Universality Classes of Fluctuation Dynamics in Hierarchical Complex Systems

Complex phenomena are known to exhibit stationary time series which often show large deviations from Gaussian statistics. A common procedure to produce non-Gaussian tails is to allow for the violation of one of the assumptions of the central limit theorem, usually the condition that the increments have finite variance which leads to stable Lévy distributions levy . The physically undesirable consequence that such distributions have infinite variance is usually dealt with, a posteriori, by imposing some sort of truncation, generating the so-called truncated Lévy distributions truncatedlevy . In other cases, heavy-tailed distributions can be accounted for by a superposition of two statistics—a procedure known in mathematics as compounding compounding and in physics as superstatistics beck . Notwithstanding their empirical successes in several areas appliedlevy ; andrews_1989 ; castaing_PhysD90 ; chabaud_PRL94 ; beck , such ad hoc treatments of non-Gaussian effects are not completely satisfactory in that they offer no clear physical explanation for the underlying dynamics driving the fluctuation phenomena. Here we propose a general stochastic dynamical framework whereby heavy-tailed distributions naturally emerge. It is shown in particular that only two classes of distributions are allowed as characterized by the nature of the tails: power law and stretched exponential, respectively.

In this work we also address another important problem that one often faces when dealing with fluctuation phenomena in complex systems, namely the fact the empirical data can almost equally well be fitted with different probability distributions, making it difficult to select between competing models beck-cohen-swinney ; sornette_review . We tackle this model selection problem by introducing a joint fitting procedure whereby we simultaneously fit the empirical distributions of both the measured quantity (signal) and an internal variable (background), which represents the changing local environment. In our formalism, the background is described by a hierarchical stochastic model whose form is severely constrained by basic physical requirements, yielding only two ‘universality classes’ of possible dynamics. Analytical expressions are obtained for the distributions of both the signal and the background in terms of certain special transcendental functions—the Meijer $G$-functions. The availability of closed form solutions within the same family of special functions not only offers a unified theoretical treatment to the problem, but more importantly it allows one to implement the joint fitting procedure described above, which in turn helps to discriminate between the allowed distributions and also provides a direct check on the assumptions of the model. Excellent agreement with the theory is found in a detailed analysis of turbulence data as well as of financial asset price fluctuations. Owing to its generality, our theory could have a wide range of applications across different areas and disciplines.

Consider a multiscale complex system in a stationary state where the probability distribution of the relevant observable, denoted by $x$, depends on the spatial or temporal scale with which the system is observed. The large-scale distribution, i.e., measured at a time scale $\tau_{0}$ above which fluctuations in $x$ are essentially uncorrelated, is assumed to be known and given by $P(x|\varepsilon_{0})$, where $\varepsilon_{0}$ is a parameter that characterizes the large-scale stationary state (global equilibrium) of the system. To be specific, we shall assume that at large scales our system follows a Gaussian distribution with variance $\epsilon_{0}$. (A Gibbsian distribution with mean $\epsilon_{0}$ could also be chosen with similar results.) We suppose furthermore that the observed quantity $x$ has a much faster dynamics than that of its local environment. This means that over short time periods (during which the environment does not change appreciably) the system follows the same distribution $P(x|\varepsilon)$, but where $\varepsilon$ now characterizes the ‘local equilibrium’. Under these assumptions, we can write

Here we assume that $\varepsilon$ is a fluctuating quantity but one that varies more slowly (in time and space) than $x$. Physically, the quantity $\varepsilon$ can be identified, for example, with the local energy flux in a turbulent flow SV1 or as a ‘local temperature’ in a hierarchical system in thermal equilibrium SV2 . If we sample the system at short intervals but over a long time span (comparable to $\tau_{0}$), then the statistics of $x$ will be described by the marginal distribution

where $f(\varepsilon)$ is the stationary probability density function of the background variable $\varepsilon$.

One distinctive aspect of our approach is that we wish to determine the possible distributions for $f(\varepsilon)$ from general physical arguments, rather than prescribe it a priori as is normally done in the compound and superstatistics formalisms. Another important ingredient is that we seek a theory that incorporates multiple time scales—a common feature of complex systems—and that clearly exhibits the connection between the local equilibrium variable $\varepsilon$ and its large scale counterpart $\varepsilon_{0}$. To this end, the Salazar-Vasconcelos model SV1 ; SV2 recently introduced to describe intermittency in multiscale fluctuation phenonema is a natural starting point. Our approach is also akin in spirit to the Palmer-Stein-Abrahams-Anderson (PSAA) hierarchical model for relaxation in spin glass anderson .

