車流動力學之連續減速模式

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(1)國立交通大學運輸科技與管理學系所碩士論文. 車流動力學之連續減速模式 The Successive Deceleration Model of Kinetic Traffic Flow. 研究生：傅昱瑄撰著指導教授：卓訓榮博士. 中華民國九十六年七月.

(2) 車流動力學之連續減速模式 The Successive Deceleration Model of Kinetic Traffic Flow 研究生：傅昱瑄. Student：Yu-Hsuan Fu. 指導教授：卓訓榮. Advisor：Hsun-Jung Cho 國立交通大學運輸科技與管理學系所碩士論文 A Thesis. Submitted to Institute of Transportation Technology and Management College of Management National Chiao Tung University In Partial Fulfillment of the Requirements for the Degree of Master In Transportation Technology and Management July 2007 Hsinchu, Taiwan, Republic of China. 中華民國九十六年七月.

(3) 車流動力學之連續減速模式研究生：傅昱瑄. 指導教授：卓訓榮. 國立交通大學運輸科技與管理學系碩士班. 摘要欲發展良好的交通控制解決交通問題需要有充足的資訊，在相關資訊中最重要且不易估算的就是車流量的預測，車流模式的發展就是朝此方向之研究，關於介觀車流的相關研究國內相關研究仍未完整，尚有許多介觀車流議題需要被討論與研究，所以本研究欲藉由回顧較具重要性的介觀的車流動力學模式，將其重要貢獻與特性整理，並且建構一可較真確描述道路車流情形之介觀車流模式。本研究引入具有物理意義之減速度考慮項目，放鬆交互影響項中瞬時速度變化與未考慮有限空間因素影響，藉由將這些特性納入並建構一新的介觀車流之動力學方程之中；並且利用動差函數方式將本研究所建構的介觀模式積分，可以獲得以巨觀參數─密度 c ( x, t ) 、平均速度 v ( x, t ) 與變異數 θ ( x, t ) 為主的巨觀方程式；而後再以特徵速度、均衡解與數值模擬加以分析該模式的特性，並獲得因本研究引入等減速度可比以往研究的模式可多加解釋的特性，比較不同超車機率下不同的加速度值，在低的超車機率下，等減速度會比較有差別，在高的超車機率下，等減速度看不出差別，因為可很自由的超車情況下，使用到等減速度的車輛相形較少，所以對整體較看不出差別。由此可知本研究所建構的介觀車流動力學模式可較真確描地述更多車流情形。另外在撰寫論文時，將本研究的想法與行為意義清楚傳達，並將過程中的數學推導詳細記錄，希冀能對於介觀車流有興趣者有所幫助，以期許將來能有利於實際應用的發展。關鍵詞：介觀車流；車流動力學；瞬時速度變化.

(4) The Successive Deceleration Model of Kinetic Traffic Flow Student：Yu-Hsuan Fu. Advisor：Hsun-Jung Cho. Department of Transportation Technology and Management National Chiao Tung University. Abstract In order to develop traffic control to solve traffic problem, we need sufficient information. The related and the most important information is to estimate quantity of traffic flow. Developing traffic flow model is one of ways to research. However, the research about mesoscopic traffic flow, gas-kinetic traffic flow, is not yet complete so we want to review some important researches and arrange the focus. And we construct a gas-kinetic traffic flow model in order to describe real traffic more. In this research, we introduce uniform deceleration with physical meaning to relax instant velocity-changing and not consider finite space in interaction term and construct a new model. Then we use momentum function to obtain three macroscopic models with density c ( x, t ) , average velocity v ( x, t ) and variance θ ( x, t ) . We use characteristic velocity, equilibrium and numerical simulation to analysis the accent of our model, and we know our model could describe more phenomenon than others. Compared with different uniform decelerations in different passing probability conditions, we could observe that different uniform decelerations make difference in smaller passing probability. Different uniform decelerations make no difference in bigger passing probability, because cars use few uniform decelerations in dilute traffic. Therefore, our research could describe more traffic phenomenon. Besides, we explain our thought and the meaning of our model clear, and record mathematic processes explicitly. We hope this research could help those who are interested in mesoscopic traffic flow and help the development of real application. Keyword: Mesoscopic Traffic Flow, Gas-Kinetic Traffic Flow, Instant velocity-changing..

(5) 誌. 謝. 細數在交大所過的日子，一轉眼間就過了六年，時間過得真快，六年的時間中，曾受過大家的照顧、關心與幫助，真的很感謝。首先從身邊的朋友開始，可愛的室友：碗偷、大鳥和芝帆，感謝你們在大學四年的時間裡陪我度過了最美好的宿舍時光，回想每次去吃好湯、唱歌，還有半夜不睡覺一直在聊吃的越聊越餓的日子，真的很有趣，也感謝韻涵，總會熱情支助我們的唱歌休閒團、幫忙我搬家，還會帶吃的給我，真的很謝謝你；也感謝大學同學：郁婷、堉慈、憲梅、佳欣、文誠與敬哲，這一路上不論在學業或者生活上的幫助，還有研究室的夥伴：叮咚、banana、小白雞、至剛、育婷學姊、健綸學長、昱光學長、黃恆學長、宜珊、胖胖還有小捲，感謝大家這段時間的照顧，很懷念實驗室大家一起出外景的日子，防曬乳總是用的特別快，每次出門總是會忘了帶點東西，一起去吃飯、打排球、一起在實驗室苦生計畫等等，雖然日子很苦，但也總覺得大家的人情味是寒冬的溫情，帶來生活的溫馨與樂趣。接著要感謝卓老師，在研究所的求學生活中，不斷的給予在研究上的指導與生活上的幫助，也感謝玉山基金會給予我獲得獎學金的機會，讓我在生活經費上無虞，最後要感謝我的家人：爸爸、媽媽和弟弟，在我覺得日子困苦時不時的關心我，也要感謝我姊和阿榮，沒有你們的幫忙我真的畢不了業，還有在我不斷搬家的過程中總是不斷的出力，謝謝。在交大的六年生活裡，感謝大家與我分享，一起度過的日子我永遠不會忘記，謝謝你們。.

(6) Contents Chapter 1. Introduction................................................................................................1. 1.1. Research Motivation ..................................................................................1. 1.2. Research Purpose .......................................................................................3. 1.3. Research Scope ..........................................................................................4. 1.4. Research Procedure....................................................................................5. Chapter 2. Literature Review.......................................................................................8. 2.1. One-Lane and One-Class ...........................................................................8. 2.2. Multilane and Multiclass.......................................................................... 11. 2.3. Rearrange Papers .....................................................................................13. 2.4. Summary ..................................................................................................15. Chapter 3. Construct Mesoscopic Traffic Flow Model .............................................17. 3.1. The Question Description ........................................................................17. 3.2. Model Construction .................................................................................18. 3.3. Summary ..................................................................................................22. Chapter 4. Macroscopic model ..................................................................................23. 4.1. Assumption ..............................................................................................23. 4.2. Density Equation......................................................................................26. 4.3. Average velocity equation........................................................................28. 4.4. Variance....................................................................................................33. 4.5. Summary ..................................................................................................40. Chapter 5. Characteristics Analysis of Macroscopic Models ....................................41. 5.1. Characteristic Velocity Analysis ..............................................................41. 5.2. Equilibrium Analysis ...............................................................................45. 5.3. Numerical Simulation Analysis ...............................................................48 i.

(7) 5.4 Chapter 6. Summary ..................................................................................................64 Contribution and Future Works................................................................66. 6.1. Contribution .............................................................................................66. 6.2. Future Works............................................................................................67. Reference… .................................................................................................................68 Appendix A ..................................................................................................................70 Appendix B - Symbols.................................................................................................71 1.. Variables.......................................................................................................71. 2.. Constants......................................................................................................72. 3.. Functions......................................................................................................72. 4.. Vectors..........................................................................................................73. ii.

(8) List of Figures Fig. 1-3-1. The relation of microscopic, mesoscopic, and macroscopic traffic flow model......................................................................................................... 4. Fig. 1-3-2. Research Scope ......................................................................................... 5. Fig. 1-4-1. The flowchart of This Research ................................................................ 7. Fig. 2-3-1. The More Important Contains of Recent Ten-Year Gas-Kinetic Traffic Flow Models ........................................................................................... 13. Fig. 2-3-2. Rearrange the Characteristics of Important Papers................................. 15. Fig. 5-1-1. Dilute and Heavy Traffic......................................................................... 44. Fig. 5-3-1. Density Changing 1................................................................................. 50. Fig. 5-3-2. Density Changing 2................................................................................. 51. Fig. 5-3-3. Density Changing 3................................................................................. 51. Fig. 5-3-4. Average Velocity Changing 1 ................................................................. 52. Fig. 5-3-5. Average Velocity Changing 2 ................................................................. 52. Fig. 5-3-6. Average Velocity Changing 3 ................................................................. 53. Fig. 5-3-7. Traffic Pressure Changing 1.................................................................... 53. Fig. 5-3-8. Traffic Pressure Changing 2.................................................................... 54. Fig. 5-3-9. Traffic Pressure Changing 3.................................................................... 54. Fig. 5-3-10 Variance Changing 1 ............................................................................... 55 Fig. 5-3-11 Variance Changing 2 ............................................................................... 55 Fig. 5-3-12 Variance Changing 3 ............................................................................... 56 Fig. 5-3-13 Passing Probability =0.7 ......................................................................... 57 Fig. 5-3-14 Passing Probability =0.8 ......................................................................... 58 Fig. 5-3-15 Passing Probability =0.9 ......................................................................... 58 Fig. 5-3-16 Passing Probability =0.99 ....................................................................... 59 Fig. 5-3-17 Density in p=0.99 and Different a........................................................... 60 iii.

