Both mechanisms can be unified within a more general simultaneously nonlinear and inhomogeous Fokker-Planck equation , essentially implying a long-range memory. The various connections of statistical mechanics either BG or nonextensive and other important approaches are schematically described in Fig. Let us illustrate the various facts mentioned above through some selected examples.
It is known that. Hence, it straighforwardly follows that. The physical interpretation of this interesting limit remains elusive. See Fig. It has been shown for this specific model, which turns out to be asymptotically scale-invariant, that. He obtained that the distribution should be a q stationart state -Gaussian with. Later on, we will come back onto this prediction. In particular, M is the amplitude of the multiplicative noise. The above examples paradigmatically illustrate how q can be determined from either microscopic or mesoscopic information. Further analytical expressions for q in a variety of other physical systems are presented in .
Connections between indices q. The full understanding of all the possible connections between these different q -indices still remains elusive. Many examples exist for which one or more of these q 's are analytically or numerically known and understood. However, the complexity of this question has not yet allowed for transparent, complete and general understanding. Nevertheless, at the light of what is presently known, the scenario appears to be that, for a given system, a countable set of q 's can exist, each of them being basically associated with a specific more or less important property of the system.
For many systems, if not all, the structure seems to be such that very few typically only one of those q 's are independent, all the others being functions of those few. Let us illustrate what we mean by assuming that only one is independent, and let us denote it by q 0. So, we typically have. The form of f m x 0 or of similar functions can be quite complex see, for instance, [71, 72]. However, intriguingly enough, very many among them seem to conform to the following simple structure.
In some case q m satisfies. In other cases it satisfies, through the socalled additive duality q 0 2 - q 0 ,. We easily see that, if we apply again this transformation, we go back to the initial value q 0. Or equivalently, the function q 0 concides with the function q 0. It respectively follows from 40 and 41 that. If we define q 0 through Eq.
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If we make an analytical extension in 40 such that m and m ' are allowed to be real numbers, we may consider the case. By replacing this into Eqs. This specific form of connection appears in very many occasions in nonextensive statistical mechanics. The families 40 and 41 can be respectively rewritten as follows:. For one of the two branches of the family 54 it is the other way around, i. It is possible to simultaneously write both families in a compact manner, namely. It is also worth stressing two interesting properties of.
It can be shown [10, 58] that, by successively and alternatively composing these two dualities, the entire basic structure of Eqs. To the best of our knowledge, the family 55 , as a set of transformations, first appeared in , and, since then, in an amazing amount of other situations. Moreover, isolated elements of the family 55 had been present in the literature even before the paper . The exhaustive list of these many situations is out of the present scope.
Let us, however, mention a few paradigmatic ones. Third, elements of these families appear in what is sometimes called the q -triplet, which we address specifically in the next Subsection.
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The sets 53 and 54 are illustrated in Figs. The q -triplet. This set is currently referred to as the q-triangle , or the q-triplet. One year later, the q -triplet was indeed found in the solar wind  through the analysis of the magnetic-field data sent to Earth NASA Goddard Space Flight Center by the spacecraft Voyager 1, at the time in the distant heliosphere.
The observations led to the following values:. The most precise q being q stationary state , it seems reasonable to fix it at its nominal value, i. By heuristically adopting simple relations belonging to the set 53 , we can conjecture .
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These assumptions are consistent with the following identification:. They are also consistent with the following identification. Although this kind of scenario is tempting, we have not yet achieved a deep understanding which would provide a physical justification about it. To make things even more intriguing, a last remarkable discovery [10, 79] deserves mention. Amazingly enough, these values satisfy. All these striking features suggest something like a deep symmetry, which eludes us.
In any case, even if notoriously incomplete, the whole story was apparently considered quite stimulating by the organizers of the United Nations International Heliophysical Year Indeed, they prepared the poster shown in Fig. One more interesting q -triplet is presently known, namely for the edge of chaos of the logistic map. It is given by. See  for the value 81 , and [84, 85] for the value No simple relations seem to hold between these three numbers.
We notice, however, that, for both q -triplets that have been presented in this Subsection, the following inequalities hold:. Central limit theorems and q -Fourier transforms. The Central Limit Theorem CLT with its Gaussian attractors in the space of probability distributions constitutes one of the main pieces mathematically grounding important parts of BG statistical mechanics.
On a different vein - still within the realm of BG statistical mechanics -, the most basic free-particle Langevin equation with additive noise , and its associated Fokker-Planck equation, provide, for all times and positions, a Gaussian distribution. As a third connection, one might argue that the velocity distribution of any particle of a system described by a classical N -particle Hamiltonian with kinetic energy and short-range two-body interactions is the Maxwellian distribution, i.
