Non-extensive statistical mechanics approach to fully developed hydrodynamic turbulence

Chaos, Solitons and Fractals 13 (2002) 499±506

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Non-extensive statistical mechanics approach to fully developed hydrodynamic turbulence Christian Beck School of Mathematical Sciences, Queen Mary and West®eld College, University of London, Mile End Road, London E1 4NS, UK

Abstract We apply non-extensive methods to the statistical analysis of fully developed turbulent ¯ows. Probability density functions of velocity di€erences at distance r obtained by extremizing the Tsallis entropies coincide well with what is measured in turbulence experiments. The coincidence is much better than that of e.g. Levy distributions. We derive a set of relations between the hyper¯atness factors Fm and the non-extensitivity parameter q, which can be used to directly extract the function q…r† from experimentally measured structure functions. We comment on various non-extensive methods to calculate the moment scaling exponents fm . Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Consider a suitable observable in a fully developed hydrodynamic turbulent ¯ow. For example, this may be a longitudinal or transverse velocity di€erence, a temperature or a pressure di€erence. The dynamics is e€ectively described by some highly non-linear set of model equations ± for example the Navier±Stokes equation. Since nobody is able to solve this equation exactly, it is desirable to ®nd some e€ective statistical description using methods from generalized statistical mechanics. The underlying idea is quite similar to what was going on more than a hundred years ago, proceeding from classical mechanics to ordinary thermodynamics. Although nobody was able to `solve' the classical Nbody problem with N P 3 exactly, one still was able to develop quite a successful e€ective theory of a gas of 1023 particles by extremizing of Boltzmann±Gibbs entropy. For fully developed turbulent spatio-temporal chaotic systems, the ordinary Boltzmann±Gibbs statistics is not sucient to describe the (non-Gaussian) stationary state. But still we can try to develop an e€ective probabilistic theory using more general information measures. The idea to start from an extremization principle for turbulent ¯ows is actually not new. For example, Cocke [1] has presented some work on turbulence where he extremizes the Fisher information. Castaing et al. [2] also start from another extremum principle to derive probability densities in fully developed turbulence. Here we will work within a new approach to turbulence [3±6] based on extremizing the Tsallis entropies [7±9] ! X q 1 Sq ˆ 1 pi : …1† q 1 i

E-mail address: [email protected] (C. Beck). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 0 3 2 - 7

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C. Beck / Chaos, Solitons and Fractals 13 (2002) 499±506

The pi are the probabilities of the various microstates of the physical system, and q is the non-extensitivity parameter. The ordinary Boltzmann±Gibbs entropy is obtained in the limit q ! 1. Generally, the Tsallis entropies are known to have several nice properties. They are positive, concave, take on their extremum for the uniform distribution and preserve the Legendre transform structure of thermodynamics. On the other hand, they are non-extensive (non-additive for independent subsystems). Extremizing Sq under suitable norm and energy constraints, one arrives at a generalized version of the canonical distribution given by pi ˆ where Zq ˆ

1 …1 ‡ …q Zq X

…1 ‡ …q

1†bi †

1=1 q

1†bi †1=1

;

…2†

q

…3†

i

is the partition function, b ˆ 1=…kT † is a suitable inverse temperature variable, and the i are the energies of the microstates i. Ordinary thermodynamics is recovered for q ! 1. One can also work with the escort P distributions [10], de®ned by Pi ˆ piq = piq . If b is allowed to depend on q, the escort distribution is of the same form as Eq. (2), with a new q0 de®ned by q=…q 1† ˆ: 1=…q0 1†. All we have to decide now is what we should take for the e€ective energy levels i in the turbulence application. This depends on the problem considered. For example, turbulence in di€erent dimensions ought to yield di€erent e€ective energy levels. Moreover, di€erent observables will also lead to di€erent e€ective energies. In a turbulent 3-dimensional ¯ow for example, temperature di€erences should be described by di€erent e€ective energies than velocity di€erences. This is clear from the fact that the experimentally observed stationary probability distributions (slightly) di€er. At this stage one has to turn to some sort of model. In the following we will consider a simple local model for longitudinal velocity di€erences that seems to reproduce the statistics of true turbulence experiments quite well [3,16].