(9) Fig. 5-3-18 Density in p=0.7 and Different a............................................................. 61 Fig. 5-3-19 Average Velocity in p=0.99 and Different a............................................ 61 Fig. 5-3-20 Average Velocity in p=0.7 and Different a.............................................. 62 Fig. 5-3-21 Variance in p=0.99 and Different a ......................................................... 62 Fig. 5-3-22 Variance in p=0.7 and Different a ........................................................... 63. iv.

(10) Chapter 1. Introduction. 1.1 Research Motivation In recent decades, the auto industry grows vigorously and the proportion of possession of the car rises year by year because of the factors, such as the demand of economic development. Accounting the materials of The Directorate General of Budget, Accounting and Statistics (DGBAS) of Executive Yuan [1], it shows that the registered car has already been up to 6,380,000 till 2004, there are 0.73 cars in average each family, and the traffic accident piece is up to 137221. There are 72.83 accidents in every ten thousand cars by accounting the vehicle, and average death rate is 1.4 people in every ten thousand cars by accounting the vehicle. Both are higher than in Japan and Britain, etc. Therefore, to develop effective method to control traffic is the important topic in order to solve the traffic problem in our country. However, it needs sufficient information to develop good traffic control. The most important and difficult to estimate one is the prediction of the flow of car in relevant information, and the development of traffic is the research in this direction. The models of mathematics could be used in the description of various kinds of the systematic physical phenomenon, and the traffic systems include drivers, vehicle and road states, etc. We could built and construct the studies of mathematical models to describe complicated driver's behaviors. Therefore, this research wants to build the way to construct a road traffic flow situation. The general traffic theory could be divided into three kinds: microscopic, macroscopic and mesoscopic. According to the descriptions of scholars, such as Hoogendoorn and Bovy (2001) [2], etc., Microscopic traffic flow model is describing the relationship between driving behavior, and uses parameters to describe the individual's behavior in detail, for instance: car-following model. Macroscopic traffic 1.

(11) flow model based on the relation of flow, velocity and density treats every car in traffic flow which is unable to identify alone with the view of the continuous fluid. The basic theories have average concepts. Mesoscopic traffic flow model is correlated with macroscopic traffic flow model and microscopic traffic flow model. Based on the relation of distance and density, mesoscopic traffic flow model builds on the distances between two cars in microscopic traffic flow model and density in macroscopic traffic flow model. Mesoscopic traffic flow model describes the individual behavior in the form of probability distribution, which basic theories are set out by the dynamics theory. Among three model of traffic flow, it is difficult to be used in dynamic simulation for carrying the simulation out wastes time. That's because macroscopic behavior is unable to catch and microscopic model has a lot of parameters. Therefore, in order to describe the micro behavior, and offer information of macroscopic behavior, we need to use mesoscopic traffic flow model. Mesoscopic traffic flow model could be said that it is a bridge connecting microscopic traffic flow model and macroscopic traffic flow model. According to the scholars, such as Hoogendoorn and Bovy (2001) [2], etc., they divide mesoscopic traffic flow model into three kinds: headway distribution model, cluster model and gas-kinetic continuum model. This research wants to use the gas-kinetic equation of mesoscopic traffic flow to develop the foundation of the traffic flow.. 2.

(12) 1.2 Research Purpose The traffic flow theory could be divided into three big classes: macroscopic, mesoscopic and microscopic traffic. Our nation has a lot of relevant research about macroscopic and microscopic traffic flow model at present, but the relevant research about mesoscopic traffic is not complete so far. There are still a lot of topics about mesoscopic traffic that need to be discussed and studied. For this reason, by reviewing the more important mesoscopic gas-kinetic traffic flow model, this research puts its important contribution and characteristic in order to make it easier to understand for those who interest mesoscopic gas-kinetic traffic flow model. By showing this research, it may offer some materials to the follow-up topic persons. The research will focus on the gas-kinetic traffic flow model of mesoscopic traffic flow with the introduction of the physical significance of considerable items. Its properties will be included in and build a new mesoscopic traffic Kinetic equation. It could be more accurate to describe the traffic flow cases, and ease the conditions of mesoscopic traffic equations which have not been relaxed or unreasonable originally. Through mesoscopic model integration, using the GMM function, this research could get Macro-parameters － density c( x, t ) 、 average velocity v ( x, t ) and variance. θ ( x, t ) combining the formula. Then we use the deterministic and numerical simulation to analysis the characteristics of the model. We hope that the establishment of mesoscopic gas-kinetic traffic flow model could be more realistic, and help those whom interested in mesoscopic traffic flow. We even hope that it could be good to real application.. 3.

(13) 1.3 Research Scope This research focuses on the gas-kinetic equation of mesoscopic traffic flow, which has a relation between microscopic traffic flow and macroscopic traffic. Its relation could be put in order as the following picture 1-3-1 by Hoogendorn and Bovy (2001) [2].. Fig. 1-3-1. The relation of microscopic, mesoscopic, and macroscopic traffic flow model. Data Source: Hoogendorn and Bovy (2001) [2] This research range is fixed on the behavior factors needed considering in Microscopic. Considering these microcosmic factors, we establish mesoscopic traffic flow model to be macroscopic model via integration. It could be convenient to study its characteristics and be used in on-line dynamic simulation in Macroscopic traffic flow model in future. Subdividing the research scope, the research scope of this research is as the following picture 1-3-2, and the research scope is fixed on the description of the non-interrupt traffic flow, for example: on freeway.. 4.

(14) Dynamic Traffic Model. Research Scope. Non-interruption Traffic Flow Model. Interruption Traffic Flow Model. Dynamic Path Flow and Travel Time Prediction. Fig. 1-3-2. Research Scope. Data Source: Cho and Lin (2004) [3]. 1.4 Research Procedure This research procedure is like Fig 1-4-1. Every step is illustrated in detail as following. 1.. Describe and define the question According to the development of mesoscopic traffic flow understands the. development condition of gas-kinetic equation in mesoscopic traffic flow at present, submit the topic of the traffic gas-kinetic equation wanted to study, and do an intact description of the topic and define motivation and objection of this research.. 5.

(15) 2.. Collect and review literature Collect relevant literature probing into traffic gas-kinetic equation of the both in. our country and abroad, and review the limiting conditions and contribution of these documents. From the velocity of vehicle assigned function to describe the traffic flow by Prigogine and Andrews (1960 ) [4] earliest, which set up the young type of gas-kinetic equation, to the follow-up research based on the Prigogine' s and Herman' s (1971) [5] models which are more expanded and improved, this research is by reviewing the literature pluses and minuses to discuss the restriction and suitability of these literature in order to help the traffic gas-kinetic equation of this research to be set up. 3.. Construct the model Build a conceptive mesoscopic gas-kinetic traffic flow model, and describe its. every parameter and each physics meaning represented by consulting and collecting the literature and materials according to the question that we describe and define. 4.. The deriving in Macroscopic traffic mode Use the zeroth-order approximation of local equilibrium, and utilize moment. function to integrate mesoscopic traffic flow model into Macroscopic traffic mode. 5.. Analyze Analyze the characteristic of the model, for example: characteristic velocity,. equilibrium, and etc. Analyze the obtained result, and then analyze the rationality of its result. Last, observe the characteristics of the model by way of numerical simulation. 6.. Conclusion and suggestion Propose conclusion and suggestion to the course and result of this research.. 6.

(16) Fig. 1-4-1. The flowchart of This Research. 7.

(17) Chapter 2. Literature Review. In this chapter, we go through the literature on the subject of kinetic traffic flow theory to understand the development of mesoscopic traffic flow at present. Then we set the topic to research. There are four sections. In the first section, we review one-lane and one-class traffic flow model. In the second section, we review multilane and multiclass traffic flow model. And we rearrange papers into figures in the third section. Finally, we conclude the results of the research topic.. 2.1 One-Lane and One-Class Gas-kinetic continuum model is proposed by Prigogine and Andrews (1960) [4] who describe traffic flow with velocity distribution function. Then Prigogine and Herman (1971) [5] arrange those papers about gas-kinetic continuum models into a book. Their model is assumed to be in an infinite freeway with low density, it does not considered interaction between drivers. There is a velocity distribution function f ( x, v, t ) given certain time t and space x . Their kinetic traffic flow model could be representing as the following. df ∂f ∂f ∂f ∂f = +v = ( ) rel + ( ) int dt ∂t ∂x ∂t ∂t. (2.1). Where, the first term of the right-hand-side of the equation of (2.1) is “Relaxation Term”. It results from the different between real and desired velocity, so it could be shown as the equation (2.2) with exponential law. (. f − f0 ∂f ) rel = − T ∂t. (2.2). Where, f0 (x,v,t) means desired velocity distribution function (Prigogine, 1961) [6], and it means the distribution of velocities that drivers want to drive. And the second term of right-hand-side is “Interaction Term”.. 8.

