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Rational extended thermodynamics (RET) is the theory that is
applicable to nonequilibrium phenomena out of local equilibrium. It
is expressed by the hyperbolic system of field equations with local
constitutive equations and is strictly related to the kinetic
theory with the closure method of the hierarchies of moment
equations. The book intends to present, in a systematic way, new
results obtained by RET of gases in both classical and relativistic
cases, and it is a natural continuation of the book "Rational
Extended Thermodynamics beyond the Monatomic Gas" by the same
authors published in 2015. However, this book addresses much wider
topics than those of the previous book. Its contents are as
follows: RET of rarefied monatomic gases and of polyatomic gases; a
simplified RET theory with 6 fields being valid far from
equilibrium; RET where both molecular rotational and vibrational
modes exist; mixture of gases with multi-temperature. The theory is
applied to several typical topics (sound waves, shock waves, etc.)
and is compared with experimental data. From a mathematical point
of view, RET can be regarded as a theory of hyperbolic symmetric
systems, of which it is possible to conduct a qualitative analysis.
The book represents a valuable resource for applied mathematicians,
physicists, and engineers, offering powerful models for many
potential applications such as reentering satellites into the
atmosphere, semiconductors, and nanoscale phenomena.
Rational extended thermodynamics (RET) is the theory that is
applicable to nonequilibrium phenomena out of local equilibrium. It
is expressed by the hyperbolic system of field equations with local
constitutive equations and is strictly related to the kinetic
theory with the closure method of the hierarchies of moment
equations. The book intends to present, in a systematic way, new
results obtained by RET of gases in both classical and relativistic
cases, and it is a natural continuation of the book "Rational
Extended Thermodynamics beyond the Monatomic Gas" by the same
authors published in 2015. However, this book addresses much wider
topics than those of the previous book. Its contents are as
follows: RET of rarefied monatomic gases and of polyatomic gases; a
simplified RET theory with 6 fields being valid far from
equilibrium; RET where both molecular rotational and vibrational
modes exist; mixture of gases with multi-temperature. The theory is
applied to several typical topics (sound waves, shock waves, etc.)
and is compared with experimental data. From a mathematical point
of view, RET can be regarded as a theory of hyperbolic symmetric
systems, of which it is possible to conduct a qualitative analysis.
The book represents a valuable resource for applied mathematicians,
physicists, and engineers, offering powerful models for many
potential applications such as reentering satellites into the
atmosphere, semiconductors, and nanoscale phenomena.
This book is dedicated to the recent developments in RET with the
aim to explore polyatomic gas, dense gas and mixture of gases in
non-equilibrium. In particular we present the theory of dense gases
with 14 fields, which reduces to the Navier-Stokes Fourier
classical theory in the parabolic limit. Molecular RET with an
arbitrary number of field-variables for polyatomic gases is also
discussed and the theory is proved to be perfectly compatible with
the kinetic theory in which the distribution function depends on an
extra variable that takes into account a molecule's internal
degrees of freedom. Recent results on mixtures of gases with
multi-temperature are presented together with a natural definition
of the average temperature. The qualitative analysis and in
particular, the existence of the global smooth solution and the
convergence to equilibrium are also studied by taking into account
the fact that the differential systems are symmetric hyperbolic.
Applications to shock and sound waves are analyzed together with
light scattering and heat conduction and the results are compared
with experimental data. Rational extended thermodynamics (RET) is a
thermodynamic theory that is applicable to non-equilibrium
phenomena. It is described by differential hyperbolic systems of
balance laws with local constitutive equations. As RET has been
strictly related to the kinetic theory through the closure method
of moment hierarchy associated to the Boltzmann equation, the
applicability range of the theory has been restricted within
rarefied monatomic gases. The book represents a valuable resource
for applied mathematicians, physicists and engineers, offering
powerful models for potential applications like satellites
reentering the atmosphere, semiconductors and nano-scale phenomena.
In 2013 several scientific activities have been devoted to
mathematical researches for the study of planet Earth. The current
volume presents a selection of the highly topical issues presented
at the workshop “Mathematical Models and Methods for Planet
Earth”, held in Roma (Italy), in May 2013. The fields of interest
span from impacts of dangerous asteroids to the safeguard from
space debris, from climatic changes to monitoring geological
events, from the study of tumor growth to sociological problems. In
all these fields the mathematical studies play a relevant role as a
tool for the analysis of specific topics and as an ingredient of
multidisciplinary problems. To investigate these problems we will
see many different mathematical tools at work: just to mention
some, stochastic processes, PDE, normal forms, chaos theory.
This book is dedicated to the recent developments in RET with the
aim to explore polyatomic gas, dense gas and mixture of gases in
non-equilibrium. In particular we present the theory of dense gases
with 14 fields, which reduces to the Navier-Stokes Fourier
classical theory in the parabolic limit. Molecular RET with an
arbitrary number of field-variables for polyatomic gases is also
discussed and the theory is proved to be perfectly compatible with
the kinetic theory in which the distribution function depends on an
extra variable that takes into account a molecule's internal
degrees of freedom. Recent results on mixtures of gases with
multi-temperature are presented together with a natural definition
of the average temperature. The qualitative analysis and in
particular, the existence of the global smooth solution and the
convergence to equilibrium are also studied by taking into account
the fact that the differential systems are symmetric hyperbolic.
