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2 Turbulent Fluid as a Physical System: A Problem in Nonequilibrium Statistical Mechanics

2 Turbulent Fluid as a Physical System: A Problem in Nonequilibrium Statistical Mechanics

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52



D. Ruelle



flowing from node 0 to (many) nodes of high level. This equivalence of turbulent

cascade to heat flown a non-standard geometry is in principle exact, although not

formulated precisely here.

Translating a nonequilibrium problem (turbulence) into another nonequilibrium

problem (heat flow) is in principle an interesting idea, but there are two obvious

difficulties:

• expressing the fluid Hamiltonian as Hamiltonian of a coupled system of “nodes”

is likely to give complicated results,

• the rigorous study of a heat flow is known to be extremely hard (see for instance

[13, 14]).

What we shall do is to use crude (but physically motivated) approximations, with

the hope that the results obtained are in reasonable agreement with experiments.

This is indeed the conclusion of our study, indicating that turbulence fits naturally

within accepted ideas of nonequilibrium statistical mechanics.



3.3 Statistical Mechanics of Turbulence Without

Fluctuations

A fundamental step forward in the understanding of turbulence has been achieved by

Kolmogorov [15–17]. He noticed that if turbulence is assumed to be spatially homogeneous and isotropic, then many features of the energy cascade are determined by

dimensional analysis.1 The experimental study of fluids has shown that turbulence

is in fact not homogeneous: this lack of homogeneity is known as intermittency.

Let us now look at the heat flow interpretation of the turbulent energy cascade.

The macroscopic description of a heat flow, ignoring the microscopic structure of

the heat conductor and the microscopic fluctuations leads to an answer in terms of

heat conductivity. We can give a heat flow equivalent version of the Kolmogorov

turbulent cascade theory: the heat flows from the site 0 towards high level sites,

respecting the nonstandard geometry of the system, and a prescribed amount of

energy (heat) leaving 0 per unit time. We have thus a complete equivalence between

the Kolmogorov turbulent energy cascade and a heat flow in a nonstandard geometry

where microscopic structure and fluctuations are ignored.

The Hamiltonian description than we have obtained in terms of interacting

nodes, each with

Ä 3 degrees of freedom has a discrete structure, and must have

fluctuations. If we had a finite temperature equilibrium state, the energy fluctuations

would be given by Boltzmann’s law. Outside of equilibrium the situation is not as



1



Dimensional analysis says how various quantities (like velocity or energy) depend on certain

variables (like spatial distance, and time): velocity is spatial distance divided by time, energy

is mass times velocity squared, etc. Dimensional analysis appears somewhat trivial, but for the

turbulent energy cascade it has led to spectacular predictions.



3 Hydrodynamic Turbulence as a Nonstandard Transport Phenomenon



53



well understood, but there are fluctuations, which must correspond to intermittency

for the turbulent cascade.



3.4 Statistical Mechanics of Turbulence with Fluctuations

To proceed with the study of fluctuations we shall make crude approximations based

on physical ideas that we have about turbulence and about the nonequilibrium

statistical mechanics of heat flows. We shall assume that a given constant Ä can

be chosen so that (allowing some approximations) the dynamical structure is

particularly simple. First we shall neglect interactions between nodes except those

between nodes corresponding to a cube of size `n and the cubes of size `nC1 that

it contains. If we consider a graph with nodes as vertices, and interacting nodes as

edges, this graph is thus a tree. A similar approximation has been made in models

of intermittency (see [18, 19]) where one would think of eddies rather than nodes.

We can assume that the flow of energy is overwhelmingly from a node of level

n towards the nodes of level n C 1 with which it interacts: this corresponds to

the differences of temperature at various levels which can be obtained from the

theory without fluctuations, and corresponds to the direction of the turbulent energy

cascade. We now make the strong but natural assumption that the nodes of level nC1

interacting with a given node of level n are in (approximate) thermal equilibrium

with the fluctuating energy of this node. We assume thus the Ä can be chosen such

that there is a Boltzmannian energy distribution at each node, with a condition

between neighboring nodes which expresses that energy flows overwhelmingly from

level n to level n C 1, and that energy is conserved.