(18) (. ∂f ∂t. )int = f ( x, v, t )c( x, t )[v ( x, t ) − v](1 − p). (2.3). Where, p means passing probability, c ( x, t ) means density, v ( x, t ) means average velocity. The interaction term shows that the faster (the latter) one car meets the other slower (the former) one without passing, the faster one needs to decelerate in order to avoid collision. Hence, the model of Prigogine and Herman (1971) [5] is the following. ∂f ∂f f − f0 +v =− + (1 − p )c (v − v ) f ∂t ∂x T. (2.4). They use velocity distribution function f ( x, v, t ) to show main functions of macroscopic traffic flow, local density and flow. Local density function ∞. c( x, t ) = ∫ dvf ( x, v, t ). (2.5). 0. Local flow function ∞. q( x, t ) = c( x, t )v ( x, t ) = ∫ dvvf ( x, v, t ). (2.6). 0. Dispersion of velocity ∞. c(v − v ) 2 = ∫ dv(v − v ) 2 f ( x, v, t ). (2.7). 0. Where, f ( x, v, t ) , c ( x, t ) , and q ( x, t ) are continue functions. We could know that velocity distribution function f ( x, v, t ) is a main role in mesoscopic traffic flow and it also relates microscopic and macroscopic traffic flow. After Prigogine and Herman (1971) [5], there are many scholars improve their model. Paveri-Fontana (1975) [7] is one of them, and he considers Phase-Space Density (PSD) g ( x, v, w, t ) . Where, w means desired velocity. He replaces velocity. 9.

(19) distribution f ( x, v, t ) and desired velocity distribution function f 0 ( x, w, t ) with PSD, showing as the followings. ∞. f ( x, v, t ) = ∫ dwg ( x, v, w, t ) 0. ∞. f 0 ( x, w, t ) = ∫ dvg ( x, v, w, t ) 0. (2.8) (2.9). In Pavery-Fontana’s (1975) [7], he proposes a lot of shortcomings of the model of Prigogine and Herman (1971) [5] as the followings. 1. The constant relaxation time of relaxation term is unreasonable. 2. In interaction term I. When the faster car passes the slower one, the velocity of the faster car does not change.. II. When the faster car passes the slower one, the velocity of the slower one does not change.. III Ignore the effect of car length. IV The faster one car meets the other slower one without passing, the instant velocity-changing is unreasonable. V. They only consider the interaction of two car, not many cars.. VI The passing process, one car passes one car or one platoon, is unreasonable. VII Assume two cars are vehicular chaos. There are many scholars try to relax those above unreasonable assumptions.. 10.

(20) 2.2 Multilane and Multiclass According to the section 2.1, we review the models of Prigogine and Herman (1971) [5] and Papery-Fontana (1975) [7]. Many researches depend on those above. Gas-kinetic traffic flow model advances to multilane and multiclass one. In this section, we first review the multilane model of Helbing (1997) [8-9]. He defines j as lane index, g j ( x, v, w, t ) as ML-PSD of j lane. ∂ t g j + v∂ x g j = (∂ t g j ) rel + (∂ t g j ) int + (∂ t g j ) vc + (∂ t g j ) lc + n +j − n −j (2.10). Where, the first term of the right-hand-side (∂ t g j ) rel is “Relaxation Term” and the second term (∂ t g j ) int is “Interaction Term”, and those meanings are the same as the previous section. The third term of the right-hand-side (∂ t g j ) vc is “Velocity Diffusion Term”, it describes the difference of velocities results to a perturbation of individual velocity. The fourth term of the right-hand-side. (∂ t g j ) lc. is. “Lane-Changing Term”, it means the change of ML-PSD results from changing lane. The most right term n +j − n −j means the rate of car flow-in and flow-out. Then, Hoogendoorn and Bovy (2000,001) [10-11] introduce attributes into models, they separate attribute r = (b, k ) into continue attribute b and discrete attribute k . Continue attribute b : 1.. The absolute value of desired velocity v 0 (not contain direction). 2.. The angle of desired moving ω 0. 11.

(21) 3.. The acceleration time T (it is also called relaxation time): It means that the time that a driver accelerate from their resent velocity or angle to their desired velocity or angle.. Discrete attribute k : 1.. Lane j : Because of traffic law, control, and geometric design of road and so on, those reasons would affect the different behavior of driver on different lanes.. 2.. Driver u : Traffic status of drivers could be car, bus, van, bicycle, pedestrian and so forth. In the other hand, it could be described travel factors, entertainment trip, commercial trip, commutation trip and so on.. 3.. Driving Status s : There are two kinds of status in this research, free driver s = 1 and driver in platoon s = 2 . This index could improve the interaction term of Pavery-Fontana (1975) [19], one car passes one car or one platoon.. 4.. Direction of Traffic Flow h : Many directions of traffic flows use the same space. For example, intersection.. 5.. Destination d : The different destination often affects driving behaviors. For example: In freeway, drivers would choose different gateways because of different destinations. It could leads to lane-changing.. Hoogendoorn and Bovy(2000,001) [10-11] introduce PSD g (t , x, v, r ) which mix continue and discrete attributes and they obtain gas-kinetic traffic flow models as the following. (I ) ( II ) III ) IV ) (V ) . ( (. . ∂ t g r + ∇ x ⋅ ( g r v) + ∇ v ⋅ ( g r Ar ) + ∇ v 0 ⋅ ( g r Br ) = (∂ t g r )event + (∂ t g r )cond. (2.11). Where, ∇ x = (∂ x1 " ∂ xn ) , ∇ v = (∂ v1 " ∂ vn ) and ∇ v 0 = (∂ v 0 1 " ∂ v 0 n ) , Ar means acceleration function, Br means acceleration behavior. I. Convection: It is fundamental to balance flow-in and flow-out cars.. II. Relaxation: It is also called ” Acceleration Term”, and it means the effect result from that current velocity accelerates to desired velocity.. 12.

(22) III Adaptation of Continuous Attributes: This term describes effect results from continue attributes changing. IV Event-based Non-continuum Processes: It means that effect results from continue and discrete attributes changing discretely by events. For instance, deceleration and lane-changing and so on. V. Condition-based Non-continuum Processes: It means that effect results from continue and discrete attributes changing discretely with certain conditions. For instance, spontaneous or postponed lane-changing, status changing (free or in platoon) and so forth.. 2.3 Rearrange Papers Since Prigogine and Herman (1971) [5] propose gas-kinetic traffic flow model, many scholars try to improve the limitations of their models step by step. Especially, Paveri-Fontana (1975) [7] introduce desired velocity as an independent variable into gas-kinetic traffic flow model and he points out the shortcomings of Prigogine and Herman (1971) [5]. The latter scholars’ research base is on his paper. The following figure 2-3-1 shows that the more important contains of recent ten-year gas-kinetic traffic flow models. Fig. 2-3-1. The More Important Contains of Recent Ten-Year Gas-Kinetic Traffic Flow Models. Scholars. Takashi Nagatani (1997)[12]. Helbing (1997) [9]. Main Contributions He improves the gas-kinetic traffic flow model of Paveri-Fontana(1975)[7] according to the density of the former space, not the space where the car is to assume desired velocity. He represents velocity distribution function as discrete form. This research shows that there is a great difference between flow-in and flow-out the heavy traffic. He constructs multilane gas-kinetic traffic and considers the behavior of lane-changing, the velocity diffusion owing to improper driving, and finite space related to car length.. He relaxes the instant velocity-changing with the average C. Wanger (1997)[13] velocity between the former car and the latter one, and obtains macroscopic model to analysis numerical simulation. 13.

(23) Klar and Wegener (1999)[14]. He considers lane-changing, braking and acceleration, and sets the reaction micro threshold to construct gas-kinetic traffic flow model. Its velocity distribution function is considered the relation between the former and the latter car. He derives macroscopic models from the gas-kinetic traffic flow model.. They consider continue and discrete attributes into constructing gas-kinetic traffic flow model which contains many factors as multilane, multiclass and so on. They also consider the situation of platoon and assume the car in platoon accelerates to the leader of platoon. They solve the Hoogendoorn and Bovy interaction times which overestimate or underestimate, no (2000, 2001) [10-11,15] relation with near cars, and the situation of dilute traffic, with considering platoon. Besides the traditional convection, relaxation and interaction terms, they also introduce the effects of adaptation of continuous attributes, event-based non-continuum and condition-based processes. They construct a self-consistent multiclass and multilane traffic flow model from traffic Boltzmann equation and traffic diffusion model ,and introduce two-dimension space Cho and Lo (2002) [16] and field. They propose that if the model that is only considered individual velocity to derive second-order momentum function has no physical meanings. It is also needed to consider equilibrium velocity. The following figure 2-3-2 shows that several important researches rearrange on dimensions, scales, processes and operations. Where, dimensions (except time and space) contain velocity v , desired velocity w , horizontal position (lane) y and the others o . Scales contain continue c and discrete d . Processes contain deterministic d and stochastic s . Helbing (1997) [9], Hoogendoorn and Bovy (2001) [11] represent lanes with low index of velocity distribution function. Cho and Lo (2002) [16] consider two-dimensions ( x, y ) to vecor space x . The other dimensions of Hoogendoorn and Bovy (2001) [11] are class and driving status (free traffic or in platoon) ,and the other dimension of Cho and Lo (2002) [16] is class.. 14.

