基于并行格子气方法的单向行人流复杂动态特性研究
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摘要
我们处在一个高速发展的社会,人们的生活节奏快,出行和参加各种集体活动的需求不断增加。如何减少拥挤,保证行人的安全越来越成为人们关注的热点问题。揭示产生拥挤的微观机制,弄清行人流复杂动态特性成为行人流研究的核心课题。近些年来,有很多学者致力于这方面的研究,已取得了一些重要研究成果,但由于行人流问题本身的复杂性,还有很多问题尚待进一步探讨。行人流研究的方法很多,本文采用的是建立模型,数值模拟和解析分析的研究方法。
由于离散模型有计算量小,方便实现等优点,因此本文用格子气方法建立离散模型对单向行人流的特性进行研究。一般地,研究行人流的格子气模型采用的是随机顺序更新规则。考虑到实际行人行走时的同步性,我们把并行更新规则引入到行人流的格子气模型中,这是本文的一个创新之处。
我们运用并行格子气方法对单向行人流进行了建模,对得到的基本图做了详细分析,并与随机顺序格子气模型对应的基本图进行比较。发现在相同的密度下,并行更新对应的流量比随机更新对应的流量小,这是因为并行更新规则下行人之间会因为争抢同一空位而发生冲突,从而影响了系统的流量。我们模型得到的基本图曲线的右分支在模型参数——向前运动趋势强度从0增大到1的过程中出现了拐点,即基本图曲线由上凹变为下凸。这表明调整我们模型中的这个参数值,能再现实测实验得到的一些不同文化背景下的行人流基本图。当这个参数值等于1时,并行更新模型的基本图有两个拥挤流分支,不同于随机更新模型的反λ结构的基本图。在不考虑相邻位子之间相关性的前提下,对并行更新模型的系统流率作了平均场解析。当模型中向前运动趋势强度参数值很小时,解析结果和模拟结果吻合得较好。但当这个值增大时,解析结果逐渐偏离了模拟结果,这是因为系统内的相关性越来越强。
考虑到实际行人在行走时会根据自己前方的视野情况对前面行人的移动有期望效应(预判能力),我们在并行更新格子气模型基础上,加入了期望效应的规则。为了便于探讨期望效应对系统性质的影响以及方便解析求解,本文研究的仅仅是单道情形。实际上,这个模型就是一个扩展的非对称排它过程(TASEP)。文中重点对这个模型开展了平均场理论解析工作。由于模型里考虑了期望效应,我们采用了与一般的对称簇平均场理论不同的非对称簇平均场方法,这也是本文的一个创新之处。通过解析分析,得到了系统流量的解析表达式。解析结果和模拟结果吻合得非常好,解析表达式有可能就是精确解,当然,这还有待今后继续加以验证。进一步对流量的解析表达式进行分析,我们还得到了最大流量对应的密度与期望效应概率之间的关系。
为了更好地解决并行更新规则下出现的同一时刻多人争抢同一空位的冲突,以及较好地体现实际行人的心理、性格特点,我们把博弈论引入到并行更新格子气行人流模型中。把行人分为两类：“合作者”与“背叛者”,合作者是指谦让、脾气好的一类人,背叛者是指霸道、急性子的一类人。给定了博弈收益矩阵和博弈参与者在博弈后改变策略的规则。通过模拟,讨论了合作频率的性质,在初始合作频率不等于1时,稳态后的合作频率与初始频率无关,并随密度增大呈现非单调性,从而系统流量也不受初始合作频率影响。我们还研究了各种博弈类型发生的比例、行人在通道里的空间分布等性质。对于模型中的向前移动趋势强度参数等于1时的特殊情况,我们重点讨论了系统在不同宽度下的动态特性。发现了合作频率在双道情形会发生一阶相变：系统密度小于临界密度时,非合作频率随时间呈指数衰减,而大于临界密度时,演化一段时间后非合作频率则保持为一常数。有趣的是,这种相变行为和其它系统发生的相变在性质上完全一致。但是这种相变行为在多道系统中就不存在了。通过对小系统的详细分析,我们给出了产生这种现象的微观机理解释。我们还发现了系统的流率大小与初始合作频率是否等于1有关。这种流率差是由两个因素产生的：不同初始合作频率下,稳态后系统里行人的空间分布状况不同；合作者与背叛者在博弈时的移动概率不同。产生流率差的这两个因素的主导地位随系统宽度的增大而改变。关于演化博弈规则对系统动态特性的影响,文中也进行了初步研究。此外,我们还运用平均场理论对各种博弈类型发生的概率进行了理论解析,解析结果和模拟结果在一定范围内吻合得比较好。
We are in a fast developing society, so people are quickening rhythm of life and work. And the demanding for traveling and participating in collective activities are increasing. Reducing congestion and ensuring pedestrian safety have become a hot issue that people pay increasingly attention to. Revealing the microscopic mechanisms resulting congestion and gaining a clear idea of complex dynamic characteristics of pedestrian flow become core issues of studying pedestrian flow. Recent years, many scholars dedicated to research in this area and they have made some important research results. But due to the complexity of pedestrian flow, there are many problems remain to be further explored. There are many methods for studying pedestrian flow. This paper employs research methods of the modeling-numerical simulation and analytical analysis.