For a node of level n, let vni be the fluctuation velocity at one of the next order

modes (box i of size `nC1 ) and write

Vn D Vni D jvni j3

The kinetic energy corresponding to the velocity vni is Ä 3.nC1/ :jvni j2 =2. The

residence time at the node of level n is `n =jvn 1 j by dimensional analysis. Energy

conservation requires that the rate of flow of energy out of the node of level n is

equal to the rate of flow into that node:

Ä



3



X jvni j3

i



`nC1



D



jvn 1 j3

`n



or



Ä



3



X

i



jvni j3 D



Vn

Ä



1



Given Vn 1 , this relation may be interpreted as a microcanonical ensemble condition

on the jvni j3 . We replace this by a canonical distribution such that each vni has a

distribution

exp



jvn j3 Á 3

d vn

Ä 1 Vn 1



54



D. Ruelle



Therefore, given Vn 1 , we find that Vn has the distribution

ÄVn Á

:

Vn 1



Ä dVn

exp

Vn 1



For a decreasing sequence of boxes of sizes `0 ; : : : ; `n we obtain that, if V0 is

fixed, we have a probability distribution

Ä dV1

e

V0



Ä dVn

e

Vn 1



ÄV1 =V0



ÄVn =Vn



1



(3.1)



for V1 ; : : : ; Vn . Note that this distribution extends naturally to a probability measure

$ on sequences .Vn /1

nD1 . Physically however, the validity of .1/ is limited by

dissipation due to the viscosity . We want `n to be larger than the length at which

dissipation due to viscosity takes place (Kolmogorov length); this is expressed by

jvni j `n >



Vn1=3 `n >



or



(3.2)



3.5 Applications of (3.1)

• (a) The exponents n .

Let us now discuss the structure functions, i.e., the moments

hjvn jp i D hVnp=3 i

for positive integer p, and the exponents

hjvn jp i



`np



or



p



ln `n



such that



p



p

ln Ä C lnhWnp=3 i

3



lnhVnp=3 i D n



where we have written Wk D Ä k Vk . We have here

Z

hWnp=3 i



D



dW1



e



W1 =W0



Z



Z

dWn



W0



1



e



Wn



1 =Wn 2



Wn



Z

dWn



2



e



Wn =Wn



Wn



1



1



Wnp=3



and also

Z



1

0



dWn



e



Wn =Wn



Wn



1



1



Z

p=3

1



Wnp=3 D Wn



1



d e



p=3



0



p

p=3

D Wn 1 €. C 1/

3



so that by induction we find

p

p=3

hWnp=3 i D Œ€. C1/n W0 ;

3



p=3



p



n p3 ln Ä C lnhWn i

n ln Ä



p 1

p

ln €. C1/

3 ln Ä

3



3 Hydrodynamic Turbulence as a Nonstandard Transport Phenomenon



55



In conclusion we have the (approximate) prediction

p



1

p

ln €. C 1/

ln Ä

3



p

3



D



(3.3)



Using either the heat propagation or the eddy cascade picture, we see that Ä

should be chosen such that the initial Wn distribution concentrated on one value

for .n; i/ thermalizes to values of WnC1 for the systems .n C 1; j/ distributed

according to

1

e

Wn



WnC1 =Wn



dWnC1



This requires Ä sufficiently large. However, if the value of Ä is too large,

several different temperatures will be present among the systems .n C 1; j/

connected with .n; j/, and the WnC1 -distribution will not be Boltzmannian. Of

course a rigorous justification of this picture is well beyond the power of current

mathematical methods. We can only claim this: Ä should be such that when an

eddy of size r has decayed to eddies of size r=Ä their energies have a thermal

distribution, after which the process can start again. In the dissipative range the

distribution of Vn should be cut off at large Vn . Numerically, one finds that the

above formula fits the experimental data [20] well with 1= log Ä D 0:32 ˙ 0:01,

i.e., Ä between 20 and 25.

Note also that (3.3) gives 3 D 1 independently of Ä. This is in agreement

with studies based on the Navier-Stokes equation.

P

• (b) Radial velocity increment u D r v D k uk .

If r

`n we have u

un

radial component of vn . Therefore, given V0 , a

rough estimate of the probability distribution F .u/ du of u is given by:

F .u/ D

D



n Z

Y



1



kD1 0



1 Ä n 1=3

. /

2 V0



Ä dVk

e

Vk 1



Z



ÄVk =Vk



1



Á



1

1=3



2Vn



Z



1=3



1=3



Œ Vn ;Vn 



n

Y

dwk e

w1 wn >.Ä n =V0 /juj3 kD1



.u/



wk



1=3

wk



(One compares with experimental data for D const.u with j j normalized by

h 2 i D 1, therefore the approximation u

radial component of vn is not as

terrible as might seem).

