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Maple Solutions for equilibrium and non-equilibrium Electromagnetism .
The Maple program, ConstCurrents.pdf, is a 38 MB Maple program that evaluates Maxwell-Faraday
intensive variables (E and B) from the topological postulate of potentials F-dA = 0
and Maxwell-Ampere intensive variables (D and H) from the topological postulate of charge currents, J-dG = 0.
Many familiar and unfamiliar examples are displayed. The topological methods are independent from a choice of
geometric coordinates, and are valid for equilibrium or non-equilibrium plasma systems on
topological spaces of Pfaff Topological Dimension 4.
The techniques are valid for Charge currents, Spin currents, Topological Torsion currents, and cubic Adjoint currents.
For simplicity, the intensive and extensive variables are assumed to be related by
the Lorentz-Minkowski vacuum constitutive equations, D = epsilon E, B = mu H.
BE AWARE. The algebraic effort required to solve by hand some of the results presented
can be overpowering, so give thanks to Maple, now.
The Category Theory of Topological Thermodyanmics.
Category theory, whose objects are T2, T0, and Not-T0 topologies, based on sets
of exterior differential forms, and whose morphisms are based upon Cartan's Magic
formula of homotopy can be used as a basis for both
the thermodynamical theory of distinguishable particles (mass),
AND the thermodynamical theory of statistical distributions (radiation)
whose complex wavelets are indistinguishable.
Boson condensates, the Arrow of time, Non-metric Gravity, and Finite Lattice
Structures.
Herein, a lattice structure, LS, is defined as a collection S of the subsets of
the power set, PS(X), of N ingredients ("points"), X={a,b,c,d...}. The elements
of the LS are, by edict, denoted as Open sets, and often are used to define a
set of lattice vertices. The collection of LS Open sets may or may not obey the
axioms of a topology, yet most of the set operations remain valid even if the LS
is not equivalent to a Topological Structure, TS. For finite LS that generate
topologies, TS, there are three topological categories based upon how the
ingredients of X may, or may not, be distinguished (the separation axioms). If
the ingredients can be distinguished by metric (geometrical) methods, then the
topologies belong to category 1 and must satisfy the Hausdorff T2 separation
axiom. If the ingredients cannot be distinguished by geometric methods, but can
be distinguished by topological methods, then the topologies belong to category
2 and must satisfy the Kolmogorov T0 (but not the T2) separation axioms.
Topologies in the third category 3 obey the topological axioms but not the
separation axioms. Hence some, if not all, of the ingredients for topologies
that belong to category 3 are indistinguishable. It appears that category 1
topologies are the basis of equilibrium thermodynamic systems, with uniquely
integrable, reversible, processes. There is evidence that Category 2 topologies
are the basis for non-equilibrium thermodynmaic systems, with non-uniquely
integrable, ireversible processes. Note that the boundary of a boundary is
not empty, for Not-T0 Topologies that have indestinguisable sets.
The Universal Effectiveness of Topological Thermodynamics.
This pdf file summarizes the ideas and important results of a theory of
non-equilibrium thermodynamics based upon sets of exterior differential forms
and the Kolomogorov-Cartan T0 (poset 3) topology. The presentation was made at
the 2009 Haceteppe Topology conference in Ankara, Turkey. An improved presentation
using ideas benerated 2009 to 2010 appears above in terms of Category theory.
Twisted Fiber Bundles, Torsion of continuous deformation, and Topological
Torsion The pdf file contains 3 short movies demonstrating the differences between
geometric torsion and Topological torsion.
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