Diffusion of finite-sized hard-core interacting particles in a
one-dimensional box: Tagged particle dynamics
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Lizana, L., and T. Ambjörnsson. “Diffusion of Finite-sized Hardcore 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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Detailed Terms
PHYSICAL REVIEW E 80, 051103 共2009兲
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 T at
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
particle T, 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 t Ⰶ ␶coll = 1 / 共2D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 t Ⰷ ␶coll but times smaller than
the equilibrium time t Ⰶ ␶eq = L2 / D, we find a single-file regime where ␳T共y T , t 兩 y T,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 t Ⰷ ␶eq,
␳T共y T , t 兩 y T,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 techniques 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 important 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 paper, the diffusion of a tagged particle immersed in a onedimensional 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 onedimensional protein diffusion along DNA molecules and
transport in nanofluidic systems.
SFD phenomena emerge in quasi-one-dimensional geometries. The particle order is under these circumstances con-
*lizana@nbi.dk
†
tobias.ambjornsson@thep.lu.se
1539-3755/2009/80共5兲/051103共16兲
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 conductivity 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 t1/2 in
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. Subsequent studies, proving the MSD law in alternative ways, are
found in 关20–23兴. Simple arguments to its origin are presented in 关24–26兴, one of which 关26兴 uses a simple relationship between the displacement of a single particle and particle 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
051103-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
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 interacting 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 phenomenological 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. However, simplified expressions for the tPDF was only considered 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 polynomials 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 system 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 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: 共A兲 short times,
t Ⰶ ␶coll = 1 / 共2D兲, where ␶coll denotes 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, ␶coll Ⰶ t Ⰶ ␶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, t Ⰷ ␶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. Section II contains 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 coordinates 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 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 hardcore interacting particles, each with a diffusion constant
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 j and initial positions y j,0 共j = 1 , . . . , N兲 and are unable to overtake. This
implies that y j+1 ⱖ y j + ⌬ 共j = 1 , . . . , N − 1兲 for all times. Also, the
particles cannot diffuse out of the box, i.e., y 1 ⬎ −L / 2 + ⌬ / 2 and
y N ⬍ L / 2 − ⌬ / 2.
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y 1 , . . . , y N兲 and yជ 0
= 共y 1,0 , . . . , y N,0兲, respectively. Due to the hard-core interaction, the particles cannot pass each other and retain their
order at all times, i.e., y j+1 ⱖ y j + ⌬ for j = 1 , . . . , N − 1. The
ends of the system are reflecting 共the particles cannot escape兲, i.e., y 1 ⬎ −共L − ⌬兲 / 2 and y N ⬍ 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
x j = y j − j⌬ +
N+1
⌬,
2
x j,0 = y j,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 R0 is also introduced for the initial coordinates
which satisfy −ᐉ / 2 ⬍ x1,0 ⬍ x2,0 . . . ⬍ xN,0 ⬍ ᐉ / 2. For convenience, we also introduce the shorthand notations xជ
= 共x1 , . . . , xN兲 and xជ 0 = 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 length ᐉ.
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 particle T 共T = 1 , . . . , N兲 is at position xT at time t, given that it
was at xT,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 calculated from the ergodicity principle: all points in the allowed
phase space R are equally probable. This leads the equilibrium NPDF
051103-2
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
N−1
Peq共xជ 兲 =
N!
兿 ␪共xi+1 − xi兲,
ᐉN i=1
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 integration technique 共see Appendix C兲 it is easy to verify that
兰Rdx1 ¯ dxN Peq共xជ 兲 = 1. Integrating Eq. 共4兲 over all coordinates leaving out one, xT, gives the tPDF
␳eq
T 共xT兲 =
冕
R
冕
ᐉ/2
xT
=
冕
xT
−ᐉ/2
⬘ ¯
dxT+1
dx1⬘ ¯
冕
ᐉ/2
冕
xT
−ᐉ/2
dxN⬘
NL
ᐉ
− xT
2
NR
,
1
␳ 共xT,0兲
eq
⫻
冕
R0
冕
R
⬘ ␦共xT,0 − xT,0
⬘ ¯ dxN,0
⬘ 兲
dx1,0
⳵t
冉
冊
⳵2
⳵2
⳵2
P共xជ ,t兩xជ 0兲,
2 +
2 + ¯+
⳵ x1 ⳵ x2
⳵ xN2
共7兲
for xជ 苸 R 关P共xជ , t 兩 xជ 0兲 ⬅ 0 outside R兴. The equation certifying
that neighboring particles cannot overtake reads
D
冉
冊
⳵
⳵
−
P共xជ ,t兩xជ 0兲兩xi+1=xi = 0.
⳵ xi+1 ⳵ xi
共9兲
= 0,
共10兲
xN=ᐉ/2
共11兲
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 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xជ 0兲 =
冕 冕
⬁
−⬁
dk1
2␲
⬁
−⬁
dk2
¯
2␲
冕
⬁
−⬁
dkN −E共k ,. . .,k 兲t
e 1 N
2␲
⫻⌸Nj=1␾共k j,x j,0兲关eik1x1+ik2x2+ik3x3+¯+ikNxN
+ S21eik2x1+ik1x2+ik3x3+¯+ikNxN
+ S32S31eik2x1+ik3x2+k1x3+¯+ikNxN
+ all other permut. of 兵k1,k2, . . . ,kN其兴,
共12兲
E共k1, . . . ,kn兲 = D共k21 + ¯ + kN2 兲,
共6兲
where ␳eq
T 共xT,0兲 is given in Eq. 共5兲.
In order to get the tPDF using Eq. 共6兲, we need to calculate the NPDF P共xជ , t 兩 xជ 0兲. It is governed by the diffusion
equation
=D
⳵ P共xជ ,t兩xជ 0兲
⳵ xN
= 0,
x1=−ᐉ/2
where E共k1 , . . . , kN兲 is the dispersion relation, Sij are the scattering coefficients, and ␾共k j , x j,0兲 denotes a function containing boundary and initial conditions. Each one of these quantities is described below.
The dispersion relation has the form
dx1⬘ ¯ dxN⬘ ␦共xT − xT⬘ 兲
⫻P共xជ ⬘,t兩xជ 0⬘兲Peq共xជ 0⬘兲,
⳵ P共xជ ,t兩xជ 0兲
冏
冏
making sure that the particles are restricted to 关−ᐉ / 2 , ᐉ / 2兴 at
all times. Finally, the initial condition is
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xT , t 兩 xT,0兲 one needs to first introduce
the N-particle joint probability density P共xជ , t ; xជ 0兲, which
gives the probability density that the system is in a state xជ
and that it initially was in a state xជ 0. The joint probability
density for the tagged particle ␳T共xT , t ; xT,0兲 is simply obtained from the joint NPDF by integration over R and R0:
␳T共xT , t ; xT,0兲 = 兰Rdx1⬘ ¯ dxN⬘ 兰R0dx1,0
⬘ ␦共xT − xT⬘ 兲␦共xT,0
⬘ ¯ dxN,0
− xT,0
⬘ 兲P共xជ ⬘ , t ; xជ 0⬘兲. Relating the conditional and joint probability densities using Bayes’ rule 关39兴, i.e., P共xជ , t ; xជ 0兲
= P共xជ , t 兩 xជ 0兲Peq共xជ 0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 =
⳵ x1
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 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.
⬘
dxT−1
冉 冊冉 冊
⳵ P共xជ ,t兩xជ 0兲
P共xជ ,0兩xជ 0兲 = ␦共x1 − x1,0兲 ¯ ␦共xN − xN,0兲.
xT
N!
