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

in a one-dimensional box: Tagged particle dynamics

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Lizana, L., and T. Ambjörnsson. “Diffusion of Finite-sized Hard-core

Interacting Particles in a One-dimensional Box: Tagged Particle

Dynamics.” Physical Review E 80.5 (2009) : 051103. © 2009 The

American Physical Society

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http://dx.doi.org/10.1103/PhysRevE.80.051103

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American Physical Society

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Final published version

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http://hdl.handle.net/1721.1/65085

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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. Ambjörnsson†

Department of Theoretical Physics, Lund University, Sölvegatan 14A, SE-223 62 Lund, Sweden

and Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

共Received 3 July 2009; published 5 November 2009兲

We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles共the particles cannot pass each other兲 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␳_T共y_T, t兩y_T,0兲 that a tagged particle T 共T=1, ... ,N兲 is at position y_Tat time t given that it at time t = 0 was at position y_T,0. Using a Bethe ansatz we obtain the N-particle probability density function and, by integrating out the coordinates共and averaging over initial positions兲 of all particles but particleT, we arrive at an exact expression for␳_T共y_T, t兩y_T,0兲 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␳_T共y_T, t兩y_T,0兲 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: 共A兲 for times much smaller than the collision time tⰆ␶coll= 1/共
2_{D兲, where}
= N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion;共B兲 for times much larger than the collision time tⰇ␶collbut times smaller than the equilibrium time tⰆ␶eq= L2_{/D, we find a single-file regime where ␳}

T共yT, t兩yT,0兲 is a Gaussian with a

mean-square displacement scaling as t1/2; and 共C兲 for times longer than the equilibrium time tⰇ␶eq, ␳T共yT, t兩yT,0兲 approaches a polynomial-type equilibrium probability density function. Notably, only regimes

共A兲 and 共B兲 are found in the previously considered infinite systems.

DOI:10.1103/PhysRevE.80.051103 PACS number共s兲: 05.40.Fb, 02.50.Ey, 05.10.Gg

I. INTRODUCTION

Recent development of single fluorophore tracking tech-niques allows experimental studies of the motion of particles in cellular environments with nanometer resolution关1兴. The

cell interior represents a crowded environment, in which the motion of an individual particle is strongly affected by the presence of other particles: crowding affects, for instance, the folding of proteins, diffusional motion 关2,3兴, as well as

rates of biochemical reactions关4,5兴. Crowding is also

impor-tant during ribosomal translation on mRNA关6兴 and binding

protein diffusion along DNA关7,8兴, where bound proteins are

hindered from passing each other. Furthermore, advances in nanofluidics allow studies of geometrically constrained nanosized particles关9,10兴. The system considered in this

pa-per, the diffusion of a tagged particle immersed in a one-dimensional bath of hard-core interacting particles—in the literature referred to as single-file diffusion 共SFD兲— represents one of the simplest systems governed by crowding effects, but with possible applications for obstructed one-dimensional protein diffusion along DNA molecules and transport in nanofluidic systems.

SFD phenomena emerge in quasi-one-dimensional geom-etries. The particle order is under these circumstances

con-served over time t, which results in interesting dynamics for a tagged particle, quite different from what is predicted from classical diffusion共governed by Fick’s law兲. Examples found in nature include ion or water transport through pores in biological membranes 关11兴, one-dimensional hopping

con-ductivity 关12兴, and channeling in zeolites 关13兴. SFD effects

have also been studied in a number of experimental setups such as colloidal systems and ringlike constructions关14–18兴.

One of the most apparent characteristics of SFD is that the mean-square displacement 共MSD兲 S共t兲=具共y_T− y_T,0兲2_{典 of a}

tagged particle is in the long-time limit proportional to t1/2in an infinite system with a fixed particle concentration共angular brackets denote an average over initial positions and noise and y_T and y_T,0 are tagged particle positions at times t and

t = 0, respectively兲. Also, the conditional probability density

function 共PDF兲 for the tagged particle position 共tPDF兲 is Gaussian.

The first theoretical study showing the t1/2 law of the MSD and that the tPDF is Gaussian is in Ref. 关19兴.

Subse-quent studies, proving the MSD law in alternative ways, are found in 关20–23兴. Simple arguments to its origin are

pre-sented in关24–26兴, one of which 关26兴 uses a simple

relation-ship between the displacement of a single particle and par-ticle density fluctuations, with the latter known to be the same as for independent particles 关19,27兴. The t1/2 law and Gaussian behavior have, in the long-time limit, been shown to be of general validity for identical strongly overdamped particles interacting via any short-range potential in which *[email protected]

†_{[email protected]}

mutual passage is forbidden关28兴. A generalized central limit

theorem for tagged particle motion has also been proven 关29兴. More recent work includes 关30兴 where particles

inter-acting via a screened Coulomb potential 共i.e., not perfectly hard core兲 were studied numerically 共see also 关31兴兲, 关32兴

deals with SFD in an external potential, and关33–35兴 address

SFD dynamics with different diffusion constants. A phenom-enological Langevin formulation of SFD was presented in 关36兴.

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 obtained. The one exception is 关37兴 where the PDF for N point particles

diffusing on a finite one-dimensional line was derived. How-ever, simplified expressions for the tPDF was only consid-ered in the thermodynamic limit共N,L→⬁, where L denotes the system length and the concentration
= N/L is kept fixed兲. 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 共NPDF兲 can be written as a Bethe ansatz solution. We obtain an exact expression for the tPDF in terms of Jacobi polyno-mials共or hypergeometric functions兲, which reduces to that in 关37兴 for the case of point particles. Second, we perform a

共nonstandard兲 large-N analysis of the tPDF, keeping the sys-tem size L finite. The expression for ␳_T共y_T, t兩y_T,0兲 in the many-particle limit is presented compactly in terms of modi-fied 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: 共A兲 short times,

tⰆ␶coll= 1/共
2D兲, where␶colldenotes the collision time and

D is the diffusion constant. In this limit the tagged particle

undergoes standard Brownian motion with a MSD S共t兲⬃t; 共B兲 for intermediate times,␶collⰆtⰆ␶eq, where␶eq= L2/D is

the equilibrium time, we get a SFD regime where S共t兲 ⬃t1/2_{; 共C兲 for long times, tⰇ␶}

eq, an equilibrium tPDF of

polynomial type is found. Notably, only regimes共A兲 and 共B兲 exist in infinite systems.

This paper has the following organization. SectionII 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.IVthe initial coordinates as well as the coor-dinates 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 共A兲–共C兲 共see above兲 are identified. More technical details are given in the appendixes. A brief summary of some of our results, corroborated with Gillespie simulations 共Monte Carlo type兲, can be found in 关38兴.

II. PROBLEM DEFINITION

In this paper we consider a system of N identical hard-core interacting particles, each with a diffusion constant

D and a linear size ⌬, diffusing in a finite one-dimensional

system extending from −L/2 to L/2. A schematic cartoon is depicted in Fig. 1. The particles each have center of mass 共CM兲 and initial coordinates yជ=共y1, . . . , yN兲 and yជ0

=共y1,0, . . . , yN,0兲, respectively. Due to the hard-core

interac-tion, the particles cannot pass each other and retain their order at all times, i.e., yj+1ⱖyj+⌬ for j=1, ... ,N−1. The ends of the system are reflecting 共the particles cannot es-cape兲, i.e., y1⬎−共L−⌬兲/2 and yN⬍共L−⌬兲/2.

The diffusion of finite-sized particles can be mapped onto a point-particle problem. Introducing the rescaled effective system length

ᐉ = L − N⌬, 共1兲

and making the coordinate transformation

xj= yj− j⌬ + N + 1 2 ⌬, xj,0= yj,0− j⌬ + N + 1 2 ⌬, 共2兲 it leads to R:− ᐉ/2 ⬍ x1⬍ x2. . . ⬍ xN⬍ ᐉ/2, 共3兲

where R denotes the phase space spanned by Eq. 共3兲. The

phase space R0is also introduced for the initial coordinates

which satisfy −ᐉ/2⬍x1,0⬍x2,0. . .⬍xN,0⬍ᐉ/2. For

conve-nience, we also introduce the shorthand notations xជ =共x1, . . . , xN兲 and xជ0=共x1,0, . . . , xN,0兲. Equations 共1兲 and 共2兲

map 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 lengthᐉ.

