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.
Copenhagen conference (supposedly to include) new ideas in thermodynamics.
This is the first conference I have attended where the venue was changed on the
day
Torsion of deformation is embeddable the plane
The first of two movie clips demonstrating the difference between the torsion of
deformation and topological torsion.
Topological Torsion cannot be embedded in the plane.
The second of two movie clips demonstrating the difference between the torsion
of deformation and topological torsion.
The Universal Effectivness of Topological Thermodynamics.
This essay is a powerpoint slide show presented in pdf format. The essay spells
out the important concepts that form the basis of Topological Thermodynamics,
with a minimum of mathematical detail. The starting point is the assumption
that a thermodynamic system can be encoded in terms of a fundamental 1form of
Action (per unit mole), and that thermodynamic processes can be encoded in terms
of an N1 form, or current.
NonEquilibrium Thermodynamics and the Kolmogorov Topology.
Thermodynamics, Hydrodynamics, and Electrodynamics have a common thread based on
Topological continuity. Topological thermodynamics can be built upon: i: a
1form of Action, A(x,y,z,t), that encodes a specific Thermodynamic System, and
ii: a set of vectorspinor direction fields, V(x,y,z,t), that define the Dynamic
Processes acting on the specific Thermodynamic System. iii: A
KolmogorovCartan T0 topology with subsets in terms of exterior differential
forms. The methods lead to precise, nonstatistical, methods for determining
when a process, V(x,y,z,t), applied to a specific thermodynamic system,
A(x,y,z,t), is 1.Thermodynamically irreversible or not, 2. Adiabatic or not,
3.Adiabatically irreversible or reversible. The details of the the
KolmogorovCartan T0 topology will be displayed, along with several applications
in terms of SpinTronics and Topological Insulators.
The Nonequilibrium Thermodynamic Environment and Prigogine's Dissipative
Structures.
This essay is based on the fundmental assumption that any physical system of
synergetic parts is a thermodynamic system. The universality of thermodynamics
is due to the fact that thermodynamic homogeneous properties, such as pressure,
temperature and their analogs, do not depend upon size or shape. That is,
thermodynamics is a topological (not a geometrical) theory. By use of Cartan’s
methods of exterior differential forms and their topological properties of
closure, it is possible to define precisley, and construct examples for, the
universal concepts of:
[1] Continuous Topological Evolution of topological properties  which in effect
is a dynamical version of the First Law. [2] Topological Torsion and Pfaff Topological Dimension  which distinguishes equilibrium (PTD < 3, TT = 0) and nonequilibrium systems (PTD > 2, TT 6= 0).
[3] A Topological Thermodynamic Environment  of PTD = 4. [4] Thermodynamic irreversible processes, which cause selfsimilar evolution in the environment, and emergence of selforganized states of PTD = 3 as topological defects in the PTD = 4 environment. These results clarify and give credence to Prigogine’s conjectures about dissipative structures.
[5] A universal thermodynamic phase function,Theta, which can have a singular
cubic factor equivalent to a deformed, universal, van der Waals gas. This van
der Waals gas admits negative pressure and dark matter properties, which are
current themes in Astronomy and GR.
Irreversible Processes and the NavierStokes equations (updated and expanded)
The concept of Continuous Topological Evolution, based upon Cartan's methods of
exterior differential systems, is used to develop a topological theory of
nonequilibrium thermodynamics, within which there exist processes that exhibit
continuous topological change and thermodynamic irreversibility. The technique
furnishes a universal, topological foundation for the partial differential
equations of hydrodynamics and electrodynamics; the topological technique does
not depend upon a metric, connection or a variational principle. Certain
topological classes of solutions to the NavierStokes equations are shown to be
equivalent to thermodynamically irreversible processes. The method
demonstrates, by example, how an irreversible dissipative process acting in an
Open nonequilibrium system of Pfaff topological dimension 4 can decay, or
create in finite time, topological defect structures, or Closed systems of Pfaff
topological dimension 3. These Closed nonequilibrium systems admit a
Hamiltonian process which can emulate the geometrical evolution of topological
stationary states far from equilibrium. The theory of Continuous Topological
Evolution gives formal credence, as well as analytic examples, to the Prigogine
conjecture of selforganization in terms of disspative (thermo)dynamics.
