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Gupta A, Pal B, Jindal A, Bhatia N, Gupta AK. Modelling of transport processes: Theory and simulations. MethodsX 2022; 10:101966. [PMID: 36578289 PMCID: PMC9791607 DOI: 10.1016/j.mex.2022.101966] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 12/11/2022] [Indexed: 12/15/2022] Open
Abstract
The transport processes, being a non-equilibrium system, have been a point of interest for physicists since many years revealing and explaining several unexpected effects. Such systems are often dealt with an archetypal model, known as totally asymmetric simple exclusion process, with two different types of boundary conditions: open and periodic. Moreover, these models are analyzed in two varieties of dynamics, random sequential and parallel updates, even at the micro level which play an important role in the global dynamics of the system. On contrary to the random sequential rule, the parallel updates introduce correlations in the system. Using theoretical and numerical methods in the framework based on mean-field approaches, the system properties are analyzed in both transient and steady state.•Both the updating rules are realized using Monte Carlo simulations.•In simplest form, mean-field approach ignores all the correlations and the results coincide with the random sequential update.•Correlations are induced in the system due to parallel update, therefore, a cluster mean-field theory is also discussed to handle them.
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Affiliation(s)
- Ankita Gupta
- Department of Mathematics, Indian Institute of Technology, Ropar Rupnagar-140001, Punjab, India
| | - Bipasha Pal
- Department of Mathematics, Indian Institute of Technology, Ropar Rupnagar-140001, Punjab, India
| | - Akriti Jindal
- Department of Computer Science, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada
| | - Nikhil Bhatia
- Department of Mathematics, Indian Institute of Technology, Ropar Rupnagar-140001, Punjab, India
| | - Arvind Kumar Gupta
- Department of Mathematics, Indian Institute of Technology, Ropar Rupnagar-140001, Punjab, India,Corresponding author.
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Song Xiao, Chen X, Liu Y. Bidirectional Totally Asymmetric Simple Exclusion Process with a Bottleneck and Different Hopping Rates. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2021. [DOI: 10.1134/s1990793120060317] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Bunzarova NZ, Pesheva NC, Brankov JG. One-dimensional discrete aggregation-fragmentation model. Phys Rev E 2019; 100:022145. [PMID: 31574637 DOI: 10.1103/physreve.100.022145] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2019] [Indexed: 11/07/2022]
Abstract
We study here one-dimensional model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time kinetics of the generalized totally asymmetric simple exclusion process (gTASEP) on open chains. The gTASEP is essentially the ordinary TASEP with backward-ordered sequential update (BSU), however, equipped with two hopping probabilities: p and p_{m}. The second modified probability p_{m} models a special kinematic interaction between the particles of a cluster in addition to the simple hard-core exclusion interaction, existing in the ordinary TASEP. We focus on the nonequilibrium stationary properties of the gTASEP in the generic case of attraction between the particles of a cluster. In this case the particles of a cluster have higher chance to stay together than to split, thus producing higher throughput in the system. We explain how the topology of the phase diagram in the case of irreversible aggregation, occurring when the modified probability equals unity, changes sharply to the one, corresponding to the ordinary TASEP with BSU, as soon as the modified probability becomes less than unity and aggregation-fragmentation of clusters appears. We estimate various physical quantities in the system and determine the parameter-dependent injection and ejection critical values by extensive computer simulations. With the aid of random walk theory, supported by the Monte Carlo simulations, the properties of the phase transitions between the three stationary phases are assessed.
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Affiliation(s)
- N Zh Bunzarova
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia.,Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
| | - N C Pesheva
- Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
| | - J G Brankov
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia.,Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
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Bunzarova NZ, Pesheva NC. One-dimensional irreversible aggregation with dynamics of a totally asymmetric simple exclusion process. Phys Rev E 2017; 95:052105. [PMID: 28618542 DOI: 10.1103/physreve.95.052105] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2016] [Indexed: 11/07/2022]
Abstract
We define and study one-dimensional model of irreversible aggregation of particles obeying a discrete-time kinetics, which is a special limit of the generalized Totally Asymmetric Simple Exclusion Process (gTASEP) on open chains. The model allows for clusters of particles to translate as a whole entity one site to the right with the same probability as single particles do. A particle and a cluster, as well as two clusters, irreversibly aggregate whenever they become nearest neighbors. Nonequilibrium stationary phases appear under the balance of injection and ejection of particles. By extensive Monte Carlo simulations it is established that the phase diagram in the plane of the injection-ejection probabilities consists of three stationary phases: a multiparticle (MP) one, a completely filled (CF) phase, and a "mixed" (MP+CF) one. The transitions between these phases are: an unusual transition between MP and CF with jump discontinuity in both the bulk density and the current, a conventional first-order transition with a jump in the bulk density between MP and MP+CF, and a continuous clustering-type transition from MP to CF, which takes place throughout the MP+CF phase between them. By the data collapse method a finite-size scaling function for the current and bulk density is obtained near the unusual phase transition line. A diverging correlation length, associated with that transition, is identified and interpreted as the size of the largest cluster. The model allows for a future extension to account for possible cluster fragmentation.