As discrete model has some advantages, such as small amount of calculation, convenient operation, this thesis establishes discrete model with the lattice gas method to study characteristics of pedestrian flow. Generally, random sequential update rule is used in lattice gas models studying pedestrian flow. Taking into account the synchronization of actual pedestrians walking, we introduce parallel update rule to the lattice gas model of pedestrian flow, which is an innovation of this thesis.
We model unidirectional pedestrian flow using the parallel lattice gas method and analyze the fundamental diagram in detail, and compare it with that of the lattice gas model with random sequential update.
We find that in the same density, the flow corresponding to parallel update is smaller than that corresponding to random sequential update. This is because the conflicts that several pedestrians intend to move to the same site happen owing to parallel update, which affects the flow of the system. The right branch of fundamental diagram curve of our model appears the inflection point with the increase of the parameter (the drift strength) in the model from0to1, namely, the fundamental diagram becomes convex from the concave curve. This indicates that our model can reproduce different fundamental diagrams obtained via realistic observations and experiments corresponding to some pedestrians under different culture backgrounds by adjusting the parameter value. When the parameter is equal to1, the fundamental diagram of parallel update model has two congested flow branches:it is different from that of random sequential update model with the reverse lambda structure. Without taking into account the correlation between adjacent sites, we analyze the flow rate of system in parallel update model by mean-field method. When the drift strength parameter in the model is small, analysis results are in approximate agreement with simulation results. But with the increase of the drift strength, analysis results gradually deviate from the simulation results because of the correlation of the system is growing.
In reality, when a pedestrian walks, he (her) has expectation effect (anticipation ability) to front pedestrian's movement according to front situations. Considering this fact, we add expectation effect rule to parallel lattice gas model. In order to facilitate to discuss how expectation effect influences on the property of the system and get analytical solution, this paper studies just one lane case. In fact, this model is an expansion of the totally asymmetric exclusion process (TASEP). Theoretical analysis for the model is carried out mainly by the mean field in the paper. As expectation effect is considered in the model, we adopt asymmetric cluster mean field method that is different from general symmetry cluster mean field theory, which is also an innovation of this thesis. Through analysis, we obtain the analytical expression of the system flow. Analysis result agrees excellently with simulation result, which indicates that the analytical expression might be the exact expression. And of course, these remain to be proven in the future. Analyzing further the analytical expression of the flow, we also obtain the relationship between the density corresponding to maximum flow and the probability of expectation effect.
In order to better solve the conflict that several pedestrians intend to move to the same site at the same moment under parallel update rule, and to better embody the actual pedestrian's psychology, personality traits, we introduce game theory to the lattice gas pedestrian flow model with parallel update. We divide pedestrians into two categories'cooperator and "defector". Cooperators are gentle with good temper and defectors are aggressive with quick temper. We give a payoff matrix of game and a rule that participant in the game change strategies after game according to. Through simulations, we discuss natures of the cooperators fraction. When an initial cooperator fraction is not equal to1, the steady-state cooperators fraction has nothing to do with it and shows non-monotonic with increase of the density. Thus initial cooperators fraction does not influence system flow. We also examine other properties of system, such as the proportions of various game types, pedestrian distribution in the channel. For the case of the drift strength being equal to1, we focus on dynamic characteristics of the system with different widths. We find a first-order phase transition of cooperators fraction in two-lane system, in which defectors fraction decays exponentially when system density is less than the critical density, otherwise it remains constant after a period of evolution. Interestingly, this phase transition behavior is same as that about nature in other system. However, this phase transition behavior does not exist in the multi-lane system. By analyzing a small system in detail, we present an interpretation about the microscopic mechanism of the phenomenon. We also find that flow rate of the system is connected with whether the initial cooperators fraction is equal to1. The flow rate difference is generated by two factors: different spatial distributions of pedestrians for different initial cooperators fraction and different transition probabilities for cooperator and defector in the game. Dominant position of the two factors producing flow rate difference changes with the increase of system width. The paper also carries out a preliminary study on how evolution rules of the game influence dynamic characteristics of the system. In addition, we analyze probabilities of various types of game using the mean-field theory. Analysis results and simulation results agree quite well in a certain range.
