Velocity correlations of a discrete-time totally asymmetric simple-exclusion process in stationary state on a circle
The discrete-time version of totally asymmetric simple-exclusion process (TASEP) on a finite one-dimensional lattice is studied with the periodic boundary condition. Each particle at a site hops to the next site with probability , if the next site is empty. This condition can be rephrased by the condition that the number of vacant sites between the particle and the next particle is positive. Then the average velocity is given by a product of the hopping probability and the probability that . By mapping the TASEP to another driven diffusive system called the zero-range process, it is proved that the distribution function of vacant sites in the stationary state is exactly given by a factorized form. We define -particle velocity correlation function as the expectation value of a product of velocities of particles in the stationary distribution. It is shown that it does not depend on positions of particles on a circle but depends only on the number . We give explicit expressions for all velocity correlation functions using the Gauss hypergeometric functions. Covariance of velocities of two particles is studied in detail and we show that velocities become independent asymptotically in the thermodynamic limit.
pacs:05.40.-a, 02.50.Ey, 02.50.-r
The one-dimensional totally asymmetric simple-exclusion process (TASEP) is a minimal statistical-mechanics model for driven diffusive systems of many particles with hardcore exclusive interaction Lig85 (); Spo91 (); SZ95 (); Lig99 (); Sch00 (); MKL09 (). In the present paper, we consider a discrete-time version of the TASEP on a circle, i.e. a one-dimensional finite lattice with periodic boundary condition, in which the parallel update rule is applied RSSS98 (). This process can be exactly mapped to another driven diffusive particle system called the zero-range process (ZRP), by regarding each number of vacant sites between successive particles in the TASEP as a number of particles at each site in the ZRP Eva00 (); EH05 ().
In the ZRP representation, particles hop from site to site on a lattice with a hopping probability which depends only on the number of particles at the departure site. We assume that there do not occur any creation and annihilation of particles in the ZRP. Then, if a site is vacant, there is no possibility that a particle hops from that empty site to other sites, as a matter of course. The configuration in the TASEP such that the next site of a particle is occupied by another particle is represented by the configuration in the ZRP such that the corresponding site is empty. Therefore the prohibition of hopping by hardcore exclusion in such a jamming situation in the TASEP is automatically satisfied in the ZRP. Moreover, the steady state of the ZRP is exactly described by the probability density function in a factorized form Eva00 (); EH05 (), and then the stationary distribution of vacant sites in the TASEP on a circle is explicitly determined as Eq.(5) given below Eva00 (); EH05 (); KNT06 (); Kan07 ().
The map from the TASEP to the ZRP is interesting from the viewpoint of quantum statistical mechanics, since it seems to be a map from an interacting Fermi gas to a free Bose gas. On the other hand, as demonstrated in the present paper, the procedure in which we analyze the TASEP/ZRP is quite different from the standard way for free Boson systems in the following sense. (i) Thermal equilibrium state of free Bose gas is usually treated in the grand canonical ensemble by introducing fugacity, while here we want to consider the driven diffusive system with a fixed number of particles and thus we treat the system in the canonical ensemble. Then the canonical partition function , where and denote the numbers of particles and vacancies in the TASEP, respectively, plays an important role in calculation. (ii) In the usual theory of free Bose gases, condensation of particles in a specified energy level is carefully studied (e.g., the Bose-Einstein condensation in a ground state). In the context of ZRP, however, distribution of vacancies (i.e., empty energy levels) should be well studied. The reason is that, if a site in the ZRP is occupied by one or more than one particle, then the velocity of the corresponding particle in the TASEP can be positive, but if the site is empty in the ZRP, the velocity of the particle in the TASEP is definitely zero. When we consider the TASEP as a simple model of traffic flow, the flux which is defined as the product of velocity and particle-density is the most important quantity. The purpose of the present paper is to study velocity correlations of particles in the stationary state of the TASEP on a circle.