1
ᐉ
+ xT
N
ᐉ N L ! N R! 2
冏
冏
D
dx1⬘ ¯ dxN⬘ ␦共xT − xT⬘ 兲Peq共xជ ⬘兲
1
N!
= N
ᐉ N L ! N R!
⫻
D
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 ki and k j, Sij = S共ki , k j兲. They are, however, independent 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8兲
Also, reflecting boundaries are placed at the system ends,
共13兲
共14兲
which means that they are independent of momenta and correspond to perfect reflection. For noninteracting particles
051103-3
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
Sij = 0, which reduces the Bethe ansatz to a standard Fourier
transform.
The quantity ␾共k j , x j,0兲 contains information about the initial and boundary conditions of the problem, which are here
defined by Eqs. 共9兲–共11兲. The form of ␾共k j , x j,0兲 satisfying
these relationships is given by
␺共xi,x j,0 ;t兲 =
兺
eik j共2m+1/2兲ᐉ , 共15兲
P共xជ ,t兩xជ 0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
共19兲
冉 冊 冉 冊
冉 冊 冉 冊
m␲xi
m␲x j,0
cos
ᐉ
ᐉ
共+兲
cos
Gm共xi,x j,0兲 = ␯m
共−兲
+ ␯m
sin
m=−⬁
which is shown explicitly in Appendix A. Notably, for an
infinite system we have ␾共k j , x j,0兲 = e−ik jx j,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 under interchange of any two particle positions, xi ↔ x j, 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 singlefile 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-ofmass 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
册
⬁
where
⬁
␾共k j,x j,0兲 = 2 cos关k j共x j,0 + ᐉ/2兲兴
冋
1
1 + 兺 Gm共xi,x j,0兲Em共t兲 ,
ᐉ
m=1
m␲xi
m␲x j,0
sin
,
ᐉ
ᐉ
Em共t兲 = e−共m␲兲
2Dt/ᐉ2
共20兲
,
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 , x j,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 particles 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 presented 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.
+ all other permut. of 兵x1,0,x2,0, . . . ,xN,0其,
共16兲
where
⬁
␺共xi,x j,0 ;t兲 =
再 冋
1
共xi − x j,0 + 2mᐉ兲2
exp
−
兺
共4␲Dt兲1/2 m=−⬁
4Dt
冋
+ exp −
关xi + x j,0 + 共2m + 1兲ᐉ兴2
4Dt
册冎
册
共17兲
is obtained from the inverse Fourier transform
2
⬁
dk j ␾共k j , x j,0兲e−Dk j teik jxi. We point out that
共2␲兲−1兰−⬁
␺共xi , x j,0 ; t兲 is the single-particle PDF for a particle in confined 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 eigenmode expansion of ␺共xi , x j,0 ; t兲, which can be done in a variety of ways. Here, we use Bromwich integration. The
Laplace transform of Eq. 共17兲 is 共see Appendix B兲
␺共xi,x j,0 ;s兲 =
冕
⬁
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 particles 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 xជ 0兲 is invariant under permutations of xi ↔ x j, extend
the integration from R to the hypercubes x j 苸 关−ᐉ / 2 , xT兴
共j = 1 , . . . , T − 1兲 and x j 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 =
xT
N L ! N R!
−ᐉ/2
⫻
ᐉ/2
⫻
xT
ᐉ/2
dxT+1,0 ¯
f L = 共ᐉ/2 + xT,0兲−1,
冑4Ds sinh共ᐉ冑s/D兲 兵cosh关共xi + x j,0兲
+ cosh关共ᐉ − 兩x j,0 − xi兩兲冑s/D兴其.
xT
dxT+1 ¯
xT,0
dx1,0 ¯
ᐉ/2
xT,0
−ᐉ/2
xT,0
−ᐉ/2
ᐉ/2
dxT−1
dxT−1,0
−ᐉ/2
dxN,0 P共xជ ,t兩xជ 0兲,
共23兲
xT,0
where
dt e−st␺共xi,x j,0 ;t兲
1
dx1 ¯
dxN
xT
0
=
冕 冕 冕
冕 冕
冕
冕
冕
f LNL f RNR
冑s/D兴
共18兲
The sought eigenvalue expansion is obtained as a sum of
residues of ␺共xi , x j,0 ; s兲 共see, e.g., Ref. 关43兴兲 and reads
f R = 共ᐉ/2 − xT,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, P共n␣,␤兲共z兲. Using the identities P共n␣,␤兲共z兲
共−␣,␤兲
= 共n + ␣兲 ! 共n + ␤兲 ! / 关n ! 共n + ␣ + ␤兲!兴关共z − 1兲 / 2兴−a Pn+
␣ 共z兲 and
共␣,␤兲
n 共␤,␣兲
Pn 共−z兲 = 共−1兲 Pn 共z兲 关48兴 leads to
051103-4
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
␳共xT,t兩xT,0兲 =
共NL + NR − 1兲! L N R N
共 ␺ L兲 L共 ␺ R兲 R
N L ! N R!
␺ R = 1 − ␺ L,
再
␺ L␺ L
NL2 L ⌽共1,0,0; ␰兲
␺L
+ N LN R
冋
+
冎
册
␺ ␺L ␺ ␺R
+ L ⌽共0,0,1; ␰兲 ,
␺LR
␺R
R
L
Km =
␺ R␺ R
NR2 R ⌽共0,1,0; ␰兲
␺R
冉 冊 冉
共25兲
Jm共z,z⬘兲 =
关NL − 共a + c兲兴 ! 共NR − b兲! c
␰ 共1 − ␰兲NL−共a+c兲
⌽共a,b,c; ␰兲 =
关NL + NR − 共a + b + c兲兴!