2. Perturbative approach to chaotically driven systems The model is based on a generalization of the Langevin equation to deterministic chaotic driving forces [11±13]. We denote the local longitudinal velocity di€erence of two points in the liquid separated by a distance r by u. Clearly u relaxes with a certain damping constant c and at the same time is driven by deterministic chaotic force di€erences Fchaot …t† in the liquid, which are very complicated. Hence a very simple local model is u_ ˆ

cu ‡ Fchaot …t†:

…4†

The force Fchaot …t† is not Gaussian white noise but a complicated deterministic chaotic forcing. It changes on a typical time scale s, which is smaller than the relaxation time c 1 , so cs is a small parameter. We may e€ectively discretize in time and consider the rescaled kick force Fchaot …t† ˆ …cs†1=2

1 X

xn d…t

ns†;

…5†

nˆ0

where the xn are the iterates of an appropriate stroboscopic chaotic map T. Integrating Eq. (4) one obtains xn‡1 ˆ T …xn †; un‡1 ˆ kun ‡

p csxn‡1 ;

…6†

where un :ˆ u…ns ‡ 0† and k :ˆ e cs . For cs ! 0, t ˆ ns ®nite, and so-called u-mixing deterministic maps T it has been shown [11] that the u-dynamics converges to the Ornstein±Uhlenbeck process, regarding the initial values x0 as random variables. Hence the invariant density of u becomes Gaussian in this limit. For

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501

®nite cs, on the other hand, the invariant density is non-Gaussian. It can have fractal and singular properties if cs is large [13]. But for small cs it approaches the Gaussian distribution provided T is u-mixing. The route to the Gaussian limit behaviour has been investigated in detail in [14] for many di€erent chaotic maps T. Within a well de®ned universality class (de®ned by square root scaling of the ®rst order correction) it was found that for small enough cs the invariant density of u is always given by 1 p…u† ˆ p e 2p



1u2 ‡cpcs 2

…u

1u 3 3

† ‡ O…cs†

…7†

provided the variable u is rescaled such that the variance of the distribution is 1. c is a non-universal constant. This means, not only the Gaussian limit distribution is universal (i.e., independent of details of T), but also the way the Gaussian is approached if the time scale ratio cs goes to 0. Much more details on this can be found in [14]. It is now reasonable to assume that also the local chaotic forces acting on longitudinal velocity di€erences in a turbulent ¯ow lie in this universality class. We can then use Eq. (7) to construct e€ective energy levels i for the non-extensive theory. 3. Constructing e€ective energy levels Eq. (7) corresponds to a Boltzmann factor pi ˆ

1 e Z

bi

p with b ˆ 1=…kT † ˆ 1, Z ˆ 2p, and energy i formally given by   p 1 1 3 u ‡ O…cs†: i ˆ u2 c cs u 2 3