The main quantity of interest in this study is the tPDF

␳T共xT, t兩xT,0兲 that is the probability density that a tagged par-ticleT 共T=1, ... ,N兲 is at position x_T at time t, given that it was at x_T,0 at t = 0 关an ensemble average over the initial 共equilibrium兲 distribution of the surrounding N−1 particles is implicit兴. The equilibrium tPDF is straightforwardly calcu-lated from the ergodicity principle: all points in the allowed phase spaceR are equally probable. This leads the equilib-rium NPDF

FIG. 1. 共Color online兲 Cartoon of the problem considered here:

N particles of linear size⌬ diffusing is a one-dimensional system of

length L. The particles have center-of-mass coordinates y_jand ini-tial positions yj,0 共j=1, ... ,N兲 and are unable to overtake. This

implies that y_j+1ⱖy_j+⌬ 共j=1, ... ,N−1兲 for all times. Also, the particles cannot diffuse out of the box, i.e., y1⬎−L/2+⌬/2 and

Peq_{共xជ兲 =}N!

ᐉN

兿

i=1 N−1

␪共xi+1− xi兲, 共4兲 where␪共z兲 is the Heaviside step function; ␪共z兲=1 for z⬎0 and zero elsewhere. Using an extended phase-space integra-tion technique 共see Appendix C兲 it is easy to verify that 兰_Rdx1¯dxNPeq共xជ兲=1. Integrating Eq. 共4兲 over all

coordi-nates leaving out one, x_T, gives the tPDF

␳Teq共xT兲 =

冕

R dx₁

⬘

¯ dxN

⬘

␦共xT− xT

⬘

兲Peq共xជ

⬘

兲 =N! ᐉN 1 NL! NR!

冕

_−ᐉ/2 x_T dx₁

⬘

¯

冕

−ᐉ/2 x_T dx_T−1

⬘

⫻

冕

x_T ᐉ/2 dx_T+1

⬘

¯

冕

x_T ᐉ/2 dxN

⬘

= 1 ᐉN N! NL! NR!

冉

ᐉ 2+ xT

冊

N_L

冉

ᐉ 2− xT

冊

N_R , 共5兲 where ␦共z兲 is the Dirac delta function and NL 共NR兲 is the number of particles to the left共right兲 of the tagged particle 共N=NL+ NR+ 1兲. 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␳_T共x_T, t兩x_T,0兲 one needs to first introduce the N-particle joint probability density P共xជ, t ; xជ0兲, which

gives the probability density that the system is in a state xជ

and that it initially was in a state xជ₀. The joint probability density for the tagged particle ␳_T共x_T, t ; x_T,0兲 is simply ob-tained from the joint NPDF by integration over R and R0:

␳T共xT, t ; xT,0兲=兰Rdx1

⬘

¯dxN

⬘

兰R0dx1,0

⬘

¯dxN,0

⬘

␦共xT− xT

⬘

兲␦共xT,0 − x_T,0

⬘

兲P共xជ

⬘

, t ; xជ₀

⬘

兲. Relating the conditional and joint proba-bility densities using Bayes’ rule 关39兴, i.e., P共xជ, t ; xជ0兲

=P共xជ, t兩xជ0兲Peq共xជ0兲 and ␳T共xT, t ; xT,0兲=␳T共xT, t兩xT,0兲␳eq共xT,0兲, leads to ␳T共xT,t兩xT,0兲 = 1 ␳eq_共x T,0兲

冕

R dx₁

⬘

¯ dxN

⬘

␦共xT− xT

⬘

兲 ⫻

冕

R0 dx_1,0

⬘

_{¯ dxN,0}

⬘

␦共x_T,0− x_T,0

⬘

兲 ⫻P共xជ

⬘

,t兩xជ₀

⬘

兲Peq_共xជ 0

⬘

兲, 共6兲 where␳_Teq_共x T,0兲 is given in Eq. 共5兲.

In order to get the tPDF using Eq.共6兲, we need to

calcu-late the NPDF P共xជ, t兩xជ0兲. It is governed by the diffusion

equation ⳵P共xជ,t兩xជ₀兲 ⳵t = D

冉

⳵2 ⳵x₁2+ ⳵2 ⳵x₂2+ ¯ + ⳵2 ⳵xN2

冊

P共xជ,t兩xជ0兲, 共7兲

for xជ苸R 关P共xជ, t兩xជ₀兲⬅0 outside R兴. The equation certifying that neighboring particles cannot overtake reads

D

冉

⳵

⳵xi+1 − ⳵

⳵xi

冊

P共xជ,t兩xជ0兲兩x_i+1=x_i= 0. 共8兲

Also, reflecting boundaries are placed at the system ends,

D

冏

⳵P共xជ,t兩xជ0兲 ⳵x1

冏

x₁=−ᐉ/2 = 0, 共9兲 D

冏

⳵P共xជ,t兩xជ0兲 ⳵xN

冏

x_N=ᐉ/2 = 0, 共10兲

making sure that the particles are restricted to关−ᐉ/2,ᐉ/2兴 at all times. Finally, the initial condition is

P共xជ,0兩xជ0兲 =␦共x1− x1,0兲 ¯␦共xN− xN,0兲. 共11兲

Summarizing this section, the problem of N hard-core in-teracting particles of size ⌬ diffusing in a one-dimensional system of a finite length L was mapped onto a point-particle problem using relationships共1兲 and 共2兲. The dynamics of the

NPDF is governed by Eqs. 共7兲–共10兲. Once they are solved

共topic of Sec.III兲, the tPDF can be calculated via Eq. 共6兲 共as

demonstrated in Sec.IV兲.

III. NPDF AS A COORDINATE BETHE ANSATZ

In this section we obtain the NPDF for the diffusion prob-lem 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 intro-duction by Bethe in 1931共see 关40兴 for a review兲. The Bethe

ansatz solution for the present problem reads

P共xជ,t兩xជ₀兲 =

冕

−⬁ ⬁ _dk 1 2␲

冕

_−⬁ ⬁ _dk 2 2␲¯

冕

_−⬁ ⬁ _dk N 2␲e −E共k1,. . .,kN兲t ⫻⌸j=1

N _{␾共kj,xj,0兲关e}ik₁x₁+ik2x2+ik3x3+¯+ikNxN

+ S21eik2x1+ik1x2+ik3x3+¯+ikNxN

+ S32S31eik2x1+ik3x2+k1x3+¯+ikNxN

+ all other permut. of 兵k1,k2, . . . ,kN其兴,

共12兲 where E共k1, . . . , kN兲 is the dispersion relation, Sijare the

scat-tering coefficients, and␾共kj, xj,0兲 denotes a function contain-ing boundary and initial conditions. Each one of these quan-tities is described below.

The dispersion relation has the form

E共k1, . . . ,kn兲 = D共k12+ ¯ + kN2兲, 共13兲

and relates “energy” to the momenta k1, . . . , kN. Equation

共13兲 is obtained by inserting the Bethe ansatz 共12兲 into the

equation of motion 共7兲.

The scattering coefficients Sij describe pairwise particle interactions and are in general functions of the momentum variables kiand kj, Sij= S共ki, kj兲. They are, however, indepen-dent of the initial positions of the particles. In Appendix A it is demonstrated that the scattering coefficients making sure that the particles cannot pass each other关i.e., satisfying Eq. 共8兲兴 are given by

Sij= 1, 共14兲

which means that they are independent of momenta and cor-respond to perfect reflection. For noninteracting particles

Sij= 0, which reduces the Bethe ansatz to a standard Fourier transform.

