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Hubble-like flows in relativistic heavy-ion collisions

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Preprint submitted to Acta Phys.Hung.A 22/1(2005)000–000HEA VY ION PHYSICS Hubble-like ?ows in relativistic heavy-ion collisions Mikolaj Chojnacki Institute of Nuclear Physics,Polish Academy of Sciences,31-342Krak′o w,Poland Received 31October 2005Abstract.We study the dynamical appearance of scaling solutions in rela-tivistic hydrodynamics.The phase transition e?ects are included through the temperature dependent sound velocity.If a pre-equilibrium transverse ?ow is included in the initial conditions,then it may reach the form of the asymptotic Hubble ?ow,r/t ,in short evolution times,7-15fm.The numerical solutions are found to support to the freeze-out models (Blast-Wave,Buda-Lund,Cracow).Keywords:ultra-relativistic heavy-ion collisions,relativistic hydrodynamics PACS:25.75.Dw,21.65.+f,14.40.-n 1.Introduction This article is a contribution to the Quark Matter 2005poster proceedings,prepared with W.Florkowski and T.Cs¨o rg?o ,and based on our recent paper [1].The successful parameterizations [2,3,4]of the freeze-out process at RHIC indicate that the system formed in Au+Au collisions is highly thermalized and exhibits strong longitudinal and transverse expansion that may be characterized by the Hubble law,v =H r .Such form of expansion is used in cosmology,with H

being a function of time (in spite of the time dependence,H is commonly called the Hubble constant).The analogy between the Hubble law used in cosmology and in relativistic hydrodynamics is related to the dependence of the four-velocity u μon the space-time position of the ?uid element x μ,

u μ=x μτ 1,x t ,

z

t 2

?x 2?y 2?z 2is the invariant time.The form of the four-velocity displayed by Eq.(1)is known as the scaling solution .In many cases it represents the asymptotic solution of the hydrodynamic equations [5,6,7].In our approach we assume that the elementary parton-parton collisions that lead to thermalization of the system also lead to collective behavior,i.e.,to the development of the pre-equilibrium transverse ?ow.This type of the ?ow has to be included in the initial

2M.Chojnacki

1?v2).The role of equation of state is played by the temperature dependent sound velocity.

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Hubble-like?ows (3)

ln(a+a?) .(7)

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In our numerical calculations we consider two forms of the sound velocity.The?rst one is obtained from the QCD lattice calculations[9]and extrapolated to lower temperatures,while the second one is the analytic parameterization of these points. The latter form allowed us to calculate functionsΦT and TΦanalytically.

2.2.Initial conditions

We use the same method for constructing initial conditions as in[8],i.e.,a single function a(r)is used to determine the initial forms of the functions a+and a?, namely a+(t=t0,r)=a(r),a?(t=t0,r)=a(?r).Such procedure insures that a cylindrically symmetrical system has vanishing velocity?eld at r=0.In addition, we assume that the initial temperature pro?le is connected with the thickness func-tion T A(r).We assume that the initially produced entropy density is proportional to the number of nucleons participating in the collision at the distance r from the collision center,σ(r)~T A(r),and at high temperature our system may be consid-ered as a system of massless particles.Since the entropy density of a massless gas is proportional to T3we?nd

T(r,t=t0)=const T1

1+v0r

,a T(r)=exp ΦT const T1

1?v0r

4M.Chojnacki

Hubble-like?ows (5)