…8†

…9†

The main contribution is the kinetic energy 12u2 , but in addition there is also a small asymmetric term with a universal u-dependence. The complicated hydrodynamic interactions and the cascade of energy dissipating from larger to smaller levels is now expected to be e€ectively described by a non-extensive theory with the above energy levels (see [15] for a related cascade model). We obtain from Eqs. (2) and (9) the formula      1=q 1 1 1 2 1 3 p u u ‡ O…cs† p…u† ˆ 1 ‡ b…q 1† : …10† c cs u Zq 2 3 This equation is in good agreement with experimentally measured probability densities. For detailed comparisons with various turbulence experiments, see [3,16±18]. The parameter b is determined by the condition that the distribution should have variance 1. For cs ˆ 0 this is achieved for b ˆ 2=…5 3q†. Note that we have identi®ed a small parameter cs in our approach. It is the ratio of two time scales ± that of the local forcing and that of the relaxation to the stationary state. One may conjecture that it is related to the inverse Reynolds number Rk 1 [3]. The turbulent statistics is determined by a kind of e€ective non-exp p tensive ®eld theory with the formal coupling constant cs. A perturbative approach is possible since cs is small. In fact, for the dynamics (6) one can work with analogues of Feynman graphs related to higher-order correlations of the chaotic dynamics [19,20]. Eq. (10) is just obtained by ®rst-order perturbation theory ± p the complete theory is the in®nite-order theory taking into account all orders of cs. 4. Tsallis versus Levy distributions It is apparent that our approach to the probability densities makes use of a central limit theorem (CLT) for the iterates of deterministic chaotic maps. The time-integrated chaotic force (5) is just a rescaled sum of iterates

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C. Beck / Chaos, Solitons and Fractals 13 (2002) 499±506

xn and it converges to a Gaussian process, the Wiener process for cs ! 0 [11]. One may now ask whether more general CLTs such as the Levy±Gnedenko generalized CLT [21] could posssibly lead to other types of relevant probability distributions for turbulent ¯ows, di€erent from the Tsallis distributions obtained in Eq. (10). For simplicity, let us just consider identically distributed independent random variables Xn . If the Xn have ®nite variance, thenp the ordinary CLT applies and the probability distribution p…u† of U :ˆ …X1 ‡ X2 ‡


‡ XN †= N converges to a Gaussian distribution for N ! 1. The Levy±Gnedenko generalized CLT applies to cases where the moments of the Xn do not exist. In this case, the limit distribution of the rescaled sum U :ˆ …X1 ‡ X2 ‡


‡ XN †=N 1=g is not Gaussian but the Levy-stable distribution [21,22]. Its characteristic function is given by    hp g G…k† ˆ exp ajkj exp i sign…k† ; …11† 2 where a is a positive constant, g is the Levy index and h satis®es jhj 6 g. The probability distribution p…u† is obtained by Fourier transformation Z 1 G…k†e iku dk: …12† p…u† ˆ 2p In particular, for h ˆ 0 one obtains Z 1 1 ak g p…u† ˆ e cos ku dk: p 0

…13†

Levy distributions asymptotically decay with a power law ± just as Tsallis distributions do. Hence a natural question is if perhaps it is more appropriate to proceed to Levy rather than Tsallis distributions in the step leading from Eqs. (7) to (10). Though this is possible in principle, it turns out that true turbulence data are much better described by Tsallis distributions, whereas Levy distributions seem not to provide good ®ts. This is illustrated in Fig. 1. It shows (a) an experimentally measured probability density in a Taylor Couette ¯ow (data from [16]) at a rather small spatial scale (b) the Tsallis distribution (10) for q ˆ 1:18; b ˆ 2=…5 3q†; cs ˆ 0 and (c) the Levy distribution (13) for g ˆ 1:85; a ˆ 0:410. The parameters are chosen in such a way that both distributions ®t the experimental data in the vicinity of the maximum. Proceeding to large values juj > 5 the Levy distribution exhibits tails that are much too large, whereas the Tsallis distribution quite well ®ts the experimental data for the entire range of u values. It seems a fully developed turbulent ¯ow really chooses Tsallis statistics rather than Levy statistics. The deeper reason why Tsallis statistics works well in turbulence is not clear at the present stage. It may have to do with the fact that in turbulent ¯ows there is a cascade of energy dissipating from larger to smaller scales. In fact, higher-dimensional cascade-like extensions of Eq. (4) have been numerically observed to

Fig. 1. Experimentally measured probability density of radial velocity di€erences in a turbulent ¯ow and comparison with Tsallis and Levy distributions.