Schwarzschild metric.
Prigogine's Thermodynamic Emergence and Continuous Topological Evolution
Irreversible processes in Open nonequilibrium thermodynamic systems, of
topological dimension 4, can decay locally to Closed nonequilibrium
thermodynamic states, of topological dimension 3, by means of continuous
topological evolution. These topologically coherent, perhaps deformable, states
of one or more components appear to "emerge" as compact 3D Contact submanifolds,
defined as topological defects in the 4D Symplectic manifold. These emergent
states are still far from equilibrium, as their topological (not geometrical)
dimension is greater than 2. The 3D Contact submanifold admits a unique
extremal Hamiltonian process (as well as fluctuation components). If the
subsequent evolution is dominated by the Hamiltonian component, the emergent
topological defects will maintain a relatively longlived, topologically
coherent, approximately nondissipative structure. These defect structures
yield an evolutionary behavior that can be associated with the idea of
"stationary states" far from equilibrium. If the fluctuation (spinor)
components are weak, but not zero, the emergent thermodynamic structure will
ultimately decay, but only after a substantial "lifetime". Analytic solutions
and examples of these processes of continuous topological evolution give
credence, and a deeper understanding, to the general theory of selforganized
states far from equilibrium, as conjectured by I. Prigogine. Moreover, in an
applied sense, universal engineering design criteria can be developed to
minimize irreversible dissipation and to improve system efficiency in general
situations. As the methods are based on universal topological, not geometrical,
ideas, the general thermodynamic results apply to all synergetic topological
systems. It may come as a surprise, but ecological applications of
thermodynamics need not be limited to the design specific hardware devices, but
apply to all synergetic systems be they mechanical, biological, economical or
political.
This article examines how the physical presence of field energy and
particulate matter can be interpreted in terms of the topological properties of
space time. The theory is developed in terms of vector and matrix equations of
exterior differential forms. The theory starts from the sole postulate that
field properties of the Cosmological Vacuum (a continuum) can be defined in
terms of a vector space domain, of maximal rank, infinitesimal, neighborhoods,
where exact differentials are mapped into exterior differential 1forms, A>, by
a Basis Frame of C2 functions, [B], with nonzero determinant. The particle
properties of the Cosmological Vacuum are defined in terms of topological
defects (or compliments) of the field vector space, where the nonzero
determinant condition fails. When the exterior differential 1forms, A>, are
not uniquely integrable, the fibers can be twisted, leading to possible Chiral
matrix arrays of certain 3forms of Topological Torsion and Topological Spin.
In addition, there exist Chiral objects constructed from vector arrays of
1forms that mimic the properties of the Einstein tensor. The abreviated theory
is backed up by a detailed example in Maple format for the isotropic
Schwarzschild metric.
Is the Chiral Universe Rotating?
In electromagnetic systems, chirality can be related to constitutive
properties that link D to B and H to E. Optical activity is associated with the
imaginary part of a chirality linkage, and FresnelFizeau phenomena can be
related to the real part. A combination of FresnelFizeau rotation and Optical
Activity can break the inboundoutbound symmetry of propagating electromagnetic
singularities, an effect that can be measured in dualpolarized ring lasers. As
the chirality concept is related to centers of symmetry, or fixed points of
expansion and rotation, and as the universe appears to be expanding, it is
natural to ask: Is the universe rotating, as well as expanding?
A Thermodynamic Explanation of why electrons in a Bohr orbit do not radiate.
The 1form of Action that encodes the thermodynamic properties of an orbiting
electron consists of two components. One component represents the acceleration
of contraction to a fixed point  the inverse r squared law. The second
component represenents the acceleration due to a rotation about the fixed point.