Now, this comes out from BG statistics. So, in what sense may we consider that the CLT enters? In the sense that Maxwellian distributions are indeed ubiquitously observed in nature, and this happens because, for not too strong perturbations, Gaussians are attractors. Since S q generalizes S BG , one expects the just mentioned connections and similar ones to be appropriately q -generalized. So should also be with the CLT itself see  and references therein for the reasons which enabled conjecturing this q -generalization.
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We do not intend in the present review to be exhaustive with regard to this rich subject. Let us nevertheless remind that the extremization of Sq under essentially the same constraints as before i. The situation changes drastically if the hypothesis of quasi- independence is violated, i. The proof of these q -generalized CLT theorems is based on the so called q-Fourier transform , defined as follows [20, 74, 75, 88, 89]:.
More precisely, the q -Fourier transform of a normalized q -Gaussian, as given by Eq. It can be rewritten as follows:. However, for a density f x not belonging to the family of q -Gaussians, the problem can be more complex. Hilhorst has introduced  two interesting and paradigmatic examples which illustrate the difficulty. One of them is presented in his article  in this same volume.
We shall here discuss his other example, which we present in what follows. Let us define the probability density. We verify that f x, a is normalized, a , i. Replacing herein Eq. Indeed, it is given by. In other words, this q -Fourier integral transform has, not one, but an entire family of functions parameterized by the real number a of pre-images. We show next that this question can be univoquely answered with some simple supplementary information.
Let us illustrate now how. We first define the following quantities :. The 2 q - 1 -variance is given by. This quantity corresponds to the standard linear variance, but with the escort distribution , instead of the original one f x, a. It is illustrated in Fig. To fully understand the implications of these results, let us remind that successive moments with appropriate escort distributions are directly related with the successive derivatives of the q -Fourier transform of f x, a .
All even such derivatives are finite, in particular. Weseein Fig. In this particular case, further information is needed. An analogy can be done at this point. As we all know, Newton's equation does not provide a single solution but a family of solutions. To have a unique solution we need to provide further information, namely x 0 and 0.
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An important difference, however, between this problem and the generic invertibility of the q -Fourier transform is that, for the nonlinear integral transform, we do not know the general form of its inverse. Would we know it, the problem would in principle be completely solvable through the use of supplementary information such as the value of nu for the above illustration which would uniquely determine the specific density within the general form. As we see, the general problem still remains open. All the above discussion about invertibility is of course relevant for the domain of validity of the q -generalized central limit theorem.
At the present moment, its proof  fully applies only within the class of densities for which invertibility is guaranteed. This class is very vast; however, as we have seen, it does not include all possible densities. An alternative q -generalized CLT can be seen in . Fittings can be extremely useful In this section we compare three analytic forms which are frequently used in the context of complex systems whenever fat tails emerge.
These forms are the q-exponential , the stretched exponential , and the Mittag-Leffler function. All three recover the exponential function as a limiting instance. Let us finally mention that the function is the solution of. The function is the solution of. Consequently, this function interpolates between the stretched exponential for small values of x and the power-law for large values of x see Fig.
However, as illustrated in Figs. This fact illustrates a well known concept: careless fittings can be dangerous. They can be however extremely useful if done meticulously. They are, in any case, inescapable whenever the analysis of experimental or numerical results is concerned. Further interesting examples along similar lines can be seen in . The q -generalized CLT, with its q -Gaussian attractors in the space of probability densities, applies for correlations of the q -independent type.
There is strong evidence that this type of correlations is deeply related to strict or asymptotic scale-invariance which might well be necessary for q -independence, although surely not sufficient. After decades of studies focusing on fractals, it is by now well established that scale-invariance is ubiquitous in natural, artificial and even social systems.
Consequently, we should expect the emergence of q -Gaussians quite often. This is indeed what happens, as we shall illustrate next needless to say that only within the error bars corresponding to each case. This has been verified both with quantum Monte Carlo methods, as well as in the laboratory with Cs atoms. Let us mention at this point that if, instead of these time averages, we do ensemble averages, we obtain other types of distributions [, ]. The q -Gaussian function is of course a q -exponential of a squared variable. The q -CLT only addresses q -Gaussians, not q -exponentials in general.
However, it is clear that the detection of a q -exponential in some central property of a system opens the door for possibly finding q -Gaussians in some other important property. There are dozens of such situations that have been reported in the literature see, for instance, . We would like to add here a very recent one which concerns spin-glasses. This system was for many years awaited to exhibit connections with nonextensive statistics due to its notorious nonergodicity and complex structure in phase space.
