Nonlinear, Topologically Coherent, and Compact Flows Far from Equilibrium

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Non-linear, Topologically Coherent, and
Compact Flows Far from Equilibrium
R.M.K. EGU Vienna 6.06 April 20, 2007
From the point of view of Continuous Topological
Evolution

• R. M. Kiehn
• University of Houston
• www.cartan.pair.com

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Part IV Macroscopic Spinors,Contact and
SymplecticThermodynamic Manifolds
R.M.K. EGU Vienna 6.06 April 20, 2007
From the point of view of Continuous Topological
Evolution

• R. M. Kiehn
• University of Houston
• www.cartan.pair.com

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Recall the encoding Axioms of Continuous
Topological Thermodynamic
Evolution
Systems Differential 1-form of Action,
Ak(x,y,z,t)
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Recall the encoding Axioms of Continuous
Topological Thermodynamic
Evolution
Systems Differential 1-form of Action,
Ak(x,y,z,t) Processes Vector Direction Fields,
Vk(x,y,z,t).
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Recall the encoding Axioms of Continuous
Topological Thermodynamic
Evolution
Systems Differential 1-form of Action,
Ak(x,y,z,t) Processes Vector Direction Fields,
Vk(x,y,z,t). Dynamics Cartans Magic Formula
(the Lie Differential) L(V) A i(V)dA
d(i(V)A) Q
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Recall the encoding Axioms of Continuous
Topological Thermodynamic
Evolution
Systems Differential 1-form of Action,
Ak(x,y,z,t) Processes Vector Direction Fields,
Vk(x,y,z,t). Dynamics Cartans Magic Formula
(the Lie Differential) L(V) A i(V)dA
d(i(V)A) Q l
l The First Law !!! W dU
Q
Work
d(Internal Energy) Heat
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Isotropic Pure Spinors (1913)
E. Cartan defined Pure Spinors, Sp gt as
Complex Vector Arrays of functions with a zero
quadratic form
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Isotropic Pure Spinors (1913)
E. Cartan defined Pure Spinors, Sp gt as
Complex Vector Arrays of functions with a zero
quadratic form lt Sp ? Sp gt 0.
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Isotropic Pure Spinors (1913)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0.
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Isotropic Pure Spinors (1913)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0. describe conjugate - pair representations
of Minimal Surfaces
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Isotropic Pure Spinors (1913)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0. describe conjugate - pair representations
of Minimal Surfaces Tangential Discontinuities
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Isotropic Pure Spinors (1913)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0. describe conjugate - pair representations
of Minimal Surfaces Tangential Discontinuities
Harmonic Vector Fields
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Isotropic Pure Spinors (1913)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0. describe conjugate - pair representations
of Minimal Surfaces Tangential Discontinuities
Harmonic Vector Fields Diffraction Waves
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Isotropic Pure Spinors (1913)
E. Cartan gave several complex mapping
constructions a Xv(-1)Y, ßZv(-1)S
Note the CHIRAL ambiguity due to the sign
! s1 a²-ß² s2 v(-1)(a²ß²) s3
2aß ltSpSpgt s1²s2²s3² 0.
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Isotropic Pure Spinors (1913)
E. Cartan gave several complex mapping
constructions a Xv(-1)Y, ßZv(-1)S
Note the CHIRAL ambiguity due to the sign
! s1 a²-ß² X²-Y²-Z²S²
v(-1)(2YX-2SZ), s2 v(-1)(a²ß²) - 2YX-2SZ
v(-1)(X²-Y²Z²-S²), s3 2aß 2XZ-2YS
v(-1)(2YZ2XS), ltSpSpgt s1²s2²s3² 0.
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Isotropic Pure Spinors (2007)
(CONJECTURE)
Pure Spinors, Sp gt ltSpSpgt s1²s2²s3²
0. describe conjugate - pair representations
of Functions that describe evolution to
Singularities in Finite Time!
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Now consider the 1-form of Work
W i(V)dA w fkdxk Pdt
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Now consider the 1-form of Work
W i(V)dA w fkdxk Pdt The Pfaff
Topological Dimension of W refines the
topological evolution
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Now consider the 1-form of Work
W i(V)dA w fkdxk Pdt The Pfaff
Topological Dimension of W refines the
topological evolution The 2-form, dA has
an Antisymmetric Matrix representation i(V)dA
Fmn ? Vngt
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Isotropic Pure Spinors and Work
In Matrix notation, the Work 1-form, W i(?V4)dA
becomes W ltdx, dy, dz, dt ? F ? ?V4gt The
Work 1-form is not identically zero, but the
skew-symmetric self product can be written as,
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Isotropic Pure Spinors and Work
In Matrix notation, the Work 1-form, W i(?V4)dA
becomes W ltdx, dy, dz, dt ? F ? ?V4gt The
Work 1-form is not identically zero, but the
skew-symmetric self product can be written
as, i(?V4)W i(?V4)i(?V4)F i(?V4)? gt Skew
product lt?V4 ? F ? ?V4gt 0 The
skew-symmetric self product is always zero by the
antisymmetry of F .
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Isotropic Pure Spinors and Work
Suppose e gt is an Eigenvector of F F ? e
gt g e gt Then (DUE TO ANTISYMMETRY lt e
?F e gt g lt e ? e gt 0 There are 2
choices Vectors ? 0, lt e ? e gt ? 0 Spinors
? ? 0, lt e ? e gt 0 Spinors have a CHIRAL
ambiguity
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Kinematic Perfection and Fluctuations
The usual perspective of a Vector field is
based on the kinematic topological
constraint dxk Vkdt 0
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Kinematic Perfection and Fluctuations
The usual perspective of a Vector field is
based on the kinematic topological
constraint dxk Vkdt 0 This is a severe
topological constraint that requires that V can
be singly parameterized. A better choice is to
consider the Fluctuation 1-form Dxk dxk
Vkdt ? 0
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Now reconsider the 1-form of Work
W i(V)dA w fmdxm Pdt Compare the
equivalent format W fk? Dxk fmdxm (fm ?
Vm)dt
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Now reconsider the 1-form of Work
W i(V)dA w fmdxm Pdt Compare the
equivalent format W fk? Dxk fmdxm (fm ?
Vm)dt If the Work W is zero, then the
fluctuation Dx is zero. If W is not zero, then
the fluctuation Dx is NOT zero. BUT if W is not
Zero it must have SPINOR COMPONENTS
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Now reconsider the 1-form of Work
Conclusion Spinors must be the source of
topological fluctuations, and the failure of the
Kinematic Hypothesis. Fluctuations in Position gt
Pressure Fluctuations in Velocity gt Temperature
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Summary Spinors and Work
Path dependent Work, irreversibility, and
topological fluctuations imply that evolutionary
processes, V, must have components proportional
to Pure Spinors. Classical historic
methods ignore the possibilities
that processes can have
Spinor, as well as Vector,
components !!!.
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Summary Spinors and Work
Spinors have a CHIRAL ambiguity Vectors do not.
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Summary Spinors and Work
Spinors have a CHIRAL ambiguity Vectors do
not. Recognition of Chiral Symmetry Breaking can
lead to many new practical applications of
non-equilibrium thermodynamics, especially in
Biological and Plasma systems.
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Isotropic Pure Spinors and Work

