Abstract
Driven diffusive systems have been a paradigm for modelling many physical, chemical, and biological transport processes. In the systems, spatial correlation plays an important role in the emergence of a variety of nonequilibrium phenomena and exhibits rich features such as pronounced oscillations. However, the lack of analytical results of spatial correlation precludes us from fully understanding the effect of spatial correlation on the dynamics of the system. Here we offer precise analytical predictions of the spatial correlation in a typical driven diffusive system, namely facilitated asymmetric exclusion process. We find theoretically that the correlation between two sites decays exponentially as their distance increases, which is in good agreement with numerical simulations. Furthermore, we find the exponential decay is a universal property of macroscopic homogeneous state in a broad class of 1D driven diffusive systems. Our findings deepen the understanding of many nonequilibrium phenomena resulting from spatial correlation in driven diffusive systems.
Introduction
Driven diffusive systems are of current interest in nonequilibrium statistical mechanics due to their rich and complex dynamic features^{1,2,3,4,5}. A simple and typical model in these systems is the asymmetric simple exclusion process (ASEP) describing particles hopping with hardcore repulsion along a one dimensional lattice unidirectionally. The ASEP was introduced in 1968 by MacDonald and Gibbs to model protein synthesis in organisms^{6}. Recently, numerous variants of ASEP have been developed to model biological transport^{7,8,9,10}, polymer dynamics in dense media^{11}, diffusion through membrane channels^{12}, traffic flow^{13,14}, and so on. Despite relatively simple rules, the ASEP and related models show a range of nontrivial macroscopic phenomena such as boundary induced and bulk induced phase transitions^{15,16,17,18}, spontaneous symmetry breaking^{19,20}, phase separation^{21,22,23,24,25}, and thus serve as basic tools to investigate the systems far from thermal equilibrium^{26,27,28}.
In driven diffusive systems, spatial correlation plays an important role in the formation of the diverse nonequilibrium phenomena^{29,30}. As an exceptional case, in the basic ASEP, the correlation is absent^{2,31}. Thus, the simple meanfield analysis is able to offer the exact current , where ρ is the system density and p is the hopping rate. In contrast, spatial correlation usually exists in general situations, which makes the traditional meanfield analysis incapable of rendering the theoretical solution. In most cases, numerical simulation is still the exclusive tool to explore the spatial and temporal correlation in driven diffusive systems. The increased use of cluster meanfield, is another method of testing the correlations^{32}. Some interesting phenomena have been observed from simulations. For instance, Gupta et al. found that density correlations display pronounced oscillations in both space and time, as a consequence of particles with extended length. The density autocorrelation has been found to decay exponentially at time increases, except at a special density when it decays as a power law^{33}.
Here we aim to offer analytical results of the spatial correlation in a representative driven diffusive system, namely facilitated asymmetric exclusion process that is subject to a generalized class of ASEP models. Specifically, in the model, the hopping probability of a particle depends on the occupancies of two neighboring sites: one ahead and one behind^{34}. The model was proposed to study nonequilibrium absorbing state phase transitions^{35}. It is relevant to several facts such as the particle mobility decreases as the local density increases in glassy dynamics^{36}, and a moving particle can exert a hydrodynamic force that pushes other particles along in molecular motor models^{37}. Moreover, due to particlehole symmetry, the facilitated exclusion process is the counterpart of the ASEP with nextnearestneighbor interaction as studied in ref. 14.
To explore the spatial correlation analytically, we first derive the joint occupancy probabilities in the facilitated asymmetric exclusion process theoretically. The formula of the joint occupancy probabilities allows us to provide the exact formula of the spatial correlation between any two sites in the model. The analytical results have been validated and are in good agreement with numerical simulations. Furthermore, we explore the spatial correlation in several other driven diffusive systems, finding that the spatial correlation decays exponentially in all the investigated systems. These observations suggest that the exponential decay of spatial correlation is a universal feature in 1D driven diffusive systems with macroscopic homogeneous state. The findings considerably deepen our understanding of the emergence of many nonequilibrium phenomena that stem from the nonlinear spatial correlation, such as the jamming in a variety of transport systems in biology and social systems.