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2011-2012 学年度上学期六年级复习题(Unit2-Unit3 ) 一、听力部分 1、听录音排序 ( ) () ()() () 2、听录音,找出与你所听到的单词属同一类的单词 () 1. A. spaceman B. pond C . tiger () 2. A.mascots B. potato C . jeans () 3. A. door B. behind C . golden () 4. A. sometimes B. shop C . prince () 5. A. chair B. who C . sell 3、听录音,将下面人物与他的梦连线 Sarah Tim Juliet Jenny Peter 4、听短文,请在属于Mr. Brown的物品下面打√ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5、听问句选答句 () 1. A. Yes, I am B. Yes, I have C . Yes, you do () 2. A.Pink B. A friendship band C . Yes. () 3. A. OK B. Bye-bye. C . Thanks, too. () 4. A. Monday. B. Some juice. C . Kitty. () 5. A. I ’ve got a shookbag. B. I ’m a student. C . It has got a round face. 6、听短文,选择正确答案 () 1. Where is Xiaoqing from? She is from . A.Hebei B. Hubei C . Hunan () 2. She goes to school at . A.7:00 B.7:30 C . 7:15 () 3. How many classes in the afternoon? classes. A. four B. three C . two () 4. Where is Xiaoqing at twelve o ’clock? She is . A. at home B. at school C .in the park () 5. What does she do from seven to half past eight? She . A.watches TV B. reads the book C. does homework

三年级下学期英语(Joinin剑桥英语)全册单元知识点归纳整理-

Starter Unit Good to see you again知识总结 一. 短语 1. dance with me 和我一起跳舞 2. sing with me 和我一起唱歌 3. clap your hands 拍拍你的手 4. jump up high 高高跳起 5.shake your arms and your legs晃晃你的胳膊和腿 6. bend your knees 弯曲你的膝盖 7. touch your toes 触摸你的脚趾8. stand nose to nose鼻子贴鼻子站 二. 句子 1. ---Good morning. 早上好。 ---Good morning, Mr Li. 早上好,李老师。 2. ---Good afternoon. 下午好。 ---Good afternoon, Mr Brown. 下午好,布朗先生。 3. ---Good evening,Lisa. 晚上好,丽莎。 ---Good evening, Bob. 晚上好,鲍勃。 4. ---Good night. 晚安。 ----Good night. 晚安。 5. ---What’s your name? 你叫什么名字? ---I’m Bob./ My name is Bob. 我叫鲍勃。 6. ---Open the window, please. 请打开窗户。 ---Yes ,Miss. 好的,老师。 7. ---What colour is it? 它是什么颜色? 它是蓝红白混合的。 ---It’s blue, red and white. 皮特的桌子上是什么? 8. ---What’s on Pit’s table? ---A schoolbag, an eraser and two books. 一个书包,一个橡皮和两本书。 9. ---What time is it? 几点钟? 两点钟。 ---It’s two. 10.---What’s this? 这是什么? ---My guitar. 我的吉他。

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★名词的复数:一般在词尾直接加s,不规则变化要牢记: fish-----fish mouse------mice 3 Sentences 1.Have you got a pet ? 你有宠物吗? Yes ,I have. 是的,我有。/No, I haven’t. 不,我没有。 2.What have you got ? 你有什么宠物吗?I’ve got a dog . / A dog. 我有一只狗。 3.What colour is the cat ? 你的猫是什么颜色的?It’s black. 它是黑色的。 What is wizard’s pet? 巫师的宠物是什么? 4.What is it ? 它是什么? It’s a rabbit .它是只兔子。 5.How many budgies /mice are there? 这里有多少只鹦鹉/老鼠? There are + 数字budgies/mice. 这里有------只鹦鹉/老鼠。 6.Fly like a budgie. 像鹦鹉一样飞。Run like a rabbit. 像兔子一样跑。 Swim like a fish. 像鱼一样游泳。Eat like a hamster. 像仓鼠一样吃东西。 Sit like a dog. 像狗一样坐。Roar like a tiger. 像老虎一样吼叫。 7.What are in the pictures. 图片里面是什么?Animals. 动物。 8. What animals? 什么动物? 9.How many pandas (elephants /bears/ giraffes/ monkeys/ budgies) are there?有多少.? How many + 可数名词的复数形式 Unit 2 The days of the week

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What’s your name? The teacher prompt the pupil by whispering, I’m (Alice). (3) Asks all the pupils the same question and help those who need it by whispering I’m…. 2. How are you today? I’m fine. I’m OK. (1) The teacher explains the meaning of How are you today? (2) Tell the class to ask the question all together and introduce two answers, I’m (not very) fine/I’m Ok. (3) Asks all the pupils the same question and make sure that all the students can reply. Step3: Speak English in class. 1. Tells the pupils to open their books at page3, look at the four photographs, and listen to the tape. 2. Asks the pupils to dramatise the situations depicted in the four photographs. A volunteer will play the part of the parts of the other pupils, answering as a group. 第二学时: The pupils learn to use these new words: Sandwich, hamburger, hot dog, puller, cowboy, jeans, cinema, walkman, snack bar, Taxi, clown, superstar.

