Entropy deduced from a non-statistical, non-phenomenological, topological
perspective.
Updated Nov 21, 2003
The concept of topological entropy is deduced (without statistics) from the
fact that Cartan-Hilbert 1-form of Action defines a symplectic system of Pfaff
Topological dimension 2n+2. The perfect differential of entropy, dS, is
composed of the interior product of the non-canonical components of momentum
with the components of the differential velocities. An equilibrium system is a
Lagrange submanifold of the 2n+2 topological space, upon which the change in
entropy is zero, dS_{equil}=0.
A Maple program to compute the properties of non-equilibrium electromagnetic
systems.
Updated Nov 9, 2003
A Maple program with more than a dozen examples of Topological Torsion and
Topological Spin in non-equilibrium electromagnetic systems"
Non-Equilibrium and Irreversible Electromagnetism from a Topological
Perspective.
Updated Nov 9, 2003
Classical electromagnetic theory of equilibrium systems can be put into
correspondence with two topological constraints placed on a 4 dimensional
variety of independent variables {x,y,z,t}. In terms of the exterior
differential systems, the two topological constraints are F-dA=0 and J-dG=0.
The topological formulation can be algebraically prolonged, leading to the
independent concepts of topological torsion, A^F, and topological spin, A^G.
Non-zero values for these 3-forms are indicators of non-equilibrium
electrodynamic systems. These 3-forms are identically zero in equilibrium
systems of Pfaff dimension 2. In the non-equilibrium systems, the direction
fields of the 3-forms A^F and A^G can exhibit linking and separation of the
systems into disconnected parts. The topological closure (exterior derivative)
of these 3-forms define the topological Poincare Invariants. When the Poincare
invariants vanish, the 3-forms have properties similar to the conservative
charge current 3-form J=dG. Closed integrals of the closed 3-forms form
rational topological period integrals (quantum numbers), even in the
non-equilibrium situations. However when the Poincare invariants do not vanish
they become the source of topological evolution, irreversible changes of phase,
topological defects, and the creation of stationary states far from equilibrium.
The possible continuous topological evolution of an equilibrium or
non-equilibrium electromagnetic system is studied in terms of Cartan's magic
formula acting on the set of exterior differential forms. This method extends
the covariant formulation of tensor analysis to include non-adiabatic processes.
Equivalence classes of systems and processes can be constructed in terms of the
Pfaff topological dimension of the set of exterior differential forms used to
define an electromagnetic system.
A Topological Perspective of Non-equilibrium Plasmas.
Updated Nov 9, 2003
A plasma is an electromagnetic physical system that is not a thermodynamic
isolated equilibrium system (which must be of Pfaff topological dimension of 1),
but can appear as an isolated physical system (of Pfaff topological dimension
2), or as a chaotic closed physical system (of Pfaff topological dimension 3),
or, in general, as an irreversibly dissipative turbulent physical system (of
Pfaff topological dimension 4). The topological properties of closed or open
plasma systems are encoded in terms of two exterior differential 3-forms of
topological spin, A^G, and topological torsion, A^F. These 3-forms are composed
from the set of exterior differential forms {A,F=dA,G,J=dG} used to define an
electromagnetic system. These 3-forms are identically zero in equilibrium
systems of Pfaff dimension 1. The topological limit sets (exterior derivative)
of these 3-forms define the topological Poincare Invariants. When the Poincare
invariants vanish, the 3-forms have properties similar to the conservative
charge current 3-form J=dG, and can exhibit topological quantization in terms of
deRham period integrals as topologically coherent sructures. However when the
Poincare invariants do not vanish they become the source of topological
evolution, irreversible changes of phase, topological defects, and stationary
states far from equilibrium.
A Topological Perspective of the Hall effect.
Updated July 30, 2003
Electromagnetic processes (currents) that preserve the closed integrals of
charge as evolutionary topological deformation invariants are necessarily of the
form J = rho V + sigma B. The result is that the topological Hall current (the
sigma B term -- not sigma x E !) is of a purely topological origin, is not
dissipative, and is valid at all scales.
A Topological Perspective of the Arrow of time and Thermodynamic
irreversibility.
Updated July 30, 2003
The Cartan dynamical equations, describing the continuous topological
evolution of physical systems whose topology may be encoded in terms of an
exterior differential 1-form of Action, demonstrate that evolution in the
direction of the topological torsion vector
is thermodynamically irreversible on domains of topological (Pfaff)
dimension of 2n+2. On the otherhand, on domains of topological dimension 2n+1
there exist unique extremal directions for which the evolution is conservative
and not thermodynamically irreversible.