The quantity␾共kj, xj,0兲 contains information about the ini-tial and boundary conditions of the problem, which are here defined by Eqs. 共9兲–共11兲. The form of ␾共kj, xj,0兲 satisfying these relationships is given by

␾共kj,xj,0兲 = 2 cos关kj共xj,0+ᐉ/2兲兴

兺

m=−⬁

⬁

eikj共2m+1/2兲ᐉ_, 共15兲

which is shown explicitly in Appendix A. Notably, for an infinite system we have␾共kj, xj,0兲=e−ikjxj,0关_41,42兴.

It is interesting to note that for SFD systems described by Eqs.共12兲–共15兲 any macroscopic quantity that is invariant

un-der interchange of any two particle positions, xi↔xj, takes the same value as for a system of noninteracting particles. This is in marked contrast to microscopic quantities such as the tPDF which in general behave very differently for single-file and independent particle systems. In Appendix F we use the Bethe ansatz to explicitly calculate two macroscopic quantities, the dynamic structure factor and the center-of-mass PDF, and show that they agree with standard results for independent particle systems.

Integration over momenta in Eq. 共12兲 关using Eqs.

共13兲–共15兲兴 leads to the NPDF

P共xជ,t兩xជ0兲 =␺共x1,x1,0;t兲␺共x2,x2,0;t兲 ¯␺共xN,xN,0;t兲 +␺共x1,x2,0;t兲␺共x2,x1,0;t兲 ¯␺共xN,xN,0;t兲 + all other permut. of 兵x1,0,x2,0, . . . ,xN,0其,

共16兲 where ␺共xi,x_j,0;t兲 = 1 共4␲Dt兲1/2_m=−

兺

_⬁ ⬁

再

exp

冋

−共xi− xj,0+ 2mᐉ兲 2 4Dt

册

+ exp

冋

−关xi+ xj,0+共2m + 1兲ᐉ兴 2 4Dt

册

冎

共17兲

is obtained from the inverse Fourier transform 共2␲兲−1_兰

−⬁ ⬁ _dk

j␾共kj, xj,0兲e−Dkj 2

t_eikjxi_{. We point out that}

␺共xi, xj,0; t兲 is the single-particle PDF for a particle in con-fined in a box of length ᐉ.

The single-particle PDF given in Eq.共17兲 is, however, not

convenient for analyzing the long-time limit t→⬁. In order to get a more suitable expression we seek instead the eigen-mode expansion of␺共xi, xj,0; t兲, which can be done in a va-riety of ways. Here, we use Bromwich integration. The Laplace transform of Eq. 共17兲 is 共see Appendix B兲

␺共xi,xj,0;s兲 =

冕

0 ⬁ dt e−st␺共xi,xj,0;t兲 =

_冑

1 4Ds sinh共ᐉ

冑

s/D兲兵cosh关共xi+ xj,0兲

冑

s/D兴 + cosh关共ᐉ − 兩x_j,0− xi兩兲

冑

s/D兴其. 共18兲

The sought eigenvalue expansion is obtained as a sum of residues of ␺共xi, xj,0; s兲 共see, e.g., Ref. 关43兴兲 and reads

␺共xi,x_j,0;t兲 =1 ᐉ

冋

1 +m=1

兺

⬁ Gm共xi,xj,0兲Em共t兲

册

, 共19兲 where Gm共xi,xj,0兲 =␯m共+兲cos

冉

m␲xi ᐉ

冊

cos

冉

m␲xj,0 ᐉ

冊

+␯m共−兲sin

冉

m␲xi ᐉ

冊

sin

冉

m␲xj,0 ᐉ

冊

, 共20兲 Em共t兲 = e−共m␲兲2_Dt/ᐉ2 , 共21兲 ␯m共⫾兲= 1⫾ 共− 1兲m. 共22兲 Elementary trigonometric identities 关44兴 were used to bring

Gm共xi, xj,0兲 onto the form in Eq. 共20兲. Equations 共19兲–共22兲

agrees with well-known results关45兴. The single-particle PDF

共19兲 is more convenient for obtaining the long-time limit as

well as for numerical computations compared to Eq.共17兲.

In summary, the many-particle NPDF for excluding par-ticles of size⌬ diffusing in a finite interval of length L with reflecting boundaries is given by Eqs.共16兲 and 共19兲 关or Eq.

共17兲兴 combined with the mapping equations 共1兲 and 共2兲. For

point particles 共⌬=0兲 these results agree with those pre-sented in 关37兴 where a different approach was used 关46兴.

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 共6兲 by integrating

out the coordinates and initial positions of all nontagged par-ticles from the NPDF given in Eq.共16兲 关47兴. As is shown in

detail in Appendix C we can, due to the property that

P共xជ, t兩xជ0兲 is invariant under permutations of xi↔xj, extend

the integration from R to the hypercubes xj苸关−ᐉ/2,x_T兴 共j=1, ... ,T−1兲 and xj苸关xT,ᐉ/2兴 共j=T+1, ... ,N兲. A similar procedure holds for integration over R0 共initial positions兲.

Using the extended phase-space technique, Eq.共6兲 becomes

␳T共xT,t兩xT,0兲 = fL N_L fR N_R NL! NR!

冕

−ᐉ/2 x_T dx1¯

冕

−ᐉ/2 x_T dx_T−1

冕

x_T ᐉ/2 dx_T+1¯ ⫻

冕

x_T ᐉ/2 dxN

冕

−ᐉ/2 x_T,0 dx_1,0¯

冕

−ᐉ/2 x_T,0 dx_T−1,0 ⫻

冕

x_T,0 ᐉ/2 dx_T+1,0¯

冕

x_T,0 ᐉ/2 dx_N,0P共xជ,t兩xជ₀兲, 共23兲 where fL=共ᐉ/2 + x_T,0兲−1, fR=共ᐉ/2 − x_T,0兲−1. 共24兲 Using a similar combinatorial analysis to the one in Ref. 关37兴, we arrive at Eq. 共D1兲 共see Appendix D兲. The tPDF is,

however, more conveniently expressed in terms of Jacobi polynomials 关44兴, Pn共␣,␤兲共z兲. Using the identities Pn共␣,␤兲共z兲

=共n+␣兲!共n+␤兲! /关n!共n+␣+␤兲!兴关共z−1兲/2兴−a_P

n+␣

共−␣,␤兲_{共z兲 and}

␳共xT,t兩xT,0兲 =共NL_N+ NR− 1兲! L! NR! 共␺L L_兲N_L共␺ R R_兲N_R ⫻

再

共NL+ NR兲␺⌽共0,0,0;␰兲 + NL 2␺ L_␺ L ␺L L ⌽共1,0,0;␰兲 + NR 2␺ R_␺ R ␺R R ⌽共0,1,0;␰兲 + NLNR

冋

␺R_␺ L ␺L R + ␺L_␺ R ␺R L

册

⌽共0,0,1;␰兲

冎

, 共25兲 where ⌽共a,b,c;␰兲 =关NL−共a + c兲兴 ! 共NR− b兲! 关NL+ NR−共a + b + c兲兴! ␰c_{共1 −␰兲}N_L−共a+c兲 ⫻PN L−共a+c兲 共c,NR−NL+a−b兲

冉

1 +␰ 1 −␰

冊

, 共26兲 ␰=␺R L_␺ L R ␺L L_␺ R R, 共27兲

were introduced. The quantities ␺L, ␺L L

, etc. are defined in Eq. 共D3兲 and are integrals of the one-particle propagator␺

=␺共xi, xj,0; t兲 关49兴. For the general case we have ␰苸关0,1兴 关50兴. By using limiting results of ␺L