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Starter Unit Let's begin 第一学时: The pupils learn to understand: How are you today? I’m fine/OK. Goodbye. The pupils learn to use: What’s your name? I’m (Alice). How are you? I’m fine. I’m OK. Activities and skills: Decoding meaning from teacher input. Using phrases for interaction in class. Greeting each other. I Introduce oneself. Asking someone’s mane. Teaching process: Step 1: Warm—up. Sing a song. Hello, what’s your name? Step 2: Presentation. 1. What’s your name? I’m…. (1) The teacher introduces herself, saying: Hello, /Good morning, I’m Miss Sun. (2) Asks a volunteer’s name: What’s your name? The teacher prompt the pupil by whispering, I’m (Alice). (3) Asks all the pupils the same question and help those who need it by whispering I’m…. 2. How ar e you today? I’m fine. I’m OK. (1)The teacher explains the meaning of How are you today? (2)Tell the class to ask the question all together and introduce two answers, I’m (not very) fine/I’m Ok. (3)Asks all the pupils the same question and make sure that all the students can reply. Step3: Speak English in class. 1. Tells the pupils to open their books at page3, look at the four photographs, and listen to the tape. 2. Asks the pupils to dramatise the situations depicted in the four photographs. A volunteer will play the part of the parts of the other pupils, answering as a group. 第二学时: The pupils learn to use these new words: Sandwich, hamburger, hot dog, puller, cowboy, jeans, cinema, walkman, snack bar, Taxi, clown, superstar. Activities and skills: Singing the song and colouring picture frames. Teaching process:

【精品】joinin英语剑桥版五年级(上)单词表

beautiful美丽的,美好的 clean(把。。。)弄干净 hard费劲地;费力地 learn学习;学会 New South Wales新南威尔士州 soon不久,很快 summer holiday暑假 take拍摄 take photos拍照 time一段时间;某段日子 the UK英国 umbrella雨伞 whose谁的 work工作,劳动 both两者,两个都 change转变;转换 come in 进来 come on快点 cook煮,烧;厨师 cousin堂(表)兄弟;表姐妹doctor医师,大夫 driver司机、驾驶员 family家庭 fantastic太好了;极好的 famer农民 grandfather祖父;外祖父 grandmother祖母;外祖母 grandparent祖父;外祖父;祖母;外祖母granny奶奶;外婆 introduce介绍,引见 Little Red Riding Hood小红帽 member成员 nurse护士 phone电话 police警察 problem问题,难题 role角色 run away逃跑 scene一场 taxi出租车 these这些 walk off离开 woods小树林,林地 word字、单词 worker工人 always总是,每次都 band圈,箍,带 because因为

bell铃 bring带来 cool冷静的,沉着的darling亲爱的,宝贝儿dollar美元 easy不担心的,不紧张的everywhere处处 friendship友谊 hurt弄伤; luck好运;幸运 mascot吉祥物 necklace项链 Olympic奥运会的 popular受欢迎的 Puma Ranch美洲狮大牧场 ring戒指 shell贝壳 silver银制的;银色的soft toy毛绒玩具 test测验 watch out小心 answer答案 athlete运动员 collect收集采集 cyclist骑自行车者 end结束;停止 go开始(做某事)kilometre千米,公里 opera house歌剧院 quarter一刻钟,十五分钟say说 special特殊的,特别的spot圆点,斑点 stamp邮票 tell告诉 time次,回 top最佳的,最好的train火车,列车 Yours您真挚的autograph亲笔签名 comic连环画 email电子邮件 for为了要;为了得到gone走了;不见了 hat帽子 postcard明信片 sticker贴纸 take接受

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