Results and Discussions
The sketch of the facilitated exclusion process studied in this paper is shown in Fig. 1. The model rules are as follows. A particle at site i moves to site i + 1 with probability p if the front site i + 1 is empty and the rear site i − 1 is also empty. Otherwise, if the rear site i − 1 is occupied, the particle at site i hops to site i + 1 with probability q if site i + 1 is empty. In the model, random update rules and periodic boundary conditions are employed. In the special case p = q, the model reduces to the basic ASEP.
We consider the four joint occupancy probabilities P(τ_{i}, τ_{i+1}). Here denoting that site i is empty (τ_{i} = 0) or occupied (τ_{i} = 1). For the convenience of expression, we also use x, y and z to denote P(1, 1), P(1, 0) and P(0, 0) respectively. Note that due to symmetry, one has P(1, 0) = P(0, 1). Via a twocluster mean field analysis of the model (see section Methods), we can obtain
where
when p ≠ q. In the special case p = q, the solution is x = ρ^{2}, y = ρ(1 − ρ), z = (1 − ρ)^{2}. As demonstrated in ref. 38, the two cluster mean field results are exact solution of the system.
Next we investigate the correlations in the system based on the exact solution. We define the correlation between two sites as
Obviously μ = 0 if and only if there is no correlation between sites i and j. Note that there are four correlation coefficients, and we let μ_{1} (μ_{2}, μ_{3}, μ_{4}) denote the one with τ_{i} = 0 and τ_{j} = 0 (τ_{i} = 1 and τ_{j} = 0, τ_{i} = 0 and τ_{j} = 1, τ_{i} = 1 and τ_{j} = 1). Thus the correlation coefficient, say μ_{1}, can be expressed as
with n denoting the distance between sites i and j. Note that the classical correlation function is related to μ_{4} via .
Using the joint occupancy probabilities, we can derive the correlation coefficients (see section Methods)
and
where
Note that when y > ρ(1 − ρ), the four correlation coefficients vary alternatively between positive and negative values. In the special case p = q, μ = 0 as expected because y = ρ(1 − ρ).
Obviously, in the case of ρ = 0.5, . Figure 2 shows the exponential relationship between , , and n. The Monte Carlo simulations and the analytical expressions are in perfect agreement.
Now we investigate physical implication of the exponential decay of correlation in the driven diffusive systems. To this end, we study four different models of driven diffusive systems.
The KatzLebowitzSpohn (KLS) model^{39,40}. In the KLS model, particle hops with rate as follows: 1100 → 1010 with rate 1 + ε, 0101 → 0011 with rate 1 − ε, 0100 → 0010 with rate 1 + δ, 1101 → 1011 with rate 1 − δ. Here “1” denotes a particle and “0” denotes an empty site.
The DierlMaassEinax (DME) model^{41}. In the DME model, particle hops from site i to site i + 1 with rate .
The bus route model^{42}. In the model, a particle (bus) hops with rate 1 if there is no passenger at the site. Otherwise the particle hops with rate p < 1. At each empty site, passengers arrive with rate λ.
The bidirectional twolane model^{25}. In the model, particles move with opposite direction on two parallel lanes and do not change lane. The interlane interaction is implemented as particles slow down when there is a particle at the same site in the other lane, which mimics narrow road section. In this case, particle hopping rate p < 1. Otherwise, particle hops with rate 1.
Although we cannot derive the exact expression of correlation, numerical simulations show that the correlation also decays exponentially in the KLS model and the DME model, see Fig. 3. Note that in the KLS model, the DME model, and the facilitated ASEP, the system is always macroscopically homogeneous.
Figure 4(a) shows the plot of average velocity versus particle density in the bus route model. Figure 4(b) shows the plot of flow rate versus particle density in the bidirectional twolane model. In the bus route model, when the density is above a critical value ρ_{c}, the system is macroscopic homogeneous, see Fig. 5(a). However, below ρ_{c}, bus bunching occurs and the system becomes macroscopically nonhomogenous, see Fig. 5(b). In the bidirectional twolane model, the system is homogenous when density is below ρ_{c1} or above ρ_{c2}, see Fig. 6(a,b). When the density is in the range ρ_{c1} < ρ < ρ_{c2}, the system is nonhomogeneous because phase separation occurs, see Fig. 6(c).
