Categorical distribution theory; heat equation (original) (raw)
Related papers
Ju l 2 00 4 Categorical distribution theory ; heat equation
2018
The simplest notion by which a theory of function spaces may be formulated is that of cartesian closed categories. To realize this concretely for spaces of smooth (= C) functions, several notions of diffeological spaces and convenient vector spaces have been developed, besides the whole body of topos theory. Topos theory in particular provides for toposes containing the category of smooth manifolds as full subcategory. In fact, Grothendieck’s “Smooth Topos” is closely related to the category of diffeological spaces. The special features of Convenient Vector Spaces were utilized by the present authors back in the 1980’s for a more elaborate topos , cf. [12], [14]. The topos there (Dubuc’s “Cahiers Topos”) in fact accomodates synthetic differential geometry. The present work is a continuation of our work from the 80’s, and is motivated by the desire to have a synthetic theory of some of the fundamental partial differential equations, like the heat equation. This forced us to sort out ...
Some Remarks on the Heat Flow for Functions and Forms
Electronic Communications in Probability, 1998
This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation dP t f = P t df is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if M is flat and f smooth of compact support, dP t f ∞ cannot be estimated on compact time intervals in terms of f or df .
On the heat potential of the double distribution
Časopis pro pěstování matematiky
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Časopis pro pěstování matematiky, roč. 98 (1973), Praha
On geometry of phenomenological thermodynamics
2018
We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symplectic geometry, and argue that it catches its geometric essence in a more profound and clearer way than the popular odd-dimensional contact structure description. Among the advantages are a number of conceptual clarifications: the geometric role of internal energy (not made as an independent variable), the lattice of potentials, and the gauge interpretation of the theory.
Symplectic Theory of Heat and Information Geometry
Preprint, 2022
We present in this chapter a new formulation of heat theory and Information Geometry through symplectic and Poisson structures based on Jean-Marie Souriau’s symplectic model of statistical mechanics, called “Lie Groups Thermodynamics”. Souriau model was initially described in chapter IV “Sta-tistical Mechanics” of his book “Structure of dynamical systems” published in 1969. This model gives an archetypal, and purely geometric, characterization of Entropy, which appears as an invariant Casimir function in coadjoint repre-sentation, from which we will deduce a geometric heat equation. The ap-proach also allows generalizing the Fisher metric of information geometry thanks to the 2-form KKS (Kirillov, Kostant, Souriau) in the affine case via the Souriau’s cocyle. In this model, the Souriau’s moment map and the coad-joint orbits play a central role. Ontologically, this model provides join geo-metric structures for Statistical Mechanics, Information Probability and Prob-ability. Entropy acquires a geometric foundation as a function parameterized by mean of moment map in dual Lie algebra, and in term of foliations. Souriau established the generalized Gibbs laws when the manifold has a symplectic form and a connected Lie group G operates on this manifold by symplectomorphisms. Souriau Entropy is invariant under the action of the group acting on the homogeneous symplectic manifold. As quoted by Souriau, these equations are universal and could be also of great interest in Mathematics. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on these surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will also explain the 2nd Principle in thermodynamics by defi-nite positiveness of Souriau tensor extending the Koszul-Fisher metric from Information Geometry. Entropy as Casimir function is characterized by Koszul Poisson Cohomology. We will finally introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau’s covariant Gibbs density by considering this space as the pure imag-inary axis of the homogeneous Siegel upper half space where Sp(2n,R)/U(n) acts transitively. Gauss density of SPD matrices is computed through Souriau’s moment map and coadjoint orbits. We will illustrate the model first for Poincaré unit disk, then Siegel unit disk and finally upper half space. For this example, we deduce Gauss density for SPD matrices.
The Eshelby tensor and the theory of continuous distributions of inhomogeneities
Mechanics Research Communications, 2002
The purpose of this note is to reaffirm the fact that there exists a natural connection between NollÕs theory of inhomogeneities and the Eshelby tensor. One way to expose this connection consists in allowing the inhomogeneity pattern to evolve in time and then exploring the thermodynamic implications.
Sciprints, 2016
We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincar&eacu...
Cosheaves and distributions on toposes
Algebra Universalis, 1995
Distributions on a Grothendieck topos were introduced by Lawvere [12] (cf. also ) as a generalization of the classical notion (cf.
The heat semigroup on configuration spaces
Publications of the Research Institute for Mathematical Sciences, 2003
In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural "Riemannian-like" structure of the configuration space Γ X over a complete, connected, oriented, and stochastically complete Riemannian manifold X of infinite volume, the heat semigroup (e −tH Γ) t∈R + was introduced and studied in [J. Func. Anal. 154 (1998), 444-500]. Here, H Γ is the Dirichlet operator of the Dirichlet form E Γ over the space L 2 (Γ X , π m), where π m is the Poisson measure on Γ X with intensity m-the volume measure on X. We construct a metric space Γ ∞ that is continuously embedded into Γ X. Under some conditions on the manifold X, we prove that Γ ∞ is a set of full π m measure and derive an explicit formula for the heat semigroup: (e −tH Γ F)(γ) = Γ∞ F (ξ) P t,γ (dξ), where P t,γ is a probability measure on Γ ∞ for all t > 0, γ ∈ Γ ∞. The central results of the paper are two types of Feller properties for the heat semigroup. The first one is a kind of strong Feller property with respect to the metric on the space Γ ∞. The second one, obtained in the case X = R d , is the Feller property with respect to the intrinsic metric of the Dirichlet form E Γ. Next, we give a direct construction of the independent infinite particle process on the manifold X, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every γ ∈ Γ ∞ , will never leave Γ ∞ , and has continuous sample path in Γ ∞ , provided dim X ≥ 2. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the P t,γ (•) above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case dim X = 1. Finally, as an easy consequence we get a "path-wise" construction of the independent particle process on Γ ∞ from the underlying Brownian motion.