L ,␺R R , ␺L R , and ␺R L from Appendix E, one concludes that ␰→0 for short times 共t→0兲, and ␰→1 in the long-time limit 共t→⬁兲. It is also

possible using standard relations for the Jacobi polynomials 关44兴 to express ⌽共a,b,c;␰兲 as a Gauss hypergeometric

func-tion ₂F₁共␣,␤,␥; z兲,

⌽共a,b,c;␰兲 =2F1共− NL+ a,− NR+ b,−关NL+ NR兴 + a + b

+ c;1 −␰兲, 共28兲

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 inte-grals of␺共xi, x_j,0; t兲 关Eq. 共D3兲兴. Integrating Eq. 共19兲 关or

inte-grating␺共xi, x_j,0; s兲 prior to Laplace inversion 共see Appendix E兲兴 yields 共arguments are left implicit兲

␺L L =1 2+ x_T ᐉ +

冉

1 2+ x_T,0 ᐉ

冊

−1

兺

m=1 ⬁ KmEm共t兲, ␺R R =1 2 − x_T ᐉ +

冉

1 2− x_T,0 ᐉ

冊

−1

兺

m=1 ⬁ KmEm共t兲, ␺L =1 2+ x_T ᐉ +m=1

兺

⬁ Jm共x_T,x_T,0兲Em共t兲, ␺L= 1 ᐉ

冋

1 +

冉

1 2+ x_T,0 ᐉ

冊

−1

兺

m=1 ⬁ Jm共x_T,0,x_T兲Em共t兲

册

, ␺R= 1 ᐉ

冋

1 −

冉

1 2− x_T,0 l

冊

−1

兺

m=1 ⬁ Jm共x_T,0,x_T兲Em共t兲

册

, ␺R = 1 −␺L, ␺L R = 1 −␺L L , ␺R L = 1 −␺R R , 共29兲 with Km= 1 共m␲兲2

冋

␯m共+兲sin

冉

m␲x_T ᐉ

冊

sin

冉

m␲x_T,0 ᐉ

冊

+␯m共−兲cos

冉

m␲x_T ᐉ

冊

cos

冉

m␲x_T,0 ᐉ

冊

册

, 共30兲 Jm共z,z

⬘

兲 = 1 m␲

冋

␯m 共+兲_sin

冉

m␲z ᐉ

冊

cos

冉

m␲z

⬘

ᐉ

冊

−␯m共−兲cos

冉

m␲z ᐉ

冊

sin

冉

m␲z

⬘

ᐉ

冊

册

, 共31兲 whereEm共t兲 and␯m共⫾兲are given by Eqs.共21兲 and 共22兲, respec-tively. To summarize, the complete expression for the tagged particle PDF is given by Eqs. 共25兲–共27兲, 共29兲, and 共30兲

together with Eqs. 共1兲 and 共2兲 for the case of finite-sized

particles. The expressions for ␳_T共x_T, t兩x_T,0兲 can straightfor-wardly be computed numerically; our MATLAB implementa-tion is available upon request.

In the remaining part of this section we derive the tagged particle PDF ␳_T共x_T, t兩x_T,0兲 valid for a large N and 共finite兲 system size ᐉ. This large-N expansion will be used in the next section for identifying different time regimes and to obtain ␳_T共x_T, t兩x_T,0兲 for short and intermediate times. From Eq.共26兲 we note that the argument in the Jacobi polynomial,

Pn共␣,␤兲共z兲, is in the interval z苸关1,⬁兲 共since ␰苸关0,1兴兲 and that the number of particles is related to the order n. A large-N expansion of⌽共a,b,c;␰兲 therefore amounts to find a large order n expansion valid for z苸关1,⬁兲 共i.e., for all times兲 for the Jacobi polynomial. One such expansion was derived in关51兴 共see also 关52,53兴兲, and applying it to Eq. 共26兲

yields ⌽共a,b,c;␰兲 ⬇ 关N − 共a + b + c兲兴1/2_␰共2c−1兲/4_{共2␲␨兲}1/2 ⫻

冉

1 −␰ 4

冊

关N−共a+b+c兲兴/2 Ic兵关N − 共a + b + c兲兴␨其 ⫻关1 + A共␨兲兴, 共32兲 where关54兴 ␨=1 2ln

冋

1 +

冑

␰ 1 −

冑

␰

册

, 共33兲

and I_␣共z兲 is the modified Bessel function of the first kind of order␣. Stirling’s formula关44兴 was also used to approximate

factorials involving NLand NR. The correction term appear-ing in Eq.共32兲 is

A共␨兲 = B0共␨兲 N −共a + b + c兲

Ic+1兵关N − 共a + b + c兲兴␨其 Ic兵关N − 共a + b + c兲兴␨其 ,

B₀共␨兲 =1

2

再

共c

2_{− 1/4兲}

冉

1

␨− coth␨

冊

−关共␦N+ a − b兲2− 1/4兴tanh共␨兲

冎

, 共34兲 where␦N= NR− NL was introduced. The expression for A共␨兲 was found by explicitly evaluating an integral in Ref. 关51兴,

and it is a straightforward matter to show that the correction term A共␨兲 is indeed always small for all ␰苸关0,1兴 provided that 1/N, 兩␦N兩/NⰆ1. Inserting Eq. 共32兲 in Eq. 共25兲 and

us-ing the same approximations as above, we obtain our final large-N result for the tPDF,

␳T共xT,t兩xT,0兲 = 共␺L L_{兲共N−␦N−1兲/2共␺} R R_{兲共N+␦N−1兲/2} ⫻共1 −␰兲共N−1兲/2

冉

␨

冑

␰

冊

1/2

再

共1 −␰兲1/2_␺I 0关N␨兴 +N 2

冉

␺L_␺ L ␺L L + ␺R_␺ R ␺R R

冊

I0关N␨兴 +N 2

冑

␰

冉

␺R_␺ L ␺L R + ␺L_␺ R ␺R L

冊

I1关N␨兴

冎

. 共35兲

We point out that this expression, in contrast to previous asymptotic expressions关20,22,37兴, is valid for a finite box of

size ᐉ, assuming only that the number of particles is large

NⰇ1 and that the tagged particle is approximately in the

center of the system:兩␦N兩/NⰆ1.

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 ␳_T共x_T, t兩x_T,0兲 are derived. Mathemati-cally, the different cases appear due to the magnitude of N␨ 关found in the argument of the Bessel functions in Eq. 共35兲兴,

and if␨is small or large. Utilizing Eqs.共E14兲 and 共33兲, these

cases can be turned into different time regimes if introducing the collision time

␶coll=

1

2_D, 共36兲

where
= N/ᐉ is the concentration of particles, and the equi-librium time

␶eq=

ᐉ2

D. 共37兲

For a particle located roughly in the middle, 兩␦N兩/NⰆ1 关i.e., Eq. 共35兲 applies兴, the three cases are given by 共A兲

short times, N␨Ⰶ1, i.e., tⰆ␶_coll,␶_eq; 共B兲 intermediate times, ␨Ⰶ1 and N␨Ⰷ1, corresponding to␶collⰆtⰆ␶eq; and

共C兲 long times,␨Ⰷ1, i.e., tⰇ␶coll,␶eq.

Time regimes 共A兲–共C兲 are analyzed in detail below.

