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Non-extensive statistical mechanics approach to fully developed hydrodynamic turbulence

We apply non-extensive methods to the statistical analysis of fully developed turbulent flows. Probability density functions of velocity differences at distance r obtained by extremizing the Tsallis entropies coincide well with what is measured in turbulence experiments. We derive a set of relations between the hyperflatness factors F_m 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 zeta_m.


Introduction
Consider a suitable observable in a fully developed hydrodynamic turbulent flow. For example, this may be a longitudinal or transverse velocity difference, a temperature or a pressure difference. The dynamics is effectively described by some highly nonlinear set of model equations-for example the Navier-Stokes equation. Since nobody is able to solve this equation exactly, it is desirable to find some effective 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 N-body problem with N ≥ 3 exactly, one still was able to develop quite a successful effective theory of a gas of 10 23 particles by extremizing the Boltzmann-Gibbs entropy. For fully developed turbulent spatio-temporal chaotic systems, the ordinary Boltzmann-Gibbs statistics is not sufficient to describe the (non-Gaussian) stationary state. But still we can try to develop an effective probabilistic theory using more general information measures.
The idea to start from an extremization principle for turbulent flows 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 [3,4,5] based on extremizing the Tsallis entropies [6,7,8] The p i 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 S q under suitable norm and energy constraints, one arrives at a generalized version of the canonical distribution given by where is the partition function, β = 1/(kT ) is a suitable inverse temperature variable, and the ǫ i are the energies of the microstates i. Ordinary thermodynamics is recovered for q → 1. One can also work with the escort distributions [9], defined by P i = p q i / p q i . If β is allowed to depend on q, the escort distribution is of the same form as eq. (2), with a new q ′ defined by q/(q − 1) =: 1/(q ′ − 1).
All we have to decide now is what we should take for the effective energy levels ǫ i in the turbulence application. This depends on the problem considered. For example, turbulence in different dimensions ought to yield different effective energy levels. Moreover, different observables will also lead to different effective energies. In a turbulent 3-dimensional flow for example, temperature differences should be described by different effective energies than velocity differences. This is clear from the fact that the experimentally observed stationary probability distributions (slightly) differ.
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 differences that seems to reproduce the statistics of true turbulence experiments quite well [3].

Perturbative approach to chaotically driven systems
The model is based on a generalization of the Langevin equation to deterministic chaotic driving forces [10]- [12]. We denote the local longitudinal velocity difference of two points in the liquid separated by a distance r by u.
Clearly u relaxes with a certain damping constant γ and at the same time is driven by deterministic chaotic force differences F chaot (t) in the liquid, which are very complicated. Hence a very simple local model iṡ The force F chaot (t) is not Gaussian white noise but a complicated deterministic chaotic forcing. It changes on a typical time scale τ , which is smaller than the relaxation time γ −1 , so γτ is a small parameter. We may effectively discretize in time and consider the rescaled kick force where the x n are the iterates of an appropriate stroboscopic chaotic map T . Integrating eq. (4) one obtains where u n := u(nτ +0) and λ := e −γτ . For γτ → 0, t = nτ finite, and so-called ϕ-mixing deterministic maps T it has been shown [10] that the u-dynamics converges to the Ornstein-Uhlenbeck process, regarding the initial values x 0 as random variables. Hence the invariant density of u becomes Gaussian in this limit. For finite γτ , on the other hand, the invariant density is non-Gaussian. It can have fractal and singular properties if γτ is large [12]. But for small γτ it approaches the Gaussian distribution provided T is ϕ-mixing. The route to the Gaussian limit behaviour has been investigated in detail in [13] for many different chaotic maps T . Within a well defined universality class (defined by square root scaling of the first order correction) it was found that for small enough γτ the invariant density of u is always given by 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 γτ goes to 0. Much more details on this can be found in [13].
It is now reasonable to assume that also the local chaotic forces acting on longitudinal velocity differences in a turbulent flow lie in this universality class. We can then use eq. (7) to construct effective energy levels ǫ i for the non-extensive theory.

