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TASEP Speed Process: An Effective Medium Approach
Aanjaneya Kumar and Deepak Dhar
Department of Physics, Indian Institute of Science Education and Research Pune
We discuss the approximate phenomenological description of the motion of a single second-class
particle in a two-species totally asymmetric simple exclusion process (TASEP) on a 1D lattice.
Initially, the second class particle is located at the origin and to its left, all sites are occupied with
first class particles while to its right, all sites are vacant. Ferrari and Kipnis proved that in any
particular realization, the average velocity of the second class particle tends to a constant, but this
mean value has a wide variation in different histories. We discuss this phenomena, here called the
TASEP Speed Process, in an approximate effective medium description, in which the second class
particle moves in a random background of the space-time dependent average density of the first class
particles. We do this in three different approximations of increasing accuracy, treating the motion
of the second-class particle first as a simple biassed random walk in a continuum Langevin equation,
then as a biased Markovian random walk with space and time dependent jump rates, and finally as
a Non-Markovian biassed walk with a non-exponential distribution of waiting times between jumps.
We find that, when the displacement at time Tis x0, the conditional expectation of displacement,
at time zT (z > 1) is zx0, and the variance of the displacement only varies as z(z−1)T. We
extend this approach to describe the trajectories of a tagged particle in the case of a finite lattice,
where there are Lclasses of particles on an L-site line, initially placed in the order of increasing
class number. Lastly, we discuss a variant of the problem in which the exchanges between adjacent
particles happened at rates proportional to the difference in their labels.
I. INTRODUCTION
There has been a lot of interest in understanding exclu-
sion processes on a line as a simple model of stochastic
evolution in systems of interacting particles [1]. These
are good models of many physical systems, such as traf-
fic on highways [2], transport in narrow channels [3] and
motion of motor proteins on microtubules [4]. Many ex-
act results are known for the simple exclusion process on
a line [5]. Several variants of the exclusion process have
been studied including multi-species exclusion processes
and the partially and totally asymmetric exclusion pro-
cesses (ASEP and TASEP) [1,5−8].
If we want to study the trajectory of individual par-
ticles in an assembly of interacting particles, one often
adopts a self-consistent mean-field kind of approxima-
tion, in which the motion of the particle occurs in an
effective field provided by the others. The best known
example being Brownian motion [9,10], that was first
studied to describe the motion of pollen grains in a liq-
uid. Other examples of self-consistent treatments include
the Hartree-Fock theory of electronic structure of atoms
[11,12], and the motion of ions in plasmas in the Vlasov
approximation [13].
In this paper, we will discuss this general approach,
called the effective medium approach here, in the specific
setting of a two-species totally asymmetric exclusion pro-
cess. We will consider a system of hard-core particles on
a 1-dimensional lattice, with two classes of particles. We
will consider the evolution from the special initial con-
dition, where there is only one second class particle at
the origin, and all sites sites to the left are occupied by
first class particles, and all sites to the right of the ori-
gin are vacant. The dynamics follows continuous-time
Markovian evolution where each first class particle ex-
changes position with a second-class particle or vacancy
to its right with rate 1. The second class particle can
jump to the left, if forced by a first class particle moving
from its left, or jump one space to an empty site on its
right, with rate 1.
For this problem, Ferrari and Kipnis found a rather
surprising observation [14]. In their own words, “a sec-
ond class particle initially added at the origin chooses
randomly one of the characteristics with the uniform law
on the directions and then moves at constant speed along
the chosen one.” This is a remarkable property as the sys-
tem undergoes Markovian evolution, and has no memory.
It happens, because if the second-class particle initially,
by chance, gets a large positive displacement, in subse-
quent times it encounters a smaller density of other par-
ticles, and hence also moves faster at later times. This
is an example of persistence, where time average of one
evolution history is very different from ensemble average
over all histories of evolution.
