Differential Geometric Foundations Of Non Equilibrium Thermodynamics

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Differential Geometric Foundations Of Non Equilibrium Thermodynamics

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Author : Marcus Hildebrandt
language : en
Publisher: BoD – Books on Demand
Release Date : 2025-02-27

Differential Geometric Foundations Of Non Equilibrium Thermodynamics written by Marcus Hildebrandt and has been published by BoD – Books on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-27 with Science categories.

While all field theories are nowadays available in a modern, differential geometric, coordinate free formulation on manifolds this has been so far only rudimentary accomplished in general non-equilibrium thermodynamics. In this work it is shown how a fitting geometric structure can be derived for arbitrary compact (discrete Schottky Systems) thermodynamic systems, such as stars and black holes, using only a few thermodynamic principles. This leads to deep geometric insights. Some central results are the following: while in the theory of relativity the energy-momentum tensor determines the geometry of the space, in non-equilibrium thermodynamics, the 1-form of the entropy production rate is responsible for the emergence of a well-known geometric structure: the contact geometry. Relaxation processes remain in the fibers in which they start and end on an attractor manifold, that can be identified with the classical equilibrium subspace of thermostatics. One then proves, that outside this attractor manifold there are no reversible process directions. As a consequence of this, the 2nd Law of thermodynamics lives mainly on the fibers of the state manifold, the so called vertical geometric structure, while the 1st Law of thermodynamics is formulated on the horizontal components of the state manifold. The internal energy provides a physical gauge for each fiber. The 1st and 2nd Law of thermodynamics are coupled via the representation of the entropy flux 1-form that can be represented in the dual basis of exchange 1-forms such as the heat 1-form. This fact can be used to provide a "coordinate free" ("invariant") definition of non-equilibrium temperature. Finally, it is shown that probably the most general geometric structure to model non-equilibrium thermodynamics of compact (discrete Schottky systems) systems is given by a composite fibred cocontact phase manifold that includes time as an explicit dimension.

Maximum Dissipation Non Equilibrium Thermodynamics And Its Geometric Structure

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Author : Henry W. Haslach Jr.
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-01-15

Maximum Dissipation Non Equilibrium Thermodynamics And Its Geometric Structure written by Henry W. Haslach Jr. and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-01-15 with Science categories.

Maximum Dissipation: Non-Equilibrium Thermodynamics and its Geometric Structure explores the thermodynamics of non-equilibrium processes in materials. The book develops a general technique created in order to construct nonlinear evolution equations describing non-equilibrium processes, while also developing a geometric context for non-equilibrium thermodynamics. Solid materials are the main focus in this volume, but the construction is shown to also apply to fluids. This volume also: • Explains the theory behind thermodynamically-consistent construction of non-linear evolution equations for non-equilibrium processes • Provides a geometric setting for non-equilibrium thermodynamics through several standard models, which are defined as maximum dissipation processes • Emphasizes applications to the time-dependent modeling of soft biological tissue Maximum Dissipation: Non-Equilibrium Thermodynamics and its Geometric Structure will be valuable for researchers, engineers and graduate students in non-equilibrium thermodynamics and the mathematical modeling of material behavior.

Partial Differential Equations And Group Theory

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Author : J.F. Pommaret
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

Partial Differential Equations And Group Theory written by J.F. Pommaret and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.

Ordinary differential control thPory (the classical theory) studies input/output re lations defined by systems of ordinary differential equations (ODE). The various con cepts that can be introduced (controllability, observability, invertibility, etc. ) must be tested on formal objects (matrices, vector fields, etc. ) by means of formal operations (multiplication, bracket, rank, etc. ), but without appealing to the explicit integration (search for trajectories, etc. ) of the given ODE. Many partial results have been re cently unified by means of new formal methods coming from differential geometry and differential algebra. However, certain problems (invariance, equivalence, linearization, etc. ) naturally lead to systems of partial differential equations (PDE). More generally, partial differential control theory studies input/output relations defined by systems of PDE (mechanics, thermodynamics, hydrodynamics, plasma physics, robotics, etc. ). One of the aims of this book is to extend the preceding con cepts to this new situation, where, of course, functional analysis and/or a dynamical system approach cannot be used. A link will be exhibited between this domain of applied mathematics and the famous 'Backlund problem', existing in the study of solitary waves or solitons. In particular, we shall show how the methods of differ ential elimination presented here will allow us to determine compatibility conditions on input and/or output as a better understanding of the foundations of control the ory. At the same time we shall unify differential geometry and differential algebra in a new framework, called differential algebraic geometry.

Scientific And Technical Aerospace Reports

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Author :
language : en
Publisher:
Release Date : 1994

Scientific And Technical Aerospace Reports written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Aeronautics categories.

Geometric Continuum Mechanics And Induced Beam Theories

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Author : Simon R. Eugster
language : en
Publisher: Springer
Release Date : 2015-03-19

Geometric Continuum Mechanics And Induced Beam Theories written by Simon R. Eugster and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-19 with Science categories.

This research monograph discusses novel approaches to geometric continuum mechanics and introduces beams as constraint continuous bodies. In the coordinate free and metric independent geometric formulation of continuum mechanics as well as for beam theories, the principle of virtual work serves as the fundamental principle of mechanics. Based on the perception of analytical mechanics that forces of a mechanical system are defined as dual quantities to the kinematical description, the virtual work approach is a systematic way to treat arbitrary mechanical systems. Whereas this methodology is very convenient to formulate induced beam theories, it is essential in geometric continuum mechanics when the assumptions on the physical space are relaxed and the space is modeled as a smooth manifold. The book addresses researcher and graduate students in engineering and mathematics interested in recent developments of a geometric formulation of continuum mechanics and a hierarchical development of induced beam theories.