Applications to shock and sound waves are analyzed together with
light scattering and heat conduction and the results are compared
with experimental data. Rational extended thermodynamics (RET) is a
thermodynamic theory that is applicable to non-equilibrium
phenomena. It is described by differential hyperbolic systems of
balance laws with local constitutive equations. As RET has been
strictly related to the kinetic theory through the closure method
of moment hierarchy associated to the Boltzmann equation, the
applicability range of the theory has been restricted within
rarefied monatomic gases. The book represents a valuable resource
for applied mathematicians, physicists and engineers, offering
powerful models for potential applications like satellites
reentering the atmosphere, semiconductors and nano-scale phenomena.
In 2013 several scientific activities have been devoted to
mathematical researches for the study of planet Earth. The current
volume presents a selection of the highly topical issues presented
at the workshop "Mathematical Models and Methods for Planet Earth",
held in Roma (Italy), in May 2013. The fields of interest span from
impacts of dangerous asteroids to the safeguard from space debris,
from climatic changes to monitoring geological events, from the
study of tumor growth to sociological problems. In all these fields
the mathematical studies play a relevant role as a tool for the
analysis of specific topics and as an ingredient of
multidisciplinary problems. To investigate these problems we will
see many different mathematical tools at work: just to mention
some, stochastic processes, PDE, normal forms, chaos theory.
Physicists firmly believe that the differential equations of nature
should be hyperbolic so as to exclude action at a distance; yet the
equations of irreversible thermodynamics - those of Navier-Stokes
and Fourier - are parabolic. This incompatibility between the
expectation of physicists and the classical laws of thermodynamics
has prompted the formulation of extended thermodynamics. After
describing the motifs and early evolution of this new branch of
irreversible thermodynamics, the authors apply the theory to
mon-atomic gases, mixtures of gases, relativistic gases, and
"gases" of phonons and photons. The discussion brings into
perspective the various phenomena called second sound, such as heat
propagation, propagation of shear stress and concentration, and the
second sound in liquid helium. The formal mathematical structure of
extended thermodynamics is exposed and the theory is shown to be
fully compatible with the kinetic theory of gases. The study closes
with the testing of extended thermodynamics through the
exploitation of its predictions for measurements of light
scattering and sound propagation.
Ordinary thermodynamics provides reliable results when the
thermodynamic fields are smooth, in the sense that there are no
steep gradients and no rapid changes. In fluids and gases this is
the domain of the equations of Navier-Stokes and Fourier. Extended
thermodynamics becomes relevant for rapidly varying and strongly
inhomogeneous processes. Thus the propagation of high frequency
waves, and the shape of shock waves, and the regression of
small-scale fluctuation are governed by extended thermodynamics.
The field equations of ordinary thermodynamics are parabolic while
extended thermodynamics is governed by hyperbolic systems. The main
ingredients of extended thermodynamics are * field equations of
balance type, * constitutive quantities depending on the present
local state and * entropy as a concave function of the state
variables. This set of assumptions leads to first order
quasi-linear symmetric hyperbolic systems of field equations; it
guarantees the well-posedness of initial value problems and finite
speeds of propaga tion. Several tenets of irreversible
thermodynamics had to be changed in subtle ways to make extended
thermodynamics work. Thus, the entropy is allowed to depend on
nonequilibrium vari ables, the entropy flux is a general
constitutive quantity, and the equations for stress and heat flux
contain inertial terms. New insight is therefore provided into the
principle of material frame indifference. With these modifications
an elegant formal structure can be set up in which, just as in
classical thermostatics, all restrictive conditions--derived from
the entropy principle-take the form of integrability conditions.
These lecture notes of the courses presented at the first CIME
session 1994 by leading scientists present the state of the art in
recent mathematical methods in Nonlinear Wave Propagation.
Questo testo, che giunge ora alla quarta edizione, e stato
concepito principalmente per le necessita delle Scuole di
Ingegneria, dove la Meccanica Razionale ha il duplice ruolo di
introdurre sia alla modellizzazione fisico-matematica rigorosa che
a specifiche applicazioni sviluppate poi in altri insegnamenti. La
trattazione che qui proponiamo vuole presentare i concetti
fondamentali mantenendo sempre l'attenzione rivolta alle
applicazioni, a volte comuni ad altre discipline, in vista di
sinergie didattiche favorite dalla presenza di corsi integrati.
Abbiamo cercato di dare al libro una impostazione il piu possibile
coerente con questa finalita, soprattutto in alcune sezioni
tradizionalmente caratterizzate da una trattazione piu astratta:
dai vincoli al Principio dei lavori virtuali, dal Principio di
d'Alembert alla Meccanica Analitica. Abbiamo comunque mantenuto la
tradizionale e, a nostro parere, irrinunciabile struttura
ipotetico-deduttiva nello svolgimento delle argomentazioni, che fa
ancora della Meccanica Razionale un disciplina formalmente
rigorosa. Sono percio presenti dimostrazioni anche complesse, sia
pure sempre motivate alla luce del contesto applicativo nel quale
si vanno a collocare. Questa edizione e frutto di un ampio lavoro
di riorganizzazione e rielaborazione rispetto alla precedente e
contiene, oltre a numerosi nuovi esempi, miglioramenti nella
presentazione dei concetti principali e nello svolgimento delle
dimostrazioni, per renderle didatticamente piu efficaci.
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