While the authors proved this result, they did not dis-
cuss how big are the fluctuations in the velocity, and how
they decrease with time. In this paper, we will describe
this process in a simple Langevin description [15], that
also allows us to estimate how the fluctuations in the
average speed decrease with time.
We will show that, when the displacement of the sec-
ond class particle at time Tis x0, the conditional expec-
tation of displacement, at time zT (z > 1) is zx0, and
the variance of the displacement only varies as z(z−1)T.
Thus the fluctuations, for fixed z, increase as √T. Equiv-
alently, we find that if v∗is the asymptotic value of ve-
locity of the second class particle, for large z, (v∗−x0/T )
has a typical spread of 1
√Twhich goes to 0as Tincreases.
The plan of this paper is as follows: in Section II, we
define the model precisely. In section III, we discuss the
2
description of the trajectory of the second-class particle
in a Langevin equation description. We use this to de-
termine the variance of the particle position at time zT ,
given the position at time T. In Section IV, we discuss
different approximations of increasing accuracy describ-
ing the trajectory as a biased random walk on the integer
lattice. We then use this approach to study the mean
trajectories of a second class particle when the lattice is
finite in Section V. We discuss the interesting end-effects
that occur in this finite lattice version, explain it using
our effective medium approach and outline an interesting
new direction that arises through a simple modification
in evolution rules. Section VI contains a summary and
concluding remarks.
II. DEFINITION OF THE PROCESS
We consider a two species TASEP with initial condi-
tions such that a single second class particle is located at
the origin of the lattice. To the left of the second class
particle, each lattice site is occupied by a first class par-
ticle and to its right, each site is vacant. We will denote
a first class particle by 1, a second class particle by 2
and a vacant site by 0. The allowed nearest neighbor
transitions in this process are:
10 →01; 20 →02; 12 →21
We assume that all these transitions occur with rate 1.
We wish to understand the dynamics of the second
class particle in this process. Ferrari and Kipnis proved
[14] that the position of the second class particle X(t)at
time tfollows:
lim
t→∞
X(t)
t=U
where Uis a uniform random variable on [−1,1].
We will call the process in which the velocity of the
second class particle tends to a random number dis-
tributed uniformly between [−1,1] as TASEP Speed Pro-
cess (TSP) and will provide a simple explanation of this
remarkable phenomena. The name TSP earlier has been
used in the study of joint distribution of the velocities of
different particles in a multi-species version of this pro-
cess [16].
III. LANGEVIN DESCRIPTION
We aim to understand the motion of the second class
particle, when all the sites to its left are occupied by first
class particles and all sites to its right are vacant, using a
simple approximation by breaking this problem into two
steps:
Figure 1. Trajectories of second class particle in the TASEP
Speed Process. 150 different trajectories consisting of 4000
steps taken by the second class particle have been plotted.
1. We first discuss how the mean density ρ(x, t)of
first class particles evolves in space and time, in
the absence of the second-class particle.
2. Then we try to describe the motion of the second
class particle moving as a random walk in a space-
time-dependent background field ρ(x, t).
We show that this simple description captures essential
features of TSP and allows for further analysis.