• Pure Spinors are associated with the non-zero
  antisymmetric matrix F of functional
  coefficients, generated by the 2-form, FdA.
• There are two cases depending on the rank of
  2-form F, or the Pfaff
  Topological Dimension of A.
• If the PTD of A is odd, then F generates a
  Contact Structure.
• If the PTD of A is even, then F generates a
  Symplectic Structure.

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Eigendirection Fields of Fmn - Fnm
i(V)dA Fmn Vngt g Vngt If the matrix is
odd dimensional then there exists One
eigendirection field has a Zero eigenvalue. All
other eigendirection fields, s, are complex with
pure imaginary eigenvalues, such that lt s s gt
0. If the matrix is even dimensional, then ALL
eigendirection fields are such that lt s s gt
0.
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Work i(V)dA 0 possible only on odd dimensions
On Odd 2n1 dimensions FdA Defines a Contact
Manifold
If Work i(V)dA ? 0 then V must be composed of
Spinors
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Work i(V)dA ltgt 0 always on even dimensions
On Even 2n2 dimensions FdA Defines a Symplectic
Manifold
If Work i(V)dA ? 0 then V must be composed of
Spinors
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Work i(V)dA ltgt 0 always on even dimensions
On Even 2n2 dimensions FdA Defines a Symplectic
Manifold
If Work i(V)dA ? 0 then V must be composed of
Spinors
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Topological Fluctuations

• On a 3n1 variety p,v,x,t encode a 1-form of
  Action as
• A L(v,x,t) dt pk (dxk - vkdt) pk dxk H dt
  ?
• A L(v,x,t) dt pk ?xk
• Define the topological fluctuation in momenta as
• ?pk dpk - (? L/?xk) dt
• ?pk is a fluctuation about the kinematic concept
  of a Newtonian force.