Instead of F .u/ du we consider the distribution Gn .y/ dy of y D

.Ä n =V0 /1=3 juj, so that

Z

Gn .y/ D



Z



n

Y

dwk e

w1 wn >y3 kD1



et Gn .et / D .



.n 1/



wk



1=3

wk



/.t/



(3.4)



56



D. Ruelle



where

.t/ D 3 exp.3t



3t



e /



Z

;



.t/ D e



1



t



e



s



.s/ ds



t



From this one obtains that Gn .y/ is a decreasing function of y.

Using the formula (3.2) for the dissipation length we obtain n D

.3=4/ln R= ln Ä

0:24 ln R. Gn .y/ gives then a reasonable fit of the numerical

data for small y. However, a comparison with the Navier-Stokes results of

Schumacher [21] shows that the behavior at large y is not as simple. This can be

understood because the dissipation length is not fixed by the Reynolds number

R, but fluctuates. In fact the probability distribution P. / d of the radial velocity

gradient (normalized by h 2 i D 1) which is computed in [2] has contributions

of various values of n (one can show that there are no contribution of n Ä j if

Ä 2j Ä R). For a study taking into account the fluctuations of the dissipation

length see the paper by Gallavotti and Garrido [3] in this volume.

• (c) Relation with the Kolmogorov-Obukhov lognormal theory.

The above formula (3.5) implies that Gn .et / as a function of et is a convolution

product of many factors for large n, which suggests an asymptotic Gaussian

distribution, i.e., a lognormal distribution with respect to t. This would be in

agreement with the well-known ideas of Kolmogorov [22] and Oboukhov for

introducing intermittency in Kolmogorov theory. However, the very explicit

forms given above for and

show that these functions do not tend very

rapidly to zero at infinity (only exponentially). This means that we do not have an

asymptotic lognormal distribution. In particular we need not trust the prediction

for the exponents n made by the lognormal theory, and it is satisfactory that (3.3)

gives a better fit to the experimental data.



References

1. D. Ruelle, Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics.

PNAS 109, 20344–20346 (2012)

2. D. Ruelle, Non-equilibrium statistical mechanics of turbulence. J. Stat. Phys. 157, 205–218

(2014)

3. G. Gallavotti, G. Garrido, Non-equilibrium statistical mechanics of turbulence: comments

on Ruelle’s intermittency theory, in The Foundations of Chaos Revisited: From Poincaré to

Recent Advancements, ed. by C. Skiadas (Springer, Heidelberg, 2016). doi:10.1007/978-3319-29701-9

4. D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

and 23, 343–344 (1971)

5. J.P. Gollub, H.L. Swinney, Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930

(1975)

6. A. Libchaber, From chaos to turbulence in Benard convection. Proc. R. Soc. Lond. A413, 63–

69 (1987)

7. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

8. P. Cvitanovi´c (ed.), Universality in Chaos, 2nd edn. (Adam Hilger, Bristol, 1989)



3 Hydrodynamic Turbulence as a Nonstandard Transport Phenomenon



57



9. B.-L. Hao (ed.), Chaos II (World Scientific, Singapore, 1990)

10. L.-S. Young, What are SRB measures, and which dynamical systems have them? J. Stat. Phys.

108, 733–754 (2002)

11. C. Bonatti, L.J. Díaz, M. Viana, Dynamics Beyond Uniform Hyperbolicity (Springer, Berlin,

2005)

12. G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–

970 (1995)

13. D. Dolgopyat, C. Liverani, Energy transfer in a fast-slow Hamiltonian system. Commun. Math.

Phys. 308, 201–225 (2011)

14. D. Ruelle, A mechanical model for Fourier’s law of heat conduction. Commun. Math. Phys.

311, 755–768 (2012)

15. A.N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very

large Reynolds number. Dokl. Akad. Nauk SSSR 30, 301–305 (1941)

16. A.N. Kolmogorov, On degeneration (decay) of isotropic turbulence in an incompressible

viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540 (1941)