Let x(t)denote the position of the second class particle
at time t. We want to discuss the stochastic properties of
this trajectory, by integrating out all the first-class parti-
cles. The hydrodynamics of TASEP was first studied by
Rost [17]. The coarse-grained evolution of the sea of first
class particles in terms of particle density ρ=ρ(x, t)can
be described by the partial differential equation:
∂ρ
∂t + (1 −2ρ)∂ρ
∂x = 0 (1)
with
ρ(x, t = 0) = θ(−x),
where θ(x)is the step function, which is 0for x < 0,
and 1, for x > 0. The solution of this partial differential
equation is obtained to be:
ρ(x, t)=1 for x < −t(2)
ρ(x, t)=0 for x > t (3)
ρ(x, t) = 1
2(1 −x
t)for −t≤x≤t(4)
The motion of the second class particle is described by a
stochastic differential equation
dx
dt =¯
V(ρ(x, t)) + η(t)(5)
3
where ¯
Vequals the mean velocity of the particle, and
η(t)takes into account all the fluctuations away from the
mean. By definition, < η(t)>= 0.¯
Vis an externally
prescribed function of ρin Eq(5). In our problem, ¯
V=
1−2ρ, where rho is given by Eq(4). Hence we write
¯
V(ρ(x, t)) = 1 −2ρ(x, t)(6)
Substituting the value of ρ(x, t)from above:
dx
dt =x
t+η(t)(7)
This is a linear differential equation, and may be solved
by using an integrating factor. Equivalently, we make a
change of variables to v(t) = x(t)/t, the mean velocity of
the particle. This satisfies the simpler equation
dv(t)
dt =η(t)
t(8)
This is easily solved to give
v(zT )−v(T) =
zT
Z
T
η(t0)
t0dt0(9)
As η, by definition has zero mean and zis a real number
greater than 1. This gives < v(z T )>=< v(T)>.
We can also determine the variance of v(T):
D(v(zT )−v(T))2E=
zT
Z
T
zT
Z
T
< η(t0)η(t00)>
t0t00 dt0dt00 (10)
We expect correlation function <η(t)η(t0)>to be short-
ranged. It was shown for TASEP in [18] that correlations
< ρ(x, t)ρ(x0, t0)>are exponentially decreasing in time
|t−t0|, unless xand x0are such that x−x0=u(t−t0),
where uis the mean velocity of the flow. In our case,
we easily see that while the second-class particle sees a
constant density, the mean velocity of first class particles
is 1−ρ, and of second class particle is 1−2ρ, and they
are not equal. So, in general, the correlation function is
short-ranged, and if D=R+∞
−∞ dτ < η(t)η(t+τ)>, we
may write < η(t)η(t0)>=Dδ(t−t0)), which gives
<[v(zT )−v(T)]2>=D(z−1)
zT (11)
which goes to 0as Tincreases. This shows that the
velocity of the second-class particle does get fixed at large
T.
IV. APPROXIMATE RANDOM WALK
DESCRIPTIONS OF THE TRAJECTORY
While the Langevin description correctly describes the
long-time behavior of trajectory correctly, the actual
walk occurs on a discrete lattice, and a more accurate
description would be as a random walk on a line in con-
tinuous time. This we will develop now.
Figure 2. A schematic representing the motion of the second
class particle (blue) in the space-time dependent background
density plotted as a heat map.
A. Simple Biased Random Walk
In the spirit of the discussion above, consider motion
of a second class particle in uniform density ρ. The tra-
jectory then has mean velocity U= 1 −2ρ, and its time
evolution for times t >> 1can be discussed as a simple
random walk. It is known that if there is no second class
particle, then in the steady state, occupation numbers of
TASEP have a product measure. Then, in the steady
state of TASEP, with a fixed density ρ, if we assume that
we place a second class particle, with prob. (1 −ρ), its
site on the right will be empty, and then it jumps with
rate 1. Similarly, with probability ρ, the site on its left
will be occupied and it will overtake the second class
particle with rate 1. So, we conclude that trajectory of
particle is a biased random walk. On a background den-
sity ρ(x, t), the second class particle jumps to the left
with rate ρ(x, t), and to the right with rate 1−ρ(x, t).
As a check, the mean velocity is U= 1 −2ρ(x, t), which
agrees with the exact asymptotic value of velocity [19].
We have simulated this walk on the background ρ(x, t)
given by eq. 2-4. The results are shown in Figure 3. We
also compare with the the simulation of the original pro-
cess (Figure 1). We see that while we do get trajectories
with velocity fixation, and the velocity Uis uniformly
distributed in the interval [−1,1], the time taken by the
walker to take 4000 steps is roughly the same while, the
time taken in TSP shows a clear ρdependence.
