Green functions for the TASEP
with sublattice parallel update
We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with the sublattice parallel dynamics describing particles moving to the right on the one-dimensional infinite chain with equal hoping probabilities. Using sequentially two mappings, we show that the model is equivalent to the TASEP with the backward-ordered sequential update in the case when particles start and finish their motion not simultaneously. The Green functions are obtained exactly in a determinant form for different initial and final conditions.
pacs:05.40.-a, 02.50.Ey, 82.20.-w
Keywords: Totally asymmetric exclusion process, sublattice parallel update, backward sequential update, Green function.
The totally asymmetric simple exclusion process (TASEP) is a stochastic interacting particle system which serves as a paradigmatic model for nonequilibrium behaviour Spoh91 (); Ligg99 (); Gunter (). The dynamics of this lattice gas model is characterized by the updating law. In one dimension on the integer lattice the most important cases of discrete-time updates are the backward-sequential, parallel and sublattice parallel updates Raje98 (). For a finite number of particles these dynamics can be defined through a master equation of the form
where describes the positions of the particles and is the transition probability to go in one time step from a configuration to a configuration . This transition probability is different for the various update schemes. For the backward-sequential update, each particle may take one step to the right with probability if the target site is vacant at the beginning of the time step or becomes vacant at the end of the time step (due to motion of the particle in front). For the parallel update, the motion to the right is allowed only if the target site is vacant at the beginning of the time step. By iterating (1) one obtains the solution of the master equation for any given initial configuration , i.e., the conditional probability to find a particle configuration at time step , given that the process started from configuration . This stochastic many-body dynamics have a natural interpretation in field-theoretic terms Matt98 (); Gunter () where specific realizations of the process correspond to paths in the path integral representation of field theoretic quantities. Therefore, in analogy to the corresponding terminology in field theory, we refer to this time-dependent conditional transition probability as the Green function.
For the first two cases, backward-sequential and parallel update, the Green functions of transition probabilities have been found by explicit solution of the master equations for the systems defined on an infinite lattice Shelest (); Rako05 (); parallel (). The Green function has a determinantal representation similar to the one first discovered for the continuous-time definition of the process Schu97 () where particles jump independently after an exponentially distributed random time with fixed rate 1 Spoh91 (); Ligg99 (); Gunter (). This representation allows for a direct derivation of the current distribution Joha00 (); Naga04 (); Rako05 () and has inspired a considerable amount of subsequent detailed analysis of dynamical properties of the TASEP and related models, see e.g. Sasa05 (); Sasa07 (); Boro07 (); Boro08 () and also of the ASEP where particles are allowed to jump in both directions Tracy1 (); Tracy2 ().
The third type of discrete-time update, sublattice parallel, was first considered in some detail in sublattice (); open () and has subsequently been studied for various applications both analytically and numerically Raje98 (); anal (). In this paper, we derive the Green function of the TASEP with sublattice parallel update which is defined as follows.
Consider the process on , i.e. the one-dimensional infinite chain. Each site labeled by an integer is occupied by at most one particle which can hop only to the right in a discrete time. At the first moment of time, we look at all pairs. In each of them if the vertex is free and the site is occupied, the particle of the vertex hops to the right with probability and doesn’t move with probability . If both sites in a pair are occupied or empty or if site is empty and site occupied, the pair remains unchanged at that time step. At the next step of time we apply this rule of hopping to all pairs . Continuing, we apply the updating rule to pairs at each odd moment of time and to pairs at each even moment111We remark that these dynamics can be interpreted as the action of the transfer matrix for the six vertex model on a diagonal lattice Kand90 (); sublattice (); open (); Hone97 ()..
Ii The equivalence of the TASEP with sublattice parallel and the backward sequential updates
As noted above, the conditional probability to find particles at positions at discrete time if these are in positions at initial moment of time is called the Green function of the process. The discrete space-time dynamics can be described by a set of trajectories on a triangle lattice which is obtained from the square lattice by adding a diagonal bond between the upper left and the lower right corners of each elementary square. Being occupied by an trajectory, diagonal bonds have a statistical weight and vertical ones can have weights or . It is convenient to draw trajectories of particles on a chessboard (Fig. 1), where black rounds show initial positions of particles. We notice that diagonal bonds of trajectories can be located only on white squares.
If we select a sublattice which contains upper left and lower right sites of white squares of the chessboard denoted by white circles in Fig. 1, we can see that particles effectively move on the sublattice of white vertices. There are some exceptions at the start and at the end of trajectories. Then, we have to consider four different cases to find a generalized determinant formula of the Green function.
