We review some of the properties of higher-dimensional superstatistical
stochastic models. As an example, we analyse the stochastic properties of a
superstatistical model of 3-dimensional Lagrangian turbulence, and compare with
experimental data. Excellent agreement is obtained for various measured
quantities, such as acceleration probability densities, Lagrangian scaling
exponents, correlations between acceleration components, and time decay of
correlations. We comment on how to proceed from superstatistics to a
thermodynamic description.Comment: 8 pages, 4 figures. To appear in the proceedings of CTNEXT07
  'Complexity, Metastability and Nonextensivity', Catania 1-5 July 2007, eds.
  S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP 200

Andrea Rapisarda

Christian Beck

Constantino Tsallis

Hans Herrmann

Piero Quarati

Sumiyoshi Abe

English

arXiv

Crossref

Correlations in superstatistical systems
Christian Beck
School of Mathematical Sciences, Queen Mary, University ofLondon, Mile End Road, London E1
4NS, UK
Abstract. We review some of the statistical properties of higher-dimensional superstatistical
stochastic models. As an example, we analyse the stochasticproperties of a superstatistical model
of 3-dimensional Lagrangian turbulence, and compare with experimental data. Excellent agreement
is obtained for various measured quantities, such as acceleration probability densities, Lagrangian
scaling exponents, correlations between acceleration components, and time decay of correlations.
We comment on how to proceed from superstatistics to a thermodynamic description.
Keywords: Superstatistics, Lagrangian turbulence, correlation functio s, scaling exponents, gen-
eralized thermodynamics
PACS: 05.40.-a; 47.27.E-
A SHORT REMINDER: WHAT IS SUPERSTATISTICS?
Complex systems often exhibit a dynamics on two time scales:A fast one as represented
by a given stochastic process and a slow one for the parameters of that process. As a
very simple example consider the following linear Langevinequation
v̇ = −γv+σL(t) (1)
with parametersγ,σ that fluctuate on a long time scale [1]. It describes the velocity v of
a Brownian particle that moves through spatial ‘cells’ withdifferent local inverse tem-
peratureβ := γ/(2σ2) in each cell (a nonequilibrium situation). Assume, for example,
that the probability distribution ofβ in the various cells is aχ2-distribution of degreen,
f (β ) ∼ β n/2−1e−
nβ
2β0 . (2)
Then the conditional probability isp(v|β )) ∼ e− 12βv2, the joint probability isp(v,β ) =
f (β )p(v|β ), and the marginal probability isp(v) =
∫ ∞
0 f (β )p(v|β )dβ . Integration yields
p(v) ∼ 1
(1+ 12β̃ (q−1)v2)1/(q−1)
(3)
i.e. we obtain power-law stationary distributions just as in Tsallis statistics [2] with
q = 1+ 2n+1, β̃ = 2β0/(3−q), whereβ0 =
∫
f (β )βdβ is the average ofβ .
All this has a very broad interpretion and can be generalizedn various ways—β
need not to be inverse temperature. One can generalize the abov example to general
probability densitiesf (β ) and general Hamiltonians in statistical mechanics. One then
has a superposition of two different statistics: that ofβ and that of ordinary statistical
mechanics. The short name for this issuperstatistics[3]. Superstatistics describes com-
plexnonequilibrium systemswith spatio-temporal fluctuations of an intensive parameter
(e.g. inverse temperature) on a large scale.
Define an effective Boltzmann factorB(E) by
B(E) =
∫ ∞
0
f (β )e−βEdβ
where f (β ) is the probability distribution ofβ andE the energy of the system. Many re-
sults can be proved forgeneral f(β ). Here we list some recent theoretical developments
of the superstatistics concept:
• Can prove superstatistical generalizations of fluctuationtheorems [4]
• Can develop a variational principle for the large-energy asmptotics of general
superstatistics [5] (depending onf (β ), one can get not only power laws for largeE
but e.g. also stretched exponentials)
• Can formally define generalized entropies for general superstatistics [6, 7]
• Can study various theoretical extensions and workouts of the superstatistics concept
[8, 9, 10, 11, 12, 13]