B. Markovian Continuous Time Random Walk
This shows that the rates of left and right jumps in
our simple approximate model do not correctly describe
the trajectories of the original problem. The difference
occurs because the average density of first class particles
near the second-class particle is not the same as in the
4
Figure 3. Trajectories of a CTRW with rate ρ(x, t)of jump-
ing to the left and 1−ρ(x, t)of jumping to the right. 150
different trajectories consisting of 4000 jumps made by the
random walker have been plotted. Notice that the time taken
by the walker to take 4000 jumps is roughly the same in each
trajectory which is not the case in TSP.
bulk, away from the second class particle. Hence our ap-
proximation of using the steady state measure of TASEP
to calculate jump rates in the problem with a second class
particle is present not adequate. The average density pro-
file near a second class particle in the steady state has
been calculated in [20] using the matrix product ansatz.
It was shown that the second class particle is attracted to
regions with a positive density gradient. More precisely,
it was found that for a second class particle on a ring
with density ρof first class particle, in the steady state,
the mean density on the site to the right is 2ρ−ρ2and
on the site to the left is ρ2. This implies the probabil-
ity of the site to the right being empty is (1 −ρ)2. If
we use a continuous time random walk model with jump
rates (1 −ρ)2to the right and ρ2to the left we still get
mean velocity = 1−2ρ. But now the agreement with the
simulations is much better as seen in Figure 4.
C. Continuous Time Random Walk with Waiting
Time Distributions
However, our description is still not sufficiently ac-
curate. If we are given a single long trajectory of the
second-class particle, with mean velocity Uin the orig-
inal process between times Tand nT , for T >> 1, and
also one generated using the Markovian jump rates de-
scibed above, can one distinguish between them? The
answer is yes. Clearly, in the Markovian approximation,
the waiting times between successive jumps are indepen-
dent random variables, with a distribution that is a sim-
ple exponential. One can easily verify that in the original
process the waiting time intervals do not have an expo-
nential distribution. This comes from the fact that since
Figure 4. Trajectories of second class particle in the CTRW
model with jump rates (1 −ρ(x, t))2to the right and (ρ(x, t))2
to the left. 150 different trajectories consisting of 4000 steps
taken by the second class particle have been plotted.
occupancy of neighbors by first class particles have non-
trivial correlations in time, so the probabilities of jump in
nearby time intervals [t, t+∆t1]and [t+∆t1, t+∆t1+∆t2]
are not uncorrelated.
The trajectories in the TASEP Speed Process show
a nearly exponential distribution of waiting times only
for velocities close to 1and −1whereas a clear depar-
ture from the exponential distributiom is observed for
smaller velocities. Figure 5 shows the distribution of
waiting times for the TSP and the CTRW model with
jump rates. The histograms were plotted for two trajec-
tories, consisting of 8000 steps taken by the second class
particle, of velocity (a) 1 and (b) 0.2. It is clearly seen
that an even more accurate modelling of the trajectory
will be as a random walk which involve a continuous time
non-Markovian walk with a prescribed distribution of res-
idence times f(τ), with probability to jump left or right
given by p(τ)and 1−p(τ). The calculation of the exact
functions f(τ)and p(τ)is rather difficult, and will not be
attempted here. We can take these to be approximately
determined from simulations.
Of course, even this modelling of the trajectory as a
continuous time random with a distribution of waiting
times is approximate. In the original process, the waiting
times between successive jumps are only approximately
uncorrelated. But going beyond this description falls out-
side our aim of providing a simple approximate descrip-
tion of the trajectories.
V. EXTENSIONS OF OUR APPROACH
This work can be extended to the case of a multi-
species partially asymmetric exclusion process (ASEP).
