Elsevier

Physics Letters A

Volume 311, Issues 2–3, 12 May 2003, Pages 126-132
Physics Letters A

Statistical complexity and disequilibrium

https://doi.org/10.1016/S0375-9601(03)00491-2Get rights and content

Abstract

We study the concept of disequilibrium as an essential ingredient of a family of statistical complexity measures. We find that Wootters' objections to the use of Euclidean distances for probability spaces become quite relevant to this endeavor. Replacing the Euclidean distance by the Wootters' one noticeably improves the behavior of the associated statistical complexity measure, as evidenced by its application to the dynamics of the logistic map.

Introduction

Jaynes [1], [2] has long ago established the relevance of information theory [3] for theoretical physics. Two essential ingredients of Jaynes' program are (i) Shannon's logarithmic information measure I [4] I=−i=1Npiln[pi], regarded as the general measure of the uncertainty associated to probabilistic physical processes described by the probability distribution {pii=1,…,N} and (ii) his celebrated maximum entropy principle (MEP) [1], [2]. Traversing a separate track, Kolmogorov and Sinai [5] converted information theory [3] into a powerful tool for the study of dynamical systems. The statistical characterization of deterministic sources of apparent randomness performed by many authors during the intervening years has shed much light into the intrincacies of dynamical behavior by describing the unpredictability of dynamical systems using such tools as metric entropy, Lyapunov exponents, and fractal dimension [6]. It is thus possible to (i) detect the presence and (ii) quantify the degree of the deterministic chaotic behavior [6].

Now, ascertaining the degree of unpredictability and randomness of a system is not automatically tantamount to adequately grasp the correlational structures that may be present, i.e., to be in a position to capture the relationship between the components of the physical system. These structures strongly influence, of course, the character of the probability distribution that is able to describe the physics one is interested in. Randomness, on the one hand, and structural correlations on the other one, are not totally independent aspects of this physics. Certainly, the opposite extremes of (i) perfect order and (ii) maximal randomness possess no structure to speak of [7], [8], [9]. In between these two special instances a wide range of possible degrees of physical structure exists, degrees that should be reflected in the features of the underlying probability distribution. One would like that they be adequately captured by some functional F[{pi}] in the fashion that Shannon's I captures randomness. A suitable candidate to this effect has come to be called the statistical complexity (see the enlightening discussion of [10]). F[{pi}] should, of course, vanish in the two special extreme instances mentioned above.

It is the goal of the present effort that of effecting some in depth analysis of one crucial ingredient shared by a family of statistical complexities that have recently received considerable attention in the pertinent literature. We are referring here to the so-called disequilibrium Q, that makes the above functional F[{pi}] to vanish in the case in which maximal randomness is present. It is clear that I vanishes in the case of perfect order [3].

Section snippets

A family of statistical complexity measures

In dealing with the concept of statistical complexity one must start by excluding processes that are certainly not complex, such as those which exhibit periodic motion. White noise random process cannot be assumed to be complex, notwithstanding its irregular and unpredictable character, since it does not contain any nontrivial structure. Statistical complexity (SC) has to do with intrincate structures hidden in the dynamics, emerging from a system which itself is much simpler than its dynamics 

Disequilibrium measure based on the Euclidean distance

As stated above, the “disequilibrium” Q is the distance between a given probability distribution P=(p1,…,pN) and the “equilibrium” one Pe=(1/N,…,1/N): Q(P)=Q0D(P,Pe), with D an appropriate measure of distance. Q0 is a normalization constant. If D is the Euclidean norm in RN we get the LMC definition, namely, QE(P)=Q0(E)DE(P,Pe)=Q0(E)PPeE=Q0(E)i=1Npi1N2, with Q0(E)=N/(N−1) so that 0⩽QE⩽1.

This straightforward definition of distance has been criticized by Wootters in an illuminating

Application

We will compare the Wootters' SC (C(LMC)W=QWH) to the LMC one (C(LMC)E=QEH) with reference to the logistic map, in order to determine whether the former allows for some more detailed grasping of the dynamics than the latter. We deal with the map F:xn→xn+1, described by the ecologically motivated, dissipative system described by the first order difference equation xn+1=rxn(1−xn)(0⩽xn⩽1, 0<r⩽4). The dynamical behavior is controlled by r. Fig. 1(a) shows the well-known bifurcation diagram for the

Conclusions

The notion of statistical complexity advanced by López-Ruiz, Mancini, and Calbet (LMC) [16] constitutes an important step towards building up an armory of measures that are both easily to compute and to intuitively grasp. It exhibits, nonetheless, some difficulties that have been nitidly pointed out in, for instance, Refs. [10], [17]. As a further step in the direction inaugurated by López-Ruiz, Mancini, and Calbet, we have here called attention to the mind-opening study of Wootters' [20]

Acknowledgements

This work was partially supported by CONICET (PIP 0029/98), Argentina and the International Office of BMBF (ARG-4-G0A-6A and ARG01-005), Germany. O.A.R. is very grateful to Prof. Dr. B. Fischer and Prof. Dr. J. Prestin for their kind hospitality at Institut für Mathematik, Medizinische Universität zu Lübeck, Germany; where part of this work was done.

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