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Affiliation(s)
- N Zh Bunzarova
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia.,Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
| | - N C Pesheva
- Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
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Aneva BL, Brankov JG. Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring. Phys Rev E 2016; 94:022138. [PMID: 27627277 DOI: 10.1103/physreve.94.022138] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2016] [Indexed: 11/07/2022]
Abstract
We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, p and p[over ̃]. The model contains as special cases the TASEP with parallel update, when p[over ̃]=0, and with sequential backward-ordered update, when p[over ̃]=p. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether p[over ̃]>p or p[over ̃]<p. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation p[over ̃]→1, when the stationary configurations contain just one cluster.
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Affiliation(s)
- B L Aneva
- Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
| | - J G Brankov
- Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia.,Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
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Hao QY, Jiang R, Hu MB, Wu QS. Mean-field analysis for parallel asymmetric exclusion process with anticipation effect. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:022103. [PMID: 20866859 DOI: 10.1103/physreve.82.022103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2010] [Indexed: 05/29/2023]
Abstract
This paper studies an extended parallel asymmetric exclusion process, in which the anticipation effect is taken into account. The fundamental diagram of the model has been investigated via cluster mean field analysis. Different from previous mean field analysis, in which the n -cluster probabilities P(σ{i},…,σ{i+n-1}) involve the (n+2) -cluster probabilities P(τ{i-1},…,τ{i+n}) , our mean-field analysis is asymmetric because the three-cluster probabilities P(σ{i},σ{i+1},σ{i+2}) involve the six-cluster probabilities P(τ{i-1},…,τ{i+4}) . We find an excellent agreement between Monte Carlo simulations and cluster mean field analysis, which indicates that the mean field analysis might give the exact expression.
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Affiliation(s)
- Qing-Yi Hao
- University of Science and Technology of China, Hefei, China
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Wölki M, Schadschneider A, Schreckenberg M. Asymmetric exclusion processes with shuffled dynamics. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/39/1/003] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Brankov J, Bunzarova N. Finite-size scaling and universality for the totally asymmetric simple-exclusion process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036130. [PMID: 15903516 DOI: 10.1103/physreve.71.036130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2004] [Indexed: 05/02/2023]
Abstract
The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.
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Affiliation(s)
- Jordan Brankov
- Institute of Mechanics, Bulgarian Academy of Sciences, Academician G. Bonchev Street 4, 1113 Sofia, Bulgaria.
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Brankov JG, Priezzhev VB, Shelest RV. Generalized determinant solution of the discrete-time totally asymmetric exclusion process and zero-range process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066136. [PMID: 15244696 DOI: 10.1103/physreve.69.066136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2003] [Indexed: 05/24/2023]
Abstract
We consider the discrete-time evolution of a finite number of particles obeying the totally asymmetric exclusion process with backward-ordered update on an infinite chain. Our first result is a determinant expression for the conditional probability of finding the particles at given initial and final positions, provided that they start and finish simultaneously. The expression has the same form as the one obtained by J. Stat. Phys. 88, 427 (1997)] for the continuous-time process. Next we prove that under some sufficient conditions the determinant expression can be generalized to the case when the particles start and finish at their own times. The latter result is used to solve a nonstationary zero-range process on a finite chain with open boundaries.
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Affiliation(s)
- J G Brankov
- Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 4, 1113 Sofia, Bulgaria
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Brankov J, Pesheva N, Bunzarova N. Totally asymmetric exclusion process on chains with a double-chain section in the middle: computer simulations and a simple theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066128. [PMID: 15244688 DOI: 10.1103/physreve.69.066128] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2003] [Indexed: 05/24/2023]
Abstract
Computer simulations of the totally asymmetric simple-exclusion process on chains with a double-chain section in the middle are performed in the case of random-sequential update. The outer ends of the chain segments connected to the middle double-chain section are open, so that particles are injected at the left end with rate alpha and removed at the right end with rate beta. At the branching point of the graph (the left end of the middle section) the particles choose with equal probability 1/2 which branch to take and then simultaneous motion of the particles along the two branches is simulated. With the aid of a simple theory, neglecting correlations at the junctions of the chain segments, the possible phase structures of the model are clarified. Density profiles and nearest-neighbor correlations in the steady states of the model at representative points of the phase diagram are obtained and discussed. Cross correlations are found to exist between equivalent sites of the branches of the middle section whenever they are in a coexistence phase.
