Information Shift Dynamics Described by Tsallis q 3 Entropy on a Compact Phase Space Jin Yan1and Christian Beck23

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Information Shift Dynamics Described by Tsallis q= 3 Entropy

on a Compact Phase Space

Jin Yan1and Christian Beck2,3

1Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

jinyan@pks.mpg.de

2School of Mathematical Sciences, Queen Mary University of London, UK

3The Alan Turing Institute, London, UK

c.beck@qmul.ac.uk

Abstract

Recent mathematical investigations have shown that under very general conditions ex-

ponential mixing implies the Bernoulli property. As a concrete example of a statistical

mechanics which is exponentially mixing we consider a Bernoulli shift dynamics by Cheby-

shev maps of arbitrary order N≥2, which maximizes Tsallis q= 3 entropy rather than

the ordinary q= 1 Boltzmann-Gibbs entropy. Such an information shift dynamics may

be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry

properties of the coupled Chebyshev systems, which are diﬀerent for even and odd N. We

show that the value of the ﬁne structure constant αel = 1/137 is distinguished as a coupling

constant in this context, leading to uncorrelated behaviour in the spatial direction of the

corresponding coupled map lattice for N= 3.

Keywords: information shift, Tsallis entropy, Chebyshev maps, ﬁne structure constant

I Introduction

The foundations of statistical mechanics are a subject of continuing theoretical interest. It is far from

obvious that a given deterministic dynamics can ultimately be described by a statistical mechanics

formalism. The introduction of generalized entropies (such as, for example, the non-additive Tsallis

entropies [1, 2, 3]) leads to a further extension of the formalism relevant for systems with long-range

interactions, or with a fractal or compactiﬁed phase space structure, or with non-equilibrium steady states

exhibiting ﬂuctuations of temperature or of eﬀective diﬀusion constants [4, 5]. Generalized entropies have

been shown to have applications for a variety of complex systems, for example, in high energy physics

[6, 7, 8, 9, 10] or for turbulent systems [11, 12, 13]. In this paper we go back to the basics and explore the

arXiv:2210.14695v2 [cond-mat.stat-mech] 15 Nov 2022

properties of a particular statistical mechanics, namely that of an information shift dynamics described

by Tsallis entropies with the entropic index q= 3 on a compact phase space.

Our work is inspired by recent mathematical work [14] that shows that the exponential mixing

property automatically implies the Bernoulli property, under very general circumstances. This theorem

is of utmost interest for the foundations of statistical mechanics. Namely, by deﬁnition the systems we

are interested in when dealing with statistical mechanics relax to an equilibrium quite quickly (under

normal circumstances). That is to say, we have the exponential mixing property. But then this implies

that somewhere (on a suitable subset of the phase space) there must exist a Bernoulli shift dynamics in

suitable coordinates.

We will work out one of the simplest example systems that is consistent with a generalized statistical

mechanics formalism, and at the same time is exponentially mixing and ultimately conjugated to a

Bernoulli shift. These are discrete-time dynamical systems on the interval [−1,1] as generated by N-th

order Chebyshev maps TN[15, 16, 17, 18]. Chebyshev maps are exponentially mixing and conjugated to

a Bernoulli shift of Nsymbols. We will review and investigate their properties in detail in the following

sections. Needless to say that Chebyshev maps do not live in ordinary physical space but just on a

compactiﬁed space, the interval [−1,1]. The simplest examples are given by the N= 2 and N= 3 cases,

i.e., T2(x) = 2x2−1 and T3(x) = 4x3−3x. Despite their simplicity, a statistical mechanics formalism can

be constructed as an eﬀective description (see also [19, 20]). In contrast to ordinary statistical mechanics

(described by states where the q= 1 Boltzmann-Gibbs-Shannon entropy has a maximum subject to

constraints), in our case the relevant entropy is the q= 3 Tsallis entropy Sq. This leads to interesting

properties. Our physical interpretation is that the above information shift dynamics may be relevant

in a pre-universe, i.e. in an extremely early stage of the universe where ordinary space-time has not

yet formed. The dynamics evolves in a ﬁctitious time coordinate (diﬀerent from physical time) which is

relevant for stochastic quantization [21, 22] (this idea has been worked out in more detail in [19, 23, 24]).

