On Applications of Orlicz Spaces to Statistical Physics

Majewski, W. A.; Labuschagne, Louis E.

doi:10.1007/s00023-013-0267-3

Cited by 40 publications

(42 citation statements)

References 30 publications

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“…We retell the basic story in order to introduce our notations and the IG background. Orlicz spaces as a setting for Boltzmann's equation have been recently proposed by [33], while the use of exponential statistical manifolds has been suggested in [18,Example 11] and sketched in [19,Section 4.4]. We start with an improvement of the latter, a few repetitions being justified by consistency between this presentation and [32, Sections 1.3, 4.5-6], compare also Proposition 10 below.…”

Section: Boltzmann Equationmentioning

confidence: 95%

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Lods

Pistone

2015

Entropy

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Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set E of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if E is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz-Sobolev spaces to include derivatives.

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Section: Boltzmann Equationmentioning

confidence: 95%

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Lods

Pistone

2015

Entropy

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“…It is important to note that this dual space is defined by the entropic-type function x → x log(x + 1). We have, [39]- [40], Proposition 2.1. The dual pair L cosh −1 , L log(L + 1) provides the basic mathematical ingredient for a description of a general (regular) classical system.…”

Section: Statistical Mechanics and Boltzmann Theorymentioning

confidence: 92%

“…It is worth pointing out that this observation can be considered as an "invitation" to Orlicz space approach -this feature of quantum statistical formalism was the starting point for developing the new approach to large systems, see [40], [41].…”

Section: 4mentioning

confidence: 99%

On Quantum Statistical Mechanics: A Study Guide

Majewski

2017

Advances in Mathematical Physics

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We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed. Basic IdeasIn this paper, we will try to give an overview and road map to the area of quantum statistical mechanics without becoming too diverted by details. In contrast, we put a strong emphasis on evolution of calculus which is used in the description of statistical mechanics. To make our exposition abundantly clear, we begin with a historical remark. Newton has given his principles for classical mechanics at the end of the 17th century. However, classical mechanics blossomed into a rich mathematical theory only in the second half of the 19th century. After a moment of reverie, we realize that although Newton and Leibniz introduced the basic principles of (classical) calculus, it was Cauchy (around 1830' , albeit the "epsilon-delta definition of limit" was first given by Bolzano in 1817) who finally clarified the concept of limit and then Riemann (around 1860') clarified the concept of integral. Consequently, in the second half of the 19th century, the principles of (classical) calculus were fully established. This gave the opportunity to transform classical mechanics into a well-developed theory (Lagrange, Hamilton, Liouville, etc.). So with a mature theory of calculus available, it took a few more decades to obtain a fully fledged theory of classical mechanics. Subsequently, (classical) statistical mechanics has appeared as a combined development of classical mechanics and probability theory.We will show that similar situation occurred also in the 20th century but in the context of quantum theory. The starting point was Heisenberg's equation of motion in quantum theory. He for the first time wrote a noncommutative derivation, a commutator. (We recall that a derivation is a unary function satisfying the Leibniz product law.) To see this, it is enough to note that a commutator satisfies the Leibniz rule! This can be considered as an analogy of Newton's introduction of (classical) differentiation to write the equations of motion for a classical system. Then Heisenberg, Born, Jordan, and Dirac realized that noncommutativity is the raison d'être of quantum mechanics and they have introduced the so-called canonical quantization. It means that the basic relations of classical mechanics, { 푖 , 푗 } ∝ 푖 푗 1, , = 1, 2, 3, . . . ,should be replaced by where {⋅, ⋅} stands for the Poisson bracket, while [ , ] = − denotes the commutator.

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“…This gives some intuition regarding the origins of these definitions. Also see [16,5,6] for some of the early literature on detailed balance, as well as [17]. More specific references will be given as we proceed.…”

Section: Definitions Of Quantum Detailed Balancementioning

confidence: 99%