• Can prove a superstatistical version of a Central Limit Theorem leading to Tsallis
statistics [14]
• Can relate it to fractional reaction equations [15]
• Can consider superstatistical random matrix theory [16]
• Can apply superstatistical techniques to networks [17] andtime series [18]
...and some more practical applications:
• Can apply superstatistical methods to analyse the statistics of 3d hydrodynamic
turbulence [1, 18, 19, 20, 21, 22]
• Can apply it to atmospheric turbulence (wind velocity fluctuations at Florence
airport [23, 24]) and defect turbulence [25]
• Can apply superstatistical methods to finance [26, 27]
• Can apply it to solar flares [28], and even to print queues [29]
• Can apply it to cosmic ray statistics [30]
• Can apply it to various scattering processes in particle physics [31, 32]
• Can apply it to hydroclimatic fluctuations [33]
• Can apply it to British train delay statistics [34]
PHYSICALLY RELEVANT SUPERSTATISTICAL
UNIVERSALITY CLASSES
Basically, there are3 physically relevant universality classes[18]:
• (a) χ2-superstatistics (= Tsallis statistics)
• (b) inverseχ2-superstatistics
• (c) lognormal superstatistics
Why? Consider, e.g., case (a). Assume there are many microscopic random variables
ξ j , j = 1, . . . ,J, contributing toβ in an additive way. For largeJ, their sum 1√J ∑
J
j=1ξ j
will approach a Gaussian random variableX1 due to the (ordinary) Central Limit The-
orem. There can ben Gaussian random variablesX1, . . . ,Xn due to various relevant de-
grees of freedom in the complex system. Sinceβ is positive we may square theXi to
obtain something positive. The sumβ = ∑ni=1X2i is thenχ2-distributed with degreen,
i.e.,
f (β ) =
1
Γ(n2)
(
n
2β0
)n/2
β n/2−1e−
nβ
2β0 , (4)
whereβ0 is the average ofβ . Integration as described in section 1 yields Tsallis statitics
as a special case of superstatistics.
(b) The same considerations can be applied if the ‘temperature’ β−1 rather thanβ
itself is the sum of several squared Gaussian random variables arising out of many
microscopic degrees of freedomξ j . The resultingf (β ) is the inverseχ2-distribution:
f (β ) =
β0
Γ(n2)
(
nβ0
2
)n/2
β−n/2−2e−
nβ0
2β . (5)
It generates superstatistical distributionsp(E) ∼
∫
f (β )e−βE that decay ase−β̃
√
E for
largeE [5].
(c) β may be generated by multiplicative random processes. Consider a local cascade
random variableX1 = ∏Jj=1 ξ j , whereJ is the number of cascade steps and theξ j are
positive microscopic random variables. By the Central Limit Theorem, 1√
J
logX1 =
1√
J
∑Jj=1 logξ j becomes Gaussian for largeJ. HenceX1 is log-normally distributed.
In general there may ben such product contributions toβ , i.e., β = ∏ni=1Xi . Then
logβ = ∑ni=1 logXi is a sum of Gaussian random variables; hence it is Gaussian aswell.
Thusβ is log-normally distributed, i.e.,
f (β ) =
1√
2πsβ
exp
{
−(ln βm)2
2s2
}
. (6)
Lognormal superstatistics is relevant in turbulence [18, 19, 20, 21, 22].
APPLICATION TO LAGRANGIAN TURBULENCE
Turbulence is a spatio-temporal chaotic state of the Navier-Stokes equation. Energy is
dissipated in a cascade-like process. Bodenschatz et al. [22, 35, 36] obtained rather
precise measurements of the acceleration~a(t) of a single tracer particle in a turbulent
flow. One can now construct a superstatistical Lagrangian model for 3-dimensional
velocity differences~u(t) :=~v(t +τ)−~v(t) of such a tracer particle (note that~a=~u/τ for
FIGURE 1. Distribution of acceleration as measured by Bodenschatz [22, 35] and as predicted by
eq. (8),s2 ≈ 3 .
smallτ). This model is given by the superstatistical stochastic differential equation [19]
~̇u = −γ~u+B~n×~u+σ~L(t). (7)
The new thing as compared to previous work is the term involving the vector product.
It describes fluctuating enstrophy (rotational energy) around the test particle. Whileγ
andB are constants, the noise strengthσ and the unit vector~n evolve stochastically on a
large time scaleTσ andT~n, respectively. One hasTσ γ ∼Rλ >> 1, whereRλ is the Taylor
scale Reynolds number. The time scaleT~n describes the average life time of a region of
given vorticity surrounding the test particle.