Consider first the case when space-time trajectories of particles start and end on the sublattice with white vertices, Fig. 2a (the case of arbitrary initial conditions will be considered in the next section). If we choose initial points on the white vertices of the first row, coordinates of particles at initial times are even.
The first transformation we use is a rotation of the set of trajectories by clockwise around the initial point . Considering the vertical axis as the new time coordinate, we obtain a new set of trajectories (Fig. 2b). This set represents a new discrete-time process on the square lattice with the unit time and space intervals corresponding to vertical and horizontal distances between neighboring sites. Starting points change their space-time coordinates as
Now vertical bonds have weights and diagonal ones have weights or . We want to map them to the space-time paths of particles of the TASEP with backward-sequential update. To this end, we shift the coordinates in each row with respect to the previous above row by . Due to the second transformation, vertical and diagonal bonds are interchanged as it is shown in Fig. 3. The transformation of coordinates (in new units) can be written as
From Fig. 3 we see that worldlines on the transformed lattice represent trajectories of particles of the TASEP with backward-sequential update with initial space-time coordinates and final coordinates . The transition probability from space-time coordinates to is given by generalized determinant formula Shelest ()
where the matrix elements of matrix are
with the function introduced in Shelest ()
with matrix elements
Iii Other cases of starting and ending points
Consider the case when the -th particle starts its motion from an odd site and ends on an even one (Fig. 4a).
As the weight of the first bond of the -th trajectory is , we can set the beginning of motion at the nearest white (sublattice) site and then rotate the sublattice as in the previous case (Fig. 4b). Applying the shift transformation, we obtain for the initial coordinates of -th particle:
where is the ceiling function. Substituting expressions of all starting and end points into the determinant formula (4), we derive the Green function for this case.
The third case is when the -th trajectory starts from the white (even) vertex and ends on a non-white one. We see, that if we add an additional vertical bond with weight to the last node of that trajectory, the total weight of the whole path will not be changed. Then we can set the endpoint of -th particle at time . Repeating two transformations, we obtain for coordinates of the end point:
The last case when the -th trajectory starts and ends on non-white vertices is an obvious combination of the second and the third case.
Generalizing all four cases of boundary conditions, we derive following transformations for the initial and final coordinates for all types of trajectories
Having explicit determinant expressions for the Green function, we can compare their relative advantages and disadvantages for the three basic updates, the backward-sequential, parallel and sublattice-parallel one. Criteria for the comparison follow from practical use of the Green function in probabilistic calculations. To find a probability distribution for a selected particle or a correlation function for several particles in the TASEP, detailed information contained in function should be reduced by summation over a part of the final coordinates for fixed initial coordinates (see e.g. Rako05 ()). Then, the first criterion for the comparison is simplicity of the summation procedure in different cases. The second criterion is simplicity of the matrix itself, because asymptotic calculations for large and need an elaborated analysis of resulting determinant expressions (see e.g. Boro07 (); Boro08 ()). The third criterion is the presence or lack of particle-hole symmetry, which is essential for the derivation of single-particle probability distributions in some particular cases Rako05 ().
(A) The backward-sequential update. The form of the matrix elements in this case is especially simple
where function is given by Eq.(6). The Green function is uniform in variables , so the summation procedure is straightforward Rako05 (). A shortcoming of this update is lack of the particle-hole symmetry. Indeed, due to possible transitions for one time step , a hole can move in the opposite direction by jumps of length .
(B)The parallel update. The form of the matrix in this case is more complicated parallel ():
The Green function obeys the particle-hole symmetry, but a drawback is in the determinant formula
which depends on the number of pairs of neighboring particles in the final configuration. Therefore, the sum over splits into groups by the number of clusters of connected particles.
(C) The sublattice parallel update. The Green function for this update is free of shortcomings of two previous cases. It is uniform in variables , obeys the particle-hole symmetry and has a relatively simple analytical form (LABEL:final). A fee for this advantage is a rather complicated time dependence in (LABEL:final) which involves both initial and final coordinates and the ceiling function . Thus, we may conclude that a proper choice of the discrete time Green function strongly depends on peculiarities of the corresponding probabilistic problem.
This work was supported by the RFBR grants 07-02-91561-a, 09-01-00271-a and the DFG grant 436 RUS 113/909/0-1(R).
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