(24) Fig. 2-3-2. Rearrange the Characteristics of Important Papers. Gas-Kinetic Traffic Flow Models. Dimensions Scales Processes v w y o. Prigogine and Herman (1971) [5]. +. c. d. Paveri-Fontana (1975) [7]. + +. c. d. Helbing (1997) [9]. + + +. c. d. C. Wanger (1997) [13]. + +. c. d. Hoogendoorn and Bovy (2001) [11]. + + + + cd. d. Cho and Lo (2002) [16]. +. d. + + c. Dimensions (Except time and space): velocity v , desired velocity v0 , horizontal position (lane) y , and the others o . Scales: continue c and discrete d . Processes: deterministic d and stochastic s .. 2.4 Summary In this chapter, we summarize researches about mesoscopic traffic flow models and we could find that recent researches have improved to multilane and multiclass traffic flow models. However, there are many shortcomings that exist in the first model of Prigogine and Herman (1971)[5] do not improve, so the application is limited. These are needed to be improved. Therefore, we want to try to relax the shortcomings of Prigogine and Herman (1971) [5]. Because the shortcoming that the faster one car meets the other slower one without passing, the instant velocity-changing is unreasonable is not completely relaxed, we try to relax in this researches. C. Wanger (1997) [13] has been tried to research on this topic, but he does not completely relax. The moving behavior of his model does not match the space changing. In order to relax the instant velocity-changing and to consider reasonable behavior, we also consider finite space in our model. This means that velocities change with the positions of cars. Hence, we 15.

(25) would construct a gas-kinetic traffic flow model to describe more completely in next chapter.. 16.

(26) Chapter 3. Construct Mesoscopic Traffic Flow Model. We could find that the model of Prigogine and Herman (1971) [5] still has some shortcoming needed to be improved with literature review. In this chapter, we want to relax assumptions of the interaction term of Prigogine and Herman (1971) [5], instant slowing-down and no consider finite space, to obtain more realistic traffic flow model.. 3.1 The Question Description Gas-kinetic traffic flow model has already been developed in decades, and a lot of them are based on Prigogine and Herman (1971) [5], and Paveri-Fontana (1975) [7]. Some research focuses on topic, multi-lane or multi-class, and the other focuses on relaxing the unreasonable assumptions of Prigogine and Herman(1971) [5], which are proposed by Paveri-Fontana (1975) [7] (As what we show in Chapter 2). This research does the latter. According to Chapter 2, literature review, we found few literatures about relaxing the instant slowing-down of interaction term. There has no more complete one. Therefore, we want to construct a model which could describe successive velocity-changing and finite space effect. We hope it could be more meaningful than previous ones. We assume the same conditions as Prigogine and Herman (1971) [5]. Assume that there is an infinite long freeway, allowing passing. If the driver could not pass the former car, its interaction term (slowing-down behavior) could be described as the following equation (3.1) ∞. ∑ (1 − p)[ fi ∫vi dv j f j (v j − vi ) − fi ∫0 i dv j f j (vi − v j )] v. j. (3.1). In the above equation (3.1), car i means the car which we care, car j means 17.

(27) other car which interact with car i (only consider two-cars interaction), and p means passing probability. So (1 − p) means the probability without passing. The equation (3.1) shows that the faster (the latter) one car meets the other slower (the former) one without passing, the faster one needs to decelerate in order to avoid collision. Hence, they assume that it decelerates its velocity to the velocity of the slower one. The first term of the equation (3.1) shows the condition the latter car (the faster), its velocity is v j , meets the former car (the slower), its velocity is vi . The second term shows the opposite condition, the latter car (the faster), its velocity is vi , meets the former car (the slower), its velocity is v j . According to the section 2.1, we could know it has some shortcoming in the equation (3.1). Our research is main to relax instant slowing-down assumption of the interaction term and to relax no consideration of finite space. If the equation does not consider instant velocity-changing in slowing-down process, the headway between two cars should change with the velocity-changing of the later car. Therefore, we need to consider finite space in constructing model of continuous velocity-changing. We would construct a model that could describe the traffic situation of non-instant slowing-down and finite space consideration in next section.. 3.2 Model Construction In this section, the assumptions we used is the same as what Prigogine and Herman (1971) [5] did. Assume that there has an infinite long freeway, allowing passing, one car type, and each driver drives its different velocity and has the same desired velocity (cars of the same type have the same desired velocity ). After passing, the car contains its origin velocity (the velocity before passing). If the faster (the latter) one car meets the other slower (the former) one without passing, the faster one needs to decelerate in order to avoid collision. We introduce the model of C. Wanger (1997) 18.

(28) [13] to improve the shortcoming, instant velocity-changing and no finite space consideration, of Prigogine and Herman (1971) [5] and to construct our model. We use velocity distribution g ( x, v, w, t ) defined by Paveri-Fontana (1975) [7]. Among it, w means desired velocity, and the relation between g ( x, v, w, t ) , f ( x, v, t ) and f 0 ( x, w, t ) could be appeared like the following equation ∞. f ( x, v, t ) = ∫ dwg ( x, v, w, t ) 0. (3.2). ∞. f 0 ( x, w, t ) = ∫ dvg ( x, v, w, t ) 0. (3.3). According to the models of Paveri-Fontana (1975) [7] and C. Wanger (1997) [13], we modify the model of Prigogine and Herman (1971) [5]. Then our model is shown as below the equation (3.4). ∂g ∂ w − v ∂g + ( g) +v ∂x ∂v T ∂t =. ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t )g ( x, v3 , w, t )δ (v − Φ(v1, v3 )). 0≤ v1 ≤ v3. − g ( x, v, w, t ). ∫∫ dv1dv2 (1 − P)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v)). 0≤ v1 ≤ v. (3.4) Where d (v1, v3 ) = τv1 +. 1 c max. +. 1 (v3 − v1) 2 2 a. (3.5). 1 (v − v1) 2 + d (v1, v) = τv1 + c max 2 a. (3.6). Φ (v3 , v1) = v3 − a ∗ λ. (3.7). Φ (v, v1) = v − a ∗ λ. (3.8). 1. Where the uniform deceleration a ≥ 0 , τ means reaction time, T means relaxation time, cmax means the heaviest density, λ means the time interval that the 19.

(29) same each time interval of the latter car in deceleration process passes through. And d (v1, v3 ) and d (v1, v) means the headway between the later and the former cars, p means passing probability, (1 − p) means probability without passing, Φ (v1, v3 ) and Φ (v1, v) means the next velocity of the later car when it starts to decelerate. Our mesoscopic traffic flow model constructs as the equation (3.4), where the left-hand side first and second terms, described the velocity-changing is caused by flow-in and flow-out, are called “convection term”. The left-hand-side third term is called “relaxation term”, it is also called “acceleration term”. Because velocities of drivers are often different with their desired velocities, they would want to accelerate to their desired velocities. The relaxation term we follow the exponential law assuming by Prigogine and Herman (1971) [5] to show its effect in velocity distribution. The right-hand-side term of equation (3.4) is called “interaction term”, and it means that the faster (the latter) one car meets the other slower (the former) one without passing, the faster one needs to decelerate in order to avoid collision. The first term of interaction term means the latter car v3 meets the former one v1 , v1 < v3 . According to the assumption, the latter car v3 should decelerate to the former car without passing. d (v1, v3 ) means the headway, so the headway should match the equation (3.5), the very two cars start to interact. By the way, the equation (3.7) shows the velocity through changing in the interaction process. It depends on the difference of velocities between two cars, the uniform deceleration and time interval. In the same way, the second term of interaction term means the latter car v meets the former one v1 , v1 < v , without passing, the faster one needs to decelerate in order to avoid collision. d (v1, v) means the headway, so the headway should match the equation (3.6), the very two cars start to interact. By the way, the equation (3.8) shows the velocity through changing in the interaction process. It depends on the difference of velocities between two cars, the uniform deceleration and time interval. 20.

(30) The second term of interaction term could also describe that successive deceleration affects. ∂g to decrease. ∂t. The main characteristics of our model are shown as the following: 1.. Headway: In this research, we express the headway between the very two car we care as the equations (3.5) and (3.6). In other researches likes [13, 17-19], they only consider the safety distance τv +. 1 cmax. or τv + A , if they introduce the. factor ” finite space” into their model. There A means car length. However, if velocity changes step by step, not instant changing, the above safety distance is not safe. The distance through decelerating process should be considered. If we take uniform deceleration formula to show the decelerating process, the headway should be shown as (reaction time*the velocity of the former car + the minimum headway + the distance required by deceleration). There “reaction time*the velocity of the former car + the minimum headway” is left to be safety distance for decelerating process. Hence, the headway should be represented the equation (3.5) and (3.6), and correspond to the safety distance in non-instant velocity-changing condition. 2.. Velocity changing: We show the process of deceleration when the later (the faster) car meets the former (the slower) one as the equations of (3.7) and (3.8). We assume cars decelerate by uniform deceleration, so the velocity changing is related to the uniform deceleration. The velocity of the later car and the headway match the assumption of uniform deceleration. Therefore, the model could more reasonably describe drivers’ behavior.. 3.. No adjust factor: In the past, scholars take adjust factor with considering finite space in their researches. Because of instant velocity-changing, they need to take 21.