C. Beck / Chaos, Solitons and Fractals 13 (2002) 499±506

503

generate invariant probability distributions similar to Tsallis distributions [15]. On the other hand, anomalous di€usion and Levy distributions have been experimentally observed in other (non-turbulent) hydrodynamic experiments, dealing with chaotic advection of passive tracer particles [23]. 5. The `free' turbulent ®eld theory Let us now understand the free turbulent ®eld theory based on Tsallis statistics with cs ˆ 0. Here almost everything can be calculated analytically. In particular the moments can be easily evaluated. In [3] we obtained  m=2 Z 1 2k B……m ‡ 1†=2; k …m ‡ 1†=2† m m
hjuj i ˆ p…u†juj du ˆ …14† b B 12 ; k 12 1 with k de®ned as k :ˆ 1=…q B…x; y† ˆ

1† (k needs not to be integer). The beta function is de®ned as

C…x†C…y† : C…x ‡ y†

…15†

We now show that this formula for the moments can be signi®cantly simpli®ed. First, we obtain from the de®nition of the beta function

 m=2 C 12 ‡ m=2 C k 12 m2 2k m

hjuj i ˆ : …16† b C 12 C k 12 Generally, one has C…x ‡ n† ˆ C…x†

n 1 Y …x ‡ j†

…17†

jˆ0

for natural numbers n. Suppose m2 ˆ: n 2 N, then it follows from Eqs. (16) and (17) m 1  m=2 Y 2 2k 2j ‡ 1 : hum i ˆ b 2j ‡ 2k m 1 jˆ0 In particular, we obtain for the ®rst few moments   2k 1 hu2 i ˆ ; b 2k 3  2 2k 1 3 ; hu4 i ˆ b 2k 5 2k 3  3 2k 1 3 5 : hu6 i ˆ b 2k 7 2k 5 2k 3

…18†

…19† …20† …21†

This can be used to evaluate the complete set of hyper¯atness factors Fm de®ned as Fm ˆ

hu2m i m: hu2 i

…22†

We obtain F1 ˆ 1; 2k F2 ˆ 3 2k

3 ; 5 2 …2k 3† F3 ˆ 3
5 ; …2k 7†…2k 5† …2k 3†3 F4 ˆ 3
5
7 …2k 9†…2k 7†…2k

…23† …24† …25† 5†

…26†

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C. Beck / Chaos, Solitons and Fractals 13 (2002) 499±506

and generally …2k 3† 1†!! Q2m‡1 jˆ5 …2k

Fm ˆ …2m

m 1

j†

…27†

(j odd). Note that all hyper¯atness factors are independent of b. 6. Extracting q…r† from experimentally measured structure functions The great advantage of the hyper¯atness factors is that they yield a simple way to estimate the rdependent non-extensitivity parameter q…r† from experimentally measured structure functions hum i…r†. Eq. (24) yields F2 ˆ

6k 2k

9 15 9q ˆ 5 7 5q

…28†

9 6

…29†

15 9

…30†

or kˆ

5F2 2F2

equivalent to qˆ

7F2 5F2

(see also [4]). Another relation follows from Eq. (25).  q 1 6F3 45  F32 ‡ 120F3 kˆ 2F3 30 equivalent to qˆ

8F3 6F3

p F32 ‡ 120F3 : p 45  F32 ‡ 120F3 75 

…31†

…32†

In fact, each hyper¯atness factor Fm with m P 2 yields a relation for k (or q), and all relations are the same in case the `free' turbulence theory is exact. Given some experimentally measured hyper¯atness structure functions Fm …r† one can now determine the corresponding curves q…r†. The less these curves di€er for the various m, the more precise is the zeroth-order (free) turbulence theory. For examples of experimentally measured curves q…r† obtained from precision ®ts of the densities, see [16]. Generally, the hyper¯atness factors are complicated functions of both the Reynolds number Rk and the separation distance r, and so is q…Rk ; r†. But one may conjecture that for Rk ! 1 one obtains a universal function q …r†. How it looks like in the entire r-range is still an open question. 7. Some remarks on the scaling exponents fm The precise values of the scaling exponents fm , which describe the scaling behaviour of the structure functions hum i  rfm in the inertial range, are still a rather controversial topic in turbulence theory. In fact, it is not at all clear whether there is exact scaling at all or just approximate scaling, whether the higher moments exist at all or not, how reliable the experimentally measured higher-order exponents are, and what the e€ects of ®nite Reynolds numbers are. On the theoretical side, a variety of models and theories have been suggested (see, e.g., [24,25]) but a true breakthrough convincing a signi®cant majority of scientists working in the ®eld seems not to have been reached at the moment.

