Dynamical foundations of nonextensive statistical mechanics

a r X i v :c o n d -m a t /0105374v 1 [c o n d -m a t .s t a t -m e c h ] 18 M a y 2001

Dynamical foundations of nonextensive statistical

mechanics

Christian Beck 1

Isaac Newton Institute for Mathematical Sciences,University of Cambridge,20Clarkson Road,Cambridge CB30EH,UK

Abstract

We construct classes of stochastic di?erential equations with ?uc-tuating friction forces that generate a dynamics correctly described by Tsallis statistics and nonextensive statistical mechanics.These systems generalize the way in which ordinary Langevin equations un-derly ordinary statistical mechanics to the more general nonextensive case.As a main example,we construct a dynamical model of velocity ?uctuations in a turbulent ?ow,which generates probability densities that very well ?t experimentally measured probability densities in Eu-lerian and Lagrangian turbulence.Our approach provides a dynamical reason why many physical systems with ?uctuations in temperature or energy dissipation rate are correctly described by Tsallis statistics.

Recently there has been considerable interest in the formalism of nonex-tensive statistical mechanics(NESM)as introduced by Tsallis[1]and further developed by many others(e.g.[2]–[4]).In the mean time there is growing evidence that the formalism,rather than being just a theoretical construc-tion,is of relevance to many complex physical systems.Applications in various areas have been reported,mainly for systems with either long-range interactions[5]–[7],multifractal behaviour[8,9],or?uctuations of temper-ature or energy dissipation rate[10]–[14].A recent interesting application of the formalism is that to fully developed turbulence[9,11,12].Precision measurements of probability density functions(pdfs)of longitudinal veloc-ity di?erences in high-Reynolds number turbulent Couette-Taylor?ows are found to agree quite perfectly with analytic formulas of pdfs as predicted by NESM[12].

Despite this apparent success of the nonextensive approach,still the ques-tion remains why in many cases(such as the above turbulent?ow)NESM works so well.To answer this question,let us?rst go back to ordinary sta-tistical mechanics and just consider a very simple well known example,the Brownian particle[15].Its velocity u satis?es the linear Langevin equation

˙u=?γu+σL(t),(1) where L(t)is Gaussian white noise,γ>0is a friction constant,andσdescribes the strength of the noise.The stationary probability density of u is Gaussian with average0and varianceβ?1,where whereβ=γ

n2?1exp ?nβ

Γ n2β0

ables X i,i=1,...,n with average0.Ifβis given by the sum

β:=

n

i=1X2i(3)

then it has the pdf(2).The average is given by

β =n X2 = ∞0βf(β)dβ=β0(4) and the variance by

β2 ?β20=2

β

2

βu2 ,(6) for the joint probability p(u,β)(i.e.the probability to observe both a certain value of u and a certain value ofβ)

p(u,β)=p(u|β)f(β)(7) and for the marginal probability p(u)(i.e.the probability to observe a certain value of u no matter whatβis)

p(u)= p(u|β)f(β)dβ.(8) The integral(8)is easily evaluated and one obtains

p(u)=

Γ n2 2 β021n u2 n2(9)

Hence the SDE(1)withχ2-distributedβ=γ/σ2generates the generalized canonical distributions of NESM[1]

p(u)～

1

2

?β(q?1)u2 1

provided the following identi?cations are made.

1

2

+1

n +1

(11)

1

n ???β

=2

?u V (u )is a nonlinear forcing.To be speci?c,let us assume

that V (u )=C |u |2α

is a power-law potential.The SDE

(13)then generates the conditional pdf

p (u |β)=

α

2α

(Cβ)

1

Z q

1

q ?1

,(15)

where

Z ?1q

=α C (q ?1)?β

1

q ?1

2α

Γ

1

2α

(16)

and

q =1+

2α

1+2α?q

β0.(18)

To generalize to N particles in d space dimensions,we may consider

coupled systems of SDEs with ?uctuating friction forces,as given by

˙ u i =?γi F i ( u 1,..., u N )+σi L

i (t )i =1,...,N

(19)

3

Suppose that a potential V( u1,..., u N)exists for this problem such that

F i =?

σ2

i

?uctuate in the same way,

i.e.are given by the same?uctuatingχ2distributed random variableβi=β. One then has for the conditional probability

p( u1,..., u N|β)=1

(1+?β(q?1)V( u1,..., u N))1

n?2x

(23) and

?β=β0

(1+?β(q?1)V s( u i))1

in[14]).The truth of what the

correct nonextensive thermodynamic descrip-

tion is will often lie inbetween the two extreme cases(22)and(25).

Let us now come to our main physical example,namely fully developed turbulence.Let u in eq.(13)represent a local velocity di?erence in a fully developed turbulent?ow as measured on a certain scale r.We de?ne

β=?τ(26) where?is the(?uctuating)energy dissipation rate(averaged over r3)and τis a typical time scale during which energy is transferred.Both?and τcan?uctuate,and we assume that?τisχ2-distributed.For power-law friction forces the SDE(13)generates the stationary pdf(15).In Fig.1 this theoretical distribution is compared with experimental measurements in two turbulence experiments,performed on two very di?erent scales.All distributions have been rescaled to variance1.Apparently,there is very good coincidence between experimental and theoretical curves,thus indicating that our simple model assumptions are a good approximation of the true turbulent statistics.