A Topological Perspective of a Non-equilibrium Cosmology
Updated July 30,2003
The topological thermodynamics of a very dilute van der Waals gas indicates
that the universe could be modeled as a turbulent non-equilibrium open state of
Pfaff topological dimension 4. Stars and galaxies appear by irreversible
thermodynamic processes as self organizing topological defects (condensations)
of Pfaff topological dimension 3, embedded in the turbulent dissipative, but
very dilute, non-equilibrium medium of Pfaff dimension 4. The defect states are
not equilibrium states, but are "stationary states" of Pfaff dimension 3, hence
far from equilibrium (equilibrium requires a Pfaff topological dimension of 2 or
less). The model gives a reason for the granularity of the night sky as well as
a reason why the density fluctuations or defects (stars) attract one another
with the Newtonian gravitational law.
A Topological Perspective of Non-equilibrium Electromagnetism.
Updated July30, 2003
Modern electromagnetic theory recognizes that the classic Maxwell-Faraday
and Maxwell-Ampere field equations belong to two thermodynamically distinct
topological categories. The classic Maxwell PDE's can be deduced from a Cartan
exterior differential system, based on two topological postulates, F-dA=0, and
J-dG=0, independent from any geometrical constraints imposed by a metric, or
connection, or gauge, and for any geometric dimension greater than 3. Both the
potentials, A, and the field excitations, G, are not uniquely defined by the
field equations, a fact which leads to topological defects, in both equilibrium
conservative and turbulent dissipative systems. The non-equilibrium topological
defects are represented by the 3-forms of Topological Torsion, A^F, and
Topological Spin, A^G.
The Third Law of Thermodynamics
Updated July 2, 2003
A topological interpretation implies that the First Law of thermodynamics is
equivalent to a cohomology on 1-forms: the difference between two non-exact
1-forms is a perfect differential, or Q-W =dU. The Third Law is equivalent to a
cohomology on 3-forms: the difference between two non-exact 3-forms is a
perfect differential. The law is useful in the study of irreversible processes
and non-equilibrium systems.
Intrinsic hydrodynamics with applications to space-time fluids.
Updated Feb 28, 2003
A download version of the a somewhat hard to find paper (1975) which
establishes hydrodynamics of continuous media in terms of evolutionary
invariance of differential forms. The methods are intrinsic in that they do not
depend upon metric or connection.
Topological Torsion, Pfaff Dimension and Coherent structures
Updated Feb 27, 2003
A download version of the first (somewhat hard to find) paper, presented at
Cambridge (1989), which introduced the thermodynamic features of topological
evolution, topological torsion and Pfaff dimension as applied to continuous
media.
Compact Dissipative flow stuctures with Topological Coherence embedded in
Eulerian environments.
Updated Feb 27, 2003
A download version of the (somewhat hard to find) paper, presented in Russia
(1990) where examples of the evolutionary creation and distruction of coherent
topological structures were described. These concepts form the prototype
descriptions of the formation of galactic and stellar condensates in a
gravitational fluid, or for Bose condensates in a quantum fluid. (This is a
large pdf file with illustrations - 1.3 mbtyes )
Topological Parity and the Turbulent State of a Navier-Stokes fluid
Updated Sept 30, 2002
Topological Parity cannot be zero in a turbulent fluid.
Topological Defects, Coherent Structures and Turbulence
Updated Sept 30, 2002
A pdf version of a 1992 article that may be hard to find in your library.
Fractals and Homogeneous Differential Forms.
Updated Sept 30, 2002
A remarkable relationship exists between homogeneous differential forms,
period integrals, fractals, and thermodynamic irreversibility.
How to survive a Nuclear Attack by Terrorists.
Updated Sept 10, 2002
If you are not turned to toast in the fireball, you have a good chance of
surviving a nuclear detonation if you know what to do. This short presentation
describes the three basic rules of what to do after a nuclear detonation.
A topological perspective of the Arrow of time and Thermodynamic
irreversibility.
Updated Feb 14, 2002
The Cartan dynamical equations, describing the continuous topological
evolution of physical systems whose topology may be encoded in terms of an
exterior differential 1-form of Action, demonstrate that evolution in the
direction of the topological torsion vector
is thermodynamically irreversible on domains of topological (Pfaff)
dimension of 2n+2. On the otherhand, on domains of topological dimension 2n+1
there exist unique extremal directions for which the evolution is conservative
and not thermodynamically irreversible. (Preliminary draft version of upcoming
presentation)
Buzz Parsec Notes 1: Hot Diamonds and Dark Matter
Updated Jan 25, 2002
The Diamond form of a Carbon lattice does not radiate below the DeBye
temperature. Could the "missing mass" associated with "dark matter" consist of
Diamond Stars?
Transformation behavior of Maxwell's PDE's.