由于离散模型有计算量小,方便实现等优点,因此本文用格子气方法建立离散模型对单向行人流的特性进行研究。一般地,研究行人流的格子气模型采用的是随机顺序更新规则。考虑到实际行人行走时的同步性,我们把并行更新规则引入到行人流的格子气模型中,这是本文的一个创新之处。
我们运用并行格子气方法对单向行人流进行了建模,对得到的基本图做了详细分析,并与随机顺序格子气模型对应的基本图进行比较。发现在相同的密度下,并行更新对应的流量比随机更新对应的流量小,这是因为并行更新规则下行人之间会因为争抢同一空位而发生冲突,从而影响了系统的流量。我们模型得到的基本图曲线的右分支在模型参数——向前运动趋势强度从0增大到1的过程中出现了拐点,即基本图曲线由上凹变为下凸。这表明调整我们模型中的这个参数值,能再现实测实验得到的一些不同文化背景下的行人流基本图。当这个参数值等于1时,并行更新模型的基本图有两个拥挤流分支,不同于随机更新模型的反λ结构的基本图。在不考虑相邻位子之间相关性的前提下,对并行更新模型的系统流率作了平均场解析。当模型中向前运动趋势强度参数值很小时,解析结果和模拟结果吻合得较好。但当这个值增大时,解析结果逐渐偏离了模拟结果,这是因为系统内的相关性越来越强。
考虑到实际行人在行走时会根据自己前方的视野情况对前面行人的移动有期望效应(预判能力),我们在并行更新格子气模型基础上,加入了期望效应的规则。为了便于探讨期望效应对系统性质的影响以及方便解析求解,本文研究的仅仅是单道情形。实际上,这个模型就是一个扩展的非对称排它过程(TASEP)。文中重点对这个模型开展了平均场理论解析工作。由于模型里考虑了期望效应,我们采用了与一般的对称簇平均场理论不同的非对称簇平均场方法,这也是本文的一个创新之处。通过解析分析,得到了系统流量的解析表达式。解析结果和模拟结果吻合得非常好,解析表达式有可能就是精确解,当然,这还有待今后继续加以验证。进一步对流量的解析表达式进行分析,我们还得到了最大流量对应的密度与期望效应概率之间的关系。
为了更好地解决并行更新规则下出现的同一时刻多人争抢同一空位的冲突,以及较好地体现实际行人的心理、性格特点,我们把博弈论引入到并行更新格子气行人流模型中。把行人分为两类：“合作者”与“背叛者”,合作者是指谦让、脾气好的一类人,背叛者是指霸道、急性子的一类人。给定了博弈收益矩阵和博弈参与者在博弈后改变策略的规则。通过模拟,讨论了合作频率的性质,在初始合作频率不等于1时,稳态后的合作频率与初始频率无关,并随密度增大呈现非单调性,从而系统流量也不受初始合作频率影响。我们还研究了各种博弈类型发生的比例、行人在通道里的空间分布等性质。对于模型中的向前移动趋势强度参数等于1时的特殊情况,我们重点讨论了系统在不同宽度下的动态特性。发现了合作频率在双道情形会发生一阶相变：系统密度小于临界密度时,非合作频率随时间呈指数衰减,而大于临界密度时,演化一段时间后非合作频率则保持为一常数。有趣的是,这种相变行为和其它系统发生的相变在性质上完全一致。但是这种相变行为在多道系统中就不存在了。通过对小系统的详细分析,我们给出了产生这种现象的微观机理解释。我们还发现了系统的流率大小与初始合作频率是否等于1有关。这种流率差是由两个因素产生的：不同初始合作频率下,稳态后系统里行人的空间分布状况不同；合作者与背叛者在博弈时的移动概率不同。产生流率差的这两个因素的主导地位随系统宽度的增大而改变。关于演化博弈规则对系统动态特性的影响,文中也进行了初步研究。此外,我们还运用平均场理论对各种博弈类型发生的概率进行了理论解析,解析结果和模拟结果在一定范围内吻合得比较好。
We are in a fast developing society, so people are quickening rhythm of life and work. And the demanding for traveling and participating in collective activities are increasing. Reducing congestion and ensuring pedestrian safety have become a hot issue that people pay increasingly attention to. Revealing the microscopic mechanisms resulting congestion and gaining a clear idea of complex dynamic characteristics of pedestrian flow become core issues of studying pedestrian flow. Recent years, many scholars dedicated to research in this area and they have made some important research results. But due to the complexity of pedestrian flow, there are many problems remain to be further explored. There are many methods for studying pedestrian flow. This paper employs research methods of the modeling-numerical simulation and analytical analysis.