Generally,in turbulent systems the entropic index q is observed to de-crease with increasing r(see[12]for precision measurements of q(r)).At the smallest scale we expect from the de?nition of the energy dissipation rate

?=5ν ?v?x2 2+ ?v

5ντ

?v

2ifα≈1.This is indeed con?rmed by

the?t of the small-scale data of the Bodenschatz group in Fig.1,yielding q=1.49and anαas given by eq.(17).

Acknowledgement

I am very grateful to Harry Swinney and Eberhard Bodenschatz for providing me with the experimental data displayed in Fig.1.

5

References

[1]C.Tsallis,J.Stat.Phys.52,479(1988)

[2]S.Abe,Phys.Lett.271A,74(2000)

[3]S.Abe,S.Mart′inez,F.Pennini,and A.Plastino,Phys.Lett.281A,126

(2001)

[4]C.Tsallis,R.S.Mendes and A.R.Plastino,Physica261A,534(1998)

[5]A.R.Plastino and A.Plastino,Phys.Lett.174A,384(1993)

[6]https://www.wendangku.net/doc/3214791423.html,vagno,G.Kaniadakis,M.Rego-Monteiro,P.Quarati and C.Tsal-

lis,Astrophys.Lett.and Comm.35,449(1998)

[7]M.Antoni and S.Ru?o,Phys.Rev.52E,2361(1995)

[8]M.L.Lyra and C.Tsallis,Phys.Rev.Lett.80,53(1998)

[9]T.Arimitsu and N.Arimitsu,J.Phys.33A,L235(2000)

[10]G.Wilk and Z.Wlodarczyk,Phys.Rev.Lett.84,2770(2000)

[11]C.Beck,Physica277A,115(2000)

[12]C.Beck,G.S.Lewis and H.L.Swinney,Phys.Rev.63E,035303(R)

(2001)

[13]I.Bediaga,E.M.F.Curado,and J.Miranda,Physica286A,156(2000)

[14]C.Beck,Physica286A,164(2000)

[15]N.G.van Kampen,Stochastic Processes in Physics and Chemistry,

North-Holland,Amsterdam(1981)

[16]G.S.Lewis and H.L.Swinney,Phys.Rev.59E,5457(1999)

[17]https://www.wendangku.net/doc/3214791423.html, Porta,G.A.Voth,A.M.Crawford,J.Alexander,and E.Boden-

schatz,Nature409,1017(2001)

6

Fig.1Histogram of longitudinal velocity di?erences as measured by Swinney et al.[12,16]in a turbulent Couette Taylor ?ow with Reynolds number R λ=262at scale r =116η(solid line),where ηis the Kolmogorov length.The experimental data are very well ?tted by the analytic formula (15)with q =1.10and α=0.90(dashed line).The square data points are a histogram of the acceleration (=velocity di?erence on a very small time scale)of a Lagrangian test particle as measured by Bodenschatz et al.for R λ=200[17].These data are well ?tted by (15)with q =1.49and α=0.92(dotted line).

1e-06

1e-050.00010.0010.01

0.11-10

-50510

p (u )

u

’Swinney’’q=1.10’’Bodenschatz’

’q=1.49’

7

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外语专业大学生职业生涯规划书范文1

姓名：蒋林 学院：外国语学院专业：英语专业 学号：12013240681

一、了解目前全国大学生就业竞争激烈的大环境 从2002年开始，随着高校扩招的毕业生陆续进入求职市场，我国大学毕业生的人数在今后一段时间内将呈现出明显的上升趋势，根据教育部提供的数据，2002年全国大学毕业生总数为145万，2003年为212万，到2004年将达到280万。2006年已达到4100万，在社会需求没有显著增加的情况下， 毕业人数的迅速增长势必导致大学生就业压力的增大。大学生就业由过去的“精英型”就业向“大众化”就业转变,已经出现农民工、和大学生竞争同一岗位的局面。目前全国的大学生就业市场出现北京上海等大城市及沿海经济发达地区就业市场明显供对于求，西部等偏远地区供小于求； 中小城市师范生的需求也已经饱和，而农村对师范类的毕业生的需求得不到满足。就业市场的需求呈现不均衡分布。发达地区大学生就业工资现为1500元-3000元/月不等，中小城市大学生工资为800元-2000元/月。由于2008年后升入高中的学生人数开始下降，师范生进入学校的难度会越来越大。 二、知道外语专业就业情况现状 外语专业的毕业生2006年就业率为95、8%，其中英语师范生一次签约率55%，专节本学生一次签约率为36%，日语专业学生一次签约率为100%，学生考研率为13%。 在城市内进入公办学校的的只有4%、少数进入公办学校的均在县城初高中任教。有37 %学生进入私立英语培训学校的，有41%的学生进入外企、三资或涉外企业，15%学生考研、考公务员、出国或进武警部队。英语专业一直是文学的报考热门专业。据教育部公布的2011年本专科专业就业状况显示，英语专业就业率≥85%;毕业生规模约在10万人。