A. Short times, t™␶coll,␶eq

For short times we have N␨Ⰶ1 and may therefore use the approximations I_␣共z兲兩z_Ⰶ1⬇共z/2兲␣/⌫共␣+ 1兲 关44兴,␨⬇

冑

␰, and

␰Ⰶ1. In this limit, one finds that the first term in Eq. 共35兲

dominates which in combination with Eq. 共2兲 leads to

␳T共yT,t兩yT,0兲 = 共4␲Dt兲−1/2exp

冋

−共yT − y_T,0兲2

4Dt

册

, 共38兲 for which the MSD is

S共t兲 = 2Dt. 共39兲

In the short-time regime, almost no collisions with the neigh-boring particles 共nor the box walls兲 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,␶coll™ t ™␶eq

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 ␨Ⰶ1 but N␨ Ⰷ1, we get the tPDF as follows. First, the first term in Eq. 共35兲 is neglected 共this is checked at the end of the

calcula-tion兲. Second, the Bessel function is approximated with

I_␣共z兲兩z_Ⰷ1⬇ez/

冑

2␲z. A straightforward expansion of Eq.共35兲

for ␺L R

, ␺R

L_{Ⰶ1 共i.e.,}

冑

_{␰Ⰶ1兲, together with Stirling’s} for-mula, gives ␳T共yT,t兩yT,0兲 ⬇ 1 2e −␦N共␺RL−␺LR兲

冑

N 2␲共␺R L_␺ L R_兲−1/4 e−共N/2兲共

冑

␺RL−

冑

␺LR兲2 ⫻

冋

␺L_␺ L+␺ R_␺ R+␺ R_␺ L

冉

␺R L ␺L R

冊

1/2 +␺L␺R

冉

␺L R ␺R L

冊

1/2

册

. 共40兲

If we furthermore assume that the average of the absolute value of␩=共x_T− x_T,0兲/

冑

4Dt is small共which is checked after the calculation兲, Eq. 共E15兲 may be used which in

combina-tion with Eqs. 共1兲, 共2兲, and 共40兲, keeping only lowest order

terms in␩, leads to the SFD result,

␳T共yT,t兩yT,0兲 = 1

冑

2␲

冢

1 4Dt ␲

冉

1 −
⌬

冊

2

冣

1/4 ⫻exp

冤

− 共yT− yT,0兲 2 2

冑

4Dt ␲

冉

1 −
⌬

冊

2

冥

, 共41兲 where the MSD is S共t兲 =1 −
⌬

冑

4Dt ␲ . 共42兲

Equation 共42兲 justifies the assumption that the expectation

value of 兩␩兩 is a small number. Also, comparing the magni-tude of the first term in Eq. 共35兲 with respect to the second

and the third shows indeed that our first assumption above was correct 关55兴. For point particles ⌬=0, Eq. 共41兲 agrees

with standard results关20,37兴, in which N,L→⬁ while

keep-ing the concentration
fixed. The result above shows that SFD behavior appears also in a finite system with reflecting ends, as an intermediate regime共for a particle roughly in the middle兲. In addition, note that the simple rescaling

→
/共1−
⌬兲 takes us from previous point-particle results to

those of finite-sized particles.

C. Long times, tš␶eq

In the long-time limit we have␨Ⰷ1, for which the tPDF can be obtained exactly for arbitrary N. Using the exact ex-pression for ␳_T共x_T, t兩x_T,0兲 found in Eqs. 共25兲 and 共28兲,

to-gether with ₂F1共␣,␤,␥; z = 0兲=1 关44兴 and Eq. 共E7兲 关also

us-ing Eqs. 共1兲 and 共2兲兴 gives

␳Teq共yT兲 = 1 共L − N⌬兲N 共NL+ NR+ 1兲! NL! NR! ⫻

冉

L 2+ yT−⌬共1/2 +NL兲

冊

N_L

冉

L 2 − yT−⌬共1/2+NR兲

冊

N_R 共43兲 where␳_Teq共y_T兲=␳_T共y_T, t→⬁兩y_T,0兲. This equation agrees with the equilibrium tPDF given in Eq.共5兲 共for ⌬=0兲, as it should

关56兴. The results above show the consistency of our analysis

and illustrate the ergodicity of the finite SFD system. Calcu-lating the second moment of the equilibrium tPDF, S共t

→⬁兲=Seq, gives Seq=

冉

1 4

冊

N_R+1

冉

L − N⌬ 2

冊

2 _{⌫共1/2兲⌫共2关NR+ 1兴兲} ⌫共NR+ 1兲⌫共NR+ 5/2兲, 共44兲 for NL= NRwhere⌫共z兲 is the gamma function. For NⰇ1 we can simplify the expression for the equilibrium tPDF as well as S_eq. If a symmetric file is assumed, i.e., NL= NR and T =共N+1兲/2, an asymptotic expansion of ␳_Teq_共y

T兲 共using Stirling’s approximation关44兴兲 gives the Gaussian PDF

␳Teq共yT兲 ⬇

冑

2N_{␲ 共L − N⌬兲}1 2exp

冋

− 2N

冉

y_T L − N⌬

冊

2

册

, 共45兲 from which we read off that

Seq⬇

共L − N⌬兲2

4N . 共46兲

Equation共46兲 can also be found directly from a large-N

ex-pansion of Eq. 共44兲 using Stirling’s formula and assuming

N⬇2NR. We point out that␳_Teq共y_T兲→0 for N,L→⬁ even if

the concentration
= N/L is kept fixed. This is consistent with the long-time limit of Eq. 共41兲, which indeed goes to

zero for large times. Finally, the analysis in this subsection also gives an estimate on the time required to reach equilib-rium, namely, tⰇ␶eq.

VI. CONCLUSIONS AND OUTLOOK

In this study we have solved exactly a nonequilibrium statistical-mechanics problem: diffusion of N hard-core inter-acting 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␳_T共y_T, t兩y_T,0兲 共denoted as tPDF兲 that a tagged particle T is at position yTat time t given that it at time t = 0 was at position y_T,0. We derived the tPDF by first finding the N-particle probability density function 共NPDF兲 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 expression for

␳T共yT, t兩yT,0兲 is found in Eqs. 共1兲, 共2兲, 共25兲, and 共26兲 and constitutes the main result of the paper. For a large number of particles and for a tagged particle located roughly in the middle of the system, an asymptotic expansion of the tPDF was derived关see Eq. 共35兲兴. Based on this equation, we found

three time regimes of interest:共A兲 for short times, i.e., times much smaller than the collision time tⰆ␶_coll= 1/共
2_D兲,

where
= N/L is the particle number concentration, the tPDF coincides with the Gaussian probability density function that characterizes a free particle 关Eq. 共38兲兴. 共B兲 For intermediate

times tⰇ␶coll, but much smaller than the equilibrium time t

Ⰶ␶eq= L2/D, a subdiffusive single-file regime was found in

which the tPDF is a Gaussian with an associated MSD pro-portional to t1/2关Eq. 共41兲兴. 共C兲 For times exceeding the

equi-librium time tⰇ␶eq, the tPDF approaches a probability

den-sity of polynomial type关Eq. 共43兲兴.

We point out that the subdiffusive behavior for a tagged particle in time regime共B兲 is of fractional Brownian motion type 关33,57,58兴 rather than that of continuous-time random

walks 共CTRWs兲 characterized by heavy-tailed waiting time densities关59–61兴. For such CTRW processes the probability

density function is not a Gaussian as for the current system. Further comparison between subdiffusion in single-file sys-tems and that occurring in CTRW theory was pursued nu-merically in 关62兴.

The Bethe ansatz is often employed in quantum mechan-ics when many-body systems are studied共e.g., quantum spin chains兲 关40,63兴, and also for stochastic many-particle lattice

problems关64,65兴. We hope that the theoretical analysis based

on the Bethe ansatz presented here will stimulate further progress in the field of single-file diffusion and that of inter-acting random walkers. For instance, it would be interesting to see whether our analysis could be extended to derive exact results also for particles interacting through potentials other than hard-core type 关31兴, and for particles having different

diffusion constants 关34兴.

From the applied point of view, our exact expression for the tPDF covers all time regimes and is straightforward to implement for numerical computations. Therefore, we be-lieve that our explicit formula for␳_T共y_T, t兩y_T,0兲, as well as the approximate results for regimes 共A兲–共C兲, will be useful for experimentalists 共see, for instance, 关18兴兲 seeking to extract

system parameters such as the particle size⌬, the system size

L, the particle’s diffusion constants共D兲, and the number 共NL

and NR兲 of particles to the left and right of the tagged par-ticle.

We finally note that the use of fluorescently labeled 共tagged兲 particles is of much use for studying biological sys-tems. The understanding of how the motion of such particles correlates with its environment is therefore the key for grasp-ing the behavior of such systems in a quantitative fashion.