While we will work out the properties of Chebyshev maps in much detail in the following sections,

what we mention right now as a prerequisite is that the invariant density p(x) of Chebyshev maps TN

of arbitrary order N≥2 is given by

p(x) = 1

π√1−x2, x ∈[−1,1].(1)

This density describes the probability density of iterates under long-term iteration. The maps TNare

ergodic and mixing. In suitable coordinates iteration of the map TNcorresponds to a Bernoulli shift

dynamics of Nsymbols. In the following we give a generalized statistical mechanics interpretation for

the above invariant density, identifying Chebyshev maps as one of the simplest systems possible for

which a generalized statistical mechanics can be deﬁned. The interesting aspect of this low-dimensional

simplicity is that q= 3 is relevant, rather than q= 1 as for ordinary statistical mechanics.

This paper is organized as follows. In section II we derive the generalized canonical distributions

obtained from the q-entropies Sq, and discuss the special distinguished features obtained for q= 3 (or

q=−1 if so-called escort distributions [20, 25] are used). In section III we discuss the exponential

mixing property, which is ﬁxed by the 2nd largest eigenvalue of the Perron-Frobenius operator. In fact,

we will present formulas for all eigenvalues and eigenfunctions, thus completely describing the exponential

mixing behaviour. In section IV we couple two maps, thus gradually expanding the phase space, and

investigate how the coupling structure induces certain symmetries in the attractor which are diﬀerent for

odd Nand even N. The degeneracy of the canonical distribution (of the invariant density) is removed by

the coupling, and all attractors become N-dependent. Finally, in section V we consider a large number

of coupled maps on a one-dimensional lattice space. This is the realm of the so-called ’chaotic strings’

which have previously been shown to have relevant applications in quantum ﬁeld theory [17, 19, 23]. We

conﬁrm, by numerical simulations, that the N= 3 string distinguishes a value of the coupling constant

given numerically by 1/137, which numerically coincides with the low-energy limit of the ﬁne structure

constant ﬁxing the strength of electric interaction. A physical interpretation for that will be given, in

the sense that we assume that the chaotic shift dynamics, described by q= 3 Tsallis entropies in a

statistical mechanics setting, is a fundamental information shift dynamics that helps to ﬁx standard

model parameters in a pre-universe, before ordinary space-time is created.

II Generalized canonical distributions from maximizing q-entropy

subject to constraints

The relevant probability density (1) mentioned above can be regarded as a generalized canonical distri-

bution in non-extensive statistical mechanics [1]. As it is well known, one deﬁnes for a dimensionless

continuous random variable Xwith probability density p(x) the Tsallis entropies as

Sq=1

q−1(1 −Zp(x)qdx).(2)

Here q∈(−∞,∞) is the entropic index. The Tsallis entropies contain the Boltzmann-Gibbs-Shannon

entropy S1=−Rp(x) log p(x)dx as a special case for q→1, as can be easily checked by writing q= 1+

and taking the limit →0 in the above equation.

We now do statistical mechanics for general q. Typically one has some knowledge on the system.

This could be, for example, a knowledge of the mean energy Uof the system. Extremizing Sqsubject

to the constraint

Zp(x)E(x)dx =U(3)

one ends up with q-generalized canonical distributions (see, e.g. [2, 12] for a review). These are given by

p(x)∼(1 + (q−1)βE(x))−1

q−1,(4)

where Eis the energy associated with microstate x, and β= 1/kT is the inverse temperature. Of course,

for q→1 one obtains the usual Boltzmann factor e−βE .

Alternatively, one can work with the so-called escort distributions, deﬁned for a given parameter q

and a given distribution p(x) as [20]

P(x) = p(x)q

Rp(x)qdx.

The escort distribution sometimes helps to avoid diverging integrals, thus ‘renormalizing’ the theory

under consideration, see [26] for more details. If the energy constraint (3) is implemented using the

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InformationShiftDynamicsDescribedbyTsallisq=3EntropyonaCompactPhaseSpaceJinYan1andChristianBeck2;31MaxPlanckInstituteforthePhysicsofComplexSystems,Dresden,Germanyjinyan@pks.mpg.de2SchoolofMathematicalSciences,QueenMaryUniversityofLondon,UK3TheAlanTuringInstitute,London,UKc.beck@qmul.ac.ukAbstractRec...

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Information Shift Dynamics Described by Tsallis q 3 Entropy on a Compact Phase Space Jin Yan1and Christian Beck23

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