(31) adjust factor to increase the frequency of interaction in considering finite space. However, our model is both considered finite space and successive velocity-changing, so it does not need to multiply an adjust factor.. 3.3 Summary We construct a standard and simple mesoscopic traffic flow model under the same assumptions as Prigogine and Herman (1971) [5], and Paveri-Fontana (1975) [7]. And we consider more, containing finite space and successive deceleration, in interaction term. If the headway satisfies the equations (3.5) or (3.6), the later (the faster) car would start to decelerate with constant uniform deceleration a . Hence, the velocity through deceleration process could be shown as the equations (3.7) and (3.8). The condition to occur interaction has been set in our model, and the deceleration process has also been regular. The situation of interaction would not happen at casual.. 22.

(32) Chapter 4. Macroscopic model. The advantage of mesoscopic traffic flow model is that it could describe behaviors of microscopic traffic flow, and it could get macroscopic model by means of integration. The previous chapter constructs the mesoscopic traffic flow model by describing microscopic cars so this chapter uses microscopic traffic flow which chapter3 constructs to obtain macroscopic traffic flow model, flow conservation, equation of average velocity, variance, by the methods like integrations. It could be good to analyze the characteristic in the later chapter.. 4.1 Assumption This research is assumed for the single car which is a simple situation. According to the assumption of Helbing (1995) [19, 20], non-equilibrium state could approach equilibrium quickly with the zeroth-order approximation of local equilibrium. According to the research [19, 21], we could know the velocity distribution of equilibrium approaching normal distribution. The velocity distribution could be expressed by the following f ( x, v, t ) ≈ f e (v ( x, t ),θ ( x, t )) c ( x, t ) = exp{−[v − v ( x, t )]2 /[ 2θ ( x, t )]} 2πθ ( x, t ). (4.1). In the above equation (4.1), f e (v ( x, t ),θ ( x, t )) shows the velocity distribution in equilibrium, v ( x, t ) shows velocity of equilibrium, and θ ( x, t ) shows variance. Using equation (4.1) to get macroscopic model, the following are common skills ∞. ∫−∞ ∞. ∫−∞. dz z 2k exp( ∞. ∫0. dz z 2k +1 exp(. [. − z2 )=0 4θ. ]. 1 − z2 ) = (−2) 2k + 4k θ k +1 / 2Γ( + k ) 4θ 2. dz z 2k +1 exp(. 23. − z2 ) = 2 ⋅ 4" 2k (2θ ) k +1 4θ.

(33) − z2 π ) = 1 ⋅ 3" (2k − 1) (2θ ) k +1 / 2 4θ 2. (4.2). Besides, we use Tayler expansion to expand space so we would use. ∂f , and we ∂x. ∞. ∫0. dz z 2k exp(. set z = v + w − 2v ( x, t ) , y = v − w , and then. ∂f could be expressed as following ∂x. c ∂f ∂ − ( z − y)2 exp[ ]} = { ∂x ∂x 2πθ 4 ⋅ 2θ =. c − 1 −3 / 2 ∂θ ∂c 1 − ( z − y)2 − ( z − y )2 exp[ ]+ ( )θ exp[ ] 8θ 8θ ∂x 2πθ ∂x 2π 2 +. c ( z − y ) 2 − 2 ∂θ − ( z − y)2 θ exp[ ] 8 8θ ∂x 2πθ. +. c − 2( z − y ) −1 ∂z − ( z − y)2 θ exp[ ] 8 8θ ∂x 2πθ. ⎧⎪ ∂c 1 − ( z − y)2 ∂θ ( z − y ) 2 − 2 ∂θ ( z − y ) ∂z −1 ⎫⎪ 1 cθ cθ ⎬ exp[ ] + − = ⎨ − cθ −1 8 4 ∂x 8θ ∂x ∂x ⎪⎭ 2πθ ⎪⎩ ∂x 2. (4.3) In order to use symbols conveniently, this chapter would use some numbers of variables to define as following Average Velocity. v ( x, t ) = ∫ dvv. f ( x, v, t ) c ( x, t ). (4.4). Velocity Variance. θ ( x, t ) = ∫ dv(v − v ( x, t ))2. f ( x , v, t ) c ( x, t ). (4.5). Traffic Pressure. Ρ ( x, t ) =. 1 dv(v − v ) f ( x, v, t ) ∫ dw(v − w) f ( x, w, t ) c ( x, t ) ∫. = ∫ dv(v − v ) 2 f ( x, v, t ). (4.6). = c( x, t )θ ( x, t ) Average Desired Velocity. v0 ( x, t ) = ∫ dv ∫ dww 24. g ( x, v, w, t ) f ( x, v, t ). (4.7).

(34) Flux of Velocity Variance. 1 dv(v − v ( x, t ))2 f ( x, v, t ) ∫ dw(v − u ) f ( x, u, t ) ∫ c ( x, t ) (4.8) 3 = ∫ dv(v − v ( x, t )) f ( x, v, t ). J ( x, t ) =. Covariance g ( x, v, w, t ) c ( x, t ) f ( x , v, t ) = ∫ dv(v − v ( x, t ))(v~0 ( x, v, t ) − v0 ( x, t )) c ( x, t ). C ( x, t ) = ∫ dv ∫ dw(v − v ( x, t ))( w − v0 ( x, t )). (4.9). where g ( x, v, w, t ) v~0 ( x, v, t ) = ∫ dww f ( x, v , t ). (4.10). The zeroth-order approximation of local equilibrium is close to flux of velocity variance J ( x, t ) ≈ J e ( x, t ) = 0 . Using the zeroth-order approximation of local equilibrium to get velocity variance J ( x, t ) , we could adapt J ( x, t ) for the following. 2 1 dv(v − v ( x, t )) f ( x, v, t ) ∫ dw(v − u ) f ( x, u, t ) ∫ c ( x, t ) ∞ ∞ 1 d (δv)(δv) 2 f ( x, v + δv, t ) ∫ d (δu )(δv − δu ) f ( x, u + δu, t ) = ∫ −∞ c ( x, t ) − ∞. J ( x, t ) =. = c ( x, t ) ∗ 2 ∫. ∞. −∞. ∞. dz ∫ dy ( 0. 1 − y2 − z2 1 y+z 2 exp( ) exp( ) ) y 4πθ 4πθ 2 4πθ 4πθ. ⎡1 1 1 1 π 1 ⎤ (4θ ) 2 = c ( x, t ) ⎢ + 0 + 2θ (4 + 4)θ 3 / 2 ⎥ 2 2 4πθ ⎦ 4πθ ⎣2 2 ⎡ 2θ 3 / 2 θ 3 / 2 ⎤ = c ( x, t ) ⎢ + ⎥ π ⎦⎥ ⎣⎢ π = c ( x, t ). 3θ 3 / 2. π (4.11). We assume v = v ( x, t ) + δv and w = v0 ( x, t ) + δw to change the first column of RHS. to. the. second. column. of. RHS. in. equation. (4.11).. We. assume z = δv + δw and y = δv − δw , where Jacobin=1/2, from the second column to. 25.

(35) the third column, and use the zeroth-order approximation of local equilibrium.. We. could get flux of velocity variance J ( x, t ) by the skills of equation (4.2). Because of J ( x, t ) ≈ J e ( x, t ) = 0 and the equation (4.11), the high order could be ignored if θ n , n ≥ 3 / 2 . In the following section, we could expect that macroscopic model we got is the Euler-like traffic equations.. 4.2 Density Equation The mesoscopic traffic flow model could get macroscopic model by the method of moment. We deal mesoscopic model with zeroth-order momentum in this section. That means we would deal mesoscopic model with integration and get macroscopic equation based on the macroscopic parameter, density c( x, t ) . Our mesoscopic mode would integrate velocity v and desired velocity w . 1.. Convection term integrates velocity v and desired velocity w . ∂g. ∫ dv ∫ dw ∂t. +v. ∂g ∂x. ∂f ∂f +v ∂x ∂t ∂c ∂ (cv ) + = ∂x ∂t = ∫ dv. 2.. Relaxation term integrates velocity v and desired velocity w .. ∂ ⎛ w−v ⎞ g⎟ = 0 T ⎠. ∫ dv ∫ dw ∂v ⎜⎝. 26.

(36) 3.. Interaction term integrates velocity v and desired velocity w .. ∫ dv ∫ dw[ ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t )g ( x, v3 , w, t )δ (v − Φ(v1, v3 )) 0≤ v1 ≤ v3. − g ( x, v, w, t ). ∫∫ dv1dv2 (1 − P)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v))]. 0≤ v1 ≤ v. = ∫ dv. ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v − Φ(v1, v3 )). 0≤ v1 ≤ v3. − ∫ dv f ( x, v, t ) = ∫ dv2. ∫∫ dv1dv2 (1 − P)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v)). 0≤ v1 ≤ v. ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v2 − Φ(v1, v3 )). 0≤ v1 ≤ v3. − ∫ dv3 f ( x, v, t ). ∫∫ dv1dv2 (1 − P)(v3 − v1) f ( x + d (v1, v3 ), v1, t )δ (v2 − Φ(v1, v3 )). 0≤ v1 ≤ v3. =0 Change the first equal sign to the second sign by integrating the whole interaction item to separately integrating the passive and negative result of the interaction. Change the section equal sign to the third sign by changing variable v to v2 of the passive result of the interaction, and changing variable v to v3 of the negative result of the interaction. Because the passive and negative results of the interaction are the same, they could cancel and the answer is zero. Use the result of integrations to velocity v and desired velocity w by convection term, relaxation term and interaction term to add, we could get macroscopic equation based on macroscopic parameter, density c( x, t ) , as following equation (4.12) ∂c ∂ (cv ) + =0 ∂t ∂x. 27. (4.12).