ACKNOWLEDGMENTS

We are grateful to Bob Silbey, Mehran Kardar, Eli Barkai, Ophir Flomenbom, and Michael Lomholt for valuable dis-cussions. L.L. acknowledges support from the Danish Na-tional Research Foundation, and T.A. from the Knut and Al-ice Wallenberg Foundation.

APPENDIX A: BETHE ANSATZ

In this section we show that the Bethe ansatz, Eq.共12兲, is

a solution to the problem defined by Eqs.共7兲–共11兲. First, it is

demonstrated that Eq.共12兲 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 dem-onstrated that Eq.共12兲 also satisfies the initial condition.

1. Boundary conditions at the ends of the box

In this subsection it is proven that Eq. 共12兲 satisfies the

reflecting boundary conditions共9兲 and 共10兲 at ⫾ᐉ/2 with an

appropriate choice for ␾共kj, xj,0兲 关Eq. 共15兲兴. The scattering

coefficients are set to Sij= 1, which is proven to be correct in the following subsection. First, we define the function

␭共k,z兲 =␾共k,z兲e−ikᐉ/2_{= 2 cos关k共z + ᐉ/2兲兴}

兺

m=−⬁ ⬁

e−2ikmᐉ, 共A1兲 which has the symmetry relation

␭共k,z兲 = ␭共− k,z兲. 共A2兲 By taking the derivative of Eq. 共12兲 with respect to x1 and

evaluating at the left boundary x1= −ᐉ/2 give

冏

⳵P共xជ,t兩xជ₀兲 ⳵x1

冏

x₁=−ᐉ/2 =

冕

−⬁ ⬁ _dk 1 2␲¯

冕

−⬁ ⬁ _dk N 2␲e −D共k1 2 +¯+kN 2_兲t ⌸j=1 N _{␾共kj,xj,0兲关ik}

1eik1x1共eik2x2+¯+ikNxN+ perm. of 兵k2, . . . ,kN其兲

+ ¯ + ikieikix1共eik2x2+¯+ik1xi+¯+ikNxN_{+ perm. of} 兵k

2, . . . ,ki−1,ki+1, . . . ,kN其兲 + ¯ + ikNeikNx1共eik2x2+¯+ik1xN + perm. of 兵k1, . . . ,kN−1其兲兴x₁=−ᐉ/2 =

冕

−⬁ ⬁ _dk 2 2␲¯

冕

_−⬁ ⬁ _dk N 2␲e −D共k2 2 +¯+kN 2 兲t_⌸ j=2 N ␾共kj,xj,0兲

冕

−⬁ ⬁ _dk 1 2␲共ik1兲e −Dk₁2t_␭共k 1,x1,0兲 + ¯ +

冕

−⬁ ⬁ _dk 1 2␲¯

冕

−⬁ ⬁ _dk i−1 2␲

冕

−⬁ ⬁ _dk i+1 2␲ ¯

冕

−⬁ ⬁ _dk N 2␲e −D共k1 2 +¯+ki−1 2 +k_i+12 +¯+kN 2_兲t ⌸j=1,j⫽i N _{␾共kj,xj,0兲} ⫻

冕

−⬁ ⬁ _dk i 2␲共iki兲e −Dk_i2t_{␭共ki,xi,0兲 + ¯ +}

冕

−⬁ ⬁ _dk 1 2␲¯

冕

_−⬁ ⬁ _dk N−1 2␲ e −D共k1 2 +¯kN−1 2 _兲t ⌸j=1 N−1_{␾共kj,xj,0兲} ⫻

冕

−⬁ ⬁ _dk N 2␲共ikN兲e −Dk_N2t_␭共kN ,xN,0兲 = 0, 共A3兲 where 兰₋⬁_⬁dkikie−Dki 2

t_{␭共ki, xi,0兲=0 关odd integrand; see Eq.} 共A2兲兴 was used in the last step. Note that this calculation

does not rely on any specific form of ␾共kj, xj,0兲. It is only required that the symmetry relation共A2兲 holds. Furthermore,

the dispersion relation E共kជ兲=D共k1 2

+¯+kN2兲 was also used in the above derivation. However, it would work equally well for any dispersion relation as long as E共kជ兲=兺iE共ki兲 with E共ki兲=E共−ki兲 is valid.

A similar analysis as just presented shows that the Bethe ansatz solution also satisfies the reflecting condition at +ᐉ/2 关Eq. 共10兲兴. In fact, since the Bethe ansatz is

in-variant under the coordinate transformation xi↔xj, it gives 关⳵P共xជ, t兩xជ₀兲/⳵xj兴x

j=⫾ᐉ/2= 0 for all xj.

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 关Eq. 共8兲兴 is satisfied

for scattering coefficients given by Sij= 1 in the Bethe ansatz solution 共12兲. We start off by expressing P共xជ, t兩xជ0兲 in two

P共xជ,t兩xជ₀兲 =

冕

−⬁ ⬁ _dk 1 2␲

冕

_−⬁ ⬁ _dk 2 2␲¯

冕

_−⬁ ⬁ _dk N 2␲e −D共k1 2 +¯+kN 2 兲t_⌸j=1N _␾共kj

,xj,0兲关eik₁x_j共eik_jx₁+¯+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+¯+ikNxN

+ all other permut. of 兵k₂, . . . ,kN其兲 + eik₂x_j共eik₁x₁+ikjx2+¯+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+¯+ikNxN

+ all other permut. of 兵k1,k3, . . . ,kN其兲 + ¯ + eikNxj共eik1x1+¯+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+¯+ikjxN

+ all other permut. of 兵k1, . . . ,kN−1其兲兴 共A4兲

and P共xជ,t兩xជ0兲 =

冕

−⬁ ⬁ _dk 1 2␲

冕

_−⬁ ⬁ _dk 2 2␲¯

冕

_−⬁ ⬁ _dk N 2␲e −D共k1 2 +¯+kN 2 兲t_⌸ j=1

N _{␾共kj,xj,0兲关e}ik₁x_j+1共eik_j+1x₁+¯+ikj−1xj−1+ikjxj+ikj+2xj+2+¯+ikNxN

+ all other permut. of 兵k2, . . . ,kN其兲 + eik2xj+1共eik1x1+ikj+1x2+¯+ikj−1xj−1+ikjxj+ikj+2xj+2+¯+ikNxN

+ all other permut. of 兵k1,k3, . . . ,kN其兲 + ¯ + eikNxj+1共eik1x1+¯+ikj−1xj−1+ikjxj+ikj+2xj+2+¯+ikj+1xN

+ all other permut. of 兵k1, . . . ,kN−1其兲兴. 共A5兲

Using the above equations one finds

冏

冉

⳵P共xជ,t兩xជ0兲 ⳵xj+1 −⳵P共xជ,t兩xជ0兲 ⳵xj

冊

冏

x_j+1=xj =

冕

−⬁ ⬁ _dk 1 2␲

冕

_−⬁ ⬁ _dk 2 2␲¯

冕

_−⬁ ⬁ _dk N 2␲e −D共k1 2 ,. . .,k_N2兲t_⌸ j=1 N _{␾共kj,xj,0兲} ⫻关ik1eik1xj共eikj+1x1+¯+ikj−1xj−1+ikjxj+1+ikj+2xj+2+¯+ikNxN

− eik_jx₁+¯+ikj−1xj−1+ikj+1xj+1+ikj+2xj+2+¯+ikNxN_{+ all other permut. of} 兵k

2, . . . ,kN其兲

+ ik2eik2xj共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} 兵k

1,k3, . . . ,kN其兲

+ ¯ + ikNeikNxj共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} 兵k

1, . . . ,kN−1其兲兴 = 0,

共A6兲

where it was used that each parenthesis is identically zero due to the cancellation of the 2共N−1兲! terms after permuta-tion over all allowed momenta. Note that this derivapermuta-tion is independent of the choice of ␾共kj, xj,0兲 and E共kជ兲.