Very recently this finally occurred in the experimentally observed relaxation curves  of paradigmatic spin-glasses such as Cu 0. The magnificent theory of Boltzmann and Gibbs has, as any other intellectual construct, limits of validity. Outside these limits a vast world exists in nature as well as in artificial systems. Some - apparently many - classes of systems of this vast world are adequately addressed by nonextensive statistical mechanics, proposed over twenty years ago  as a generalization of the standard theory. The complexity of this extended theory certainly is much higher than that of the BG statistical mechanics.
Another feature which also reflects this complexity is the fact that the involved mathematics is definitively nonlinear , whereas that of the BG theory is, in many aspects, a linear one. Under such circumstances, one could expect to have an extremely slow technical and epistemological progress. The intensive and world-wide developments that we have witnessed during the last two decades neatly show that it has not been so.
Indeed, an amazingly large field of knowledge is now available, demanding for further analytical, experimental, and computational efforts to advance. Sem pressa e sem pausa Without hurry and without stops The idea of comparing, together with the q -exponential and the stretched exponential functions, the Mittag-Leffler function emerged during an interesting conversation with Ryszard Kutner, to whom I am indebted. Along this line, I also thank M.
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Jauregui Rodriguez for the construction of Fig. I thank as well G. Ruiz for the construction of Figs. I am indebted to H. Hilhorst for authorizing me to present his unpublished example, which enables the analysis of the invertibility of the q -Fourier transform out from the family of the q -Gaussians, although within a specific larger family of densities. I am also indebted to him, as well as to S. Umarov and E. Curado, for long and fruitful discussions, always on the subject of the nontrivial difficulties related with the q -Fourier transform possible invertibility.
My thanks also go to R. Kutner and E. Lenzi for useful discussions about fractional derivatives. I am deeply grateful to R. Mendes, L. Evangelista, L.
Malacarne and E. Planck, Uber irreversible Strahlugsforgange , Ann d. Okun, The fundamental constants of physics , Sov. Tsallis and S. Okun, Cube or hypercube of natural units? Duff, L. Okun and G.
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Tsallis, Possible generalization of Boltzmann-Gibbs statistics , J. But it was published only in , particularly stimulated by the fruitful discussions held on the subject with E. Curado and H. Tsallis, R. Mendes and A.
Plastino, The role of constraints within generalized nonextensive statistics , Physica A , Curado and C. Tsallis, Generalized statistical mechanics: connection with thermodynamics , J. A 24 , L69 ; Corrigenda: 24, and 25, Abe, Instability of q-averages in nonextensive statistical mechanics , Europhys. Hanel, S. Thurner and C. Hanel and S. Thurner, Stability criteria for q-expectation values , Phys. A , Abe and G.
Bagci, Necessity of q-expectation value in nonextensive statistical mechanics , Phys. E 71 , Abe, Why q-expectation values must be used in nonextensive statistical mechanics , Astrophys. Space Sci. Thistleton, J. Marsh, K. Nelson and C. Umarov, C. Steinberg, On a q-central limit theorem consistent with nonextensive statistical mechanics , Milan J.
Bercher, Tsallis distribution as a standard maximum entropy solution with 'tail' constraint , Phys. Naudts, The q-exponential family in statistical physics , Central Eur. Tsallis, A. Plastino and R. Alvarez-Estrada, Escort mean values and the characterization of power-law-decaying probability densities , J. Many of these phenomena seem to be susceptible to description using approaches drawn from thermodynamics or statistical mechanics, particularly approaches involving the maximization of entropy and of Boltzmann-Gibbs statistical mechanics and standard laws in a natural way.
The book addresses the interdisciplinary applications of these ideas, and also on various phenomena that could possibly be quantitatively describable in terms of these ideas. Applications relate to dynamical, physical, geophysical, biological, economic, financial, and social systems, and to networks, linguistics, and plectics. A dripping faucet as a nonextensive system, the pricing of stock options, and spatial patterns in forest ecology are some subjects discussed. Material originated at an April workshop held at the Santa Fe Institute. Help Centre. My Wishlist Sign In Join.
Be the first to write a review. Add to Wishlist. Ships in 10 to 15 business days. Link Either by signing into your account or linking your membership details before your order is placed. Description Table of Contents Product Details Click on the cover image above to read some pages of this book! Industry Reviews "Gell-Mann Science Board, Santa Fe Institute and Tsallis Brazilian Center for Physics Research present material on interdisciplinary applications of ideas related to the nonextensive generalization of entropy, Boltzmann- Gibbs statistical mechanics, and standard thermodynamics.
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