(37) 4.3 Average velocity equation This section will do the first-order momentum to mesoscopic model. That means that mesoscopic model times velocity and then integrates velocity. We could get macroscopic equation based on macroscopic parameter, average velocity v ( x, t ) . Our mesoscopic model times velocity v and integrates velocity v and desired velocity w . 1.. Convection term times velocity v and then integrates velocity v and desired velocity w . ⎡ ∂g. ∫ dv ∫ dwv ⎢⎣ ∂t. +v. ∂g ⎤ ∂x ⎥⎦. ∂f ∂f + v2 ∂t ∂x ∂f ∂f = ∫ dvv + (v − v + v )(v − v + v ) ∂t ∂x ∂f ∂f = ∫ dvv + (v − v ) 2 + 2(v − v )v + v 2 ∂t ∂x = ∫ dvv. [. ]. ∂ (cv ) ∂ (cθ ) ∂ (cv 2 ) + + ∂x ∂x ∂t ∂v ∂c ∂ (cθ ) ∂v ∂ (cv ) =c +v + + cv +v ∂t ∂t ∂x ∂x ∂x ∂v ∂ (cθ ) ∂v =c + + cv ∂t ∂x ∂x. =. The second column is expanded to the form of the third column in order to make it become the integration of the forth column. We could get the above result by means of the definitions of equation (2.2), equation (2.3), equation (4.5), and equation (4.6).. 28.

(38) 2.. Relaxation term times velocity v and then integrates velocity v and desired velocity w . ∂ ⎛w−v ⎞ g⎟ T ⎠ ~ v −v = − ∫ dv 0 f T (v − v ) = −c 0 T. ∫ dv ∫ dwv ∂v ⎜⎝. 3.. Interaction term times velocity v and then integrates velocity v and desired velocity w. ∫ dv ∫ dwv[ ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t )g ( x, v3 , w, t )δ (v − Φ(v1, v3 )) 0≤ v1 ≤ v3. − g ( x, v, w, t ). ∫∫ dv1dv2 (1 − p)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v))]. 0≤ v1 ≤ v. = ∫ dv. ∫∫ dv1dv3v(1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v − Φ(v1, v3 )). 0≤ v1 ≤ v3. − ∫ dvvf ( x, v, t ). = ∫ dv2. =. 0≤ v1 ≤ v. ∫∫ dv1dv3v2 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v2 − Φ(v1, v3 )). 0≤ v1 ≤ v3. − ∫ dv3 f ( x, v, t ) = ∫ dv2. ∫∫ dv1dv2 (1 − p)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v)). ∫∫ dv1dv2v3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t )δ (v2 − Φ(v1, v3 )). 0≤ v1 ≤ v3. ∫∫ dv1dv3 (1 − p)(v2 − v3 )(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v2 − Φ(v1, v3 )). 0≤ v1 ≤ v3. ∫∫ dv1dv3 (1 − p)[(v3 − aλ ) − v3 ](v3 − v1) f ( x, v3 , t )[ f ( x, v1, t ) + d (v1, v3 ). 0≤ v1 ≤ v3. = −(1 − p)aλ. ∫∫ dv1dv3 (v3 − v1) f ( x, v3 , t ). 0≤ v1 ≤ v3. ∗ { f ( x, v1, t ) + [τv1 +. 1 c max. +. 1 (v3 − v1) 2 ∂f ( x, v1, t ) ] } a ∂x 2. Where v1 = v ( x, t ) + δv1 , v3 = v ( x, t ) + δv3 , and Jacobin=1. 29. ∂f ( x, v1, t ) ] ∂x.

(39) = −(1 − p)aλ ∫. ∞. −∞. d (δv3 ). ∫ d (δv1)(δv3 − δv1) f ( x, v + δv3 , t ). δv1 <δv3. ∗ { f ( x, v + δv1, t ) + [. 1 c max. Set z = δv3 + δv1 and y = δv3 − δv1 , the δv3 =. + τ (v + δv1) +. 1 (δv3 − δv1) 2 ∂f ( x, v + δv1, t ) ] } 2 a ∂x. z+ y z−y , δv1 = , and Jacobin=1/2 2 2. use the zeroth-order approximation of local equilibrium, equation (4.1) and equation(4.3) ∞ ∞ = −(1 − p )aλ{c 2 dz dy y 0 −∞. ∫. + c∫. ∞. −∞. ∫. ∞. 1. 0. c max. dz ∫ dy y[. −z2 − y2 1 1 exp( ) exp( ) 4θ 4θ 4πθ 4πθ. + τ (v +. −z2 − y2 z−y y2 1 1 )+ ] exp( ) exp( ) 4θ 4θ 2 2a 4πθ 4πθ. ⎡ ∂c 1 ∂θ ( z − y ) 2 − 2 ∂θ ( z − y ) ∂z −1 ⎤ ∗ ⎢ − cθ −1 + − cθ cθ ⎥} ∂x ∂x 8θ 4θ ∂x ⎣⎢ ∂x 2 ⎦⎥. Set A =. 1 c max. + τv , B =. ∂c 1 −1 ∂θ 1 ∂θ 1 ∂z −1 − cθ cθ . , E=− , D = cθ − 2 ∂x 2 ∂x 4θ ∂x 8 ∂x. We take A , B , D , and E to the original equation and change its form as following. = −(1 − p )aλ{c 2 ( + c∫. ∞. 1 2θ ) 4πθ. ∞. dz ∫ dy y[ A + τ. −∞. 0. − z2 − y2 1 ( z − y) y 2 1 + ] exp( ) exp( ) 2 2a 4πθ 4θ 4θ 4πθ. ⋅ [ B + D( z − y ) 2 − E ( z − y )]. Set ∞. ∞. −∞. 0. ∆ = ∫ dz ∫ dyy[A + τ. (z − y) y 2 1 −z 2 1 −y 2 exp( ) exp( ) + ] 2 2a 4πθ 4θ 4θ 4πθ. ∗ ⎡⎣ B + D(z − y) 2 − E(z − y) ⎤⎦}. 30.

(40) The calculation as following. ∞. ∞. −∞. 0. ∆ = ∫ dz ∫ dy[ABy + AD(z 2 y − 2y 2 z + y3 ) − AE(zy − y 2 ) +. B τ(zy − y 2 ) 2. D 3 E By3 τ(z y + 3zy3 − 3y 2 z 2 − y 4 ) − τ(z 2 y − 2y 2 z + y3 ) + 2 2 2a 2 D E 1 1 −z −y 2 exp( ) exp( ) + (z 2 y3 − 2y 4 z + y5 ) − (y3z − y 4 )] 2a 2a 4θ 4θ 4πθ 4πθ +. We could cancel the above because of the first column of equation (4.2). ∞. ∞. −∞. 0. B D τ(− y 2 ) + τ(−3y 2 z 2 − y 4 ) 2 2 3 E By D E + τ(z 2 y − 2y 2 z + y3 ) + + (z 2 y3 − 2y 4 z + y5 ) − (y3z − y 4 )] 2 2a 2a 2a 2 2 1 1 −z −y exp( ) exp( ) ∗ 4θ 4θ 4πθ 4πθ. ∆ = ∫ dz ∫ dy[ABy + AD(z 2 y + y3 ) − AE(− y 2 ) +. ⎡ 3/ 2 π 1 ⎞ 1 ⎛ 2θ = AB ⎜ 2θ + 2(2θ) 2 ⎟ + AD ⎢8θ 2 4 πθ 4 πθ ⎝ ⎠ ⎣. = AB. ⎛ π 1 ⎞ 1 ⎤ (2θ)3/ 2 ⎟ ⎥ − AE ⎜⎜ − 2 4πθ ⎟⎠ 4πθ ⎦ ⎝. +. π π 1 π B ⎛ π 1 ⎞ D ⎛ 1 ⎞ 3/ 2 3/ 2 τ ⎜⎜ − − 3 (2θ)5/ 2 (2θ)3/ 2 ⎟⎟ + τ ⎜⎜ −3 (2θ) 8θ ⎟ 2 ⎝ 2 2 2 4πθ 2 4πθ ⎠ 2 ⎝ 4πθ ⎠⎟. −. π 1 E ⎛ 1 ⎞ B⎛ 1 ⎞ 2 τ ⎜⎜ 2θ ⋅ 8θ3/ 2 − 2(2θ) 2 ⎟⎟ + ⎜ 2(2θ) ⎟ 2 ⎝ 2 4πθ 4πθ ⎠ 2a ⎝ 4πθ ⎠. +. π 1 π D⎛ 1 ⎞ E⎛ 5/ 2 2 3/ 2 − 2 ⋅ 4(2θ)3 ⎜⎜ −2(2θ) 8θ ⎟⎟ − ⎜⎜ −1⋅ 3(2θ) 2 2a ⎝ 2 4πθ 4πθ ⎠ 2a ⎝. 1 ⎞ ⎟ 4πθ ⎟⎠. 6 B 3 2 20 3 θ1/ 2 ADθ3/ 2 + AEθ − τθ − 6τDθ2 − Bθ3/ 2 + Dθ5/ 2 + Dθ 2 + τEθ3/ 2 + 2 a π π π a π a π. Because we had set B =. ∂c 1 −1 ∂θ 1 ∂θ 1 ∂z −1 − cθ ， D = cθ − 2 ，E = − cθ , we ∂x 2 ∂x 4θ ∂x 8 ∂x. could decrease the numbers of symbols and join ∆ as following ∆=(. 3 τ ∂c ∂θ A −1 / 2 τ θ1/ 2 − θ ) + ( θ θ 1 / 2 )c − + 2 ∂x 4 π 2 2a π ∂x π 3τ 1 / 2 3 A ∂z θ +( − + θ )c 4 4 π 4a ∂x. A. 31.