Defineβ := 2γ/σ2, then in this modelβ−1 ∼ ν1/2〈ε〉−1/2ε, whereν is the kinematic
viscosity and〈ε〉 the average energy dissipation. The probability density ofhe stochastic
processβ (t) is assumed to be a lognormal distribution as given in eq. (6).For very small
τ an acceleration component of the particle is given byax = ux/τ and one gets the
following prediction for the stationary distribution:
p(ax) =
τ
2πs
∫ ∞
0
dβ β−1/2exp
{
−(log βm)2
2s2
}
e−
1
2βτ
2a2x (8)
This compares very well with the experimentally measured probability distribution of
acceleration, see Fig. 1.
CORRELATIONS INDUCED BY SUPERSTATISTICS
3-dimensional superstatistics induces correlations betwe n the componentsax,ay,az of
the acceleration vector~a. Consider the ratioR := p(ax,ay)/(p(ax)p(ay)). For indepen-
dent acceleration components this ratio would always be givn byR= 1. However, our
FIGURE 2. The quantitityRas given by eq. (9) for lognormal superstatistics.
3-d superstatistical model yields the prediction
R=
∫ ∞
0 β f (β )e
− 12βτ2(a2x+a2y)dβ
∫ ∞
0 β 1/2 f (β )e−
1
2βτ2a2xdβ
∫ ∞
0 β 1/2 f (β )e
− 12βτ2a2ydβ
. (9)
This is a very general formula, it is also valid for Tsallis statistics, wheref (β ) is the
χ2-distribution. Note thatR= 1 for f (β ) = δ (β −β0), i.e. if there are no fluctuations in
β then all components are independent random variables.
Fig. 2 showsR = p(ax,ay)/(p(ax)p(ay)) as predicted by lognormal superstatistics.
The figure strongly resemblesR as experimentally measured by Bodenschatz et al. [22]
in a turbulent flow, see Fig. 3.
Besides correlations between components one can also look at temporal correlations.
The superstatistical model [19] allows for the calculationof temporal correlation func-
tions as well. In particular, we may be interested in temporal correlation functions of
single componentsux of velocity differences, i.e.C(t) = 〈ux(t ′+ t)ux(t ′)〉. By averaging
over the possible random vectors~n one arrives at the formula
C(t) =
1
3
〈u2x〉e−γt(2cosBt+1), (10)
i.e. there is rapid (exponential) decay with a zero-crossing at t∗ = 23πB
−1. Exponential
decay and zero-crossings are also observed for the experimental data. The model [19]
also correctly reproduces the experimentally observed fact that the correlation function
of the absolute value|~a| decays very slowly as compared to that of the single compo-
nents. Moreover, it correctly describes the fact that enstrophy lags behind dissipation
[37].
FIGURE 3. log10Ras measured by Bodenschatz et al. [22].
LAGRANGIAN SCALING EXPONENTS
The moments of velocity differences of a single Lagrangian test particle that is embed-
ded in a turbulent flow scale differently from those measuredin a fixed laboratory frame.
Our superstatistical model [19] allows for the analytic evaluation of the Lagrangian scal-
ing exponents. The moments of velocity difference componentsux on a time scaleτ are
obtained as
〈u jx〉 = ( j −1)!!m−
j
2w
1
8 j
2
. (11)
Assuming simple scaling laws of the form∼ τa, w = es2 ∼ τb, wherea andb are
so far arbitrary real numbers, on gets〈u jx〉 ∼ τζ j ∼ τ−a
j
2+b
1
8 j
2
. Hence the Lagrangian
scaling exponents are given byζ j = −a2 j + b8 j2. Usually one assumesζ2 = 1, hence we
geta = 12b−1 thus
ζ j = (
1
2
+λ 2) j − 1
2
λ 2 j2, (12)
whereλ 2 := −14b. This prediction is in good agreement with the recent measurements
of Bodenschatz et al. [36], see Fig. 4.
FROM SUPERSTATISTICS TO (GENERALIZED)
THERMODYNAMICS
We end this paper with some more general thoughts. Can we proced from superstatistics
as a merely statistical technique to a properth rmodynamic description? There are some
early attempts in this direction by Tsallis and Souza [6]. Here we want to follow a
somewhat different approach [7]: One starts quite generally from two random variables
E andB (representing energy and invere temperature) and then considers the following
FIGURE 4. Lagrangian scaling exponents. Data points: Measurements of Bodenschatz et al. [36]. Solid
line: Theoretical prediction of the superstatistical model (λ 2 = 0.085) [19]. Dashed lines: Some other
competing models [38].
effective entropy for a superstatistical system
