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.