It was conjectured in [17] that even in partially asym-
metric case, the asymptotic velocity tends to a uniformly
5
Figure 5. Waiting time distributions for trajectories with
speeds (a) 1 and (b) 0.2 in the CTRW model (black circle)
and the TASEP Speed Process (red circle). For speeds close
to 1, the waiting time distribution for the TASEP Speed Pro-
cess matches closely with the exponential waiting times of
the CTRW model. However, for intermediate speeds, a clear
departure from exponential waiting time distribution is ob-
served.
distributed random variable. More precisely, in this case,
if we start with the initial conditions as before and look
at the motion of the second class particle, then:
lim
t→∞
X(t)
t=Up
where X(t)is the position of the second class particle
at time tand Upis a uniform random variable between
[−(2p−1),(2p−1)] where pis the rate of jumping to
the right (1−pbeing the rate of jumping to the left).
An Langevin description can be developed for this as the
evolution of density for first class particles is given by:
∂ρ(x, t)
∂t + (2p−1)(1 −2ρ(x, t)) ∂ρ(x, t)
∂x = 0 (12)
Even in the case of multi-species ASEP, our analysis goes
through and the fluctuations about the average velocity
die out as t−1/2.
The finite lattice version of the multi-species prob-
lem [21] offers an interesting extension to the effective
medium approach. The system considered is a finite lat-
tice with nsites in which, each site is occupied by a
particle and its class is labeled by its initial position on
the lattice. The time evolution of the system is given by
the stochastic nearest neighbor exchange rule:
ij rate 1
−−−−−→ ji for all i<j
If we wish to study the dynamics of a tagged particle
of the k-th class, it is clear that the problem is, again,
reducible to a two species problem with particles of l-th
class (l < k) being equivalent to first class particles, par-
ticles of m-th class (m > k) being holes and the tagged
particle being the second class particle.
The motion of a tagged particle is strongly affected by
the ends and displays an interesting behaviour - initially,
its dynamics of the tagged particle mimics the dynamics
of a tagged particle in TSP on an infinite line. However,
at later times, the particle reaches a growing impenetra-
ble region of density 1 (0) on the right (left) and travels
along with it, remaining at the moving end of this re-
gion at subsequent times, till its absorbing position. This
is expected as the lattice is finite and after some time,
clearly the first class particles (holes) start to get accu-
mulated at the right (left) boundary. It is interesting to
note that the absorbing position of a particle whose ini-
tial position was kis always n−k. This behaviour can
be described by CTRW model with jump rates given by
the following background density:
ρ(x, t) = 0 for x≤l(t)
1for x≥n−r(t)
1
2(1 −x−k
t)for −t+l(t)< x −k < t −r(t)
1for l(t)≤x<k−t+l(t)
0for k+t−r(t)≤x<n−r(t)
(13)
l(t)and r(t)are the mean widths of impenetrable re-
gions of density 1 and 0 on the right and left boundary
respectively and they satisfy:
r(t) = 0 for t ≤(n−k)
t+ (n−k)−2pt(n−k)for ts≥t > (n−k)
k for t > ts
l(t) = 0 for t ≤k
t+k−2√tk for ts≥t>k
n−k for t > ts
(14)
where tsis the mean time taken by the tagged particle
to reach its absorbing position and is given by ts=n+
2pk(n−k).
We simulated the process on a lattice of 1000 sites and
looked at the various trajectories of the particle labelled
200. Figure 6 shows 150 such trajectories and Figure 7
shows analogous results for the effective medium descrip-
tion using a continuous time random walker on a line with
jump rates determined by the background density given
in Eq(13).
6
Figure 6. 150 different trajectories of the particle labelled 200
in the finite lattice version of TSP with 1000 particles.
Figure 7. 150 different trajectories of a continuous time ran-
dom walker with space time dependent jump rates deter-
mined by the evolving density of first class particles defined
by Eq(13).
As an interesting variation to the multi-species prob-
lem, consider a 1D lattice where each lattice site is oc-
cupied with a particle and the class of each particle is
labeled by its position on the lattice with only the fol-
lowing nearest neighbor transitions allowed:
ij rate (j−i)α
−−−−−−−−−−→ ji for all i<j
This model is clearly a better model for traffic flow if
we visualize the x-coordinate of particles to not be their
position in real space, but their relative order on the road
as the overtakings between two particles happen at rates
proportional to the difference between their labels (which
is a proxy for the velocity).
