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)