R−NL+a−b兲
⫻PN共c,N−共a+c兲
L
␰=
冉 冊
1+␰
,
1−␰
共27兲
were introduced. The quantities ␺L, ␺LL, etc. are defined in
Eq. 共D3兲 and are integrals of the one-particle propagator ␺
= ␺共xi , x j,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 ␺LL, ␺RR, ␺LR, and ␺RL 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 function 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 integrals of ␺共xi , x j,0 ; t兲 关Eq. 共D3兲兴. Integrating Eq. 共19兲 关or integrating ␺共xi , x j,0 ; s兲 prior to Laplace inversion 共see Appendix
E兲兴 yields 共arguments are left implicit兲
冉
冊
冉
冊
1 xT
1 xT,0
− +
−
2 ᐉ
2
ᐉ
⫻
KmEm共t兲,
−1 ⬁
␺L =
冋
冋
1
1 xT,0
1+
+
2
ᐉ
ᐉ
冉
冊
冉
冊
1
1 xT,0
−
␺R =
1−
2
ᐉ
l
兺 Jm共xT,0,xT兲Em共t兲
m=1
−1 ⬁
兺 Jm共xT,0,xT兲Em共t兲
m=1
冋 冉 冊 冉 冊
冉 冊 冉 冊册
1
m␲z
m␲z⬘
␯m共+兲 sin
cos
m␲
ᐉ
ᐉ
m␲z
m␲z⬘
sin
ᐉ
ᐉ
,
共31兲
冉 冊
1−␰
4
关N−共a+b+c兲兴/2
Ic兵关N − 共a + b + c兲兴␨其
共32兲
where 关54兴
兺 KmEm共t兲,
m=1
−1 ⬁
共30兲
,
⫻关1 + A共␨兲兴,
m=1
1 xT
+ + 兺 Jm共xT,xT,0兲Em共t兲,
2 ᐉ m=1
冊册
冊
⌽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
冋 册
1
1 + 冑␰
␨ = ln
,
2
1 − 冑␰
⬁
␺L =
共29兲
共⫾兲
are given by Eqs. 共21兲 and 共22兲, respecwhere Em共t兲 and ␯m
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xT , t 兩 xT,0兲 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 ␳T共xT , t 兩 xT,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xT , t 兩 xT,0兲 for short and intermediate times. From
Eq. 共26兲 we note that the argument in the Jacobi polynomial,
P共n␣,␤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
−1 ⬁
兺
m␲xT
m␲xT,0
cos
ᐉ
ᐉ
共−兲
− ␯m
cos
共26兲
␺RL␺LR
,
␺LL␺RR
1 xT
1 xT,0
␺LL = + +
+
2 ᐉ
2
ᐉ
冋 冉 冊冉
1
m␲xT
m␲xT,0
␯共+兲 sin
sin
共m␲兲2 m
ᐉ
ᐉ
共−兲
+ ␯m
cos
where
␺RR =
␺RL = 1 − ␺RR ,
with
⫻ 共NL + NR兲␺⌽共0,0,0; ␰兲
+
␺LR = 1 − ␺LL,
册
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 NL and NR. The correction term appearing in Eq. 共32兲 is
A共␨兲 =
,
051103-5
Ic+1兵关N − 共a + b + c兲兴␨其
B 0共 ␨ 兲
,
N − 共a + b + c兲 Ic兵关N − 共a + b + c兲兴␨其
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
B 0共 ␨ 兲 =
再
冉
1
1
共c2 − 1/4兲 − coth ␨
2
␨
冊
冎
− 关共␦N + a − b兲2 − 1/4兴tanh共␨兲 ,
␰ Ⰶ 1. 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
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 / N Ⰶ 1. Inserting Eq. 共32兲 in Eq. 共25兲 and using the same approximations as above, we obtain our final
large-N result for the tPDF,
␳T共xT,t兩xT,0兲 = 共␺LL兲共N−␦N−1兲/2共␺RR兲共N+␦N−1兲/2
冉冑 冊 再
冉
冊
冑冉
冊
␨
⫻共1 − ␰兲共N−1兲/2
␰
1/2
N ␺ ␺L ␺ ␺R
+ R I0关N␨兴
2 ␺LL
␺R
+
␺ R␺ L ␺ L␺ R
N
␰
+ L I1关N␨兴 .
2
␺LR
␺R
R
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
N Ⰷ 1 and that the tagged particle is approximately in the
center of the system: 兩␦N兩 / N Ⰶ 1.
S共t兲 = 2Dt.
In the short-time regime, almost no collisions with the neighboring 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.
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 ␨ Ⰶ 1 but N␨
Ⰷ 1, 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 calculation兲. Second, the Bessel function is approximated with
I␣共z兲 兩zⰇ1 ⬇ ez / 冑2␲z. A straightforward expansion of Eq. 共35兲
for ␺LR , ␺RL Ⰶ 1 共i.e., 冑␰ Ⰶ 1兲, together with Stirling’s formula, gives
L R
1
␳T共y T,t兩y T,0兲 ⬇ e−␦N共␺R−␺L 兲
2
冋
+ ␺ L␺ R
冑
N L R −1/4 −共N/2兲共冑␺L − 冑␺R兲2
R
L
共␺ ␺ 兲 e
2␲ R L
共36兲
␳T共y T,t兩y T,0兲 =
A. Short times, t ™ ␶coll , ␶eq
For short times we have N␨ Ⰶ 1 and may therefore use the
approximations I␣共z兲 兩zⰆ1 ⬇ 共z / 2兲␣ / ⌫共␣ + 1兲 关44兴, ␨ ⬇ 冑␰, and
␺RL
␺LR
1/2
1/2
共40兲
.
1
冢
冤
1
冑2␲ 4Dt 1 − ⌬
␲
冉
冊冣
1/4
2
共y T − y T,0兲2
⫻exp −
共37兲
For a particle located roughly in the middle, 兩␦N兩 / N Ⰶ 1
关i.e., Eq. 共35兲 applies兴, the three cases are given by 共A兲
short times, N␨ Ⰶ 1, i.e., t Ⰶ ␶coll , ␶eq; 共B兲 intermediate
times, ␨ Ⰶ 1 and N␨ Ⰷ 1, corresponding to ␶coll Ⰶ t Ⰶ ␶eq; and
共C兲 long times, ␨ Ⰷ 1, i.e., t Ⰷ ␶coll , ␶eq.
Time regimes 共A兲–共C兲 are analyzed in detail below.
冉 冊册
␺LR
␺RL
冉 冊
If we furthermore assume that the average of the absolute
value of ␩ = 共xT − xT,0兲 / 冑4Dt is small 共which is checked after
the calculation兲, Eq. 共E15兲 may be used which in combination with Eqs. 共1兲, 共2兲, and 共40兲, keeping only lowest order
terms in ␩, leads to the SFD result,
where = N / ᐉ is the concentration of particles, and the equilibrium time
ᐉ2
␶eq = .
D
共39兲
⫻ ␺ L␺ L + ␺ R␺ R + ␺ R␺ L
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xT , t 兩 xT,0兲 are derived. Mathematically, 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
1
,
2D
共38兲
for which the MSD is
V. THREE DIFFERENT TIME REGIMES
␶coll =
册
共y T − y T,0兲2
,
4Dt
B. Intermediate times, ␶coll ™ t ™ ␶eq
共1 − ␰兲1/2␺I0关N␨兴
+
L
冋
␳T共y T,t兩y T,0兲 = 共4␲Dt兲−1/2exp −
共34兲
2
冑 冉
4Dt 1 − ⌬
␲
冊冥
2
,
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 magnitude 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
051103-6
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
with standard results 关20,37兴, 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 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, t š ␶eq
In the long-time limit we have ␨ Ⰷ 1, for which the tPDF
can be obtained exactly for arbitrary N. Using the exact expression for ␳T共xT , t 兩 xT,0兲 found in Eqs. 共25兲 and 共28兲, together with 2F1共␣ , ␤ , ␥ ; z = 0兲 = 1 关44兴 and Eq. 共E7兲 关also using Eqs. 共1兲 and 共2兲兴 gives
␳eq
T 共y T兲 =
共NL + NR + 1兲!
1
N
N L ! N R!
共L − N⌬兲
⫻
冉
L
+ y T − ⌬共1/2 + NL兲
2
冊冉
NL
冊
L
− y T − ⌬共1/2 + NR兲
2
NR
共43兲
where ␳eq
T 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. Calculating the second moment of the equilibrium tPDF, S共t
→ ⬁兲 = Seq, gives
Seq =
冉冊 冉
1
4
NR+1
L − N⌬
2
冊
2
⌫共1/2兲⌫共2关NR + 1兴兲
, 共44兲
⌫共NR + 1兲⌫共NR + 5/2兲
for NL = NR where ⌫共z兲 is the gamma function. For N Ⰷ 1 we
can simplify the expression for the equilibrium tPDF as well
as Seq. If a symmetric file is assumed, i.e., NL = NR and T
= 共N + 1兲 / 2, an asymptotic expansion of ␳eq
T 共y T兲 共using
Stirling’s approximation 关44兴兲 gives the Gaussian PDF
␳eq
T 共y T兲 ⬇
冑
冋 冉
1
2N
yT
exp − 2N
␲ 共L − N⌬兲2
L − N⌬
冊册
2
,
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 expansion of Eq. 共44兲 using Stirling’s formula and assuming
N ⬇ 2NR. We point out that ␳eq
T 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 equilibrium, namely, t Ⰷ ␶eq.