It is easy to show that the Pfaff Topological DImension of the 1form is in
general 3 for the specified Action 1form of two components. Hence the
nonequilibrium thermodynamic system is closed, and will exchange radiation to
its environment. However, if the second term representing the rotation is
homogeneous of degree zero, it then corresponds to deRham period integral, which
when integrated around a closed cycle has values with integer ratios. The Pfaff
topological dimension of the thermodynamic 1form becomes equal to 2, and the
resulting thermodynamic system is an isolated equilibrium system that does not
exchange matter or radiation with its environment. The "quantized" orbiting
electron does not radiate even thought it is accelerated.
Turbulence and the Navier Stokes Equations
The concept of Continuous Topological Evolution, based upon Cartan’s methods of
exterior differential systems, is used to develop a topological theory of
nonequilibrium thermodynamics, within which there exist processes that exhibit
continuous topological change and thermodynamic irreversibility. The technique
furnishes a universal, topological foundation for the partial differential
equations of hydrodynamics and electrodynamics; the topological technique does
not depend upon a metric, connection or a variational principle. Certain
topological classes of solutions to the NavierStokes equations are shown to be
equivalent to thermodynamically irreversible processes. The method demonstrates,
by example, how an irreversible dissipative process acting in an Open
nonequilibrium system of Pfaff topological dimension 4 can decay, or create in
finite time, topological defect structures, or Closed systems of Pfaff
topological dimension 3. These Closed nonequilibrium systems admit a
Hamiltonian process which can emulate the geometrical evolution of topological
stationary states far from equilibrium. The theory of Continuous Topological
Evolution gives formal credence, as well as analytic examples, to the Prigogine
conjecture of selforganization in terms of disspative (thermo)dynamics (written
in response to the Clay Institute millenium challenge).
Prigogine'a Thermodynamic Emergence and Continuous Topological Evolution
Irreversible processes can be described in Open nonequilibriumthermodynamic
systems, of topological dimension 4. By means of Continuous Topological
evolution, such processes can cause local decay to Closed nonequilibrium
thermodynamic states, of topological dimension 3. These topologically coherent,
perhaps deformable, regions or states of one or more components appear to
"emerge" as compact 3D Contact submanifolds that can be defined as topological
defects in the 4D Symplectic manifold. These emergent states are still far from
equilibrium, as their topological (not geometrical) dimension is greater than 2.
The 3D Contact submanifold admits evolutionary processes with a unique extremal
Hamiltonian vector component, as well as fluctuation spinor components. If the
subsequent evolution is dominated by the Hamiltonian component, the emergent
topological defects will maintain a relatively longlived, topologically
coherent, approximately nondissipative structure. These topologically coherent,
"stationary states" far from equilibrium ultimately will decay, but only after a
substantial "lifetime". Analytic solutions and examples of these processes of
Continuous Topological Evolution give credence, and a deeper understanding, to
the general theory of selforganized states far from equilibrium, as conjectured
by I. Prigogine. Moreover, in an applied sense, universal engineering design
criteria can be developed to minimize irreversible dissipation and to improve
system efficiency in general nonequilibrium situations. As the methods are
based on universal topological, not geometrical, ideas, the general
thermodynamic results apply to all synergetic topological systems. It may come
as a surprise, but ecological applications of thermodynamics need not be limited
to the design of specific hardware devices, but apply to all synergetic systems,
be they mechanical, biological, economical or political. (submitted for
consideration of the WTI Prigogine award)
The powerpoint presentations of a talk, "Nonlinear, Topologically Coherent, and Compact Flows Far from Equilibrium" given at the EGU 2007 conference are made available for those interested.
The fundamental idea is that topological thermodynamics predicts the production
of topological defect structures
The Pfaff Topological dimension 3 subdomains are thermodynamic systems that are
far from equilibrium,
Vienna EGU
April 20, 2007 "Part1"