Figures 7 and 8 show the correlation in the bus route model and in the bidirectional twolane model. One can see that when the system is homogenous, the correlation decays exponentially (Figs 7(a) and 8(a,b)). However, when the system is not homogenous, the correlation does not decay exponentially, which bends upward in the semilog plane (Figs 7(b) and 8(c)).
Our studies thus demonstrate that the exponential decay behavior of correlation might be a universal property in a broad class of 1D driven diffusive systems with macroscopic homogeneous state. This might be because there is a specific correlation length, which should be the same for homogeneous cases. However, such one length does not exist for inhomogeneous cases. Of course further efforts are needed upon this issue in the future work.
Methods
Mean Field Analysis
In the mean field analysis, the two equations
can be written easily. The third equation can be obtained via the master equation for P(1, 0) according to the evolution configurations as shown in Fig. 9, which presents the configurations at t and t + 1 as well as the corresponding transition probabilities. The first column shows all those configurations which can give rise to the configurations shown in the second column. The second column lists exhaustive clusters configurations with τ_{i} = 1, τ_{i+1} = 0. The third column presents the corresponding transition probabilities from the configurations in the first column to the corresponding configurations in the second column. Thus,
In the 2cluster mean field analysis, can be expressed mathematically as^{13,43,44,45}
where
is 2cluster conditional probability. Similarly, and can be expressed as
where
is also 2cluster conditional probability. So the probabilities of 4clusters and 3clusters involved in the righthandside of Eq. (11) can be expressed as follows
Note that the first P (1, 1, 0) in Eq. (11) corresponds to , thus
The second P (1, 1, 0) in Eq. (11) corresponds to , thus
which is identical to Eq. (20). This feature can be easily proved in the general case, since
where and t = m − 2 − s. This is independent of the location of i.
Substituting Eqs (17)–(21) into Eq. (11), we have the third equation about x, y, z
Solving the three Eqs (9), (10) and (22), we can obtain x, y, z as shown in Eqs (1)–(3).
Correlation Coefficient Analysis
Now we derive the correlation coefficient μ_{1}. We denote
which can be written as
Since
and
One has
Substituting Eq. (27) into Eq. (5) and simplifying, we can obtain
Since z = 1 − ρ − y and x = ρ − y, one can easily prove
Thus
where
From Eq. (30), we can easily prove that
via mathematical induction method. Substituting and into Eq. (32), one can derive μ_{1}(n), μ_{2}(n), μ_{3}(n) and μ_{4}(n) as shown in Eqs (6)–(8).
Additional Information
How to cite this article: Hao, Q.Y. et al. Exponential decay of spatial correlation in driven diffusive system: A universal feature of macroscopic homogeneous state. Sci. Rep. 6, 19652; doi: 10.1038/srep19652 (2016).
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Acknowledgements
This work is funded by the National Basic Research Program of China (No. 2012CB725404), the National Natural Science Foundation of China (Grant Nos 11422221 and 71222101), the Key Project of Natural Science Research in University of Anhui Province (Grant No. KJ2014A139), and Outstanding Young Talent Support Program in University of Anhui Province. RJ acknowledges the support of Senior Visiting Program of Shanghai Key Laboratory for Contemporary Applied Mathematics.
Author information
Affiliations
School of Mathematics and Computational Science, Anqing Teachers College, Anqing 246133, P. R. China
 QingYi Hao
School of Systems Science, Beijing Normal University, Beijing, 100875, P.R. China
 QingYi Hao
 & WenXu Wang
School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, P.R. China
 Rui Jiang
 & Bin Jia
School of Engineering Science, University of Science and Technology of China, Hefei 230026, P. R. China
 Rui Jiang
 & MaoBin Hu
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
 WenXu Wang
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Contributions
Conceived and designed the research: Q.Y.H., R.J., M.B.H., W.X.W. and B.J. Performed the research: Q.Y.H. and R.J. Wrote the paper: Q.Y.H., R.J. and W.X.W.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Rui Jiang or WenXu Wang.
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