17. A.N. Kolmogorov, Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk

SSSR 32, 16–18 (1941)

18. G. Parisi, U. Frisch, On the singularity structure of fully developed turbulence, in Turbulence

and Predictability in Geophysical Fluid Dynamics, ed. by M. Ghil, R. Benzi, G. Parisi (NorthHolland, Amsterdam, 1985), pp. 84–88

19. R. Benzi, G. Paladin, G. Parisi, A. Vulpiani, On the multifractal nature of fully developed

turbulence and chaotic systems. J. Phys. A 17, 3521–3531 (1984)

20. F. Anselmet, Y. Gagne, E.J. Hopfinger, R.A. Antonia, High-order velocity structure functions

in turbulent shear flows. J. Fluid Mech. 140, 63–89 (1984)

21. J. Schumacher, J. Scheel, D. Krasnov, D. Donzis, K. Sreenivasan, V. Yakhot, Small-scale

universality in turbulence. Proc. Natl. Acad. Sci. USA 111(30), 10961–10965 (2014)

22. A.N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of

turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85

(1962)



Chapter 4



Non-equilibrium Statistical Mechanics

of Turbulence

Comments on Ruelle’s Intermittency Theory

Giovanni Gallavotti and Pedro Garrido



Abstract The recent proposal by D. Ruelle for a theory of the corrections to the

OK theory (“intermittency corrections”) is to take into account that the Kolmogorov

scale itsef should be regarded as a fluctuating variable. Some quantitative aspects of

the theory can be quite easily studied also via computer and will be presented.



4.1 A Hierarchical Turbulence Model

The proposal [7, 8] for a theory of the corrections to the OK theory (“intermittency

corrections”) is to take into account that the Kolmogorov scale itself should be

regarded as a fluctuating variable.

The OK theory is implied by the assumption, for n large, of zero average work

due to interactions between wave components with wave length < Ä n `0 Á `n

and components with wave length > Ä Ä n `0 (`0 being the length scale where the

energy is input in the fluid and Ä a scale factor to be determined) together with the

assumption of independence of the distribution of the components with inverse wave

length (“momentum”) in the shell ŒÄ n ; ÄÄ n `0 1 , [5, p. 420].

It is represented by the equalities

v3.nC1/i0

v3ni

D

;

`n

`nC1



v D jvj; v 2 R3



(4.1)



G. Gallavotti ( )

INFN-Roma1, Roma, Italy

Rutgers University, New Brunswick, NJ, USA

e-mail: giovanni.gallavotti@roma1.infn.it

P. Garrido

Physics Department, University of Granada, Avda. del Hospicio, S/N, 18010 Granada, Spain

e-mail: garrido@onsager.ugr.es

© Springer International Publishing Switzerland 2016

C. Skiadas (ed.), The Foundations of Chaos Revisited: From Poincaré to Recent

Advancements, Understanding Complex Systems,

DOI 10.1007/978-3-319-29701-9_4



59



60



G. Gallavotti and P. Garrido



interpreted as stating an equality up to fluctuations of the velocity components of

scale Ä n `0 , i.e. of the part of the velocity field which can be represented by the

Fourier components in a basis of plane waves localized in boxes, labeled by i D

1; : : : Ä 3n , of size Ä n `0 into which the fluid (moving in a container of linear size `0 )

is imagined decomposed (a wavelet representation) so that .n C 1; i0 / labels a box

contained in the box .n; i/.

The length scales are supposed to be separated by a suitably large scale factor Ä

(i.e. `n D Ä n `0 D Ä 1 `n 1 ) so that the fluctuations can be considered independent,

however not so large that more than one scalar quantity (namely v3n;i ) suffices to

describe the independent components of the velocity (small enough to avoid that

“several different temperatures will be present among the systems .n C 1; j0 /”

inside the containing box labeled .n; j/, and the vnC1;j distribution “will not be

Boltzmannian for a constant temperature inside”,[7, p. 2]).

The distribution of v3nC1;j is then simply chosen so that the average of the v3nC1;j

is the value v3n;i Ä if the v3nC1;j on scale n C 1 gives a finer description of the field in

a box named j contained in the box named i of scale larger by one unit.

