Vanishing largest Lyapunov exponent and Tsallis entropy

We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamics aspects of which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison in metric terms by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed with respect to the corresponding Euclidean metric to zero. This conclusion is in agreement with all currently known results about systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.


Introduction
The Tsallis entropy is a single-parameter family of functionals, first introduced in the Physics literature in 1988 [1], providing an alternative to the Boltzmann/Gibbs/Shannon (BGS) entropy used in the statistical description of a system. Consider a probability distribution {p i }, i ∈ I in a discrete sample space indexed by I ⊂ N. Its Tsallis entropy is defined as It may be worth comparing (1) with the BGS entropy where we immediately observe as the non-extensive/entropic parameter q → 1, we get An analogous definition of the Tsallis entropy can be given for continuous sample spaces.
Henceforth, we will be setting the Boltzmann constant k B = 1, for simplicity.
The Tsallis entropy, conjecturally, describes collective phenomena with long-range spatial and temporal correlations [2], [3], systems whose phase portraits exhibit a fractal-like behavior etc. for which there is no reason or justification why their description by the BGS entropy should be accurate or even valid [3] (and references therein). Following the approach and viewpoint of Boltzmann [4], [5], one can state that the dynamical basis of the Tsallis entropy remains unclear so far [3]. This mirrors the existing difficulties in deriving the BGS entropy [3]- [6] from dynamical principles. As in the latter case, in the case of the Tsallis entropy, some progress has been made in identifying characteristics of systems effectively described by it. One class of such systems are ones exhibiting "weak chaos" or being at the "edge of chaos" for some subset of their parameter space [2], [3], [7]- [10]. This is more accurately expressed by referring to them as dynamical systems having vanishing largest Lyapunov exponent [3], [7]- [10].
A general formal justification of why such systems are described by the Tsallis entropy has been lacking so far [3], [11]. We attempted to outline an explanation in our previous work [12]. In the present work we make things more concrete and also show how such results are independent of and can be extended to the more general case of CAT(k) configuration/phase spaces. We would like point out some important predecessors of applications of (differential-) geometric methods in Statistics [13]- [19] and Probability [20], [21]. These references may be consulted for some relevant background as well as for providing a far broader and deeper perspective on the geometric methods employed in the present work, from the viewpoint of Statistics [13]- [19] and Probability [20], [21].
In Section 2, we argue that if a system is described by the Tsallis entropy and moreover if the Tsallis induced generalised addition is a direct result of its configuration/phase space dynamics, then the largest Lyapunov exponent of the underlying system should vanish. This statement is in agreement with all currently known numerical as well as the very few available analytical results [22]- [36]. In general, it is probably fair to say that few things are known about the analytical basis of such results, and this the issue that we partly intend to address in the present work. A recently pointed out mismatch between numerical predictions [37]- [39] and analytical results [40] shows that we have to be careful when performing numerical extrapolations. On the other hand, [41] shows that both the numerical predictions and the analytical results can be in agreement with each other. In Section 3, we extend this conclusion to interacting systems having different values of q. Such interacting systems should be modelled on CAT(k), k < 0 spaces, as we pointed out in [42]. Section 4, contains some comments and points toward further implications of the Tsallis entropy composition property (6), employing the underlying concept of hyperbolicity.
For brevity, we do not provide the required background in Riemannian geometry or the metric geometry of CAT(k), k < 0 spaces, referring instead to some of the excellent references such as [43], [44] and [21], [45] respectively, on these topics, in addition to the ones provided above.

