arXiv subjects

Nikolaos Kalogeropoulos

Publications and source records attributed to Nikolaos Kalogeropoulos.

At least 19 recordsLinked to original sources

arXiv2024

On the origin of escort distributions for q-entropies

We present an argument about the origin of escort distributions used in conjunction with the q-entropy in non-additive thermo-statistics. The origin of the escort distributions is ascribed to the fact that the effective statistical description of the underlying system is provided by the measured Gromov-Hausdorff limit of a sequence of manifolds having a warped product metric structure. We interpret the entropic parameter \ q \ appearing in escort distributions as a "dimension" related to the fibration structure of the underlying phase spaces.

cond-mat.stat-mech↗

arXiv2024

Composition of q-entropies and hyperbolic orthogonality

We point out that the q-entropy composition for independent events has exactly the same form as the Pythagorean theorem in hyperbolic geometry. We justify the formal relation of hyperbolic geometry with the q-entropy through the use of the $κ$-entropy, which is directly related to the hyperboloid model of hyperbolic space. We comment on the relation between orthogonality in this form of the Pythagorean theorem and the independence of the probability distributions appearing in the q-entropy composition through the use of the Dvoretzky-Rogers lemma.

cond-mat.stat-mech↗

arXiv2024

Logarithmic and power-law entropies from convexity

In an attempt to understand the origin and robustness of the Boltzmann/Gibbs/Shannon entropic functional, we adopt a geometric approach and discuss the implications of the Johnson-Lindenstrauss lemma and of Dvoretzky's theorem on convex bodies for the choice of this functional form. We contrast these results with a more recent result on flowers of balls, which may be interpreted as suggesting the use of power-law entropies for some systems.

cond-mat.stat-mech↗

arXiv2022

Quantization in Cartesian coordinates and the Hofer metric

P.A.M. Dirac had stated that the Cartesian coordinates are uniquely suited for expressing the canonical commutation relations in a simple form. By contrast, expressing these commutation relations in any other coordinate system is more complicated and less obvious. The question that we address in this work, is the reason why this is true. We claim that this unique role of the Cartesian coordinates is a result of the existence and uniqueness of the Hofer metric on the space of canonical transformations of the phase space of the system getting quantized.

math-ph↗

arXiv2021

Riemannian submersions for q-entropies

In an attempt to find the dynamical foundations for $q$-entropies, we examine the special case of Lagrangian/Hamiltonian systems of many degrees of freedom whose statistical behavior is conjecturally described by the $q$-entropic functionals. We follow the spirit of the canonical ensemble approach. We consider the system under study as embedded in a far larger total system. We explore some of the consequences that such an embedding has, if it is modelled by a Riemannian submersion. We point out the significance in such a description of the finite-dimensional Bakry-Émery Ricci tensor, as a local mesoscopic invariant, for understanding the collective dynamical behavior of systems described by the $q$-entropies.

cond-mat.stat-mech↗

arXiv2021

Coarse-graining and symplectic non-squeezing

We address aspects of coarse-graining in classical Statistical Physics from the viewpoint of the symplectic non-squeezing theorem. We make some comments regarding the implications of the symplectic non-squeezing theorem for the BBGKY hierarchy. We also see the cubic cells appearing in coarse-graining as a direct consequence of the uniqueness of Hofer's metric on the group of Hamiltonian diffeomorphisms of the phase space.

cond-mat.stat-mech↗

arXiv2020

Systolic aspects of black hole entropy

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3+1)-dimensional spacetimes. We ascribe this entropy to the non-trivial topology of the space-like sections $Σ$ of the horizon. This is not forbidden by topological censorship, since all the known energy inequalities needed to prove the spherical topology of $Σ$ are violated in quantum theory. We choose the systoles of $Σ$ to encode its complexity, which gives rise to the black hole entropy. We present hand-waving reasons why the entropy of the black hole can be considered as a function of the volume entropy of $Σ$. We focus on the limiting case of $Σ$ having a large genus.

gr-qc↗

arXiv2019

Non-linear Fokker-Planck equations from conformal metrics and scalar curvature

We present an argument which intends to explore a potential geometric origin of a class of non-linear Fokker-Planck equations related to the mesoscopic behavior of systems conjecturally described by the $q$-entropy. We argue that the appearance of the non-linear term(s) in such equations can be ascribed to the fact that the effective mesoscopic metric describing the behavior of the underlying system may not be the originally chosen one, but a conformal deformation of it. Motivated by Liouville's theorem, we highlight the role played by the scalar curvature of conformally related metrics in establishing such a non-linear Fokker-Planck equation.