Let and . For , we consider a one-dimensional lattice . Each site is either occupied by a particle, which is denoted by , or vacant denoted by . The following discrete-time stochastic process is considered for simulating the TASEP on a circle. Let . At each time , given a particle configuration , let , where the periodicity is assumed and the nearest-neighbor pair of sites is identified with . Every particle at site such that has chance to move to its next site , since the site is vacant; . But, in general, only a part of such particles move depending on the parameter as follows. We choose a subset of randomly, in the sense that each pair of nearest-neighbor sites is chosen independently with probability . The obtained subset of is written as . In the present paper, the total number of elements included in a set is denoted by and, for , the complementary set of in the set is expressed by . (By definition .) Then the probability that is chosen from is given by . Only the particles at sites such that indeed move to their next sites. That is, the particle configuration at time , , is given by
Here note that by definition of , if , then . The parameter is called the hopping probability and the above procedure is said to be the parallel update rule. The total number of particles is conserved in the process, which we write in the present paper. We assume .
Given a particle configuration , let and define
i.e., is the site occupied by the -th particle, . Then we put
We should note that does not determine uniquely, however, since the information on the position of the first particle, , is missing in the map . This information may be, however, not important, since here we consider the TASEP on a circle. The stochastic process , obtained from by this map, is a special case of the ZRP Eva00 (); EH05 (). As a consequence of general theory of ZRP Eva97 (); EH05 (); KNT06 (); Kan07 (), the probability distribution function in the stationary state of configuration of vacancies is uniquely determined as follows. Since the lattice size and the total number of particles are conserved, the total number of vacant sites is also a constant. We fix . Then the configuration space of is given by
where Remark1 ()
with the Gauss hypergeometric function AS72 ()
We write the expectation with respect to the stationary distribution (5) as in this paper.
For the -th particle, , if , that is, if , then that particle can move to the next site with probability in a time-step. Then if the velocity of -th particle is denoted by , the expectation of this random variable in the stationary distribution is given by
where is an indicator of an event ; if occurs, otherwise. The average velocity (9) is independent of , since the system is homogeneous in space, and it has been explicitly calculated as KNT06 (); Kan07 ()
The density of particles is given by
and the flux is defined by
If we plot versus , we obtain a fundamental diagram as demonstrated by Kanai Kan07 () (see Fig.1 in the present paper). Moreover, Kanai et al.KNT06 () determined the thermodynamic limit, i.e. the scaling limit of with keeping be a constant, for the average velocity and obtained (see Fig.1)
This result coincides with the exact solution for an infinite system obtained by Schadschneider and Schreckenberg SS93 ().
In the present paper, we study velocity correlation functions and show that, as extensions of the formula (10), they are generally expressed by using the Gauss hypergeometric functions. In particular, the covariance of velocities and of two particles at different sites,
is studied in detail and it is shown that
That is, velocities of particles are correlated in the stationary state in any finite systems, but it is proved that they become independent asymptotically in the thermodynamic limit in the discrete-time TASEP on a circle.
The paper is organized as follows. In Sec.II.A velocity correlation functions are defined and the general formula is derived. As special cases, the obtained expressions of average velocity and covariance of velocities of two particles are studied in detail in Sec.II.B and C, respectively. Sec.III is devoted to proving asymptotic independence of velocities (15) in the thermodynamic limit. Concluding remarks are given in Sec.IV. Appendix A is given for showing the derivation of a Riccati equation, which governs the covariance of velocities and is used in the proof in Sec.III.
Ii Velocity Correlation Functions
ii.1 General Formula
Let . From the particles, we pick up distinct particles arbitrarily; let , and consider a set of particles such that the -th particle in this set, , is originally the -th particle in the whole particle systems. Write the velocity of the -th particle by . The velocity correlation function for the particle is then defined by
Since the stationary distribution function is given by the factorized form (5), it is written as
where and .