Riemannian spaces: vanishing largest Lyapunov exponent
One of the features distinguishing the Tsallis (1) from the BGS (2) entropy is their different composition properties. This distinction has profound consequences for the definitions of independence and additivity/extensivity that pervade Statistical Mechanics and Thermodynamics [3] (and references therein). Two systems A and B are conventionally defined to be "independent" if in their statistical description, the corresponding probability distributions p A , p B have as composition law the ordinary multiplication Here and henceforth A + B indicates the compound system formed by combining A and B. For such independent systems (4), the BGS entropy is additive as can be immediately seen from (2). By contrast, under the same definition of independence (4), the Tsallis entropy (1) obeys the generalized additivity/composition property This composition motivated the introduction of a generalized addition [46], [47] by The generalised addition (7) stemming form the Tsallis entropy composition property (6) is arguably the biggest difference between the BGS and the Tsallis entropies. It is worth observing that x ⊕ q y ∼ x + y, |x| ≪ 1, |y| ≪ 1 (8) and that asymptotically x ⊕ q y ∼ xy, |x| → ∞, |y| → ∞ Therefore the difference between (7) and the ordinary addition should becomes significant in the asymptotic regime |x|, |y| → ∞ describing very highly entropic systems, such as black holes, for instance. So, we expect that the differences between the BGS and the Tsallis entropies express the different ways that they describe highly entropic systems. For such systems, the Tsallis entropy induces an essentially "exponential way" of combining interacting systems as is evident from (9). This should be contrasted with the "linear" fashion in which the BGS entropy performs the same task.
It took sometime to find a generalised product that would be distributive with respect to (7), so together they would form a nice algebraic structure expressing the non-trivial composition properties of the Tsallis entropy versus the corresponding properties of the BGS entropy. Actually, two such generalized multiplications were introduced independently in [48], [49]. Although conjecturally independent, an explicit equivalence between them is still lacking. In these works a field isomorphism was, essentially, introduced that was indicated by τ q in [49] where some of its rudimentary metric and measure-theoretical properties on R were examined. In our subsequent work [42], we developed further some of the metric consequences of τ q , 0 ≤ q < 1, initially for R n . Motivated by an analogy with the translation invariance of the Euclidean metric, we constructed in [42] a Riemannian metric g, induced by (6), with components and corresponding line element The BGS composition property ("ordinary addition") is expressed, in this formalism, via the effective Euclidean metric whose corresponding line element is The two metrics (10), (13) were used in [42] to encode and compare, in metric terms, the composition properties of the BGS and the Tsallis entropies.
The metric tensor (10) turned out [42] to have a constant negative sectional curvature which provided a geometric interpretation of the non-extensive parameter q ∈ [0, 1). Endowing the plane R 2 with (7), turned it into a re-scaled version of the hyperbolic plane H 2 . Then, we concluded in [42], that due to (7) which gave rise to (10), the Tsallis entropy can be thought as a "hyperbolic analogue" of the BGS entropy.
The subsequent discussion will assume that the generalised addition (7) is somehow manifest at the level of configuration/phase space of the system. Although such spaces are Riemannian manifolds, therefore locally Euclidean, to understand the difference between the Tsallis and the BGS entropies, we should be actually looking at their large-scale/ large-distance, asymptotic properties [45], [21], [50]. This is suggested by a comparison between (9) and the ordinary addition. In a sense it is the inverse of what Statistical Mechanics attempts to do: here we use the thermodynamic additivity to determine the effective concept of additivity that is applicable in the configuration/phase space of the underlying dynamical system. Naturally the present assumption is not valid if the additivity properties of the thermodynamic quantities are "emergent" in some non-trivial way from the underlying dynamics.