cond-mat.stat-mech↗

arXiv2019

Toward a relative q-entropy

We address the question and related controversy of the formulation of the $q$-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an $L_p$ normalized functional proposed by Lutwak-Yang-Zhang (LYZ), which is essentially a variation of a properly normalized relative Rényi entropy up to a logarithm, has extremal properties that make it an attractive candidate which can be used to construct such a relative $q$-entropy. We comment on the extremizing probability distributions of this LYZ functional, its relation to the escort distributions, a generalized Fisher information and the corresponding Cramér-Rao inequality. We point out potential physical implications of the LYZ entropic functional and of its extremal distributions.

cond-mat.stat-mech↗

arXiv2018

Irreversibility from staircases in symplectic embeddings

We present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of Hamitonian systems of many degrees of freedom. We use recent flexibility and rigidity results of symplectic embeddings, quantified via the (stabilized) Fibonacci and Pell staircases, to encode the underlying breadth of the possible initial conditions, which alongside the multitude of degrees of freeedom of the underlying system give rise to time-irreversibility.

cond-mat.stat-mech↗

arXiv2018

Power-law entropies for continuous systems and generalized operations

We present our view in a standing debate about the definition and meaning of power-law entropies for continuous systems. Our suggestion is that such arguments should take into account the generalized operations of addition and multiplication induced by the power-law entropies' composition properties. To be concrete, we highlight our view using the case of the $q$- also known as "Tsallis", entropic functionals.

cond-mat.stat-mech↗

arXiv2018

Embolic aspects of black hole entropy

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in (3+1)-dimensional spacetimes. We treat the case of horizons having space-like sections $Σ$ which are topological spheres, following Hawking's and the Topological Censorship theorems. We use the injectivity radius of the induced metric on $Σ$ to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of $Σ$ as the fundamental quantity expressing the complexity of $Σ$ on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.

gr-qc↗

arXiv2018

The $τ_q$-Fourier transform: covariance and uniqueness

We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "$τ_q$-Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the $τ_q$-Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the $τ_q$-Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.

physics.gen-ph↗

arXiv2017

Time irreversibility from symplectic non-squeezing

The issue of how time reversible microscopic dynamics gives rise to macroscopic irreversible processes has been a recurrent issue in Physics since the time of Boltzmann whose ideas shaped, and essentially resolved, such an apparent contradiction. Following Boltzmann's spirit and ideas, but employing Gibbs's approach, we advance the view that macroscopic irreversibility of Hamiltonian systems of many degrees of freedom can be also seen as a result of the symplectic non-squeezing theorem.

cond-mat.stat-mech↗

arXiv2017

An entropy for groups of intermediate growth

One of the few accepted dynamical foundations of non-additive "non-extensive") 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 $δ$-entropy. This entropy is a one-parameter variation of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from the power-law entropies that have been recently studied. We use the first Grigorchuk group for our purposes. We comment on the connections of the above construction with the conjectured evolution of the underlying system in phase space.

cond-mat.stat-mech↗

arXiv2017

The Legendre Transform in Non-additive Thermodynamics and Complexity

We present an argument which purports to show that the use of the standard Legendre transform in non-additive Statistical Mechanics is not appropriate. For concreteness, we use as paradigm, the case of systems which are conjecturally described by the (non-additive) Tsallis entropy. We point out the form of the modified Legendre transform that should be used, instead, in the non-additive thermodynamics induced by the Tsallis entropy. We comment on more general implications of this proposal for the thermodynamics of "complex systems".

cond-mat.stat-mech↗

arXiv2017

Convexity and the Euclidean metric of space-time

We address the question about the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

physics.gen-ph↗

arXiv1994

Geometry of the Frenkel-Kac-Segal cocycle

We present an analysis of the cocycle appearing in the vertex operator representation of simply-laced, affine, Kac-Moody algebras. We prove that it can be described in the context of $R$-commutative geometry, where $R$ is a Yang-Baxter operator, as a strong $R$-commutative algebra. We comment on the Hochschild, cyclic and dihedral homology theories that appear in non-commutative geometry and their potential relation to string theory.

hep-th↗