Non Commuting Variations In Mathematics And Physics

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Author : Serge Preston
language : en
Publisher: Springer
Release Date : 2016-03-02

Non Commuting Variations In Mathematics And Physics written by Serge Preston and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-02 with Technology & Engineering categories.

This text presents and studies the method of so –called noncommuting variations in Variational Calculus. This method was pioneered by Vito Volterra who noticed that the conventional Euler-Lagrange (EL-) equations are not applicable in Non-Holonomic Mechanics and suggested to modify the basic rule used in Variational Calculus. This book presents a survey of Variational Calculus with non-commutative variations and shows that most basic properties of conventional Euler-Lagrange Equations are, with some modifications, preserved for EL-equations with K-twisted (defined by K)-variations. Most of the book can be understood by readers without strong mathematical preparation (some knowledge of Differential Geometry is necessary). In order to make the text more accessible the definitions and several necessary results in Geometry are presented separately in Appendices I and II Furthermore in Appendix III a short presentation of the Noether Theorem describing the relation between the symmetries of the differential equations with dissipation and corresponding s balance laws is presented.

Classical And Geometrical Theory Of Chemical And Phase Thermodynamics

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Author : Frank Weinhold
language : en
Publisher: John Wiley & Sons
Release Date : 2009-02-17

Classical And Geometrical Theory Of Chemical And Phase Thermodynamics written by Frank Weinhold and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-02-17 with Science categories.

Because it is grounded in math, chemical thermodynamics is often perceived as a difficult subject and many students are never fully comfortable with it. The first authoritative textbook presentation of equilibrium chemical and phase thermodynamics in a reformulated geometrical framework, Chemical and Phase Thermodynamics shows how this famously difficult subject can be accurately expressed with only elementary high-school geometry concepts. Featuring numerous suggestions for research-level extensions, this simplified alternative to standard calculus-based thermodynamics expositions is perfect for undergraduate and beginning graduate students as well as researchers.

Data Driven Modelling And Scientific Machine Learning In Continuum Physics

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Author : Krishna Garikipati
language : en
Publisher: Springer Nature
Release Date : 2024-07-29

Data Driven Modelling And Scientific Machine Learning In Continuum Physics written by Krishna Garikipati and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-29 with Mathematics categories.

This monograph takes the reader through recent advances in data-driven methods and machine learning for problems in science—specifically in continuum physics. It develops the foundations and details a number of scientific machine learning approaches to enrich current computational models of continuum physics, or to use the data generated by these models to infer more information on these problems. The perspective presented here is drawn from recent research by the author and collaborators. Applications drawn from the physics of materials or from biophysics illustrate each topic. Some elements of the theoretical background in continuum physics that are essential to address these applications are developed first. These chapters focus on nonlinear elasticity and mass transport, with particular attention directed at descriptions of phase separation. This is followed by a brief treatment of the finite element method, since it is the most widely used approach to solve coupled partial differential equations in continuum physics. With these foundations established, the treatment proceeds to a number of recent developments in data-driven methods and scientific machine learning in the context of the continuum physics of materials and biosystems. This part of the monograph begins by addressing numerical homogenization of microstructural response using feed-forward as well as convolutional neural networks. Next is surrogate optimization using multifidelity learning for problems of phase evolution. Graph theory bears many equivalences to partial differential equations in its properties of representation and avenues for analysis as well as reduced-order descriptions--all ideas that offer fruitful opportunities for exploration. Neural networks, by their capacity for representation of high-dimensional functions, are powerful for scale bridging in physics--an idea on which we present a particular perspective in the context of alloys. One of the most compelling ideas in scientific machine learning is the identification of governing equations from dynamical data--another topic that we explore from the viewpoint of partial differential equations encoding mechanisms. This is followed by an examination of approaches to replace traditional, discretization-based solvers of partial differential equations with deterministic and probabilistic neural networks that generalize across boundary value problems. The monograph closes with a brief outlook on current emerging ideas in scientific machine learning.

Thermodynamics Of Energy Conversion And Transport

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Author : Stanislaw Sieniutycz
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Thermodynamics Of Energy Conversion And Transport written by Stanislaw Sieniutycz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.

It is becoming more and more important to manage energy resources effectively: to maximize their benefits while minimizing the negative environmental impacts. Scientist and engineers are thus faced with the problem of optimizing complex systems subject to constraints from, ecology, economics, and thermodynamics. It is chiefly to the last that the present volume is addressed. Nonequilibrium thermodynamic approaches, such as finite-time thermodynamics and Second-Law analyses, can provide realistic models and analyses that can be used to search for optimum ways to operate machines and processes. Intended for physicists, chemists, and engineers, this volume reviews the state of the art in the thermodynamics of energy conversion and transmission. Using examples from solar, thermal, mechanical, chemical, and environmental engineering, the book focuses on the use of thermodynamic criteria for optimizing energy conversion and transmission. The first set of chapters focuses on solar energy conversion; the second set discusses the transfer and conversion of chemical energy (as in internal combustion engines or distillation columns); a concluding set of chapters deals with geometric methods in thermodynamics.

Foundations Of Finsler Geometry And Special Finsler Spaces

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Author : 松本誠
language : en
Publisher:
Release Date : 1986

Foundations Of Finsler Geometry And Special Finsler Spaces written by 松本誠 and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986 with Finsler spaces categories.