VI. CONCLUSIONS AND OUTLOOK
In this study we have solved exactly a nonequilibrium
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 ␳T共y T , t 兩 y T,0兲 共denoted as
tPDF兲 that a tagged particle T is at position y T at 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y T , t 兩 y T,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 t Ⰶ ␶coll = 1 / 共2D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 t Ⰷ ␶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 proportional to t1/2 关Eq. 共41兲兴. 共C兲 For times exceeding the equilibrium time t Ⰷ ␶eq, the tPDF approaches a probability density 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 systems and that occurring in CTRW theory was pursued numerically in 关62兴.
The Bethe ansatz is often employed in quantum mechanics 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 interacting 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 believe 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 particle.
We finally note that the use of fluorescently labeled
共tagged兲 particles is of much use for studying biological systems. The understanding of how the motion of such particles
correlates with its environment is therefore the key for grasping the behavior of such systems in a quantitative fashion.
051103-7
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
ACKNOWLEDGMENTS
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k j , x j,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
We are grateful to Bob Silbey, Mehran Kardar, Eli Barkai,
Ophir Flomenbom, and Michael Lomholt for valuable discussions. L.L. acknowledges support from the Danish National Research Foundation, and T.A. from the Knut and Alice Wallenberg Foundation.
⬁
␭共k,z兲 = ␾共k,z兲e
APPENDIX A: BETHE ANSATZ
⳵ P共xជ ,t兩xជ 0兲
⳵ x1
冏
=
x1=−ᐉ/2
冕
⬁
−⬁
dk1
¯
2␲
冕
⬁
−⬁
= 2 cos关k共z + ᐉ/2兲兴
兺
e−2ikmᐉ ,
m=−⬁
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 demonstrated that Eq. 共12兲 also satisfies the initial condition.
冏
−ikᐉ/2
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
dkN −D共k2+¯+k2 兲t N
ik1x1 ik2x2+¯+ikNxN
1
N ⌸
e
共e
+ perm. of 兵k2, . . . ,kN其兲
j=1␾共k j,x j,0兲关ik1e
2␲
+ ¯ + ikieikix1共eik2x2+¯+ik1xi+¯+ikNxN + perm. of 兵k2, . . . ,ki−1,ki+1, . . . ,kN其兲 + ¯ + ikNeikNx1共eik2x2+¯+ik1xN
+ perm. of 兵k1, . . . ,kN−1其兲兴x1=−ᐉ/2
=
冕
⬁
−⬁
dk2
¯
2␲
+ ¯+
⫻
冕
⬁
−⬁
⫻
冕
⬁
−⬁
2
冕
冕
⬁
−⬁
⬁
−⬁
dkN −D共k2+¯+k2 兲t N
2
N ⌸
e
j=2␾共k j,x j,0兲
2␲
dk1
¯
2␲
冕
⬁
−⬁
dki−1
2␲
冕
⬁
−⬁
dki+1
¯
2␲
2
dki
共iki兲e−Dki t␭共ki,xi,0兲 + ¯ +
2␲
冕
⬁
−⬁
冕
⬁
−⬁
冕
−⬁
2
dk1
共ik1兲e−Dk1t␭共k1,x1,0兲
2␲
dkN −D共k2+¯+k2 +k2 +¯+k2 兲t N
1
i−1 i+1
N ⌸
e
j=1,j⫽i␾共k j,x j,0兲
2␲
dk1
¯
2␲
2
dkN
共ikN兲e−DkNt␭共kN,xN,0兲 = 0,
2␲
⬁
冕
⬁
−⬁
dkN−1 −D共k2+¯k2 兲t N−1
1
N−1 ⌸
e
j=1 ␾共k j,x j,0兲
2␲
共A3兲
⬁
where 兰−⬁
dki kie−Dki 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k j , x j,0兲. It is only
required that the symmetry relation 共A2兲 holds. Furthermore,
the dispersion relation E共kជ 兲 = D共k21 + ¯ +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 ↔ x j, it gives
关⳵P共xជ , t 兩 xជ 0兲 / ⳵x j兴x j=⫾ᐉ/2 = 0 for all x j.
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 xជ 0兲 in two
alternative ways:
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PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
P共xជ ,t兩xជ 0兲 =
冕 冕
⬁
−⬁
dk1
2␲
⬁
−⬁
dk2
¯
2␲
冕
⬁
−⬁
dkN −D共k2+¯+k2 兲t N
ik1x j ik jx1+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ikNxN
1
N ⌸
e
共e
j=1␾共k j,x j,0兲关e
2␲
+ all other permut. of 兵k2, . . . ,kN其兲 + eik2x j共eik1x1+ik jx2+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ikNxN
+ all other permut. of 兵k1,k3, . . . ,kN其兲 + ¯ + eikNx j共eik1x1+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ik jxN
+ all other permut. of 兵k1, . . . ,kN−1其兲兴
共A4兲
and
P共xជ ,t兩xជ 0兲 =
冕 冕
⬁
−⬁
dk1
2␲
⬁
−⬁
dk2
¯
2␲
冕
⬁
−⬁
dkN −D共k2+¯+k2 兲t N
ik1x j+1 ik j+1x1+¯+ik j−1x j−1+ik jx j+ik j+2x j+2+¯+ikNxN
1
N ⌸
e
共e
j=1␾共k j,x j,0兲关e
2␲
+ all other permut. of 兵k2, . . . ,kN其兲 + eik2x j+1共eik1x1+ik j+1x2+¯+ik j−1x j−1+ik jx j+ik j+2x j+2+¯+ikNxN
+ all other permut. of 兵k1,k3, . . . ,kN其兲 + ¯ + eikNx j+1共eik1x1+¯+ik j−1x j−1+ik jx j+ik j+2x j+2+¯+ik j+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xជ 0兲
⳵ x j+1
−
⳵ P共xជ ,t兩xជ 0兲
⳵xj
冊冏
冕 冕
⬁
=
x j+1=x j
−⬁
dk1
2␲
⫻关ik1e
−e
⬁
−⬁
ik1x j
共e
dk2
¯
2␲
冕
⬁
−⬁
dkN −D共k2,. . .,k2 兲t N
1
N ⌸
e
j=1␾共k j,x j,0兲
2␲
ik j+1x1+¯+ik j−1x j−1+ik jx j+1+ik j+2x j+2+¯+ikNxN
ik jx1+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ikNxN
+ all other permut. of 兵k2, . . . ,kN其兲
+ ik2eik2x j共eik1x1+ik j+1x2+¯+ik j−1x j−1+ik jx j+1+ik j+2x j+2+¯+ikNxN
− eik1x1+ik jx2+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ikNxN + all other permut. of 兵k1,k3, . . . ,kN其兲
+ ¯ + ikNeikNx j共eik1x1+¯+ik j−1x j−1+ik jx j+1+ik j+2x j+2+¯+ik j+1xN
− eik1x1+¯+ik j−1x j−1+ik j+1x j+1+ik j+2x j+2+¯+ik jxN + all other permut. of 兵k1, . . . ,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 permutation over all allowed momenta. Note that this derivation is
independent of the choice of ␾共k j , x j,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 ik x
e j ␾共k j,y兲,
2␲
共A7兲
⍀共x,y兲 =
␦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 of R and R0 where, per definition, P共xជ , t 兩 xជ 0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 xជ 0兲 = 0 outside R, one
obtains
P共xជ ,t → 0兩xជ 0兲 = ␦共x − x0兲 ¯ ␦共xN − xN,0兲
共A10兲
as t → 0, which is the desired result.
