. (F4)
give the same contribution, leads to
−ℓ/2
dxi
?ℓ/2
?ℓ/2
−ℓ/2
dxj,0ψ(xi,xj,0;t)eiQ(xi−xj,0)
F(Q) =
1
ℓ2
−ℓ/2
dxi
−ℓ/2
dxj,0
?ℓ/2
−ℓ/2
dxk
?ℓ/2
−ℓ/2
dxl,0
(F5)
S(Q,t) = C(Q)C(0)N−1+ (N − 1)F(Q)C(0)N−2. (F6)
This result is identical to that of non-interacting particles
(obtained e.g. by putting, Sij≡ 0, in the Bethe Ansatz
Eq. (12) and allowing the particles to be in the full phase
space xj∈ [−ℓ/2,ℓ/2]). The expression above is simpli-
fied by noticing that C(0) = 1, due to the normalization
of ψ(xi,xj,0;t).
As an example, we consider the case ℓ → ∞ and
∆ = 0 where ψ(xi,xj,0;t) = (4πDt)−1/2exp[−(xi−
xj,0)2/(4Dt)] can be used. A straightforward calculation
of Eq. (F5) for this case gives
C(Q) = e−DQ2t
(F7)
F(Q) =
2π
ℓδ(Q)e−DQ2t, (F8)
where δ(z) = (2π)−1?∞
−∞dαeiαzwas used. Inserting the
results above into Eq. (F6) gives
S(Q,t) = e−DQ2t, Q ?= 0 (F9)
which is in agreement with standard results for non-
interacting particles in one dimension (see e.g. Ref. [68]).
2. Center-of-mass dynamics
The probability density function for the center-of-mass
(CM) coordinate X is given by
P(X,t|? x0) =
?
×P(? x,t|? x0),
where the NPDF P(? x,t|? x0) is given in Eq. (12), and the
integration region R is defined in Eq. (3). Since the inte-
grand is invariant under xi↔ xj, it is possible to extend
the integration region to xj ∈ [−ℓ/2,ℓ/2] (provided we
divide by N!), as was done in previous subsection. Also,
using the integral representation of the δ-function from
previous subsection, Eq. (F10) can be rewritten as
R
dx1dx2···dxNδ
?
X −x1+ x2+ ... + xN
N
?
(F10)
P(X,t|? x0) =
1
N!
?∞
−∞
dQ
2π
?ℓ/2
−ℓ/2
dx1···
?ℓ/2
??
−ℓ/2
dxN
×exp
?
iQ
?
X −x1+ x2+ ... + xN
N
P(? x,t|? x0).
(F11)
Combining this equation with Eqs. (12) and (13), and
defining g(kj,Q) =?ℓ/2
P(X,t|? x0) =
−∞
×e−D(k2
×g(k1,Q)···g(kN,Q).
Integrating over kjleads to
−ℓ/2dxieixi(kj−Q/N)gives
?∞
1+...+k2
?∞
dQ
2πeiQX
−∞
N)tφ(k1,x1,0)···φ(kN,xN,0)
dk1
2π···
?∞
−∞
dkN
2π
(F12)
P(X,t|? x0) =
?∞
−∞
dQ
2πeiQXh(Q,x1,0)···h(Q,xN,0),
(F13)
where
h(Q,xj0) =
?∞
?ℓ/2
×φ(kj,xj,0)eixi(kj−Q/N)
(F13) is identical to that of non-
interacting particles, as can be shown straightforwardly
−∞
dkj
2πe−Dk2
?∞
jtφ(kj,xj0)g(kj,Q)
=
−ℓ/2
dxi
−∞
dkj
2πe−Dk2
jt
(F14)
The result in Eq.

Page 15

15
by setting Sij ≡ 0 in Eq. (12) and not restricting the
particles to the phase space region R.
As an example, Eq. (F13) is evaluated for an infinite
system (ℓ → ∞) and ∆ = 0 where φ(kj,xj,0) = e−ikjxj,0.
For this case Eq. (F14) becomes
h(Q,xj,0) = exp
?
−Q2Dt
N2
−ixj,0Q
N
?
(F15)
which when inserted in Eq. (F13) and integrated over Q
gives the well known Gaussian for the CM
P(X,t|? x0) =
1
(4πDCMt)1/2exp
?
−(X − X0
4DCMt
CM)2
?
(F16)
where X0
the CM initial position, and diffusion constant, respec-
tively.
CM=
??N
i=1xi,0
?
/N and DCM= D/N denote
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can be obtained from the solution in [65] by the replace-
ments?2π
in a power series in momenta to lowest order.
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of discrete wave vectors [64]. Using the Poisson sum-
mation formula gives the identity?∞
ansatzsolution, Eqs.(12)-(15),
in term of discrete wave vectors mjπ/ℓ (mj
−∞,...,−1,0,1,...,∞) if desired.
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[46] Note that our convention, with coordinates in the range
[−L/2,L/2], is different to Ref. [37] where the left (right)
end of the system is at 0 (L).
[47] Alternatively, one may be interested in the tPDF for
a fixed initial particle distribution ? x0: ˜ ρT(xT,t|? x0) =
?
ward integration yields, using Eq. (16), a result for
0
dkj/2π →?∞
−∞dkj/2π and by expanding the
energy E(k1,...,kN) and the scattering coefficients Sij
m=−∞e2imkjℓ=
(π/ℓ)?∞
m=−∞δ(kj + mπ/ℓ), from which the Bethe
canbe rewritten
=
Rdx′
all particles start at the same position a straightfor-
1···dx′
Nδ(xT − x′
T)P(? x′,t|? x0). For the case where