We perform the binomial expansion
where the first summation in the RHS is taken over all subsets of and is the number of elements in the complementary set of in . Note that and implies , since the weight is assigned in the expansion. Therefore we can set in (17). Since we set as (6), we obtain
When , and thus . Therefore
Since the number of distinct subsets in satisfying is , (19) is equal to . The result does not depend on the choice of particle positions , but depends only on the total number of particles, whose velocity correlation is calculated. This special property comes from the factorized form (5) of the stationary distribution in the present system, in which the factor are independent of the system sizes, and , as given by (6). We summarize the result by the following formula,
It should be noted that still velocities are correlated in the sense that . In other words,
ii.2 Average Velocity
By setting in the general formula (20), we obtain
If we use the recurrence relation of the Gauss hypergeometric series AS72 ()
Next we apply the formula AS72 ()
to the first and the fourth terms in (23). Then the sum of them becomes
ii.3 Covariance of Velocity
By the first expression in (20) for , we obtain
Then the covariance of velocities (14) is given by
By the expression of partition function (7) using the hypergeometric function, it is written as
If we use the recurrence relation of the Gauss hypergeometric function AS72 ()
the second expression of (26) is rewritten as
Figure 2 shows given by (26) for with as a function of the density of particles . This figure shows that velocities are negatively correlated. We performed computer simulations for a variety of systems with and by changing the number of particles . For each trial, we discarded first time-step data and used time-step data for evaluating the covariance of velocities. In Fig.2, each dot indicates the averaged value of trials. We can find that the simulation data coincide very well with the exact solution (26).
Iii Asymptotic Independence of Velocities
iii.1 Riccati Equations
Here we consider the average velocity (10) as a function of the hopping probability and express it by
Kanai et al. KNT06 () proved that it solves a Riccati equation
iii.2 Large-size expansion and thermodynamic limit
Following the procedure given by KNT06 (), we consider the power expansion of the quantities with respect to the inverse of system size, with ,
where the coefficients are assumed to be functions of and . Putting (33) and (34) into (30) and its modification obtained by setting , and taking the thermodynamic limit with const., the first terms in (33) and (34) are determined as KNT06 ()
This result implies (13).
for , which is solved as
On the other hand, in the similar way, we can show that (27) gives
Vanishing of the covariance implies that velocities of particles of the discrete-time TASEP become independent asymptotically in the thermodynamic limit in the stationary state on a circle.
Figure 3 shows the numerical data demonstrating as . Here we set and and performed computer simulations by increasing from 100 to 1000 by 150, in which with and are evaluated. The linear fitting of versus gives
Iv Concluding Remarks
In the present paper, we study a discrete-time version of the TASEP on a circle developed by the parallel update rule. This version is easily simulated by a computer. As a matter of fact, we have checked the validity of our exact solutions for finite-size systems by comparing them with the numerical simulation data as shown by Fig.2.
From the viewpoint of statistical mechanics, it is important to discuss the thermodynamic limit for non-equilibrium steady states realized in the present model. It is rather difficult, however, to define parallel update dynamics for a system with an infinite number of particles, since the number of updated sites can be infinity, i.e., using the notation in Sec.I. Kanai et al.KNT06 () overcame this difficulty and determined the thermodynamic limit of average velocity (13). They obtained the differential equation which governs the average velocity and took the thermodynamic limit in the equation. In the present paper, we extend their procedure for the covariance of two-particle velocities and showed that it becomes asymptotically zero in the thermodynamic limit (39).
It should be remarked that both equations which govern the average velocity and the covariance are given by the Riccati-type differential equations with respect to the hopping probability . As a matter of course, the Riccati equation (31) for , which governs through (27), is coupled with the equations (30) for the average velocities (see (32)), and thus it becomes much complicated. Further study of hierarchy in the coupled system of differential equations which determine the third and higher-order moments of velocities will be an interesting future problem.
Acknowledgements.The present authors would like to thank M. Kanai for useful discussion on the problem. This work is supported in part by the Grant-in-Aid for Scientific Research (C) (No.21540397) of Japan Society for the Promotion of Science.
Appendix A Derivation of Riccati equation (31)
where we put
By the following formula of the Gauss hypergeometric function AS72 ()
(41) is written as
Here we consider the generalized hypergeometric differential equation