To continue and without actually adding any essential complexity to the subsequent argument, we will consider in the formalism instead of just R 2 , a general Riemannian manifold (M, g) with tangent bundle indicated by T M. This M could represent the configuration or the phase space of a physical system as noticed in the previous paragraph. Let the Levi-Civita connection compatible with g be indicated by ∇. Such a connection is expressed in terms of g by the Koszul formula where X, Y, Z ∈ T M. The geodesic equation is [43] - [45] ∇ X X = 0 (17) for X ∈ T M tangent to the geodesic. If J ∈ T M indicates a Jacobi field then it satisfies the Jacobi/geodesic deviation equation [43], [44] Here R(X, Y )Z indicates the Riemann tensor which is defined by [43], [44] R It is extremely difficult to explicitly solve either the geodesic (17) or the Jacobi equations (18), except in a few particularly simple cases, our case of interest being one of them. Let e 1 , e 2 be orthonormal vectors, with respect to g, spanning a 2-dimensional subspace of T M in a neighborhood of x ∈ M. The sectional curvature of M in this 2-plane subspace of T M is defined by [43], [44] k = g(R(e 1 , e 2 )e 2 , e 1 ) In our case of interest, initially at least, k < 0 is constant. Then the Jacobi equation (18) reduces to the ordinary differential equation which has the general solution where a i , b i are constants, {e i (s)} are parallel orthonormal vectors (Fermi basis) and s is the arc-length parameter of the geodesic whose tangent is X(s). The summation takes place over the n − 1 directions orthogonal to X(s). Substituting (15) into (22), we find that Hence, we see that in a Riemannian manifold of constant negative curvature (15) Let's rephrase the above argument in an alternative, coordinate-dependent, way. Consider a unit vector with respect to (13). This vector will have a smaller magnitude with respect to (10), as can be immediately seen. So (10) does not increase the magnitudes of the unit vectors. Instead, it exponentially decreases their y-component. As a result, it will increase distances of any rectifiable curve in the y/transversal direction to x, by an exponential factor, which is exactly the statement contained in (22).
We turn our attention to Lyapunov exponents. We will only be needing their definition for the case of uniformly hyperbolic dynamical systems [51], [52], a standard example of which is the geodesic flow on a Riemannian manifold of, generally variable, negative sectional curvature. To be slightly more general than that, let f t : M → M be a flow on the Riemannian manifold (M, g), whose generating vector field is Let Y ∈ T M and let its norm with respect to the Riemannian metric g be indicated The Lyapunov exponent of the perturbation in the direction of Y ∈ T x M along the evolution/trajectory of the flow f t is defined by The Lyapunov exponent measures the asymptotic rate of change of the magnitude of a perturbation in the direction of Y . Is is clear that if someone is interested in a stability analysis of a flow, the most pertinent Lyapunov exponent is the largest positive one in some transversal direction. Consider, as a special case, f t to be the geodesic flow on R 2 endowed with (8) that was discussed above. We see from (23) therefore, the definition (25) gives To summarize: we started from a dynamical system modelled on a Riemannian manifold M equipped with a metric induced by (14). We assumed that this dynamical system's effective statistical description is provided by the Tsallis entropy (1). In turn, the Tsallis entropy composition property (6) indicated that it is more suitable to use the effective hyperbolic metric (10) on M manifold, instead of the original one (13). It was the collective description of the underlying dynamics that dictated this "hyperbolization" in producing the effective metric (10) from (13). This "hyperbolization" was concretely implemented as the "warping" by a convex function, the exponential in the present case, in transitioning from (13) to (10). The instabilities of the dynamical system were expressed through its positive Lyapunov exponents, initially with respect to (13), the largest of them being the most important. When the effective behavior described by the hyperbolic metric (10) is taken into account, the largest positive Lyapunov exponent of the underlying dynamical system becomes zero, as can be seen in (26) and (27). The reason is that the perturbations of the underlying dynamical system and the distances with respect to the hyperbolic metric (10) grow at the same rate as seen in (26). Hence their relative growth rate is linear, or could more generally be polynomial/power-law, as seen in (27), so the Lyapunov exponent of the underlying dynamical system relative to (10) is zero. This is the main conclusion of the present work and it is in agreement with all known results reached by analyzing particular models [3] and references therein, as well as [22]- [36] for some, mostly numerical, results in this direction.