3. Initial condition

In this subsection we show that the Bethe ansatz 共12兲

agrees with the initial condition共11兲 when t→0. By defining

⍀共x,y兲 =

冕

−⬁ ⬁ _dk j 2␲e ik_jx_{␾共kj,y兲, 共A7兲} Eq. 共12兲 reads P共xជ,t→ 0兩xជ0兲 = ⍀共x1,x1,0兲⍀共x2,x2,0兲 ¯ ⍀共xN,xN,0兲 +⍀共x1,x2,0兲⍀共x2,x1,0兲 ¯ ⍀共xN,xN,0兲

+ all other permut. of 兵x1,0, . . . ,xN,0其. 共A8兲

Using the explicit expression for ␾共k,y兲 found in Eq. 共15兲

and that␦共x−z兲=共2␲兲−1兰_−⬁⬁ dk eik共x−z兲 leads to

⍀共x,y兲 =

兺

m=−⬁

⬁

␦共x − y + 2mᐉ兲 +␦共x + y + 关2m + 1兴ᐉ兲. 共A9兲 For all m⫽0 the ␦ functions are nonzero for coordinates lying outside ofR and R₀ where, per definition,P共xជ, t兩xជ₀兲 ⬅0 共see discussion in Sec.II兲. For m=0, it is only the first␦

function in the sum, ␦共x−y兲, that contributes to the NPDF. Furthermore, since all terms except the first one in Eq.共A8兲

are zero due to the fact that P共xជ, t兩xជ0兲=0 outside R, one

obtains

P共xជ,t→ 0兩xជ0兲 =␦共x − x0兲 ¯␦共xN− xN,0兲 共A10兲

as t→0, which is the desired result.

APPENDIX B: LAPLACE TRANSFORM OF␺(x_i, x_j,0; t)

In this section it is shown explicitly how one can go from the one-particle PDF ␺共xi, xj,0; t兲 for a particle in a box expressed in terms of Gaussians 关Eq. 共17兲兴, to the

eigen-mode expansion 关Eqs. 共19兲–共22兲兴 used in this paper. First,

兺m=−⬁ ⬁fm= fm=0+兺m=1⬁ 关fm+ f−m兴. Then, using the Laplace

transform L关共␲t兲−1/2e−a/共4t兲兴=s−1/2e−a冑s _{共a⬎0兲 关44兴 one} finds ␺共xi,xj,0;s兲 = Q共s兲 + e−共xi+xj,0+ᐉ兲冑s/D

冑

4Ds +

冋

e −共xi−xj,0兲冑s/D₊关_e共xi−xj,0兲冑s/D兴

冑

4Ds +e −共xi+xj,0+ᐉ兲冑s/D_{+ e}共xi+xj,0+ᐉ兲冑s/D

冑

4Ds

册

m=1

兺

⬁ e−2mᐉ冑s/D, 共B1兲 where Q共s兲 =

再

共4Ds兲−1/2e−共xi −xj,0兲冑s/D_{, xi}ⱖ xj,0 共4Ds兲−1/2_e−共xj,0−xi兲冑s/D_{, xi}ⱕ xj,0_.

冎

共B2兲

Considering the cases xi⬎xj,0 and xi⬍xj,0 separately, and that 兺_m=1⬁ e−2mᐉ冑s/D=共e2ᐉ冑s/D− 1兲−1, leads to

␺共xi,xj,0;s兲 =

1

冑

sD sinh关ᐉ

冑

s/D兴

再

cosh关共ᐉ/2 + xi兲

冑

s/D兴cosh关共ᐉ/2 − xj,0兲

冑

s/D兴, xiⱕ xj0

cosh关共ᐉ/2 − xi兲

冑

s/D兴cosh关共ᐉ/2 + xj,0兲

冑

s/D兴, xiⱖ xj,0,

冎

共B3兲 which after elementary trigonometric manipulations result in

Eq. 共18兲.

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共xT兲 =

冕

R\xT

dx₁

⬘

¯ dxT−1

⬘

dxT+1

⬘

¯ dxN

⬘

P共xជ

⬘

,t兩xជ0兲 共C1兲

over the region R 关Eq. 共3兲兴 with the tagged particle

coordi-nate x_Tleft out. As pointed previously in the paper, the Bethe ansatz solution关Eq. 共12兲兴 is symmetric under the

transforma-tion xi↔xjwhen all scattering coefficients are given by Sij = 1. This allows the integration ofP共xជ, t兩xជ0兲 in Eq. 共C1兲 to be

extended to the whole hyperspace xj苸关−ᐉ/2,x_T兴, j = 1 , . . . ,T−1 and xj苸关xT,ᐉ/2兴, 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 3 is taggedT=3,

I共x3兲 =

冕

−ᐉ/2⬍x1⬍x2⬍x3⬍ᐉ/2

dx1dx2P共x1,x2,x3,t兩xជ0兲.

共C2兲

The integration area in the共x1, x2兲 plane is sketched in Fig.2

共upper dark triangle兲. Since P共xជ, t兩xជ0兲 is invariant under

x1↔x2, integration over the lower triangle gives the same

result, i.e.,

I共x3兲 =

冕

−ᐉ/2⬍x2⬍x1⬍x3⬍ᐉ/2

dx1dx2P共x1,x2,x3,t兩xជ0兲.

共C3兲 If Eqs.共C2兲 and 共C3兲 are added, I共x3兲 can be expressed as an

integral over the full rectangle

I共x3兲 = 1 2

冕

_−ᐉ/2 x_T dx1

冕

−ᐉ/2 x_T dx2P共x1,x2,x3,t兩xជ0兲. 共C4兲

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 −ᐉ/2⬍x₁⬍ ... ⬍x_T−1⬍x_T to −ᐉ/2⬍xj⬍x_T for j = 1 , . . . ,T − 1 provided that we divide by NL!. This holds also for NR particles to the right and Eq.共C1兲 can in general be written

as 1

x

x

2

x

1

=

x

2

x

1

x > x

2

x

1

<

2

FIG. 2. Integration area共the darker upper area, x1⬍x2兲 for the integral defined in Eq.共C2兲. For any function which is symmetric

under x₁↔x₂the integration over the lower triangle共x1⬎x2兲 yields the same result as integration over the same function over the upper triangle共x1⬍x2兲. One may therefore extend the integration area to the full rectangle above provided one divides the corresponding result by 2.

I共xT兲 = 1 NL! NR!

冕

−ᐉ/2 x_T dx₁

冕

−ᐉ/2 x_T dx₂¯ ⫻

冕

−ᐉ/2 x_T dx_T−1

冕

x_T ᐉ/2 dx_T+1¯

冕

x_T ᐉ/2 dxNP共xជ,t兩xជ0兲. 共C5兲 Since P共xជ, t兩xជ0兲 is also invariant under xi,0↔xj,0, a similar

extended phase-space technique is valid for integrations over the initial particle positionsR0.

APPENDIX D: FROM THE NPDF TO THE tPDF

In this appendix the tPDF is obtained from integral 共6兲

using the Bethe ansatz NPDF共16兲 explicitly. A similar

com-binatorial analysis to that found in 关37兴 gives

␳共xT,t兩xT,0兲 = 1 NL! NR!