Eq. 共12兲 reads
P共xជ ,t → 0兩xជ 0兲 = ⍀共x1,x1,0兲⍀共x2,x2,0兲 ¯ ⍀共xN,xN,0兲
APPENDIX B: LAPLACE TRANSFORM OF ␺(xi , xj,0 ; t)
+ ⍀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.
兺
m=−⬁
共A8兲
Using the explicit expression for ␾共k , y兲 found in Eq. 共15兲
⬁
dk eik共x−z兲 leads to
and that ␦共x − z兲 = 共2␲兲−1兰−⬁
In this section it is shown explicitly how one can go from
the one-particle PDF ␺共xi , x j,0 ; t兲 for a particle in a box
expressed in terms of Gaussians 关Eq. 共17兲兴, to the eigenmode expansion 关Eqs. 共19兲–共22兲兴 used in this paper. First,
the summation in Eq. 共17兲 is divided schematically into
051103-9
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
⬁
⬁
兺m=−⬁
f m = f m=0 + 兺m=1
关f m + 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,x j,0 ;s兲 = Q共s兲 +
冋
+
e−共xi+x j,0+ᐉ兲
x1 = x2
冑s/D
冑4Ds
e−共xi−x j,0兲
冑s/D
e−共xi+x j,0+ᐉ兲
+
x2
x1 < x2
+ 关e共xi−x j,0兲
冑s/D
冑4Ds
冑s/D
+ e共xi+x j,0+ᐉ兲
冑4Ds
兴
冑s/D
x1
册兺
⬁
e−2mᐉ
冑s/D
x1 > x2
,
m=1
共B1兲
where
Q共s兲 =
再
共4Ds兲
共4Ds兲
−1/2 −共xi−x j,0兲冑s/D
, xi ⱖ x j,0
−1/2 −共x j,0−xi兲冑s/D
, xi ⱕ x j,0 .
e
e
冎
共B2兲
Considering the cases xi ⬎ x j,0 and xi ⬍ x j,0 separately, and
⬁
e−2mᐉ冑s/D = 共e2ᐉ冑s/D − 1兲−1, leads to
that 兺m=1
␺共xi,x j,0 ;s兲 =
1
冑sD sinh关ᐉ冑s/D兴
再
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 ↔ x2 the 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.
cosh关共ᐉ/2 + xi兲冑s/D兴cosh关共ᐉ/2 − x j,0兲冑s/D兴,
cosh关共ᐉ/2 − xi兲冑s/D兴cosh关共ᐉ/2 + x j,0兲冑s/D兴,
which after elementary trigonometric manipulations result in
Eq. 共18兲.
APPENDIX C: EXTENDED PHASE-SPACE
INTEGRATION
xi ⱖ x j,0 ,
冎
共B3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 xជ 0兲 is invariant under
x1 ↔ x2, integration over the lower triangle gives the same
result, i.e.,
When the tPDF is integrated out from the NPDF we need
to resolve the following type of integral:
冕
xi ⱕ x j0
I共x3兲 =
冕
−ᐉ/2⬍x2⬍x1⬍x3⬍ᐉ/2
dx1dx2 P共x1,x2,x3,t兩xជ 0兲.
⬘ dxT+1
⬘ ¯ dxN⬘ P共xជ ⬘,t兩xជ 0兲 共C1兲
dx1⬘ ¯ dxT−1
共C3兲
over the region R 关Eq. 共3兲兴 with the tagged particle coordinate xT left out. As pointed previously in the paper, the Bethe
ansatz solution 关Eq. 共12兲兴 is symmetric under the transformation xi ↔ x j when all scattering coefficients are given by Sij
= 1. This allows the integration of P共xជ , t 兩 xជ 0兲 in Eq. 共C1兲 to be
extended to the whole hyperspace x j 苸 关−ᐉ / 2 , xT兴 , j
= 1 , . . . , T − 1 and x j 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 tagged T = 3,
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xT兲 =
I共x3兲 =
R\xT
冕
−ᐉ/2⬍x1⬍x2⬍x3⬍ᐉ/2
dx1dx2 P共x1,x2,x3,t兩xជ 0兲.
共C2兲
I共x3兲 =
1
2
冕
xT
−ᐉ/2
dx1
冕
xT
dx2 P共x1,x2,x3,t兩xជ 0兲.
共C4兲
−ᐉ/2
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 ⬍ . . . ⬍ xT−1 ⬍ xT to −ᐉ / 2 ⬍ x j ⬍ xT 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
051103-10
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
I共xT兲 =
冕
1
N L ! N R!
⫻
冕
xT
dx1
−ᐉ/2
xT
dxT−1
−ᐉ/2
冕
冕
xT
−ᐉ/2
ᐉ/2
dxT+1 ¯
xT
extended phase-space technique is valid for integrations over
the initial particle positions R0.
dx2 ¯
冕
ᐉ/2
dxN P共xជ ,t兩xជ 0兲.
APPENDIX D: FROM THE NPDF TO THE tPDF
xT
共C5兲
Since P共xជ , t 兩 xជ 0兲 is also invariant under xi,0 ↔ x j,0, a similar
1
␳共xT,t兩xT,0兲 =
N L ! N R!
再
In this appendix the tPDF is obtained from integral 共6兲
using the Bethe ansatz NPDF 共16兲 explicitly. A similar combinatorial analysis to that found in 关37兴 gives
min兵NL−1,NR其
min兵NL,NR其
兺
H
0
共q兲共␺LL兲NL−q共␺RL兲q␺共␺LR兲q共␺RR兲NR−q
+
兺
HLL共q兲共␺LL兲NL−q−1共␺RL兲q␺L␺L共␺LR兲q共␺RR兲NR−q
min兵NL−1,NR−1其
min兵NL,NR−1其
+
兺
q=0
q=0
HRR共q兲共␺LL兲NL−q共␺RL兲q␺R␺R共␺LR兲q共␺RR兲NR−q−1
+
兺
HLR共q兲
冎
q=0
q=0
min兵NL−1,NR−1其
兺
q=0
⫻共␺LL兲NL−q−1共␺RL兲q+1␺R␺L共␺LR兲q共␺RR兲NR−q−1 +
with the combinatorial factors
冉 冊冉
冉 冊冉
冉 冊冉
冉 冊冉
冉 冊冉
NL
H 共q兲 =
q
0
HRR共q兲
q
HLR共q兲 =
HRL共q兲 =
q
NL − 1
NL
NR − 1
q+1
q
NL ! NR ! NR ,
冕 冕
xT
−ᐉ/2
␺RR共t兲 = f R
dx j,0 ␺共xi,x j,0 ;t兲,
−ᐉ/2
冕 冕
ᐉ/2
dx j,0 ␺共xi,x j,0 ;t兲,
xT,0
冕 冕
ᐉ/2
xT,0
dxi
xT
␺RL共t兲 = f R
ᐉ/2
dxi
xT
␺LR共t兲 = f L
xT,0
dxi
冕 冕
xT
ᐉ/2
dxi
−ᐉ/2
dx j,0 ␺共xi,x j,0 ;t兲,
−ᐉ/2
xT,0
dx j,0 ␺共xi,x j,0 ;t兲
xT
dxi ␺共xi,x j,0 ;t兲,
冕
ᐉ/2
dxi ␺共xi,x j,0,t兲,
xT
␺L共t兲 = f L
NL ! NR ! NR ,
and integrals 共leaving arguments xT,0 and xT implicit兲
␺LL共t兲 = f L
␺R共t兲 =
冕
xT,0
dx j,0 ␺共x j,x j,0 ;t兲,
−ᐉ/2
NR
NL ! NR ! NL ,
q+1
q
冕
共D1兲
−ᐉ/2
NR
NL ! NR ! NL ,
q
NR − 1
NL
=
q
␺L共t兲 =
NR
NL ! NR ! ,
q
NL − 1
HLL共q兲 =
冊
冊
冊
冊
冊
HRL共q兲共␺LL兲NL−q−1共␺RL兲q␺L␺R共␺LR兲q+1共␺RR兲NR−q−1 ,
␺R共t兲 = f R
冕
ᐉ/2
dx j,0 ␺共x j,x j,0 ;t兲.