Constructing effective energy levels
Eq. (7) corresponds to a Boltzmann factor with β = 1/(kT ) = 1, Z = √ 2π, and energy ǫ i formally given by The main contribution is the kinetic energy 1 2 u 2 , 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 effectively described by a non-extensive theory with the above energy levels (see [14] for a related cascade model). We obtain from eq. (2) and (9) the formula This equation is in very good agreement with experimentally measured probability densities. For detailed comparisons with various turbulence experiments, see [3,15,16,17]. The parameter β is determined by the condition that the distribution should have variance 1. For γτ = 0 this is achieved for β = 2/(5 − 3q). Note that we have identified a small parameter γτ 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 R −1 λ [3]. The turbulent statistics is determined by a kind of effective non-extensive field theory with the formal coupling constant √ γτ . A perturbative approach is possible since √ γτ 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 [18,19]. Eq. (10) is just obtained by first-order perturbation theory-the complete theory is the infinite-order theory taking into account all orders of √ γτ . But first we have to understand the 'free' turbulent field theory obtained for γτ = 0. Here almost everything can be calculated analytically.

The 'free' turbulent field theory
If γτ = 0 the moments can be easily evaluated. In [3] we obtained with k defined as k := 1/(q − 1) (k needs not to be integer). The beta function is defined as We now show that this formula for the moments can be significantly simplified. First, we obtain from the definition of the beta function Generally, one has for natural numbers n. Suppose m 2 =: n ∈ N, then it follows from eq. (13) and (14) In particular, we obtain for the first few moments This can be used to evaluate the complete set of hyperflatness factors F m defined as We obtain and generally (j odd). Note that all hyperflatness factors are independent of β.
5 Extracting q(r) from experimentally measured structure functions The great advantage of the hyperflatness factors is that they yield a direct way to estimate the r-dependent non-extensitivity parameter q(r) from experimentally measured structure functions u m (r). Eq. (22) yields (see also [4]).
In fact, each hyperflatness factor F m with m ≥ 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 hyperflatness structure functions F m (r) one can now determine the corresponding curves q(r). The less these curves differ for the various m, the more precise is the zeroth-order (free) turbulence theory. For examples of such experimentally measured curves q(r), see [15]. Generally, the hyperflatness factors are complicated functions of both the Reynolds number R λ and the separation distance r, and so is q(R λ , r). But one may conjecture that for R λ → ∞ one obtains a universal function q * (r). How it looks like in the entire r-range is still an open question.
6 Some remarks on the scaling exponents ζ m The precise values of the scaling exponents ζ m , which describe the scaling behaviour of the structure functions u m ∼ r ζm 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 effects of finite Reynolds numbers are. On the theoretical side, a variety of models and theories have been suggested (see, e.g., [20,21]) but a true breakthrough convincing a significant majority of scientists working in the field seems not to have been reached at the moment.
To 'derive' values for the scaling exponents using methods from nonextensive 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 Herek = 1/(q − 1) is the average value of the non-extensitivity parameter in the inertial range. On the other hand, Kolmogorov's lognormal model (the K62-theory [22]) predicts where µ is the intermittency parameter. Now, asuming thatk is rather large we can expand the logarithms in eq. (31). The linear terms cancel, and the first non-trivial terms are the quadratic ones. Neglecting the higher-order cubic terms, one precisely obtains from eq. (31) the result (32), identifying 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 (µ ≈ 0.2) if one choosesk ≈ 6, and the same k fits the experimentally measured probability densities correctly. On the other hand, starting in the derivation leading to eq. (31) from a different dependence of k on r rather than the logarithmic one, one also obtains different formulas for the scaling exponents. So the final answer to the question of the scaling exponents is still open.
Actually, in [3] eq. (31) was derived using the 'free' turbulence theory, but clearly the correct theory is the infinite-order theory. As one can easily see, the infinite-order theory yields probability densities p(u) living on a compact support for arbitrary small but finite γτ . Indeed, iterating eq. (6) one obtains u n = λ n u 0 + √ γτ n j=1 λ n−j x j . Hence, for any chaotic dynamics T bounded on some finite phase space (say [-1,1]) one obtains for n → ∞ from the geometric series the rigorous bound |u| ≤ √ γτ /(1 − λ) ∼ 1/ √ γτ . Thus for the infinite-order theory all moments exist for arbitrary small but finite γτ -in contrast to the 0-th order theory, where only moments u m with m < 2k − 1 exist, since here the densities live on a non-compact support and there is polynomial decay for large |u|. This once again shows how delicate the problem of the existence of the moments and of the scaling exponents in general is. For densities living on a compact support, as provided by the infinite-order theory, one expects a linear asymptotics of ζ m for large m.
Generally, the way non-extensive statistics can be used to 'derive' the scaling exponents ζ m is not unique. An alternative approach, based on an extension of the multifractal model and q-values smaller than 1, has been suggested by T. and N. Arimitsu [5]. In their model asymptotically there is a logarithmic dependence og ζ m on m.