71,148 research outputs found
Stochastic embedding of dynamical systems
Most physical systems are modelled by an ordinary or a partial differential
equation, like the n-body problem in celestial mechanics. In some cases, for
example when studying the long term behaviour of the solar system or for
complex systems, there exist elements which can influence the dynamics of the
system which are not well modelled or even known. One way to take these
problems into account consists of looking at the dynamics of the system on a
larger class of objects, that are eventually stochastic. In this paper, we
develop a theory for the stochastic embedding of ordinary differential
equations. We apply this method to Lagrangian systems. In this particular case,
we extend many results of classical mechanics namely, the least action
principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain
a Hamiltonian formulation for our stochastic Lagrangian systems. Many
applications are discussed at the end of the paper.Comment: 112 page
Spacetime deployments parametrized by gravitational and electromagnetic fields
On the basis of a "Punctual" Equivalence Principle of the general relativity
context, we consider spacetimes with measurements of conformally invariant
physical properties. Then, applying the Pfaff theory for PDE to a particular
conformally equivariant system of differential equations, we make explicit the
dependence of any kind of function describing a "spacetime deployment", on
n(n+1) parametrizing functions, denoting by n the spacetime dimension. These
functions, appearing in a linear differential Spencer sequence and determining
gauge fields of spacetime deformations relatively to a "substrat spacetime",
can be consistently ascribed to unified electromagnetic and gravitational
fields, at any spacetime dimensions n greater or equal to 4.Comment: 26 pages, LaTeX2e, file macro "suppl.sty", correction in the
definition of germs and local ring
A parametrix for quantum gravity?
In the sixties, DeWitt discovered that the advanced and retarded Green
functions of the wave operator on metric perturbations in the de Donder gauge
make it possible to define classical Poisson brackets on the space of
functionals that are invariant under the action of the full diffeomorphism
group of spacetime. He therefore tried to exploit this property to define
invariant commutators for the quantized gravitational field, but the operator
counterpart of such classical Poisson brackets turned out to be a hard task. On
the other hand, the mathematical literature studies often an approximate
inverse, the parametrix, which is, strictly, a distribution. We here suggest
that such a construction might be exploited in canonical quantum gravity. We
begin with the simplest case, i.e. fundamental solution and parametrix for the
linear, scalar wave operator; the next step are tensor wave equations, again
for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear
Einstein equations are studied, relying upon the well-established
Choquet-Bruhat construction, according to which the fifth derivatives of
solutions of a nonlinear hyperbolic system solve a linear hyperbolic system.
The latter is solved by means of Kirchhoff-type formulas, while the former
fifth-order equations can be solved by means of well-established parametrix
techniques for elliptic operators. But then the metric components that solve
the vacuum Einstein equations can be obtained by convolution of such a
parametrix with Kirchhoff-type formulas. Some basic functional equations for
the parametrix are also obtained, that help in studying classical and quantum
version of the Jacobi identity.Comment: 27 page
Continuum thermodynamics of chemically reacting fluid mixtures
We consider viscous, heat conducting mixtures of molecularly miscible
chemical species forming a fluid in which the constituents can undergo chemical
reactions. Assuming a common temperature for all components, we derive a closed
system of partial mass and partial momentum balances plus a mixture balance of
internal energy. This is achieved by careful exploitation of the entropy
principle and requires appropriate definitions of absolute temperature and
chemical potentials, based on an adequate definition of thermal energy
excluding diffusive contributions. The resulting interaction forces split into
a thermo-mechanical and a chemical part, where the former turns out to be
symmetric in case of binary interactions. For chemically reacting systems and
as a new result, the chemical interaction force is a contribution being
non-symmetric outside of chemical equilibrium. The theory also provides a
rigorous derivation of the so-called generalized thermodynamic driving forces,
avoiding the use of approximate solutions to the Boltzmann equations. Moreover,
using an appropriately extended version of the entropy principle and
introducing cross-effects already before closure as entropy invariant couplings
between principal dissipative mechanisms, the Onsager symmetry relations become
a strict consequence. With a classification of the factors in the binary
products of the entropy production according to their parity--instead of the
classical partition into so-called fluxes and driving forces--the apparent
anti-symmetry of certain couplings is thereby also revealed. If the diffusion
velocities are small compared to the speed of sound, the Maxwell-Stefan
equations follow in the case without chemistry, thereby neglecting wave
phenomena in the diffusive motion. This results in a reduced model with only
mass being balanced individually. In the reactive case ..
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