共D3兲
xT,0
共D2兲
The prefactors f L and f R are found in Eq. 共24兲 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 ␺L共t兲 + ␺R共t兲 = 1, ␺LL共t兲 + ␺LR共t兲 = 1, and ␺RR共t兲
+ ␺RL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 follows. First, ␺LL 共␺RR兲 is the probability that a single particle is
to the left 共right兲 of xT at time t given that it started, with an
equal probability, anywhere to the left 共right兲 of xT,0. Similar
interpretations hold also for ␺LR and ␺RL. The quantity ␺L 共␺R兲
is the probability that a single particle is at position xT given
that the particle started somewhere to the left 共right兲 of xT,0.
Finally, ␺L 共␺R兲 is the probability that a single particle is to
the left 共right兲 of xT at time t given that it started at position
xT,0.
051103-11
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
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
␺L共s兲 =
␺L共s兲 =
␺R共s兲 =
␺LL共s兲 =
␺RR共s兲
=
1
sinh关共ᐉ/2 + xT兲冑s/D兴cosh关共ᐉ/2 − xT,0兲冑s/D兴,
fL
再
再
sinh关ᐉ冑s/D兴 − cosh关共ᐉ/2 + xT兲冑s/D兴sinh关共ᐉ/2 − xT,0兲冑s/D兴, xT ⱕ xT,0
s sinh关ᐉ冑s/D兴 cosh关共ᐉ/2 − xT兲冑s/D兴sinh关共ᐉ/2 + xT,0兲冑s/D兴,
fR
s sinh关ᐉ冑s/D兴
s sinh关ᐉ冑s/D兴
s sinh关ᐉ冑s/D兴
冦
冦
xT ⱖ xT,0 ,
cosh关共ᐉ/2 + xT兲冑s/D兴sinh关共ᐉ/2 − xT,0兲冑s/D兴,
sinh关ᐉ冑s/D兴 − cosh关共ᐉ/2 − xT兲冑s/D兴sinh关共ᐉ/2 + xT,0兲冑s/D兴,
关1 − 共xT,0 − xT兲f L兴sinh关ᐉ冑s/D兴 − f L
sinh关ᐉ冑s/D兴 − f L
冑
冑
xT ⱕ xT,0
xT ⱖ xT,0 ,
冎
共E1兲
冎
冎
共E2兲
共E3兲
D
sinh关共ᐉ/2 + xT兲冑s/D兴sinh关共ᐉ/2 − xT,0兲冑s/D兴, xT ⱕ xT,0
s
D
sinh关共ᐉ/2 − xT兲冑s/D兴sinh关共ᐉ/2 + xT,0兲冑s/D兴,
s
xT ⱖ xT,0 ,
冧
共E4兲
sinh关ᐉ冑s/D兴 − f R
冑
D
sinh关共ᐉ/2 + xT兲冑s/D兴sinh关共ᐉ/2 − xT,0兲冑s/D兴,
s
关1 − 共xT − xT,0兲f R兴sinh关ᐉ冑s/D兴 − f R
冑
1
␺LL + ␺LR = ,
s
xT ⱕ xT,0
D
sinh关共ᐉ/2 − xT兲冑s/D兴sinh关共ᐉ/2 + xT,0兲冑s/D兴, xT ⱖ xT,0 .
s
冧
共E5兲
The remaining three integrals follow from the normalization
conditions 共arguments are left implicit兲
1
␺L + ␺R = ,
s
xT ⱕ xT,0
s sinh关ᐉ冑s/D兴 sinh关ᐉ冑s/D兴 − sinh关共ᐉ/2 − xT兲冑s/D兴cosh关共ᐉ/2 + xT,0兲冑s/D兴, xT ⱖ xT,0 ,
1
1
再
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
xT and xT,0 left out兲 are given by
1
␺RR + ␺RL = .
s
␺ = ␺L = ␺R =
1
,
sᐉ
␺L = ␺LL = ␺RL =
共E6兲
␺R = ␺RR = ␺LR =
The Laplace inversion of the above relationships, using, e.g.,
residue calculus 关43兴, gives Eq. 共29兲. In the following subsections we give asymptotic results in the 共1兲 long- and 共2兲
short-time limits for the expressions above.
冉 冊
1 1 xT
,
+
s 2 ᐉ
冉 冊
1 1 xT
−
.
s 2 ᐉ
共E7兲
The inverse transforms are found from L−1共s−1兲 = 1 关44兴.
2. Short-time behavior
1. Long-time behavior
The long-time behavior of Eqs. 共E1兲–共E16兲 is obtained
from a series expansion for ᐉ冑s / D Ⰶ 1 and reads 共arguments
are left implicit兲
Short
times
is
defined
here
as
共ᐉ ⫾ xT兲冑s / D , 共ᐉ ⫾ xT,0兲冑s / D Ⰷ 1, 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
051103-12
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
␺ 共s兲 ⬇
L
␺L共s兲 ⬇
␺R共s兲 ⬇
␺LL共s兲 ⬇
␺RR共s兲
⬇
冦
冦
冦冉
冉
冦
冦冉
1 −共x −x 兲冑s/D
e T,0 T
,
2s
冊
冊
1
1
冑
1 − e−共xT−xT,0兲 s/D , xT ⱖ xT,0 ,
2
s
1
fL
冑
1 − e−共xT,0−xT兲 s/D , xT ⱕ xT,0
2
s
f L −共x −x 兲冑s/D
e T T,0
,
2s
xT ⱖ xT,0 ,
f R −共x −x 兲冑s/D
e T,0 T
,
2s
xT ⱕ xT,0
冊
xT − xT,0
冑4Dt
冧
冧
冧
冉
冉 冑
冉 冑
冉
1
fL
1 − 共xT,0 − xT兲f L −
s
2
冑
fL
共1 − erf ␩兲,
2
1
␺L共t兲 = 共1 + erf ␩兲,
2
␺LL共t兲
fL
=1−
2
␺RR共t兲 = 1 −
fR
2
冑
冑
共E10兲
冊
冊
冊
1
fL
1−
s
2
D −共x −x 兲冑s/D
,
e T T,0
s
xT ⱖ xT,0 ,
1
fR
1−
s
2
D −共x −x 兲冑s/D
e T,0 T
,
s
xT ⱕ xT,0
1
fR
1 − 共xT − xT,0兲f R −
s
2
冑
冊
D −共x −x 兲冑s/D
e T T,0
, xT ⱖ xT,0 .
s
␺L共t兲 =
␺L共t兲 =
共E13兲
,
␺R共t兲 =
共E9兲
D −共x −x 兲冑s/D
e T,0 T
, xT ⱕ xT,0
s
冉 冊
冉 冑冊
fR
共1 + erf ␩兲,
2
1
␺R共t兲 = 共1 − erf ␩兲,
2
冧
冧
共E11兲
共E12兲
冉 冊
冉 冑冊
2␩
fL
1−
冑␲ ,
2
␺R共t兲 =
2␩
fR
1+
冑␲ ,
2
2␩
1
1+
,
2
␲
␺R共t兲 =
1
2␩
1−
,
2
␲
it leads to
␺L共t兲 =
共E8兲
1
fR
冑
1 − e−共xT−xT,0兲 s/D , xT ⱖ xT,0 ,
2
s
The remaining integrals follow from Eq. 共E16兲.