In case the largest Lyapunov exponent is zero, as in the case discussed above, the instabilities of the dynamical system grow at a milder than exponential rate with respect to the hyperbolic metric (10), and consequently their asymptotic behavior has to be encoded differently, if we want to obtain non-trivial results. One way to quantify the growth of such perturbations by modifying the definition of Lyapunov exponents (28) tõ These modified exponents describe perturbations obeying an asymptotic, power-law evolution tλ x (Y ) . This is essentially the definition adopted in [3], [7]- [9] dξ dt = λ q ξ q (29) for systems described by the Tsallis entropy, in just different notation. Naturally, the modified Lyapunov exponents (29) will explicitly depend on the value of the entropic parameter q of the dynamical system. Since, as is conjectured, the systems described by the Tsallis entropy may possess more than just one value of non-extensive parameter q [3], [11], depending on which property of the system is described by them, the modified Lyapunov exponents (29) would evidently depend on the value of q that determines the sensitivity of the system to infinitesimal perturbations, indicated as q sen in [3].
The use of a definition like (29) would be of limited interest for applications, if explicit constructions/examples did not exist for which it could provide non-trivial information.
Motivated by the linear (exponential) increase of the geodesic distances in Riemannian manifolds of zero (negative resp.) sectional curvature (23), one may be wondering whether an intermediate behavior such as described by (29) is even possible. The answer turns out to be affirmative, but it is not possible in the context of Riemannian manifolds. An example of a quadratic separation of geodesics in a 2-complex endowed with a CAT(0) metric was constructed * in [53]. This quadratic geodesic deviation in a CAT(0) space is quite different from the case of CAT(k), k < 0, whose geodesics deviate exponentially from each other, as will be explained in the next Section. Whether the property of quadratic * We are grateful to Professor Panos Papasoglou for bringing this work to our attention.
or, more generally, polynomial divergence of geodesics encoded in (29) is "typical", or even common, for CAT(0) spaces, and whether in such a case the definition (29) exhausts all possible geodesic deviation behavior, does not seem to be known at this time. More importantly, it is unclear, to us at least, exactly what, if any, physical system the CAT(0) space construction of [53] can be used to describe. For these reasons, we refer to [53] for the construction itself and its mathematical aspects, aiming to re-visit this topic, if physical reasons warrant it in the future.

Geodesic deviation in CAT(k), k < 0 spaces
We continue with the generalization of the above results to the case of CAT(k), k < 0 spaces. The need for using CAT(k), k < 0 spaces, motivated by the composition of the Tsallis entropy (6), was explained in [12], [42]. The major obstacle in repeating the Riemannian approach verbatim, is that CAT(k) spaces do not possess a differential structure [21], [43]- [45] so one has to dispense with statements relying on regularity properties, such as ones formulated via vector fields, the geodesic (17) and the Jacobi (18) equations etc. The only option left is to employ the triangle inequality, which when combined with the CAT(k), k < 0 condition proves to be sufficient for attaining the sought-after goal [21], [45], [50]. One should notice, that due to lack of smoothness, most statements in the present Section can only be formulated via inequalities, as contrasted to the equalities, such as (22), derived in the Riemannian case. As a result, the arguments in this Section, applied to the case of CAT(k), k < 0 spaces are, inevitably, synthetic as opposed to the analytic ones in the Riemannian case of the previous section. We will, largely, follow [50] in the sequel.
One begins by realizing that a geodesic space (X, d) which is CAT(k), k < 0 is necessarily hyperbolic. There are several definitions of hyperbolicity at various levels of generality [50], [45] and several equivalences among them [50], [45]. The following definition, ascribed to E. Rips, is the most useful for our purposes: a metric space (X, d) is δ-hyperbolic, for δ > 0, if for any three points x, y, z ∈ X, any side of the triangle having x, y, z as vertices lies in a δ-neighborhood of the union of the two others. Given this definition, the hyperbolicity of the CAT(k), k < 0 space (X, d) follows immediately from the definition of the CAT(k) condition.
