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One of the few accepted dynamical foundations of nonadditive (“nonextensive”) statistical mechanics is that the choice of the appropriate entropy functional describing a system with many degrees of freedom should reflect the rate of growth of its configuration or phase space volume. We present an example of a group, as a metric space, that may be used as the phase space of a system whose ergodic behavior is statistically described by the recently proposed

The idea of entropies that have a functional form which is different from that of Boltzmann/Gibbs/Shannon (BGS),

One of the basic problems that all the non-BGS functionals face is their dynamical foundations. Many conjectures exist about the dynamical foundations of such functionals and their scope of applicability. However, little is actually known or reasonably well-established, especially if one is interested in analytical/nonnumerical approaches and results [

Let

Not too long ago, a proposal was made [

We have attempted to make the present work somewhat self-contained and readable to our intended audience of physicists. In Section

One would, ideally, like a dynamically derived answer to the above question. Namely, pick a conventional phase space (which is an even-dimensional symplectic manifold), or a set of states in the Hilbert or Fock space of a quantum system, and determine the evolution, or probability distribution, that would give rise to (

To proceed we modify the question and ask the following: assume that we are given a dynamical system which is ergodic. What properties should its phase space have to allow for a growth rate given by (

If we use a classical rather than a quantum system, for simplicity, the author knows of no results in the category of even-dimensional symplectic manifolds of high dimension that would allow for the subexponential increase in the phase space volume. We recall [

Alternatively, one can take Riemannian/metric viewpoint (see, e.g., [

The advantage of the metric approach is that Riemannian manifolds have local structure: the sectional curvature, or equivalently the Riemann tensor, quantifies locally the deviation of the manifold from flatness. To proceed, one may directly solve the differential equation (geodesic equation, in the simplest cases) describing the evolution of the system if the underlying manifold is simple enough. Alternatively, one can use comparisons with appropriately chosen simply connected manifolds of constant sectional curvature (space forms) to attain bounds on the growth rate of the volume of the phase space. The former method is much more desirable but practically impossible to explicitly implement in almost any case of physical interest. Hence someone may use extremely simplified models of phase spaces, such as symmetric or homogeneous spaces, whose additional group theoretical structure makes them amenable to further analysis [

We have come to realize by using comparison methods [

If someone insists on having a metric and a symplectic structure which are compatible with each other on the phase space and also integrable, then one is naturally led to demanding the phase space to be a Kähler manifold [

As a result, one may have to broaden the class of spaces that should be considered as phase spaces of ergodic systems described by (

A class of objects that have been extensively studied in Mathematics over the last two centuries and have also been extensively used in Physics are groups. One would be hard pressed to find a field of Physics in which groups and group theoretic arguments play a minor role. In this work we will be interested in discrete rather than continuous (topological, Lie, etc.) groups. It turns out that such (usually infinite) groups, which have a finite number of generators, have strong geometric properties that have been explored for more than an century. In particular, one can use geometric constructions and the concomitant arguments based on these groups or acting on metric and measure spaces to infer or elucidate some of their algebraic aspects. This is the goal of combinatorial [

During the last three decades there has been a revival of the activity on the geometric properties of such groups or their geometric actions on metric measure spaces, due in no small part to the influence of [

Consider a finitely generated group

Let

Having defined a distance function

By the definitions of the above paragraph one can classify the growth rate of groups into polynomial, intermediate, and exponential. We can immediately see that the growth of a group

The power-law functions

All exponential functions

All functions of intermediate type

One can look at the above equivalences from the viewpoint of entropies regarding the proposal of [

It is also worth seeing that something similar is true for groups of intermediate growth: the mapping from an exponential to an intermediate volume growth rate which is induced by an entropic functional such as (

We proceed by stating the construction of the first Grigorchuk group [

To go straight to the heart of the matter, the first Grigorchuk group is the finitely generated infinite group

We try to provide a more “physical interpretation” of some of the constructions above. First of all, the fact that we use an infinite tree can be interpreted as indicating that the system under study has infinite memory. A binary option can be interpreted as a fermionic state which is either empty (0) or full

The conventional creation and annihilation operators are composite in this description. To fill an empty single fermionic slot, someone would have to move back one step toward the root of the tree and then forward one more step along the complementary branch of the subtree. The same applies to annihilating a single fermionic state in such a multiparticle analogy. This is, in essence, what the automorphism

This lack of a familiar interpretation of the generators

Physics extensively uses “infinite” groups: the gauge groups in Particle Physics; the diffeomorphism group, the Bondi-Metzner-Sachs group, and so on in general relativity; the Kac-Moody and loop groups in string theories; and so on are all infinite dimensional. However,

The fact that

It may be of physical interest to see whether any of the more familiar phase spaces (finite dimensional Riemannian manifolds) can possibly have intermediate volume growth. A theorem of A. Avez, responding to a conjecture of E. Hopf, seems to exclude such a possibility, for the compact case at least: a compact, connected Riemannian manifold

In this work we argued that an entropic functional (

The present work can be placed in the wider context of examining the dynamical foundation of statistical mechanics, particularly in determining the basis and range of applicability of the nonadditive entropic forms that have been developed over the last three decades in the Physics community. A common weakness of these approaches is the complete lack of analytically tractable models of systems with many degrees of freedom that can be studied as examples and for obtaining physically relevant results. The present work also suffers from such a lack of concrete examples. On the other hand, it expands the range of systems and techniques at which one may usually look for properties of systems that may be described by such nonadditive entropies. The connection between groups of intermediate growth, self-similar groups, and fractals may be potentially quite promising in addressing some of the important issues that have arisen since the introduction of (

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author is greatly indebted and grateful to the Director of CRANS, Professor Anastasios Bountis, for his constant inspiration, encouragement, and support as well as for many fruitful conversations without which this work would have never been possible.