(41) Because of equation (4.11), we could ignore the high terms when θ n , n ≥ 3 / 2 . The interaction term could be expressed as following. Interaction term. τ ∂c A −1 / 2 θ θ1/ 2 − θ ) + ( 2 ∂x 4 π π π τ 3 3τ 1 / 2 3 A ∂θ ∂z θ 1 / 2 )c θ − + +( − + θ )c } 2 2a π 4 4 π 4a ∂x ∂x. = −(1 − p)aλc 2. 1. θ 1 / 2 − (1 − p)aλc{(. A. Use the result of doing multiplication to v and integrations to velocity v and desired velocity w by convection term, relaxation term and interaction term to add, we could get as following. (v − v0 ) ∂v ∂v ∂P +c + cv + ∂x T ∂t ∂x 1 1/ 2 A 1 / 2 τ ∂c A −1 / 2 θ θ θ = −(1 − p)aλc 2 − (1 − p)aλc{( − θ) + ( 2 ∂x 4 π π π 3τ 1 / 2 3 3 A τ ∂z ∂θ θ θ 1 / 2 )c +( − + θ )c } − + 4 4 π 4a 2 2a π ∂x ∂x. c. For the above divides by density c( x, t ) , we could get the macroscopic equation based on macroscopic parameter, average number v ( x, t ) . ∂v (v − v0 ) ∂v 1 ∂P + +v + ∂x ∂t c ∂x T 1 1/ 2 A 1 / 2 τ ∂c A −1 / 2 θ θ θ = −(1 − p) aλc − (1 − p) aλ{( − θ) + ( 2 ∂x 4 π π π 3τ 1 / 2 3 3 A τ ∂z ∂θ θ θ 1 / 2 )c +( − + θ )c } − + 4 4 π 4a 2 2a π ∂x ∂x. (4.13) In the derivation process, we assume z = δv3 + δv1 = v3 + v1 − 2v ( x, t ) so. ∂z ∂v = −2 . ∂x ∂x. Take it to equation (4.13), and we could get macroscopic equation (4.14) based on average number v ( x, t ) .. 32.

(42) ∂v 1 ∂P ∂v (v0 − v ) + +v − T ∂t c ∂x ∂x 1 1/ 2 A 1 / 2 τ ∂c A −1 / 2 θ θ θ = −(1 − p )aλc − (1 − p )aλ{( − θ) + ( 2 ∂x 4 π π π 3 τ 3τ 1 / 2 3 ∂θ ∂v A θ 1 / 2 )c θ − + + (− + − θ )c } 2 2a π 2 2 π 2a ∂x ∂x. (4.14) Where A =. 1 c max. + τv. 4.4 Variance This section will do the second-order momentum to mesoscopic model. That means that mesoscopic model times the square of velocity and then integrates velocity. We could get macroscopic equation based on macroscopic parameter, variance θ ( x, t ) . Our mesoscopic model (3.4) times the square of velocity v 2 and then integrates velocity v and desired velocity w . 1.. Convection term times the square of the velocity v 2 and then integrates velocity v and desired velocity w .. ∫ dv ∫ dwv = ∫ dv. 2 ∂g. (. ∂t. +v. ∂g ) ∂x. ∂ ( fv 2 ) ∂ ( fv 3 ) + ∂x ∂t. = ∫ dv{. ∂ ( fvv 2 ) ∂ ( fvv ) ∂ ( fv 2 ) ∂[ f (v − v )3 ] ∂[ f (v − v ) 2 ] −3 + − +2 ∂x ∂x ∂t ∂t ∂t +3. ∂ ( fv 2v ) ∂ ( fv 3 ) } + ∂t ∂x. ∂ (cθ ) ∂ (cv 2 ) ∂ ( J ) ∂ (cv θ ) ∂ (cv 3 ) + + +3 + ∂x ∂x ∂x ∂t ∂t ∂v ∂ (cθ ) ∂θ ∂c ∂c ∂v ∂ (cJ ) ∂θ ∂ (cv ) + v 2 + 2cv + + 2cθ + 2v =c +θ + cv +θ ∂t ∂t ∂t ∂t ∂x ∂x ∂x ∂x ∂x ∂ (cv ) ∂v + v2 + 2cv 2 ∂x ∂x ∂θ ∂v ∂ ( J ) ∂v ∂ (cθ ) ∂θ ∂v =c + 2cv + + 2cθ + 2v + cv + 2cv 2 ∂x ∂x ∂t ∂t ∂x ∂x ∂x =. 33.

(43) ∂θ ∂ ( J ) ∂v ∂θ (v − v ) 1 1/ 2 − (1 − p )aλc + + 2cθ + cv + 2cv{ 0 θ ∂t ∂x ∂x ∂x T π 3 A 1 / 2 τ ∂c A 1/ 2 τ ∂θ − (1 − p)aλ[( − θ) + ( − + θ 1 / 2 )c θ θ 2 ∂x 4 π 2 2a π ∂x π 3τ 1 / 2 3 A ∂z θ +( − + θ )c ]} 4 4 π 4a ∂x (v − v ) 1 1/ 2 ∂θ ∂v ∂θ θ =c + 2cθ + cv + 2cv { 0 − (1 − p )aλc T ∂t ∂x ∂x π 3 A 1 / 2 τ ∂c A τ ∂θ θ 1 / 2 )c θ θ −1 / 2 − + − (1 − p)aλ[( − θ) + ( 2 ∂x 4 π 2 2a π ∂x π 3τ 1 / 2 3 A ∂z θ +( − + θ )c ]} 4 4 π 4a ∂x =c. The second column is expanded to the form of the third column in order to make it become the integration of the forth column. We could get the integration of the forth column by means of the definitions of equation (2.2), equation (2.3), equation (4.4), equation (4.5), and equation (4.8). We could change it to the result of the sixth column by equation (4.12), and the result of the seventh column by equation (4.13). 2.. Relaxation term times the square of the velocity v 2 and then integrates velocity v and desired 2 ∂ ⎛w−v. ⎞ g⎟ ⎜ ∂v ⎝ T ⎠ w−v g = −2 ∫ dv ∫ dwv T. ∫ dv ∫ dwv. [(v − v )( w − v~0 ) − v v~0 + wv + vv~0 ] − [(v − v ) 2 + 2vv − v 2 ] f = −2 ∫ dv ∫ dw T − 2c [C + v v0 − θ − v 2 ] = T. 34.

(44) 3.. Interaction term times the square of the velocity v 2 and then integrates velocity v and desired. ∫ dv ∫ dwv. 2. ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t )g ( x, v3 , w, t )δ (v − Φ(v1, v3 )). [. 0≤ v1 ≤ v3. ∫∫ dv1dv2 (1 − p)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v))]. − g ( x, v, w, t ). 0≤ v1 ≤ v. = ∫ dvv 2[. ∫∫ dv1dv3 (1 − p)(v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v − Φ(v1, v3 )). 0≤ v1 ≤ v3. − ∫ dv f ( x, v, t ). = ∫ dv2. 0≤ v1 ≤ v. ∫∫ dv1dv3 (1 − p)v2 (v3 − v1) f ( x + d (v1, v3 ), v1, t ) f ( x, v3 , t )δ (v2 − Φ(v1, v3 )) 2. 0≤ v1 ≤ v3. − ∫ dv3 f ( x, v, t ) =. ∫∫ dv1dv2 (1 − p)(v − v1) f ( x + d (v1, v), v1, t )δ (v2 − Φ(v1, v))]. ∫∫ dv1dv2 (1 − p)v3 (v3 − v1) f ( x + d (v1, v3 ), v1, t )δ (v2 − Φ(v1, v3 )) 2. 0≤ v1 ≤ v3. ∫∫ dv1dv3 (1 − p)[(v3 − aλ ). 2. − v32 ](v3 − v1) f ( x, v3 , t )[ f ( x, v1, t ) + d (v1, v3 ). 0≤ v1 ≤ v3. ∂f ( x, v1, t ) ] ∂x. ∫∫ dv1dv3 (v3 − v1)(2v3 − aλ ) f ( x, v3 , t ). = −(1 − p )aλ. 0≤ v1 ≤ v3. ∗ { f ( x, v1, t ) + [τv1 +. 1 c max. +. 1 (v3 − v1) 2 ∂f ( x, v1, t ) ] } 2 a ∂x. Set v1 = v ( x, t ) + δv1 , v3 = v ( x, t ) + δv3 , and Jacobin=1. = −(1 − p )aλ ∫. ∞. −∞. d (δv3 ). ∫ d (δv1)(δv3 − δv1)(2δv3 + 2v − aλ ) f ( x, v + δv3 , t ). δv1 <δv3. ∗ { f ( x, v + δv1, t ) + [. 1 c max. + τ (v + δv1) +. Set z = δv3 + δv1 and y = δv3 − δv1 , then δv3 =. 1 (δv3 − δv1) 2 ∂f ( x, v + δv1, t ) ] } a ∂x 2. z+ y z−y , δv1 = , and Jacobin=1/2 use 2 2. the zeroth-order approximation of local equilibrium, equation(4.1)and equation (4.3). 35.