Equations 共E9兲–共E12兲 can be inverted exactly into time
domain. Using standard formulas 关44兴 and introducing
␩=
xT ⱕ xT,0
␺LR共t兲 =
fL
2
␺RL共t兲 =
fR
2
冑
冑
4Dt
共1 − 冑␲␩兲,
␲
4Dt
共1 + 冑␲␩兲,
␲
共E15兲
and 共arguments are left implicit兲
␺L + ␺R = 1,
4Dt −␩2
关e + 冑␲␩共erf ␩ − 1兲兴,
␲
4Dt −␩2
关e + 冑␲␩共erf ␩ + 1兲兴, 共E14兲
␲
␺LL + ␺LR = 1,
␺RR + ␺RL = 1.
共E16兲
The limit where t → 0 limit 共s → ⬁兲 is most conveniently
found from Eqs. 共E1兲–共E16兲 which, in combination with
L−1共s−1兲 = 1, read
where erf z is the error function 关44兴. The expressions above
are valid for times such that 4Dt / 共ᐉ / 2 ⫾ xT,0兲2 Ⰶ 1 and
4Dt / 共ᐉ / 2 ⫾ xT兲2 Ⰶ 1. Expanding the result in Eq. 共E14兲 for
small ␩ and using the normalization conditions 共E6兲 yield
051103-13
␺L共t → 0兲 =
␺L共t → 0兲 =
再
再
0, xT ⱕ xT,0
1, xT ⱖ xT,0 ,
f L , xT ⱕ xT,0
0, xT ⱖ xT,0 ,
冎
冎
共E17兲
共E18兲
PHYSICAL REVIEW E 80, 051103 共2009兲
L. LIZANA AND T. AMBJÖRNSSON
␺R共t → 0兲 =
␺LL共t → 0兲 =
␺RR共t
→ 0兲 =
再
再
再
0,
xT ⱕ xT,0
f R , xT ⱖ xT,0 ,
冎
1 − 共xT,0 − xT兲f L , xT ⱕ xT,0
1,
xT ⱖ xT,0 ,
1,
xT ⱕ xT,0
1 − 共xT − xT,0兲f R , xT ⱖ xT,0 ,
共E19兲
冎
冎
S共Q,t兲 =
1 1 1
N N! ᐉN
⫻
共E20兲
共E21兲
冕
APPENDIX F: MACROSCOPIC DYNAMICS—THE
DYNAMIC STRUCTURE FACTOR AND
CENTER-OF-MASS MOTION
+
S共Q,t兲 =
1
N
冕
R
dx1 ¯ dxN
冕
R0
冕
⫻ 兺 eiQ共xi−x j,0兲P共xជ ,t兩xជ 0兲Peq共xជ 0兲,
−ᐉ/2
ᐉ/2
dxN
冕
ᐉ/2
dx1,0
−ᐉ/2
−ᐉ/2
dxN,0 兺 eiQ共xi−x j,0兲P共xជ ,t兩xជ 0兲,
i,j
冕
冕
ᐉ/2
ᐉ/2
dx1
−ᐉ/2
−ᐉ/2
ᐉ/2
dx2 ¯
i
dxN,0
冋冉 兺
dxN
冕
ᐉ/2
dx1,0
−ᐉ/2
−ᐉ/2
eiQ共xi−xi,0兲
i=1
−ᐉ/2
j,0兲
ᐉ/2
N
ᐉ/2
dx2,0 ¯
冕
冊
册
共F4兲
+ remaining N ! − 1 terms .
Defining
C共Q兲 =
1
ᐉ
冕 冕
1
ᐉ2
ᐉ/2
ᐉ/2
dxi
−ᐉ/2
dx j,0 ␺共xi,x j,0 ;t兲eiQ共xi−x j,0兲 ,
−ᐉ/2
冕 冕
ᐉ/2
ᐉ/2
dxi
−ᐉ/2
−ᐉ/2
冕 冕
ᐉ/2
dx j,0
ᐉ/2
dxk
−ᐉ/2
dxl,0
−ᐉ/2
⫻␺共xi,x j,0 ;t兲␺共xk,xl,0 ;t兲eiQ共xi−xl,0兲 ,
共F5兲
and noticing that all the N! terms corresponding to permutations of the initial particle positions in Eq. 共F4兲 give the same
contribution lead to
S共Q,t兲 = C共Q兲C共0兲N−1 + 共N − 1兲F共Q兲C共0兲N−2 .
共F1兲
共F6兲
This result is identical to that of noninteracting particles
共obtained, e.g., by putting, Sij ⬅ 0, in the Bethe ansatz 共12兲
and allowing the particles to be in the full phase space x j
苸 关−ᐉ / 2 , ᐉ / 2兴兲. The expression above is simplified by noticing that C共0兲 = 1 due to the normalization of ␺共xi , x j,0 ; t兲.
As an example, we consider the case ᐉ → ⬁ and ⌬ = 0
where ␺共xi , x j,0 ; t兲 = 共4␲Dt兲−1/2exp关−共xi − x j,0兲2 / 共4Dt兲兴 can be
used. A straightforward calculation of Eq. 共F5兲 for this case
gives
共F2兲
2
C共Q兲 = e−DQ t ,
i,j
where we averaged over the 共equilibrium兲 initial positions.
Since the integrand above is invariant under xi ↔ x j, we can
apply the technique explained in Appendix C to extend the
integrations over coordinates as well as initial positions to
关−ᐉ / 2 , ᐉ / 2兴, which leads to
冕
eiQ共x −x
兺
i,j,i⫽j
F共Q兲 =
dx1,0 ¯ dxN,0
ᐉ/2
冕
⫻␺共x1,x1,0 ;t兲␺共x2,x2,0 ;t兲 ¯ ␺共xN,xN,0 ;t兲
1. Dynamic structure factor
where the angular brackets denote an average over different
realizations of the noise. In terms of the NPDF and the equilibrium NPDF 共see Sec. II兲, it is given by
−ᐉ/2
dx2,0 ¯
1 1 1
N N! ᐉN
−ᐉ/2
1
兺 具eiQ关Xi共t兲−X j,0兴典,
N i,j
−ᐉ/2
dx2 ¯
where also Eq. 共4兲 was used. Inserting explicitly the NPDF
from Eq. 共16兲 yields
⫻
S共Q,t兲 =
ᐉ/2
dx1
共F3兲
Again, the remaining integrals follow from the normalization
condition 共E16兲.
The dynamic structure factor is tractable via scattering
experiments 共see, e.g., Ref. 关66兴兲 and is widely used in
condensed-matter physics and crystallography. Considering a
set of noise-driven stochastic trajectories X1共t兲 , . . . , XN共t兲, the
dynamic structure factor is 关67兴
冕
冕
ᐉ/2
ᐉ/2
−ᐉ/2
S共Q,t兲 =
Macroscopic quantities for a single-file system are the
same as for a system consisting of noninteracting particles
关19,27兴. By “macroscopic” we here refer to any quantity
which is invariant under xi → x j, i.e., any one which does not
“notice” if two particles are interchanged in the system. In
this appendix we explicitly evaluate two such macroscopic
quantities: 共1兲 the dynamic structure factor and 共2兲 the PDF
for the center-of-mass coordinate. We demonstrate that indeed they agree with known results for noninteracting systems.
