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 differential equations with fluc-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 fluctuations in a turbulent flow,which generates probability densities that very well fit experimentally measured probability densities in Eu-lerian and Lagrangian turbulence.Our approach provides a dynamical reason why many physical systems with fluctuations in temperature or energy dissipation rate are correctly described by Tsallis statistics.

Despite this apparent success of the nonextensive approach,still the ques-tion remains why in many cases(such as the above turbulentflow)NESM works so well.To answer this question,let usfirst go back to ordinary sta-tistical mechanics and just consider a very simple well known example,the Brownian particle[15].Its velocity u satisfies 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 n2β0ables 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 identifications are made.

1

2

+1

n +1

(11)

1

n ⇐⇒˜β

=2

∂u V (u )is a nonlinear forcing.To be specific,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 fluctuating friction forces,as given by

˙ u i =−γi F i ( u 1,..., u N )+σi L

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

(19)

3

F i =∂

σ2

i

fluctuate in the same way,

i.e.are given by the samefluctuatingχ2distributed 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))1in[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 difference in a fully developed turbulentflow as measured on a certain scale r.We define

β=ǫτ(26) whereǫis the(fluctuating)energy dissipation rate(averaged over r3)and τis a typical time scale during which energy is transferred.Bothǫand τcanfluctuate,and we assume thatǫτisχ2-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 different 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 definition of the energy dissipation rate

ǫ=5ν ∂v∂x2 2+ ∂v

5ντ

∂v

2ifα≈1.This is indeed confirmed by

thefit 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

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(2001)

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North-Holland,Amsterdam(1981)

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6

Fig.1Histogram of longitudinal velocity differences as measured by Swinney et al.[12,16]in a turbulent Couette Taylor flow with Reynolds number R λ=262at scale r =116η(solid line),where ηis the Kolmogorov length.The experimental data are very well fitted 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 difference 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 fitted by (15)with q =1.49and α=0.92(dotted line).

1e-061e-05

0.0001

0.001

0.01

0.1

1

-10-50

510

p (u )u ’Swinney’’q=1.10’’Bodenschatz’’q=1.49’7下载本文