(45) = −(1 − p )aλ{c 2 ∫. ∞. ∞. −∞. dz ∫ dy y ( z + y + 2v − aλ ) 0. −z2 − y2 1 1 exp( ) exp( ) 4θ 4θ 4πθ 4πθ. z− y y2 1 1 −z2 − y2 )+ ] exp( ) exp( ) + c ∫ dz ∫ dy y ( z + y + 2v − aλ )[ + τ (v + −∞ 0 c max 2 2a 4πθ 4θ 4θ 4πθ ⎡ ∂c 1 ∂θ ( z − y ) 2 − 2 ∂θ ( z − y ) ∂z −1 ⎤ cθ cθ ⎥} + − ∗ ⎢ − cθ −1 x 2 x 8 θ x 4 θ x ∂ ∂ ∂ ∂ ⎦⎥ ⎣⎢ ∞. Set. ∞. 1. A=. c max. 1. + τv ,. B=. ∂c 1 −1 ∂θ , − cθ ∂x ∂x 2. 1 ∂θ , D = cθ − 2 8 ∂x. E=−. 1 ∂z −1 cθ , 4θ ∂x. and F = 2v − aλ . We take A , B , D , and E to the original equation and change its form as following. = −(1 − p)aλ{c 2 ∫. ∞. −z2 − y2 1 1 exp( ) exp( ) 4θ 4θ 4πθ 4πθ. ∞. −∞. dz ∫ dy ( yz + y 2 + Fy ) 0. ∞ τ y2 −z2 − y2 1 1 dz ∫ dy y ( yz + y 2 + Fy )[ A + ( z − y ) + ] exp( ) exp( ) 0 −∞ 2 2a 4πθ 4θ 4θ 4πθ. + c∫. ∞. [. ]. ∗ B + D( z − y ) 2 − E ( z − y ) } = −(1 − p)aλ{c 2 [. π 2. (2θ )3 / 2. 1 1 + F (2θ ) ] 4πθ 4πθ. + c[ AB( yz + y 2 + Fy ) + AD(− y 2 z 2 + y 4 − y 3 z + yz 3 + Fyz 2 − 2 Fy 2 z + Fy 3 ) − AE (− yz 2 − y 3 + Fyz − Fy 2 ) + +. τ 2. τ 2. B( yz 2 − y 3 + Fyz − Fy 2 ). D(−2 y 2 z 3 + 2 y 4 z − y 5 + yz 4 − Fy 4 − 3Fy 2 z 2 + 3Fy 3 z + Fyz 3 ). τ B 3 − E (− y 2 z 2 + y 4 − y 3 z + yz 3 + Fyz 2 − 2 Fy 2 z + Fy 3 )} + ( y z + y 4 + Fy 3 ) 2 2a. +. D (− y 4 z 2 + y 6 − y 5 z + y 3 z 3 + Fy 3 z 2 − 2 Fy 4 z + Fy 5 ) 2a. −z2 − y2 E 1 1 5 3 3 4 − (− y + y z + Fy z − Fy )]} exp( ) exp( ) 4θ 4θ 2a 4πθ 4πθ. 36.

(46) = −(1 − p )aλc 2 [θ +. Fθ 1/ 2. − (1 − p )aλc{ AB[ +3. π 2. π π 2. ]. π 1 1 1 ] + AD[− (2θ )3 / 2 8θ 3 / 2 + F (2θ ) 4πθ 2 4πθ 4πθ. (2θ )3 / 2. π 1 1 1 ] + F (2θ )8θ 3 / 2 + F 2(2θ ) 2 2 4πθ 4πθ 4πθ. (2θ )5 / 2. − AE[(2θ )8θ 3 / 2. π. 1. 2 4πθ. − 2(2θ ) 2. π 1 1 (2θ )3 / 2 ] −F 2 4πθ 4πθ. τ π 1 1 π 1 (2θ )3 / 2 ] + B[(2θ )8θ 3 / 2 − 2(2θ ) 2 −F 2 2 4πθ 2 4πθ 4πθ τ π π π 1 1 (2θ )5 / 2 (2θ )3 / 2 8θ 3 / 2 + D[− F 3 − 3F ⋅ 3 2 2 2 2 4πθ 4πθ 1 3 π 1 − 2 ⋅ 4(2θ )3 ] 2 2 4πθ 4πθ. + 2θ ⋅ 32θ 5 / 2. τ π π 1 π 1 − E[− +3 (2θ )3 / 2 8θ 3 / 2 (2θ )5 / 2 2 2 2 4πθ 2 4πθ π. + F (2θ )8θ 3 / 2 + +. 2 4πθ. + F ⋅ 2(2θ ) 2. 1 ] 4πθ. π B 1 1 + F 2(2θ ) 2 ] [3 (2θ )5 / 2 2a 2 4πθ 4πθ. π π 1 π D 1 + 1⋅ 3 ⋅ 5 [−3 (2θ )5 / 2 8θ 3 / 2 (2θ )7 / 2 2a 2 2 4πθ 2 4πθ π. + F ⋅ 2 ⋅ (2θ ) 2 8θ 3 / 2 −. 1. 1. + F ⋅ 2 ⋅ 4(2θ )3. 2 4πθ. 1 ] 4πθ. E π 1 π 1 1 − 2 ⋅ 4(2θ )3 − F ⋅3 ]} (2θ )5 / 2 [2 ⋅ (2θ ) 2 8θ 3 / 2 2 2a 2 4πθ 4πθ 4πθ. = −(1 − p)aλc 2[θ +. F. π. θ1/ 2 ]. − (1 − p)aλc{ AB[θ + − AE[− Fθ + + −. τ 2. τ 2. F. π 2. π. θ 1 / 2 ] + AD[−2θ 2 + 6θ 2 +. π. π. τ 2 3/ 2 4 3/ 2 θ 3 / 2 ] + B[− Fθ + θ − θ ] 2 π π π. 4. θ 3/ 2 −. D[−6 Fθ 2 − 6 Fθ 2 + E[−2θ 2 + 6θ 2 +. 2F 3 / 2 F 3 / 2 θ + θ ]. 2. π. 12 5 / 2 32 5 / 2 θ − θ ]. π. Fθ 3 / 2 +. 37. π. 4. π. θ 3/ 2] +. B 4F 3 / 2 [ 2θ + 6θ 2 ] 2a π.

(47) D 8F 5 / 2 32 F 5 / 2 [−12θ 3 + 60θ 3 + θ + θ ] 2a π π E 8 5 / 2 32 5 / 2 − [−6 Fθ 2 + θ − θ ]} 2a π π F 1/ 2 = −(1 − p)aλc 2[θ + θ ] +. π. − (1 − p)aλc{ AB[θ + +. τ. B[− Fθ −. F. π. θ 1 / 2 ] + AD[4θ 2 +. 6F 3 / 2 2 3/ 2 θ ] − AE − θ − Fθ ]. π. π. τ 20 5 / 2 τ 6 Fθ 3 / 2 ] θ 3 / 2 ] + D[−12 Fθ 2 − θ ] − E[4θ 2 + 2 2 π π π. 2. 2 B 4F 3 / 2 D E 40 F 5 / 2 24 5 / 2 + [ 2θ + 6θ 2 ] + [48θ 3 + θ ] − [−6 Fθ 2 − θ ]} 2a π 2a 2a π π Because we set B =. ∂c 1 −1 ∂θ 1 ∂z −1 ∂θ 1 − cθ ，D = cθ − 2 ，E = − cθ ，we make the ∂x 2 ∂x 4θ ∂x 8 ∂x. number of symbols simple and take them into above. = −(1 − p )aλc 2 [θ +. F. π. θ 1/ 2 ]. 2 F 3 / 2 3 2 ∂c AF 1/ 2 τ τ )θ θ + ( A − F )θ + (− + + θ ] 2 a ∂x π π a π 3τ 3F 3 AF −1/ 2 F ∂θ )θ 1/ 2 + θ ]c θ +[ − + (− + 2 2a ∂x 4 π 4 π 2a π 3τ 1 τ 3F 3 3 / 2 ∂z A )θ − θ ]c } F )θ 1/ 2 + AF + ( − + + [( − 4 2 4a ∂x 2 π 4 π a π. − (1 − p)aλc{[ +. Because. of. equation. (4.11),. we. could. ignore. the. high. terms. when θ n and n ≥ 3 / 2 .The interaction term could be expressed as following. Interaction term = −(1 − p)aλc 2[θ +. F. π. θ1/ 2 ]. AF 1 / 2 AF −1 / 2 F ∂c 3τ 3F τ + ( A − F )θ ] + [ − + (− + )θ 1 / 2 θ θ ∂x 4 π 2 2 4 π 2a π π ∂θ ∂z 3 1 τ 3F A 3τ + θ ]c + [( − )θ ]c } F )θ 1 / 2 + AF + (− + ∂x ∂x 2a 4 2 4a 2 π 4 π. − (1 − p)aλc{[ +. Use the result of doing multiplication to v 2 and integrations to velocity v and desired velocity w by convection term, relaxation term and interaction term to add, we 38.