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Affiliation(s)
- Jordan Brankov
- Institute of Mechanics, Bulgarian Academy of Sciences, Academician G. Bonchev Street 4, 1113 Sofia, Bulgaria.
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Boutillier C, Fran ois P, Mallick K, Mallick S. A matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/46/301] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Brankov J. Finite-size scaling in the steady state of the fully asymmetric exclusion process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046111. [PMID: 12005930 DOI: 10.1103/physreve.65.046111] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2001] [Indexed: 05/23/2023]
Abstract
Finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition are found in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and discrete-time dynamics. The corresponding finite-size scaling variables are identified as the ratio of the chain length to the localization length of the relevant domain wall.
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Affiliation(s)
- Jordan Brankov
- Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street 4, 1113 Sofia, Bulgaria
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Brankov J, Pesheva N. Exact density profiles for the fully asymmetric exclusion process with discrete-time dynamics on semi-infinite chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:046111. [PMID: 11308916 DOI: 10.1103/physreve.63.046111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2000] [Indexed: 05/23/2023]
Abstract
Exact density profiles in the steady state of the one-dimensional fully asymmetric simple-exclusion process on a semi-infinite chain are obtained in the case of forward-ordered sequential dynamics by taking the thermodynamic limit in our recent exact results for a finite chain with open boundaries. The corresponding results for sublattice-parallel dynamics follow from the relationship obtained by Rajewsky and Schreckenberg [Physica A 245, 139 (1997)], and for parallel dynamics from the mapping found by Evans, Rajewsky, and Speer [J. Stat. Phys. 95, 45 (1999)]. Our analytical expressions involve Laplace-type integrals, rather than complicated combinatorial expressions, which makes them convenient for taking the limit of a semi-infinite chain, and for deriving the asymptotic behavior of the density profiles at large distances from its end. By comparing the asymptotic results appropriate for parallel update with those published in the above cited paper by Evans, Rajewsky, and Speer, we find complete agreement except in two cases, in which we correct technical errors in the final results given there.
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Affiliation(s)
- J Brankov
- Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 4, 1113 Sofia, Bulgaria.
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Cheybani S, Kertész J, Schreckenberg M. Stochastic boundary conditions in the deterministic Nagel-Schreckenberg traffic model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:016107. [PMID: 11304314 DOI: 10.1103/physreve.63.016107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/1999] [Indexed: 05/23/2023]
Abstract
We consider open systems where cars move according to the deterministic Nagel-Schreckenberg rules [K. Nagel and M. Schreckenberg, J. Phys. I 2, 2221 (1992)] and with maximum velocity v(max)>1, which is an extension of the asymmetric exclusion process (ASEP). It turns out that the behavior of the system is dominated by two features: (a) the competition between the left and the right boundary, (b) the development of so-called "buffers" due to the hindrance that an injected car feels from the front car at the beginning of the system. As a consequence, there is a first-order phase transition between the free flow and the congested phase accompanied by the collapse of the buffers, and the phase diagram essentially differs from that for v(max)=1 (ASEP).
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Affiliation(s)
- S Cheybani
- Theoretische Physik, Gerhard-Mercator Universität, D-47048 Duisberg, Germany
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Fouladvand ME, Lee HW. Exactly solvable two-way traffic model with ordered sequential update. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:6465-79. [PMID: 11970562 DOI: 10.1103/physreve.60.6465] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/1999] [Indexed: 04/18/2023]
Abstract
Within the formalism of the matrix product ansatz, we study a two-species asymmetric exclusion process with backward and forward site-ordered sequential updates. This model, which was originally introduced with the random sequential update [J. Phys. A 30, 8497 (1997)], describes a two-way traffic flow with a dynamic impurity and shows a phase transition between the free flow and the traffic jam. We investigate characteristics of this jamming and examine similarities and differences between our results and those with a random sequential update.
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Affiliation(s)
- M E Fouladvand
- Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran.
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Mallick K, Mallick S, Rajewsky N. Exact solution of an exclusion process with three classes of particles and vacancies. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/48/303] [Citation(s) in RCA: 40] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Sasamoto T. One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/41/306] [Citation(s) in RCA: 103] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Krebs K, Sandow S. Matrix product eigenstates for one-dimensional stochastic models and quantum spin chains. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/9/024] [Citation(s) in RCA: 59] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Evans MR. Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/16/011] [Citation(s) in RCA: 96] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Hinrichsen H, Sandow S. Deterministic exclusion process with a stochastic defect: matrix-product ground states. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/8/020] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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