S[E,B] = S[E|B]+S[B]
=
∫
dβ f (β )(βU(β )+ lnZ(β ))−
∫
dβ f (β ) ln f (β ),
whereU is the local internal energy andZ the local partition function. One can do ther-
modynamics with this extended entropy function. It reducesto ordinary thermodynamics
for f (β ) = δ (β −β0). For sharply peaked distributionsf (β ) this is a slightly deformed
thermodynamics, which can be evaluated in a perturbative way. One can also maximize
this entropy with respect to appropriate constraints inβ to get e.g. a lognormal distribu-
tion for f (β ), or generally some other distributionf (β ) depending on the constraints.
For more details, see [7].
SUMMARY
• Superstatistics (a ‘statistics of a statistics’) providesa physical reason why more
general types of Boltzmann factors (e.g. of power-law type)are relevant for
nonequilibrium systems with fluctuations of an intensive parameter.
• There is evidence for three major physically relevant universality classes:χ2-
superstatistics= Tsallis statistics, inverseχ2-superstatistics, and lognormal super-
statistics. These arise as universal limit statistics for many different complex sys-
tems.
• Superstatistical techniques have been successfully applied to a variety of complex
systems.
• A superstatistical model of Lagrangian turbulence [19] is in excellent agreement
with the experimental data for probability densities, correlations between compo-
nents, decay of correlations, and Lagrangian scaling exponents.
• The long-term aim is to find a good thermodynamic descriptionfor general super-
statistical systems.
REFERENCES
1. C. Beck, Phys. Rev. Lett.87, 180601 (2001)
2. C. Tsallis, J. Stat. Phys.52, 479 (1988)
3. C. Beck and E.G.D. Cohen, Physica A322, 267 (2003)
4. C. Beck and E.G.D. Cohen, Physica A344, 393 (2004)
5. H. Touchette and C. Beck, Phys. Rev. E71, 016131 (2005)
6. C. Tsallis and A.M.C. Souza, Phys. Rev. E67, 026106 (2003)
7. S. Abe, C. Beck, and E.G.D. Cohen, Phys. Rev. E76, 031102 (2007)
8. P.-H. Chavanis, Physica A359, 177 (2006)
9. C. Vignat, A. Plastino and A.R. Plastino, Nuovo Cimento B120, 951 (2005)
10. G.E. Crooks, Phys. Rev. E75, 041119 (2007)
11. J. Naudts, arXiv:0709.0154
12. R.F. Rodriguez and I. Santamaria-Holek, arXiv:0706.0436
13. I. Lubashevsky et al., arXiv:0706.0829
14. C. Vignat and A. Plastino, arXiv:0706.0151
15. A.M. Mathai and H.J. Haubold, Physica A375, 110 (2007)
16. A.Y. Abul-Magd, Physica A361, 41 (2006)
17. S. Abe and S. Thurner, Phys. Rev. E72, 036102 (2005)
18. C. Beck, E.G.D. Cohen, and H.L. Swinney, Phys. Rev. E72, 026304 (2005)
19. C. Beck, Phys. Rev. Lett.98, 064502 (2007)
20. A. Reynolds, Phys. Rev. Lett.91, 084503 (2003)
21. C. Beck, Europhys. Lett.64, 151 (2003)
22. A.M. Reynolds, N. Mordant, A.M. Crawford, and E. Bodenschatz, New Journal of Physics7, 58
(2005)
23. S. Rizzo and A. Rapisarda,Environmental atmospheric at Florence airport, Proceedings of the 8th
Experimental Chaos Conference, Florence, AIP Conf. Proc.742, 176 (2004)
24. S. Rizzo and A. Rapisarda,Application of superstatistics to atmospheric turbulence, in Complex-
ity, Metastability and Nonextensivity, eds. C. Beck, G. Benedek, A. Rapisarda, C. Tsallis, World
Scientific (2005)
25. K. E. Daniels, C. Beck, and E. Bodenschatz, Physica D193, 208 (2004)
26. J.-P. Bouchard and M. Potters,Theory of Financial Risk and Derivative Pricing, Cambridge Univer-
sity Press, Cambridge (2003)
27. M. Ausloos and K. Ivanova, Phys. Rev. E68, 046122 (2003)
28. M. Baiesi, M. Paczuski and A.L. Stella, Phys. Rev. Lett.96, 051103 (2006)
29. U. Harder and M. Paczuski, cs/PF/0412027
30. C. Beck, Physica A331, 173 (2004)
31. G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett.84, 2770 (2000)
32. G. Wilk and Z. Wlodarczyk, Physica A376, 279 (2007)
33. A. Porporato, G. Vico, and P.A. Fay, Geophys. Res. Lett.33, L15402 (2006)
34. K. Briggs and C. Beck, Physica A278, 498 (2007)
35. A. La Porta, G.A. Voth, A.M. Crawford, J. Alexander, and E. Bodenschatz, Nature409, 1017 (2001)
36. H. Xu, N.T. Ouellette, and E. Bodenschatz, Phys. Rev. Lett 96, 114503 (2006)
37. B. Zeff et al., Nature421, 146 (2003)
38. L. Chevillard et al., Phys. Rev. Lett.91, 214502 (2003)
