Editorial

6 pages, 533 KiB

Open AccessEditorial

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

by Frédéric Barbaresco and Jean-Pierre Gazeau

Entropy 2019, 21(3), 250; https://doi.org/10.3390/e21030250 - 06 Mar 2019

Cited by 1 | Viewed by 3289

Abstract

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern [...] Read more.

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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Research

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38 pages, 2450 KiB

Open AccessArticle

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs

by Remco Duits, Erik J. Bekkers and Alexey Mashtakov

Entropy 2019, 21(1), 38; https://doi.org/10.3390/e21010038 - 08 Jan 2019

Cited by 4 | Viewed by 5349

Abstract

Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D [...] Read more.

Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on

S E (2)

. Here, we extend these approaches to 3D using Fourier transform on the Lie group

S E (3)

of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations

R^{3} ⋊ S^{2} : = S E (3) / ({0} \times S O (2))

as the quotient in

S E (3)

. In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker–Planck PDEs of

α

-stable Lévy processes on

R^{3} ⋊ S^{2}

. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for

α = 1

(the diffusion kernel) to the kernel for

α = \frac{1}{2}

(the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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25 pages, 6614 KiB

Open AccessArticle

Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

by Adam Brus, Jiří Hrivnák and Lenka Motlochová

Entropy 2018, 20(12), 938; https://doi.org/10.3390/e20120938 - 06 Dec 2018

Cited by 5 | Viewed by 2882

Abstract

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the [...] Read more.

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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23 pages, 355 KiB

Open AccessArticle

Geometry of Thermodynamic Processes

by Arjan Van der Schaft and Bernhard Maschke

Entropy 2018, 20(12), 925; https://doi.org/10.3390/e20120925 - 04 Dec 2018

Cited by 46 | Viewed by 4593

Abstract

Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, [...] Read more.

Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

15 pages, 259 KiB

Open AccessArticle

Short-Time Propagators and the Born–Jordan Quantization Rule

by Maurice A. De Gosson

Entropy 2018, 20(11), 869; https://doi.org/10.3390/e20110869 - 10 Nov 2018

Cited by 4 | Viewed by 3019

Abstract

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct [...] Read more.

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

60 pages, 6291 KiB

Open AccessFeature PaperArticle

Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

by Frédéric Barbaresco

Entropy 2018, 20(11), 840; https://doi.org/10.3390/e20110840 - 02 Nov 2018

Cited by 10 | Viewed by 4900

Abstract

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted [...] Read more.

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology). Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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13 pages, 2041 KiB

Open AccessArticle

Emergence of Non-Fourier Hierarchies

by Tamás Fülöp, Róbert Kovács, Ádám Lovas, Ágnes Rieth, Tamás Fodor, Mátyás Szücs, Péter Ván and Gyula Gróf

Entropy 2018, 20(11), 832; https://doi.org/10.3390/e20110832 - 30 Oct 2018

Cited by 29 | Viewed by 3176

Abstract

The non-Fourier heat conduction phenomenon on room temperature is analyzed from various aspects. The first one shows its experimental side, in what form it occurs, and how we treated it. It is demonstrated that the Guyer-Krumhansl equation can be the next appropriate extension [...] Read more.

The non-Fourier heat conduction phenomenon on room temperature is analyzed from various aspects. The first one shows its experimental side, in what form it occurs, and how we treated it. It is demonstrated that the Guyer-Krumhansl equation can be the next appropriate extension of Fourier’s law for room-temperature phenomena in modeling of heterogeneous materials. The second approach provides an interpretation of generalized heat conduction equations using a simple thermo-mechanical background. Here, Fourier heat conduction is coupled to elasticity via thermal expansion, resulting in a particular generalized heat equation for the temperature field. Both aforementioned approaches show the size dependency of non-Fourier heat conduction. Finally, a third approach is presented, called pseudo-temperature modeling. It is shown that non-Fourier temperature history can be produced by mixing different solutions of Fourier’s law. That kind of explanation indicates the interpretation of underlying heat conduction mechanics behind non-Fourier phenomena. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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14 pages, 266 KiB

Open AccessArticle

Hermite Functions, Lie Groups and Fourier Analysis

by Enrico Celeghini, Manuel Gadella and Mariano A. Del Olmo

Entropy 2018, 20(11), 816; https://doi.org/10.3390/e20110816 - 23 Oct 2018

Cited by 7 | Viewed by 2530

Abstract

In this paper, we present recent results in harmonic analysis in the real line

R

and in the half-line

R^{+}

, which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done [...] Read more.

In this paper, we present recent results in harmonic analysis in the real line

R

and in the half-line

R^{+}

, which show a closed relation between Hermite and Laguerre functions, respectively, their symmetry groups and Fourier analysis. This can be done in terms of a unified framework based on the use of rigged Hilbert spaces. We find a relation between the universal enveloping algebra of the symmetry groups with the fractional Fourier transform. The results obtained are relevant in quantum mechanics as well as in signal processing as Fourier analysis has a close relation with signal filters. In addition, we introduce some new results concerning a discretized Fourier transform on the circle. We introduce new functions on the circle constructed with the use of Hermite functions with interesting properties under Fourier transformations. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

16 pages, 314 KiB

Open AccessArticle

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

by Hervé Bergeron and Jean-Pierre Gazeau

Entropy 2018, 20(10), 787; https://doi.org/10.3390/e20100787 - 13 Oct 2018

Cited by 9 | Viewed by 3201

Abstract

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary [...] Read more.

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

Review

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39 pages, 2007 KiB

Open AccessReview

From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

by François Gay-Balmaz and Hiroaki Yoshimura

Entropy 2019, 21(1), 8; https://doi.org/10.3390/e21010008 - 23 Dec 2018

Cited by 41 | Viewed by 7148

Abstract

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite-dimensional case of discrete systems, as well as for the infinite-dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton’s [...] Read more.

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite-dimensional case of discrete systems, as well as for the infinite-dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton’s principle, we show, with the help of thermodynamic systems with gradually increasing complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat, and mass transfer in both the adiabatically closed cases and open cases. On the continuum side, we illustrate our theory using the example of multicomponent Navier–Stokes–Fourier systems. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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