Some special cases of this model have been studied
before [6 −8]. It is known that the steady state of such
a model on a 1D lattice with open boundary conditions
and α= 1, in which there was an additional feature that
particles could enter the system from the left end and
leave from the right at rates depending on their labels,
can be obtained by a matrix product ansatz. This was
Figure 8. 150 trajectories consisting of 3000 jumps made by a
tagged particle in the modified multi-species exclusion process
with α= 0.5.
later generalized to obtain the steady state properties of
this system on a ring.
We consider the dynamics of a tagged particle in this
modified multi-species exclusion process on an infinite
line for a general αwhere particles do not enter or exit the
system. This problem cannot be reduced simply to the
2-species problems as each particle interacts with every
other particles differently. However, we find that some-
thing analogous to the “velocity selection” in TSP hap-
pens in this process as well. The trajectories of the tagged
particle in this process seem to belong to the family of
curves t=ax1−αfor α6= 1 and t= ln ax for α= 1 where
aparametrizes the trajectories. A heuristic argument for
this is as follows:if a particle has moved distance xin
time t, then the typical change in velocity it encounters
with its neighbor is proportional to x. Then dx/dt ∼xα
implies that x∼t1/(1−α). for α6= 1 and t∼ln xfor
α= 1.
Figure 9. Trajectories of the tagged particle in the modified
multi-species exclusion process with α= 0.5after time coor-
dinate tis scaled as t
2√x. 150 different trajectories are plotted
showing that all the trajectories belong to a family of curves
given by t∼√x.
We demonstrate numerical results of our scaling in Fig-
ures 8 and 9 for α=1
2. Figure 8 shows 150 different
7
space-time trajectories of the process and Figure 9 shows
the trajectories when the time coordinate tis scaled as
t
2√xwhere we have only chosen the trajectories whose
displacement always remains positive. We see that t/√x
is nearly constant for each trajectory, but different trajec-
tories have very different values of this variable. Finding
the distribution of the asymptotic value of aover different
trajectories remains an open problem.
VI. SUMMARY AND CONCLUDING
REMARKS
In summary, we discussed the effective medium ap-
proach to describe the motion of a tagged particle in the
time-evolving background other particles. We provided
a simple Langevin description of the dynamics, that cap-
tures the key features of the large-scale behavior of TSP,
and also calculated the variance of the average velocity
within one history, and for different histories. We dis-
cussed how to improve the effective medium description
to take into account different additional features of the
trajectories. These were approximating the trajectory
as a biassed random walk, with rates of walk calculated
from the steady state of the TASEP, in the absence of
the second-class particle. We found that to get a quanti-
tative agreement with the original process, one has to in-
corporate the modification of the average density profile
that occurs near the second-class particle. Also, while
the time evolution of the original process was Marko-
vian by definition, the evolution of the projected process
is non-Markovian. This is most easily seen in the non-
exponential distribution of waiting times between jumps
in the motion of the second-class particle. We proposed
a non-Markovian continuous time random walk with a
distribution of waiting times between jumps as a good
description of this. We later extended our approach to a
finite lattice version of the TSP and studied the trajec-
tories of a tagged particle in this process. In this case,
interesting end-effects are seen which we explained us-
ing the effective medium approach. Lastly, we looked
at a modified multi-species exclusion process in which
exchanges between adjacent particles happened at rates
proportional to the difference in their labels. We showed
that in this process too, the motion of a tagged particle
shows a behaviour in which it initially “chooses” a trajec-
tory from a family of curves and sticks to it asymptoti-
cally. A better understanding of our heuristic arguments,
and numerical observations about this process for a gen-
eral αseems to be an interesting problem for further
study.
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