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Introduction: A Foundational PerspectiveFormal Group Theory

Strong and weak composabilityGroups, Difference Operators and Entropies

LECTURES ONGROUPS, ENTROPIES AND NUMBER

THEORY

Piergiulio Tempesta

Universidad Complutense de Madridand

Instituto de Ciencias Matematicas (ICMAT),Madrid, Spain.

TEMPLETON SCHOOL ON FOUNDATIONS OF COMPLEXITY

Rio de Janeiro, October 14 - 15, 2015

Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY

LECTURE I

On the Mathematical Foundations of theTheory of Generalized Entropies

OutlineIntroduction: A Foundational Perspective

Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom

Formal Group TheoryDefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory

Strong and weak composabilityStrong and weak composabilityGeneral properties of the trace-form class

Groups, Difference Operators and EntropiesDelta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operatorsPiergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY

SK and

Composability

axioms Generalized Entropies

Group-theoretical

Structure

Delta Operators Formal group

theory

A Foundational Perspective

Statement of the problem

I What are the mathematical foundations of the notion ofentropy?

I The main idea is to propose a group-theoretical approach to thetheory of generalized entropies

Different foundational approaches

This problem has been addressed from many different perspectives:

I Large deviation theory

I Maximum entropy principle

I Superstatistics

I Information Theory:

Divergences (Kullback-Leibler, Renyi, Bregman, Csiczar, α-,β-, etc.)

Exponential families

Information Geometry (Riemannian manifolds, conformal geometry),

I Superstatistics

Characterization theorems

Let W ∈ N/0. Let piWi=1, with pi ≥ 0 for all i = 1, . . . ,W ,∑Wi=1 pi = 1 denote a discrete probability distribution, where PW

represents the set of all discrete probability distributions with W entries.The Boltzmann-Gibbs entropy

SBG : PW −→ R+ ∪ 0

SBG [p] :=W∑i=1

pi ln1

p1, piWi=1 ∈ PW

has been characterized by several existence and uniqueness theorems,proved in the last century.

Khinchin’s theorem

(A. I. Khinchin, Mathematical foundations of information theory, 1957).Given two systems A and B, let pij(A ∪ B) denote its joint probabilitydistribution, i = 1, . . . ,WA, j = 1, . . . ,WB . Also,

pi (A) :=

WB∑j=1

pij(A ∪ B).

Let S(B | A) denote the conditional entropy associated with theconditional probability distribution pij(B | A) = pij(A ∪ B)/pi (A).Theorem 1.1.i) Let S be a continuous function of all its arguments (p1, . . . , pW ).i) Assume that it takes its maximum for the equiprobability distribution,i.e. when pi = 1/W , for i = 1, . . . ,W .iii) Also, assume that S(p1, . . . , pw , 0) = S(p1, . . . , pw ).iv) Given two systems A and B, S(A ∪ B) = S(A) + S(B | A).Then the unique entropy satisfying (i)-(iv) is Boltzmann’s entropy.

Abe’s theorem

S. Abe, Phys. Lett. A, 271, 74-79 (2000).Theorem 1.2.i) Let S be a continuous function of all its arguments (p1, . . . , pW ).i) Assume that it takes its maximum for the equiprobability distribution,pi = 1/W , for i = 1, . . . ,W .iii) Also, assume that S(p1, . . . , pw , 0) = S(p1, . . . , pw ).iv) Given two systems A and B,S(A ∪ B) = S(A) + S(B | A) + (1− q)S(A)S(B | A).Then the unique entropy satisfying the axioms (i)-(iv) is Tsallis’entropy:

Sq[p] :=

∑Wi=1 pq

i − 1

1− q, q > 0.

dos Santos’ theorem

R. J. V. dos Santos, J. M. Phys. 38, 4104 (1997)This theorem generalizes the original Shannon’s uniqueness theorem forthe Boltzmann entropy.

Theorem 1.3i) Let S be a continuous function of all its arguments (p1, . . . , pW ).ii) For the equiprobability distribution, pi = 1/W , for i = 1, . . . ,W , S isa monotonic increasing function of W , i.e.

S =W 1−q − 1

1− q.

iii) Given two statistically independent systems A and B,S(A ∪ B) = S(A) + S(B) + (1− q)S(A)S(B).

iv) If W = WL + WM , with

WL∑i=1

pi , (WL terms)

PM =W∑

i=WL+1

pi , (WM terms)

(hence PL + PM = 1)we have:

S [p] = S(pL, pM) + (pL)qS

)+ (pM)qS

Then, the unique entropy satisfying the axioms (i)-(iv) is Tsallis’entropy Sq.

The modern Shannon-Khinchin formulation

I (SK1) (Continuity). The function S(p1, . . . , pW ) is continuous withrespect to all its arguments

I (SK2) (Maximum principle). The function S(p1, . . . , pW ) takes itsmaximum value over the uniform distribution pi = 1/W ,i = 1, . . . ,W .

I (SK3) (Expansibility). Adding an impossible event to a probabilitydistribution does not change its entropy:S(p1, . . . , pW , 0) = S(p1, . . . , pW ).