Consider a polygon in such (X, d) with n ∈ N vertices x 1 , x 2 , . . . x n . Let m = n 2 + 1 (30) where the square brackets indicate the integer part of their argument. Because (X, d) is a geodesic space, every segment has a midpoint. Consider the midpoint x m and the triangle x 1 x m x n . Since (X, d) is δ-hyperbolic, there is a point y in the union of the two other sides x 1 x m ∪ x m x n such that d(y, z) ≤ δ where z ∈ x n x 1 . Without loss of generality, we assume that y ∈ x 1 x m . Using induction on the m-polygon x 1 , x 2 , . . . , x m , we find that the distance of z from the union of all other sides of this m-polygon is where p ∈ N satisfies p ≥ log(n − 1) log 2 We conclude then, that in (X, d) each side of the n-polygon is contained in a pδneighborhood of the union of its other sides.
The second, and last, step is the quantification of the concept of the exponential separation of geodesics in (X, d). Consider two segments emanating from x ∈ X toward y, z ∈ X, respectively. Let two objects move with unit speed, one in each of these two arc-length parametrized segments. We are interested in the separation of x and y after time t. An obvious way to measure such a separation would be to start at the location of y, move back along the segment joining it with x by a distance t, switch segment at x and then continue from x alongside the other segment for an additional distance t until reaching z. This is not however, the analogue of a curve joining y and z of Section 2. What we want is to measure the "direct" separation between y and z, without having to go back close to the intersection x of the two segments. So, we want to determine the length of a path from y to z, in the complement of a ball B r (x) of radius r > 0 centered at x. Consider a path x 1 x 2 . . . x n , n ∈ N, such that d(x i , x i+1 ) ≤ ǫ, i = 1, . . . n − 1 which lies outside the ball B r (x), with x ∈ x n x 1 . Let p ∈ N be as in (30). Using the conclusion of the previous paragraph, there exists a point w ∈ x i x i+1 , i = 1, . . . , n − 1 such that d(x, w) ≤ pδ. Since we find where The conclusion that we reach from (35), is that the number of points with a uniform distance upper bound ǫ between two consecutive ones, making up the path between y and z outside B r (x) increases exponentially with r in the CAT(k), k < 0 space (X, d). This is what we wanted to show. We see that (35) is the analogue of (23) for (X, d). Naturally, the argument leading to (35) is already applicable to the case of a Riemannian manifold (M, g) of negative sectional curvature, since in this case (M, g) is a CAT(k), k < 0 space. So, the conclusions drawn in the Riemannian case, about the effective metric behavior of systems described by the Tsallis entropy, can be extended unaltered to the case of any number of interacting systems described by different values of q.

Conclusions and Outlook
In the present work, we attempted to justify why dynamical systems whose statistical behavior is described by the Tsallis entropy, have vanishing largest Lyapunov exponent. This was essentially ascribed to employing the effective negative curvature metric (10), which is the "hyperbolization" of the Euclidean initially employed metric (13), as was pointed out in [12], [42]. Moreover we made the very strong assumption that the additivity properties of the configuration/phase space of the system are directly reflected on its thermodynamic additivity (7) which is not emergent in any non-trivial manner. Our conclusion is in agreement with all currently known results.
The process of generalizing this conclusion to the case of CAT(k), k < 0 spaces presented in Section 3, is of interest, as it points out to the underlying reason behind such behavior. The argument of Section 3 shows that the key in understanding consequences of the Tsallis entropy composition property (7) is the concept of hyperbolicity, which was also alluded to in [12], [42]. Knowing this, is should not come as a surprise that the argument of Section 3 is a small part of a well-known proof of Morse's Lemma, which establishes the stability of geodesics in hyperbolic geodesic spaces under quasi-isometries [21], [45], [50]. In particular, since Riemannian manifolds of negative sectional curvature (M, g) are a subset of CAT(k), k < 0 spaces, the vanishing of the highest Lyapunov exponent of the dynamical systems modelled by (M, g) just expresses their underlying hyperbolicity in a very compact way. The general framework of this hyperbolicity and its implications for systems described by the Tsallis entropy will be examined further in a future work.