Among the distributions with this property is selected the one which maximizes

entropy1 and is:

Ä

n Y

Y

dWi;mC1

m



def



3



Wni Djvni j ;



mD0 iD1



Wi0 m



Äe



Ä



Wi;mC1

Wi0 ;m



(4.2)



with W0 a constant that parameterizes the fixed energy input at large scale: the

motion will be supposed to have a 0 average total velocity at each point; hence

1



W03 can be viewed as an imposed average velocity gradient at the largest scale `0 .

1



The vin D Win3 is then interpreted as a velocity variation on a box of scale `0 Ä n

or Ä n as `0 will be taken 1. The index i will be often omitted as we shall mostly

be concerned about a chain of boxes, one per each scale Ä n ; n D 0; 1; : : :, totally

ordered by inclusion (i.e. the box labeled .i; n/ contains the box labeled .i0 ; n C 1/).

def



The distribution of the energy dissipation Wn;i Dv3n;i in the hierarchically arranged

sequence of cells is therefore close in spirit to the hierarchical models that have

been source of ideas and so much impact, at the birth of the renormalization group

approach to multiscale phenomena, in quantum field theory, critical point statistical

mechanics, low temperature physics, Fourier series convergence to name a few,

and to their nonperturbative analysis, either phenomenological or mathematically

rigorous, [1–4, 10–12].

The present turbulent fluctuations model can therefore be called hierarchical

model for turbulence in the inertial scales. It will be supposed to describe the



0

If the box

D .n; j/

D .n 1; j0 / then the distribution ˘.WjW 0 / of W Á v3 is

conditioned

to

be

such

that

h

W

i D ÄR 1 W 0 ; therefore the maximum entropy condition is that

R

W˘.WjW 0 /dW, where

is a Lagrange multiplier, is

˘.WjW 0 / log ˘.WjW 0 /dW

maximal under the constraint that h W i D W 0 Ä 1 : this gives the expression, called Boltzmannian

in [7], for ˘.WjW 0 /.



1



4 Non-equilibrium Statistical Mechanics of Turbulence



61



velocity fluctuations at scales n at which the Reynolds number is larger than 1, i.e.

n

as long as vn Ä `0 > 1.

The description will of course be approximate, [8, Sect. 3]: for instance the

correlations of the velocity gradient components are not considered (and skewness

will still rely on the classic OK theory, [6, Sect. 34]).

Given the distribution (and the initial parameter W0 ) it “only” remains to study its

properties assuming the distribution valid for velocity profiles such that vn Ä n `0 >

after fixing the value of Ä in order to match data in the literature (as explained

in [8, Eq. (12)]). As a first remark the scaling corrections proposed in [12] can be

rederived.

The average energy dissipation in a box of scale n can be defined as the average

def



of "n DWn ` n ; `n D `0 Ä n : the latter average and its p”th order moments can be

readily computed to be, for p > 0:

log h "pn i

log `n

h.



! p

n!1



Wn p

/3 i

`n



Ä



n



p

3



;



D



log .1 C p/

;

log Ä

h vpn i



p



`03 Ä



n



h "pn i

p



;



p



Än p ;

D



p

C

3



(4.3)

p



1



The Wn3 being interpreted as a velocity variation on a box of scale `0 Ä n , the last

p

log . 3 C1/

.

formula can also be read as expressing the h . j rr vj /p i r p with p D 13

log Ä

The p is the intermittency correction to the value 13 : the latter is the standard

value of the OK theory in which there is no fluctuation of the dissipation per unit

time and volume W`nn ; this gives us one free parameter, namely Ä, to fit experimental

data: its value, universal within Ruelle’s theory, turns out to be quite large, Ä

22:75, [7], fitting quite well all experimental p-values (p < 18).

Other universal predictions are possible. In [8] a quantity has been studied for

which accurate simulations are available.

If W is a sample .W0 ; W1 ; : : :/ of the dissipations at scales 0; 1; : : : for the

distribution in the hierarchical turbulence model, the smallest scale n.W/ at which

1



Wn3 `0 Ä



n



'



occurs is the scale at which the Kolmogorov scale is attained (i.e. the

1

3



Reynolds number Wn `n becomes < 1).

Taking `0 D 1; D 1, at such (random) Kolmogorov scale the actual dissipation

is D Wn.W/ Ä n.W/ with a probability distribution with density P . /. If wk D WWk k 1

then Wn D W0 w1 wn and the computation of P . / can be seen as a problem on

extreme events about the value of a product of random variables. Hence is natural

that the analysis of P involves the Gumbel distribution .t/ (which appears with

parameter 3), [8].