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505

To `derive' values for the scaling exponents using methods from non-extensive statistical mechanics, one needs additional assumptions ± just as in all other models and theories dealing with the scaling exponents. In [3] a logarithmic depence of k on r was suggested to derive the following formula for the scaling exponents      m 3 m 1 log2 1 fm ˆ ‡ log2 1 : …33† 3 2k 1 2k 1 Here k ˆ 1=…q 1† represents the average value of the non-extensitivity parameter in the inertial range. On the other hand, Kolmogorov's lognormal model (the K62-theory [26]) predicts m l 1 1‡ lm2 : …34† fm ˆ 3 2 18 where l is the intermittency parameter. Now, assuming that k is rather large we can expand the logarithms in Eq. (33). The linear terms cancel, and the ®rst non-trival terms are the quadratic ones. Neglecting the higher-order cubic terms, one obtains from Eq. (33) the result (34), identifying lˆ

9 ln 2…2k

1†

2

:

…35†

It is encouraging that the simplest non-extensive model assumptions lead to an old theory that has a long tradition in turbulence theory (though it is known that K62 cannot be correct ± it is, however, a good approximation for m not too large). The value of the intermittency parameter comes out with the correct order of magnitude (l  0:2) if one chooses k  5. On the other hand, starting in the derivation leading to Eq. (33) from a di€erent dependence of k on r rather than the logarithmic one, one also obtains di€erent formulas for the scaling exponents. So the ®nal answer to the question of the scaling exponents is still open. Actually, in [3] Eq. (33) was derived using the free turbulence theory, but clearly the correct theory is the in®nite-order theory. As one can easily see, the in®nite-order theory yields probability densities p…u† living on a compact P support for arbitrary small but ®nite cs. Indeed, iterating Eq. (6) one obtains p un ˆ kn u0 ‡ cs njˆ1 kn j xj . Hence, for any chaotic dynamics T bounded on some ®nite phase space (say p p [)1,1]) one obtains for n ! 1 from the geometric series the rigorous bound juj 6 cs=…1 k†  1= cs. Thus for the in®nite-order theory all moments exist for arbitrary small but ®nite cs ± in contrast to the 0-th order theory, where only moments hum i with m < 2k 1 exist, since here the densities live on a noncompact support and there is polynomial decay for large juj. This once again shows how delicate the problem of the existence of the moments and of the scaling exponents is. For densities living on a compact support, as provided by the in®nite-order theory, one expects a linear asymptotics of fm for large m. Generally, the way non-extensive statistics can be used to `derive' the scaling exponents fm is not unique. An alternative approach, based on an extension of the multifractal model and a constant q smaller than 1, has been suggested by Arimitsu and Arimitsu [5,6]. In their model asymptotically there is a logarithmic correction term for the fm . Acknowledgements A large part of this research was performed during the author's stay at the Institute for Theoretical Physics, University of California at Santa Barbara, supported in part by the National Science Foundation under Grant No. PHY94-07194. The author also gratefully acknowledges support by a Leverhulme Trust Senior Research Fellowship of the Royal Society. References [1] Cocke WJ. Phys Fluids 1996;8:1609. [2] Castaing B, Gagne Y, Hop®nger EJ. Physica 1990;46D:177. [3] Beck C. Physica 2000;277A:115.

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