再

兺

q=0 min兵NL,NR其 H0共q兲共␺L L_兲N_L−q_共␺ R L_兲q_␺共␺ L R_兲q_共␺ R R_兲N_R−q₊

兺

q=0 min兵NL−1,NR其 HL L_{共q兲共␺} L L_兲N_L−q−1_共␺ R L_兲q_␺L_␺L共␺ L R_兲q_共␺ R R_兲N_R−q +

兺

q=0 min兵NL,NR−1其 HR R_{共q兲共␺} L L_兲N_L−q_共␺ R L_兲q_␺R_␺R共␺ L R_兲q_共␺ R R_兲N_R−q−1₊

兺

q=0 min兵NL−1,NR−1其 HL R_共q兲 ⫻共␺L L_兲N_L−q−1_共␺ R L_兲q+1_␺R_␺L共␺ L R_兲q_共␺ R R_兲N_R−q−1₊

兺

q=0 min兵NL−1,NR−1其 HR L_{共q兲共␺} L L_兲N_L−q−1_共␺ R L_兲q_␺L_␺R共␺ L R_兲q+1_共␺ R R_兲N_R−q−1

冎

_{, 共D1兲}

with the combinatorial factors

H0_{共q兲 =}

冉

NL q

冊冉

NR q

冊

NL! NR! , HL L_{共q兲 =}

冉

NL− 1 q

冊冉

NR q

冊

NL! NR! NL, HR R_{共q兲 =}

冉

NL q

冊冉

NR− 1 q

冊

NL! NR! NR, HL R_{共q兲 =}

冉

NL− 1 q

冊冉

NR q + 1

冊

NL! NR! NL, HR L_{共q兲 =}

冉

NL q + 1

冊冉

NR− 1 q

冊

NL! NR! NR, 共D2兲

and integrals共leaving arguments x_T,0 and x_Timplicit兲

␺L L_{共t兲 = fL}

冕

−ᐉ/2 x_T dxi

冕

−ᐉ/2 x_T,0 dxj,0␺共xi,xj,0;t兲, ␺R R_{共t兲 = fR}

冕

x_T ᐉ/2 dxi

冕

x_T,0 ᐉ/2 dx_j,0␺共xi,x_j,0;t兲, ␺L R_{共t兲 = fL}

冕

x_T ᐉ/2 dxi

冕

−ᐉ/2 x_T,0 dxj,0␺共xi,xj,0;t兲, ␺R L_{共t兲 = fR}

冕

−ᐉ/2 x_T dxi

冕

x_T,0 ᐉ/2 dxj,0␺共xi,xj,0;t兲 ␺L_{共t兲 =}

冕

−ᐉ/2 x_T dxi␺共xi,xj,0;t兲, ␺R_{共t兲 =}

冕

x_T ᐉ/2 dxi␺共xi,xj,0,t兲, ␺L共t兲 = fL

冕

−ᐉ/2 x_T,0 dx_j,0␺共xj,x_j,0;t兲, ␺R共t兲 = fR

冕

x_T,0 ᐉ/2 dxj,0␺共xj,xj,0;t兲. 共D3兲

The prefactors fL and fR are found in Eq. 共24兲 and corre-spond 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 nor-malization that ␺L_共t兲+␺R_{共t兲=1, ␺} L L_共t兲+␺ L R_{共t兲=1, and ␺} R R_共t兲 +␺R L_共t兲=1.

If considering a single particle in a box of length ᐉ, the integrals defined in Eq. 共D3兲 are easily interpreted as

fol-lows. First,␺L L_共␺

R

R_{兲 is the probability that a single particle is} to the left共right兲 of x_Tat time t given that it started, with an equal probability, anywhere to the left共right兲 of x_T,0. Similar interpretations hold also for␺_LRand␺_RL. The quantity␺L共␺R兲 is the probability that a single particle is at position x_Tgiven that the particle started somewhere to the left共right兲 of x_T,0. Finally, ␺L_共␺R_{兲 is the probability that a single particle is to} the left共right兲 of x_Tat time t given that it started at position

APPENDIX E: INTEGRALS OF␺(xi, xj,0; s)

IN LAPLACE SPACE

In this appendix, exact expressions as well as limiting forms for the integrals appearing in Eq.共D3兲 are given in the

Laplace domain. Using the Laplace-transformed one-particle PDF ␺共xi, x_j,0; s兲 found in Eq. 共B3兲, the integrals defined in

Eq. 共D3兲 共with time t replaced with Laplace variable s, and

x_Tand x_T,0left out兲 are given by

␺L_{共s兲 =} 1

s sinh关ᐉ

冑

s/D兴

再

sinh关共ᐉ/2 + x_T兲

冑

s/D兴cosh关共ᐉ/2 − x_T,0兲

冑

s/D兴, x_Tⱕ x_T,0

sinh关ᐉ

冑

s/D兴 − sinh关共ᐉ/2 − x_T兲

冑

s/D兴cosh关共ᐉ/2 + x_T,0兲

冑

s/D兴, xTⱖ xT,0,

冎

共E1兲

␺L共s兲 = fL

s sinh关ᐉ

冑

s/D兴

再

sinh关ᐉ

冑

s/D兴 − cosh关共ᐉ/2 + x_T兲

冑

s/D兴sinh关共ᐉ/2 − x_T,0兲

冑

s/D兴, x_Tⱕ x_T,0

cosh关共ᐉ/2 − xT兲

冑

s/D兴sinh关共ᐉ/2 + xT,0兲

冑

s/D兴, xTⱖ xT,0,

冎

共E2兲

␺R共s兲 = fR

s sinh关ᐉ

冑

s/D兴

再

cosh关共ᐉ/2 + x_T兲

冑

s/D兴sinh关共ᐉ/2 − x_T,0兲

冑

s/D兴, x_Tⱕ x_T,0

sinh关ᐉ

冑

s/D兴 − cosh关共ᐉ/2 − x_T兲

冑

s/D兴sinh关共ᐉ/2 + x_T,0兲

冑

s/D兴, x_Tⱖ x_T,0,

冎

共E3兲

␺L L_{共s兲 =} 1 s sinh关ᐉ

冑

s/D兴

冦

关1 − 共xT,0− xT兲fL兴sinh关ᐉ

冑

s/D兴 − fL

冑

D s sinh关共ᐉ/2 + xT兲

冑

s/D兴sinh关共ᐉ/2 − xT,0兲

冑

s/D兴, xTⱕ xT,0 sinh关ᐉ

冑

s/D兴 − fL

冑

D s sinh关共ᐉ/2 − xT兲

冑

s/D兴sinh关共ᐉ/2 + xT,0兲

冑

s/D兴, xTⱖ xT,0,

冧

共E4兲 ␺R R_{共s兲 =} 1 s sinh关ᐉ

冑

s/D兴

冦

sinh关ᐉ

冑

s/D兴 − fR

冑

D s sinh关共ᐉ/2 + xT兲

冑

s/D兴sinh关共ᐉ/2 − xT,0兲

冑

s/D兴, xTⱕ xT,0

关1 − 共xT− xT,0兲fR兴sinh关ᐉ

冑

s/D兴 − fR

冑

D_s sinh关共ᐉ/2 − xT兲

冑

s/D兴sinh关共ᐉ/2 + xT,0兲

冑

s/D兴, xTⱖ xT,0.

冧

共E5兲

The remaining three integrals follow from the normalization conditions 共arguments are left implicit兲

␺L +␺R=1 s, ␺L L +␺L R =1 s, ␺R R +␺R L =1 s. 共E6兲

The Laplace inversion of the above relationships, using, e.g., residue calculus 关43兴, gives Eq. 共29兲. In the following

sub-sections we give asymptotic results in the共1兲 long- and 共2兲 short-time limits for the expressions above.

1. Long-time behavior

The long-time behavior of Eqs. 共E1兲–共E16兲 is obtained

from a series expansion forᐉ

冑

s/DⰆ1 and reads 共arguments

are left implicit兲

␺=␺L=␺R= 1 sᐉ, ␺ L =␺L L =␺R L =1 s

冉

1 2+ x_T ᐉ

冊

, ␺R =␺R R =␺L R =1 s

冉

1 2− x_T ᐉ

冊

. 共E7兲

The inverse transforms are found fromL−1_共s−1_{兲=1 关44兴.}

2. Short-time behavior

Short times is defined here as

共ᐉ⫾x_T兲

冑

s/D, 共ᐉ⫾x_T,0兲

冑

s/DⰇ1, i.e., times shorter than the

time it takes to diffuse across the entire box. The short-time behavior of Eqs.共E1兲–共E16兲 is given by