arXiv subjects

Piergiulio Tempesta

Publications and source records attributed to Piergiulio Tempesta.

At least 37 records · Page 2Linked to original sources

arXiv2021

Higher Haantjes Brackets and Integrability

We propose a new, infinite class of brackets generalizing the Frölicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.

math.DG↗

arXiv2021

Haantjes Algebras of Classical Integrable Systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $ω\mathscr{H}$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post-Winternitz system and of a stationary flow of the KdV hierarchy.

nlin.SI↗

arXiv2021

New computable entanglement monotones from formal group theory

We present a mathematical construction of new quantum information measures that generalize the notion of logarithmic negativity. Our approach is based on formal group theory. We shall prove that this family of generalized negativity functions, due their algebraic properties, is suitable for studying entanglement in many-body systems. Under mild hypotheses, the new measures are computable entanglement monotones. Also, they are composable: their evaluation over tensor products can be entirely computed in terms of the evaluations over each factor, by means of a specific group law. In principle, they might be useful to study separability and (in a future perspective) criticality of mixed states, complementing the role of Rényi's entanglement entropy in the discrimination of conformal sectors for pure states.

math-ph↗

arXiv2020

Multivariate Group Entropies, Super-exponentially Growing Complex Systems and Functional Equations

We define the class of multivariate group entropies as a novel set of information - theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to formulate this correspondence is offered by the theory of formal groups and rings.

math-ph↗

arXiv2020

Haantjes Algebras and Diagonalization

We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form.

math-ph↗

arXiv2020

A generalized permutation entropy for random processes

Permutation entropy measures the complexity of deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or just permutations. The reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations, and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as is usually the case when time series are output by random processes. In this Letter we propose a generalized permutation entropy that is finite for random processes, including discrete-time dynamical systems with observational or dynamical noise.

physics.data-an↗

arXiv2019

Universality Classes and Information-Theoretic Measures of Complexity via Group Entropies

We introduce a class of information measures based on group entropies, allowing us to describe the information-theoretical properties of complex systems. These entropic measures are nonadditive, and are mathematically deduced from a series of natural requirements. In particular, we introduce an extensivity postulate as a natural requirement for an information measure to be meaningful. The information measures proposed are suitably defined for describing universality classes of complex systems, each characterized by a specific phase space growth rate function.

cond-mat.stat-mech↗

arXiv2019

Formal Rings

A notion of one-dimensional formal ring is presented. It consists of a triple $(A,Φ,Ψ)$ where $A$ is a unital ring and $Φ$ and $Ψ$ are two formal power series in $2$ variables ${Φ(x,y),Ψ(x,y)\in A\llbracket x,y\rrbracket}$, the first one defining a one-dimensional formal group law over $A$ and the second one providing a second composition law satisfying axiomatic properties of compatibility with the first one. For a characteristic-zero ring $A$, a large class of one-dimensional formal rings can be obtained by constructing a new composition law, defined in terms of the group logarithm associated with a given formal group law $Φ(x,y)$, and associative and distributive with respect to it. A natural $n$-dimensional generalization of the previous construction is also proposed; curves on $n$-dimensional formal rings are introduced. The higher-dimensional theory allows us to define a generalization of the ring $W(A)$ of Witt vectors over a ring $A$, which is recovered by means of a specific choice of the associated group logarithm. The composition laws of our generalized Witt ring are defined in terms of an underlying formal ring structure. Examples of formal rings related to Hirzebruch's theory of genera are explicitly computed. Finally, we also propose the examples of Euler's and Abel's formal rings.

math.AT↗

arXiv2018

Group entropies: from phase space geometry to entropy functionals via group theory

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume $W$ on $N$. We review the ensuing entropies, discuss the composability axiom, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective.

cond-mat.stat-mech↗

arXiv2018

Statistical mechanics of exploding phase spaces: Ontic open systems

The volume of phase space may grow super-exponentially ("explosively") with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create new emergent states. Standard ensemble theory can break down as we demonstrate in a simple model reminiscent of complex systems where new collective states emerge. We present an axiomatically defined entropy and argue that it is extensive in the micro-canonical, equal probability, and canonical (max-entropy) ensemble for super-exponentially growing phase spaces. This entropy may be useful in determining probability measures in analogy with how statistical mechanics establishes statistical ensembles by maximising entropy.

cond-mat.stat-mech↗

arXiv2018

A new class of entropic information measures, formal group theory and information geometry

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.

math-ph↗

arXiv2017

A duality principle for the multi-block entanglement entropy of free fermion systems

The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set $A$ of sites of the subsystem considered and the set $K$ of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets $A$ and $K$ to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets $A$ and $K$ consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and $r$-partite information generalizing the known ones for the single block case.

quant-ph↗

arXiv2017

A foundational approach to the Lie theory for fractional order partial differential equations

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

math.AP↗

arXiv2017

Uniqueness and characterization theorems for generalized entropies

The requirement that an entropy function be composable is key: it means that the entropy of a compound system can be calculated in terms of the entropy of its independent components. We prove that, under mild regularity assumptions, the only composable generalized entropy in trace form is the Tsallis one-parameter family (which contains Boltzmann-Gibbs as a particular case). This result leads to the use of generalized entropies that are not of trace form, such as Rényi's entropy, in the study of complex systems. In this direction, we also present a characterization theorem for a large class of composable non-trace-form entropy functions with features akin to those of Rényi's entropy.

math-ph↗

arXiv2017

Formal Groups and $Z$-Entropies

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the $Z$-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every $Z$-entropy is composable [1]. This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required, to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the $Z$-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

math-ph↗

arXiv2016

Generalized Sobolev orthogonal polynomials, matrix moment problems and integrable systems

We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear forms is also proposed. We consider both polynomial deformations and a class of transformations related to the action of linear operators on the entries of a given bilinear form. Transformation formulae among new and old polynomial sequences are determined. Finally, integrable hierarchies of evolution equations arising from the factorization of a time deformation of the moment matrix are presented.

math.CA↗

arXiv2016

Groups, Information Theory and Einstein's Likelihood Principle

We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.

cond-mat.stat-mech↗

arXiv2016

Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions

We introduce a general class of su$(1|1)$ supersymmetric spin chains with long-range interactions which includes as particular cases the su$(1|1)$ Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su$(1|1)$ permutation operator, and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low energy excitations and the low temperature behavior of the free energy, which coincides with that of a $(1+1)$-dimensional conformal field theory (CFT) with central charge $c=1$ when the chemical potential lies in the critical interval $(0,\mathcal E(π))$, $\mathcal E(p)$ being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson $(1+1)$-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge $c=1$. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su$(1|1)$ elliptic chain.

quant-ph↗