Let us assume that the system has $N$ well-separated time scales $\tau_{i}$, $i=1,...,N$, in addition to the large scale $\tau_{0}$, and let us order them from smallest to largest, i.e., $\tau_{i}\ll\tau_{i-1}$. Our arguments will be presented in a general setting, but to fix the ideas one may think of the system as a particle diffusing in a slowly changing environment. The relevant observable $x$ (say, the particle velocity) is measured at the shortest time scale $\tau_{N}$, while the environment seen by the particle is described by a slower degree of freedom (related to the local diffusion coefficient) represented by a stochastic variable $\varepsilon_{N}$ with distribution $f_{N}(\varepsilon_{N})$. In order to find $f_{N}(\varepsilon_{N})$, we use a hierarchical dynamical model where at each level $i$ $(i=1,...,N$) of the hierarchy the corresponding variable $\varepsilon_{i}$ is described by the following stochastic differential equation (SDE):

where $F(\varepsilon_{i},\varepsilon_{i-1})$ represents the deterministic driving term, $G(\varepsilon_{i},\varepsilon_{i-1})$ is the noise amplitude, and the $dW_{i}$’s are independent Wiener processes. That the functions $F$ and $G$ depend only on $\varepsilon_{i}$ and $\varepsilon_{i-1}$ encodes the hierarchical nature (local interactions) of the system. It expresses the fact that the physical constraints imposed on the system at the large scale are not directly felt at the small scales but rather are transferred down the hierarchy through the intervening scales.

The possible functional forms of $F(\varepsilon_{i},\varepsilon_{i-1})$ and $G(\varepsilon_{i},\varepsilon_{i-1})$ are severely constrained by three general physical requirements: (i) equilibrium condition, which states that $\left\langle\varepsilon_{i}(t)\right\rangle=\varepsilon_{0}$ for $t\to\infty$; (ii) invariance under change of scale, which requires that $F_{i}(\lambda\varepsilon_{i-1},\lambda\varepsilon_{i})=\lambda F_{i}(\varepsilon_{i-1},\varepsilon_{i})$ and $G_{i}(\lambda\varepsilon_{i-1},\lambda\varepsilon_{i})=\lambda G_{i}(\varepsilon_{i-1},\varepsilon_{i})$; and (iii) positivity of $\varepsilon_{i}$, meaning that ${\rm Prob}(\varepsilon_{i}(t)<0)=0,\;\forall t,\;{\rm if}\;\varepsilon_{j}(t=0)\geq 0,\forall j$, which in turn entails that $G_{i}(0,0)=0$.

The most general model that satisfies the three requirements above is of the form

where $\gamma_{i}$ and $\kappa_{i}$ are positive constants, and $s$ is, in principle, an arbitrary real number $0\leq s\leq 1$. The positivity of $\varepsilon_{i}(t)$ is proved in Appendix A. The stationary distribution $f(\varepsilon_{i}|\varepsilon_{i-1})$ can be computed from the Fokker-Planck equation associated with the SDE (4), for $\varepsilon_{i-1}$ kept fixed. If we now impose the (physically reasonable) condition that the corresponding Fokker-Planck equation should have analytic coefficients, then this further restricts the model to only two possible cases: $s=1$ or $s=1/2$.

In the case $s=1$ (which was first treated in SV1 ), the stationary conditional distribution $f(\varepsilon_{i}|\varepsilon_{i-1})$ is given by an inverse-gamma distribution

whereas for $s=1/2$ the result is a gamma distribution:

where $\beta_{i}={2\gamma_{i}}/{k^{2}_{i}}$ in both cases.

Describing fluctuation phenomena in multiscale complex systems is an admittedly difficult task because (among other reasons) one does not usually have direct access to the interscale dynamics, and hence indirect inferences have to be made about its effect on the measured quantities. For instance, non-Gaussian statistics is usually seen as an evidence of complex interactions between scales, but a general dynamical framework to explain such deviations from Gaussianity has not yet been established. Here we have shown, from a rather minimal set of assumptions on the interscale stochastic dynamics, that there exist two general classes of heavy-tailed distributions for the statistics of multiscale fluctuations. The distributions in both classes are given in terms of the same family of special functions (Meijer $G$-function) but differ regarding the nature of the tail: power law and modified stretched exponential, respectively. Good agreement was found with experimental data on classical fluid turbulence as well as financial data—both sets of data analyzed here were shown to belong to the power-law class. Further development of the theory presented here and additional applications, including of the stretched-exponential class, will be discussed in forthcoming publications.