Updated Dec 3, 2001
The transformation properties of Maxwell's Electrodynamics, when written in
terms of exterior differential forms, are carried out in detail for combinations
of Euclidean translations and rotations. Contrary to current dogma, when the
field intensities E and B are transformed as covariant tensors, and when the
field excitations D and H are transformed properly as contravariant tensor
densities, then the PDE's that represent Maxwell's equations are INVARIANT in
form for all reference systems. The relative motions do not change the PDE's
but do modify the constitutive relations between E,B and D,H.
Differential forms are NOT necessarily tensors.
Updated Oct 22, 2001
Differential Forms are well defined with respect to the Pullback even when
the mapping is not a diffeomorphism. Hence differential forms can carry
topological information and can be used to describe topological evolution.
Tensors (defined with respect to diffeomorphisms) cannot be used to describe
topological evolution.
Charge as a Pseudo-Scalar.
Updated Oct 16, 2001
E. J. Post presented arguments (1977) that charge should be considered as a
pseudoscalar, with enantiomorphic pairs of plus and minus values. Topological
arguments are presented which yield results in agreement with Post's derivation,
which was based on the Newmann principle of crystallographics, and in
disagreement with the conventional dogma of Henley and Sakurai.
Frame and Metric perturbations in 4D - A Schwarzshild solution with Torsion.
Updated July 9, 2001
G. Shipov's proposal that the physical vacuum is an An space, with inertial
terms coming from torsion, is extended to the conjecture: The Universe is an An
space, with inertial terms coming from both Frame and metric perturbations. An
example is given that produces a Schwarzshild metrical solution with a Frame
field torsion component.
Frame vs. Metric Connections, and their Curvatures.
Updated July 6, 2001
For An spaces, non-zero metric curvature can be balanced by non-zero Ricci
rotation curvature (which can include torsion effects). The "An" condition of
zero total curvature establishes an equivalence between gravitational (metric)
mass and inertial (Ricci rotation and torsion) mass.
Nanometer Vortex Defects
Updated 03/08/01 - 08/25/2001
A correlation is established between nanometer vorticity in quantum
mechanics, Landau Ginsburg superconductors, and a Navier-Stokes fluid. "
The Hopf Map and Minimal surfaces of imaginary principal curvatures
Updated 12/29/00 - 08/25/2001
Implicit Hypersurface theory indicates that the Hopf Map represents a
Minimal 2-surface with Positive Gauss curvature.
Curvature of Implicit Hypersurfaces and the Origin of Charge and Internal Energy
Updated 12/27/00 - 08/25/2001
Cartan techniques indicate an equivalence between the cubic Adjoint
curvature of an Implicit Hypersurface in 4D (a 3 brane), internal energy density
and the interaction energy density of electromagnetic systems.
The Photon Spin and other Topological Features of Classical Electromagnetism
Updated 10/25/00 - 08/25/2001
A contribution to the Vigier 2000 conference in San Franciso, August 2000
(PDF format)
An Intrinsic Transport Theorem
Updated 10/15/00 - 08/23/2001
The original research (1969) article that gave birth to Cartan's Corner
Spinors, Minimal Surfaces, Torsion, Helicity, Chirality, Spin, Twistors,
Orientation, Continuity, Fractals, Point Particles, Polarization, the Light Cone
and the Hopf Map.
Updated 08/01/00 - 08/23/2001
A connection is established between all the concepts in the title!
Chemistry, Topological Parity and Chiral separation
Updated 06/12/00 - 08/23/2001
The interplay between irreversible processes and electromagnetic fields of
Pfaff dimension 4.
Curvature and Coriolis
Updated 04/12/00 - 08/23/2001
Coriolis effects can be abstractly defined in terms of Cartan's connection
on arbitrary manifolds
Electromagnetic Waves in the Vacuum with Torsion and Spin
Updated 03/21/00 - 08/23/2001
New time dependent solutions to MAxwell's Equations that exhibit torsion and
spin, and are not time reversible.
Topological Torsion and Spin form Coherent Structures in Plasmas and
electromagnetic media.
(Updated 03/14/00 - 08/23/2001 )
The general theory with numerous example closed form solutions.
Chirality and Helicity vs Topological Spin and Topological Torsion
(Updated 03/07/00 -- 08/23/2001 )
Optical Activity <=> Topological Spin; Faraday rotation <=> Topological
Torsion
Optical Vortices and Topological Torsion
Updated 03/07/00 - 08/23/2001
The vortex defects observed in Laser beams <=> Falaco Solitons
Introducing Buzz Parsec, a new contributor to Cartan's Corner
Updated 12/29/00 - 08/23/2001
Buzz is developing a new book entitled "Cosmological Strings, Falaco
Solitons, and the Origin of Life"
PDF file Notes on Holonomic and Anholonomic Constraints, Frobenius
intgrability, and Torsion of various types.
Updated 12/29/99 - 08/23/2001
A pdf file that explains the differences and utility of holonomic and
anholonomic constraints and differential forms.