I (SK4) (Additivity). Given two subsystems A, B of a statisticalsystem,

S(A ∪ B) = S(A) + S(B | A).

In particular, if they are statistically independent,

S(A ∪ B) = S(A) + S(B).

S(A ∪ B) = S(A) + S(B | A).

S(A ∪ B) = S(A) + S(B).

S(A ∪ B) = S(A) + S(B | A).

S(A ∪ B) = S(A) + S(B).

S(A ∪ B) = S(A) + S(B | A).

S(A ∪ B) = S(A) + S(B).

A possible unifying principle: The group-theoreticalapproach

We wish to find, possibly, a unifying principle, that would be responsibleof the many properties of a generalized entropy:

Thermodynamics: Legendre structure, Extensivity, SK axioms, etc.

Classical and Quantum Information Theory: Generalized divergences,etc.This principle should allow to classify all the known entropies and allowto design new ones.

Beyond the SK formulation: the Composability Axiom

P. T., Beyond the SK formulation: the Composability axiom and theUniversal-group entropy, arXiv: 1407.3807 Annals of Physics (in press)

I The notion of entropy should be properly defined just in terms ofmacroscopic configurations of a system (without the need for amicroscopic description of the associated dynamics).

I Foundational property: an entropy should be a coarse-grainedquantity (Gell-Mann); if not so, the concept of entropy would besimply empty.

I Even the second law of thermodynamics loses any meaning if notreferred to the evolution of macroscopic subsystems: the entropywould stay invariant if defined on microscopic configurations.

I Therefore, when composing two different systems, defined overany probability distribution piWi=1, we should have:

S(A ∪ B) = Φ(S(A),S(B); η) (1)

where η is a possible set of parameters.

I In additionS(A ∪ B) = S(B ∪ A).

I Another crucial property:

S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )

I Finally, when we compose a system A with another system B, suchthat S(B) = 0, we should have that

S(A ∪ B) = S(A).

S(A ∪ B) = Φ(S(A),S(B); η) (1)

S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )

S(A ∪ B) = S(A).

S(A ∪ B) = Φ(S(A),S(B); η) (1)

S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )

S(A ∪ B) = S(A).

S(A ∪ B) = Φ(S(A),S(B); η) (1)

S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )

S(A ∪ B) = S(A).

The Shannon-Khinchin axioms (SK1)-(SK3) are basic properties, nonnegotiable.

However, they are not sufficient for thermodynamical purposes.

We need a further requirement, that replaces the axiom (SK4) in fullgenerality: the Composability Axiom.

The Composability Axiom

Definition 1.4. An entropy S is (strictly) composable if there exists asmooth function of two real variables Φ(x , y) such that (C1)

S(A ∪ B) = Φ(S(A),S(B); η) (2)

where A ⊂ X and B ⊂ X are two statistically independent systems,defined over any probability distribution piWi=1, with the furtherproperties(C2) Symmetry:

Φ(x , y) = Φ(y , x). (3)

(C3) Associativity:

Φ(x ,Φ(y , z)) = Φ(Φ(x , y), z). (4)

(C4) Null-composability:

Φ(x , 0) = x . (5)

Composition of two arbitrary systems

Remark. Condition (2) can be formulated in more general terms.Consider the case of a composite system A ∪ B arising from two systemsnot statistically independent, with a conditional probability distributionpij(B | A) := pij(A ∪ B)/pi (A).Here pij(A ∪ B), i = 1, . . . ,WA, j = 1, . . . ,WB denotes the jointprobability distribution for the composite system A ∪ B, and pi (A) is

the marginal probability distribution pi (A) =∑WB

j=1 pij(A ∪ B).In this general context we postulate the relation

S(A ∪ B) = Φ(S(A),S(B | A); η), (6)

where S(B | A) denotes the conditional entropy associated with theconditional distribution pij(B | A). Equation (6) reduces to the relation(2) in the case of statistically independent subsystems. The relation (6)generalizes the original axiom (SK4) for the case of systems notindependent.

How to realize the composability axiom?

According to the Composability Axiom, we are interested to the generalclass of functions

Φ : R2 −→ R

such that the properties (C 1)-(C 4) apply. Surprisingly enough, there is afull mathematical theory that deals with such functions: the FormalGroup Theory. It was developed in the context of algebraic topologystarting from the seminal work of Bochner in 1946, and is still an activeresearch area.

DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory

Formal Group Theory: a crash course

The theory of formal groups offers a natural language for formulatingthe theory of generalized entropies.S. Bochner, Formal Lie groups, Annals of Mathematics 47, 192–201(1946).M. Hazewinkel, Formal Groups and Applications, Academic Press, NewYork, 1978.Many important mathematicians have been working in this area: S. P.Novikov, D. Quillen, J. P. Serre, G. Faltings, etc.Nowadays, it has a prominent role in fields as

I Algebraic topology,

I The theory of elliptic curves

I Arithmetic number theory.

Definitions

I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.

I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that

i) Φ (x , 0) = Φ (0, x) = x