冕
F共Q兲 =
2␲
2
␦共Q兲e−DQ t ,
ᐉ
共F7兲
共F8兲
⬁
d␣ ei␣z was used. Inserting the results
where ␦共z兲 = 共2␲兲−1兰−⬁
above into Eq. 共F6兲 gives
051103-14
PHYSICAL REVIEW E 80, 051103 共2009兲
DIFFUSION OF FINITE-SIZED HARD-CORE…
2
S共Q,t兲 = e−DQ t,
Q ⫽ 0,
共F9兲
P共X,t兩xជ 0兲 =
P共X,t兩xជ 0兲 =
h共Q,x j0兲 =
冊
冕 冕
⬁
−⬁
dQ
2␲
冋冉
ᐉ/2
−ᐉ/2
⫻exp iQ X −
dx1 ¯
冕
=
⬁
−⬁
dkN −D共k2+¯+k2 兲t
1
N
e
2␲
⬁
dxi
−⬁
dk j −Dk2t
e j ␾共k j,x j,0兲eixi共k j−Q/N兲 .
2␲
共F14兲
The result in Eq. 共F13兲 is identical to that of noninteracting
particles, as can be shown straightforwardly by setting Sij
⬅ 0 in Eq. 共12兲 and not restricting the particles to the phasespace region R.
As an example, Eq. 共F13兲 is evaluated for an infinite system 共ᐉ → ⬁兲 and ⌬ = 0, where ␾共k j , x j,0兲 = e−ik jx j,0. For this
case Eq. 共F14兲 becomes
冉
h共Q,x j,0兲 = exp −
dxN
x1 + x2 + ¯ + xN
N
冕
dk j −Dk2t
e j ␾共k j,x j0兲g共k j,Q兲
2␲
−ᐉ/2
ᐉ/2
−ᐉ/2
⬁
ᐉ/2
where the NPDF P共xជ , t 兩 xជ 0兲 is given in Eq. 共12兲, and the
integration region R is defined in Eq. 共3兲. Since the integrand is invariant under xi ↔ x j, it is possible to extend the
integration region to x j 苸 关−ᐉ / 2 , ᐉ / 2兴 共provided we divide by
N!兲, as was done in the previous subsection. Also, using the
integral representation of the ␦ function from previous subsection, Eq. 共F10兲 can be rewritten as
1
N!
−⬁
dk1
¯
2␲
dQ iQX
e h共Q,x1,0兲 ¯ h共Q,xN,0兲, 共F13兲
2␲
冕
冕 冕
−⬁
x1 + x2 + ¯ + xN
P共xជ ,t兩xជ 0兲,
N
共F10兲
P共X,t兩xជ 0兲 =
冕
⬁
−⬁
where
冉
⬁
共F12兲
dx1dx2 ¯ dxN
⫻␦ X −
冕
⫻␾共k1,x1,0兲 ¯ ␾共kN,xN,0兲g共k1,Q兲 ¯ g共kN,Q兲.
The probability density function for the CM coordinate X
is given by 关69兴
R
dQ iQX
e
2␲
Integrating over k j leads to
2. Center-of-mass dynamics
冕
⬁
−⬁
which is in agreement with standard results for noninteracting particles in one dimension 共see, e.g., Ref. 关68兴兲.
P共X,t兩xជ 0兲 =
冕
冊册
P共xជ ,t兩xជ 0兲.
冊
Q2Dt ix j,0Q
−
,
N2
N
共F15兲
which when inserted in Eq. 共F13兲 and integrated over Q
gives the well-known Gaussian for the CM,
P共X,t兩xជ 0兲 =
共F11兲
冋
册
0
兲2
共X − XCM
1
exp
−
, 共F16兲
1/2
共4␲DCM t兲
4DCM t
Combining this equation with Eqs. 共12兲 and 共13兲 and definᐉ/2
dxi eixi共k j−Q/N兲 give
ing g共k j , Q兲 = 兰−ᐉ/2
0
N
where XCM
= 共兺i=1
xi,0兲 / N and DCM = D / N denote the CM initial position and the diffusion constant, respectively.
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The Bethe ansatz solution for particles jumping on a infinite
lattice was given in 关65兴. Our solution for the infinite system,
Eqs. 共12兲–共14兲 and ␾共k j , x j,0兲 = e−ik jx j,0, can be obtained from
the solution in 关65兴 by the replacements 兰20␲dk j / 2␲
⬁
→ 兰−⬁
dk j / 2␲ and by expanding the energy E共k1 , . . . , kN兲 and
the scattering coefficients Sij in a power series in momenta to
lowest order.
The Bethe ansatz solution is often written in terms of discrete
wave vectors 关64兴. Using the Poisson summation formula
⬁
⬁
gives the identity 兺m=−⬁
e2imk jᐉ = 共␲ / ᐉ兲兺m=−⬁
␦共k j + m␲ / ᐉ兲,
from which the Bethe ansatz solution, Eqs. 共12兲–共15兲, can
be rewritten in term of discrete wave vectors m j␲ / ᐉ 共m j
= −⬁ , . . . , −1 , 0 , 1 , . . . , ⬁兲 if desired.
A. D. Wunsch, Complex Application, 2nd ed. 共AddisonWesley, Reading, MA, 1994兲.
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Note that our convention, with coordinates in the range
关−L / 2 , L / 2兴, is different to Ref. 关37兴 where the left 共right兲 end
of the system is at 0 共L兲.
关47兴 Alternatively, one may be interested in the tPDF for a fixed
initial particle distribution xជ 0: ˜␳T共xT , t 兩 xជ 0兲 = 兰Rdx⬘1 ¯ dxN⬘ ␦共xT
− xT⬘ 兲P共xជ ⬘ , t 兩 xជ 0兲. For the case where all particles start at the
same position a straightforward integration yields, using Eq.
共16兲, a result for ˜␳T共xT , t 兩 xជ 0兲 which in the limit ᐉ → ⬁ is identical to that obtained in 关69兴, as it should.
关48兴 G. Szegö, Orthogonal Polynomials 共American Mathematical
Society, New York, 1959兲.
关49兴 The expression for the tPDF, Eq. 共25兲, is 共for ⌬ = 0兲 related to
Eq. 共61兲 in 关37兴 via elementary recursion relations for the Jacobi polynomial 关44兴.
关50兴 The variable ␰ defined in Eq. 共27兲 is related to the variable y in
Ref. 关37兴 according to ␰ = 1 / y.
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关54兴 We inverted the definition 共see Ref. 关51兴兲 cosh 2␨ = 共1 + ␰兲 / 共1
− ␰兲 to arrive at Eq. 共33兲.
关55兴 In more detail, in arriving at the SFD expression 共41兲
for ␳T共y T , t 兩 y T,0兲 we assumed 1 / N Ⰶ 1, ␦N / N Ⰶ 1,
共4Dt兲 / 共ᐉ / 2 ⫾ xT兲2 Ⰶ 1, 共4Dt兲 / 共ᐉ / 2 ⫾ xT,0兲2 Ⰶ 1, 兩␩兩 Ⰶ 1, xT,0 / ᐉ
Ⰶ 1, and 兩␦N共xT − xT,0兲兩 / ᐉ Ⰶ 1.
关56兴 The result in Eq. 共43兲 共large N兲 can also be obtain from Eq.
共35兲. Using I␣共z兲 兩zⰇ1 ⬇ ez / 冑2␲z 共but not assuming that
␺LR , ␺LR are small like in Sec. V B兲 one shows that the large-N
long-time result 共␰ → 1兲, as determined by Eq. 共35兲, agrees
with Eq. 共43兲.
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