The P is a distribution (universal once the value of Ä has been fixed to fit the

mentioned intermittency data) which is interesting because it can be related to a

quantity studied in simulations.

It has been remarked, [8], that, assuming a symmetric distribution of the velocity

increments on scale Ä



n



1



3

whose modulus is Wn;i

, the hierarchical turbulence model



62



G. Gallavotti and P. Garrido



can be applied to study the distribution of the velocity increments: for small velocity

increments the calculation can be performed very explicitly and quantitatively

precise results are derived, that can be conceivably checked at least in simulations.

The data analysis and the (straightforward) numerical evaluation of the distribution

P is described below, following [8].



4.2 Data Settings

Let `0 ; D 1 and let W D .W0 ; W1 ; : : :/ be a sample chosen with the distribution

p.dW/ D



1

Y

Ä dWi



Wi



iD1



1



e



Wi

i 1



ÄW



(4.4)

1



1



with W0 ; Ä given parameters; and let v D .v0 ; v1 ; : : :/ D .W03 ; W13 ; : : :/.

1



Define n.W/ D n as the smallest value of i such that Wi3 Ä i Á vi Ä i < 1: n.W/

will be called the “dissipation scale” of W.

Imagine to have a large number N of p-distributed samples of W’s. Given h > 0

let

def



Pn . / D



Á

1 1

1

.# W with n.W/ D n/ \ . < .Wn =W0 / 3 Ä n < C h/

hN



(4.5)



hence hPn . / is the probability that the dissipation scale n is reached with in

def P

Œ ; Ch. Then P . / D 1

nD0 Pn . / is the probability density that, at the dissipation

scale, the velocity gradient vvn0 Ä n is between and C h.

The velocity component in a direction is vn cos #: so that the probability that it is

in d with gradient vvn0 Ä n and that this happens at dissipation scale D n is d times

Z

Pn .



vn n

Ä D

v0



0 /d 0 ı. 0 j cos #j



/



sin #d#d'

D

4



Z



1



Pn . 0 /

0



d



0



(4.6)



Let

def



Z



P. / D



d



0



1

X



0 nD1



0>



Pn . 0 /



(4.7)



that is the probability distribution of the (normalized radial velocity gradient) and

Z

m



D

0



1



d P. /



m



(4.8)



4 Non-equilibrium Statistical Mechanics of Turbulence



63



its momenta. To compare this distribution to experimental data [9] it is convenient

to define

p.z/ D



1

2



1=2

1=2

2 P. 2 jzj/



(4.9)



We have used the following computational algorithm to P. /:

1.

2.

3.

4.



Build a sample .i/ W.i/ D .W0 ; W1 ; : : : ; Wn ; : : :/

1=3

1=3

Stop when n D nN i such that Wn 1 Ä .n 1/ > 1 > Wn Ä n

nN i

Evaluate m

N i D int. i =h/ C 1 where i D Ä .WnN i =W0 /1=3

goto to (1) during N times

Then, the distribution P. / is given by

P.mh



N

h=2/ D h 1 P.m/



;



N

1 X 1

N

.m

Ni

P.m/

D

N iD1 m

Ni



m/



(4.10)



where .A/ D 1 if A is true and 0 otherwise. It is convenient to define the probability

to get a given m value as

N

1 X

Q.m/ D

ı.m

N i ; m/

N iD1



(4.11)



where ı.n; m/ is the Kronecker delta. Once obtained Q.m/, we can get recursively

N

P.m/:

N C 1/ D P.m/

N

P.m



1

Q.m/ ;

m



N

P.1/

D



1

N

X

1

1 X 1

Q.m/ D

m

N iD1 m

Ni

mD1



(4.12)



and the momenta distribution is then given by:



m



D hm



m

Ni

N

1 X 1 X

lm

N iD1 m

N i lD1



(4.13)



N are computed by

Finally, the error bars of a probability distribution (for instance P)

considering that the probability that in N elements of a sequence there are n in the

box m is given by the binomial distribution:

!

N N

Dm .n; N/ D

P.m/n 1

n



N

P.m/



N n



(4.14)



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2 Turbulent Fluid as a Physical System: A Problem in Nonequilibrium Statistical Mechanics

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