As discrete model has some advantages, such as small amount of calculation, convenient operation, this thesis establishes discrete model with the lattice gas method to study characteristics of pedestrian flow. Generally, random sequential update rule is used in lattice gas models studying pedestrian flow. Taking into account the synchronization of actual pedestrians walking, we introduce parallel update rule to the lattice gas model of pedestrian flow, which is an innovation of this thesis.
We model unidirectional pedestrian flow using the parallel lattice gas method and analyze the fundamental diagram in detail, and compare it with that of the lattice gas model with random sequential update.
We find that in the same density, the flow corresponding to parallel update is smaller than that corresponding to random sequential update. This is because the conflicts that several pedestrians intend to move to the same site happen owing to parallel update, which affects the flow of the system. The right branch of fundamental diagram curve of our model appears the inflection point with the increase of the parameter (the drift strength) in the model from0to1, namely, the fundamental diagram becomes convex from the concave curve. This indicates that our model can reproduce different fundamental diagrams obtained via realistic observations and experiments corresponding to some pedestrians under different culture backgrounds by adjusting the parameter value. When the parameter is equal to1, the fundamental diagram of parallel update model has two congested flow branches:it is different from that of random sequential update model with the reverse lambda structure. Without taking into account the correlation between adjacent sites, we analyze the flow rate of system in parallel update model by mean-field method. When the drift strength parameter in the model is small, analysis results are in approximate agreement with simulation results. But with the increase of the drift strength, analysis results gradually deviate from the simulation results because of the correlation of the system is growing.
In reality, when a pedestrian walks, he (her) has expectation effect (anticipation ability) to front pedestrian's movement according to front situations. Considering this fact, we add expectation effect rule to parallel lattice gas model. In order to facilitate to discuss how expectation effect influences on the property of the system and get analytical solution, this paper studies just one lane case. In fact, this model is an expansion of the totally asymmetric exclusion process (TASEP). Theoretical analysis for the model is carried out mainly by the mean field in the paper. As expectation effect is considered in the model, we adopt asymmetric cluster mean field method that is different from general symmetry cluster mean field theory, which is also an innovation of this thesis. Through analysis, we obtain the analytical expression of the system flow. Analysis result agrees excellently with simulation result, which indicates that the analytical expression might be the exact expression. And of course, these remain to be proven in the future. Analyzing further the analytical expression of the flow, we also obtain the relationship between the density corresponding to maximum flow and the probability of expectation effect.
In order to better solve the conflict that several pedestrians intend to move to the same site at the same moment under parallel update rule, and to better embody the actual pedestrian's psychology, personality traits, we introduce game theory to the lattice gas pedestrian flow model with parallel update. We divide pedestrians into two categories'cooperator and "defector". Cooperators are gentle with good temper and defectors are aggressive with quick temper. We give a payoff matrix of game and a rule that participant in the game change strategies after game according to. Through simulations, we discuss natures of the cooperators fraction. When an initial cooperator fraction is not equal to1, the steady-state cooperators fraction has nothing to do with it and shows non-monotonic with increase of the density. Thus initial cooperators fraction does not influence system flow. We also examine other properties of system, such as the proportions of various game types, pedestrian distribution in the channel. For the case of the drift strength being equal to1, we focus on dynamic characteristics of the system with different widths. We find a first-order phase transition of cooperators fraction in two-lane system, in which defectors fraction decays exponentially when system density is less than the critical density, otherwise it remains constant after a period of evolution. Interestingly, this phase transition behavior is same as that about nature in other system. However, this phase transition behavior does not exist in the multi-lane system. By analyzing a small system in detail, we present an interpretation about the microscopic mechanism of the phenomenon. We also find that flow rate of the system is connected with whether the initial cooperators fraction is equal to1. The flow rate difference is generated by two factors: different spatial distributions of pedestrians for different initial cooperators fraction and different transition probabilities for cooperator and defector in the game. Dominant position of the two factors producing flow rate difference changes with the increase of system width. The paper also carries out a preliminary study on how evolution rules of the game influence dynamic characteristics of the system. In addition, we analyze probabilities of various types of game using the mean-field theory. Analysis results and simulation results agree quite well in a certain range.
引文
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