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Topological Fluctuations

• Recall The kinematic assumption is a Topological
  Constraint
• ?xk dxk - vkdt ? 0.
• The Pfaff Topological Dimension of ?xk lt 3
• Define a transverse topological fluctuation in
  position as
• ?xk dxk - vkdt ? 0. (Pressure)
• The Pfaff Topological Dimension of ?x gt 2
• Define a transverse topological fluctuation in
  velocity as
• ?vk dvk - Akdt ? 0. (Temperature)
• The Pfaff Topological Dimension of ?v gt 2

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Topological Fluctuations
Then compute the elements of the Pfaff sequence
(for k1) Action A L(v,x,t)
dt p ?x Vorticity dA (? L/?v-p)
?vdt ?p?x Torsion AdA L ?p?xdt
- p(? L/?v-p) ?v?xdt Parity dAdA
- 2 (? L/?v-p) ?p?v?xdt These remarks can be
extended for any value of k.
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Topological Fluctuations
If there is no Pressure fluctuation, ?x 0
Action A L(v,x,t) dt Vorticity
dA (? L/?v-p) ?vdt Torsion
AdA 0 Parity dAdA 0 The system
is an isolated system of PTD 2, if there is a
finite temperature fluctuation, ?v ? 0, and the
momentum are not canonical, (? L/?v-p) ? 0. If
either condition is zero, then the system is an
equilibrium system of PTD 1.
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Topological Fluctuations
Assume the Heisenberg fluctuation is zero ?p?x
0 Then, Action A L(v,x,t) dt
p ?x Vorticity dA (? L/?v-p)
?vdt Torsion AdA - p(? L/?v-p)
?v?xdt Parity dAdA - 0 If p is
canonical, (? L/?v-p) 0, or if there is no
temperature, ?v0, the equilibrium system is of
PTD 1 If p is not canonical, and both ?v ? 0 and
?x ? 0, the non-equilibrium system is of PTD 3
(or PTD2 if there is no pressure term, ?x ? 0).
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Topological Fluctuations
Assume the momenta are canonical, (? L/?v-p) gt
0 Then, Action A L(v,x,t) dt
p ?x Vorticity dA ?p?x Torsion
AdA L ?p?xdt Parity dAdA
0 If the Heisenberg fluctuation is zero, the
equilibrium system is of PTD1 If the Heisenberg
fluctuation is not zero, the non-equilibrium
system is of PTD3
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Topological Fluctuations
NOW compute the elements of the Pfaff sequence
Action A L(v,x,t) dt p ?x
Vorticity dA (? L/?v-p) ?vdt
?p?x Torsion AdA L ?p?xdt - p(?
L/?v-p) ?v?xdt Parity dAdA - 2 (?
L/?v-p) ?p?v?xdt
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Topological Evolution

• Topological evolution (change) is a necessary
  condition for both logical time asymmetry (the
  Arrow of Time) and Thermodynamic Irreversibility.
• A unique Extremal direction field which
  represents a conservative reversible Hamiltonian
  process always exists on subspaces of topological
  dimension,
• 2n1 contact manifolds.
• A unique Torsional direction field which
  represents a thermodynamically irreversible
  process always exists on subspaces of even
  topological dimension,
• 2n2 symplectic manifolds.

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Topological Evolution

• Continuous Topological Evolution can describe the
  irreversible evolution on an
• Open symplectic domain of Pfaff dimension 4,
  with evolutionary orbits being irreversibly
  attracted to a
• Closed contact domain of Pfaff dimension 3,
  with topological defects (stationary states and
  coherent structures), and a possible ultimate
  decay to the
• Isolated-Equilibrium domain of Pfaff dimension
  2 or less (integrable Caratheodory surface).

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Contact Manifolds, n 2k1 .

• On a Contact Manifold there exists a
• unique Extremal process VE (the null eigenvector
  of dA).
• Such processes are Hamiltonian and reversible.
• The evolution obeys the Helmholtz-Poincare
  constraint (conservation of vorticity)
• L(VE)dA dQ 0.
• and all such evolutionary processes are therefore
• Thermodynamically Reversible.
• If U i(VE)A 1, then VE is called a Reeb field.

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Symplectic Manifolds n 2k2.

• On symplectic manifolds extremal fields do not
  exist.
• However, a unique direction field T can be
  defined in terms of the topological features of
  the physical system, A
• i(T)dxdydzdtAdA.
• Processes in the direction of the Torsion Vector,
  T, are
• Thermodynamically Irreversible,
  as
• L(T) A L(T) dA QdQ ?2 AdA ? 0.

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Lagrangian Example

• A Cartan-Hilbert 1-form of Action, A, for a
  physical system can be written as
• A L(v,q,t) dt pk (dqk - vkdt)
• The k1 base variables are t, q.
• The 2k fiber variables are v,p.
• The Lagrange function L(t,q,v,p) is a function of
  the 3k1 variables, (t,x,v,q) .

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Lagrangian Example

• However, direct computation shows that the
  maximum Pfaff topological dimension is 2k2, not
  3k1.
• and the top Pfaffian on the
• symplectic manifold is equal to
• (dA)k1 (k1)!(?L/?vk-pk)dvk?p?qdt
• ?p dp1.. dpn ?qdq1.. dqn
• Note that the symplectic momenta are NOT
  canonically defined
• (?L/?vk-pk) ? 0.

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Lagrangian Example

• As the top Pfaffian is a 2n2 Volume element, the
  coefficient must be a perfect differential.
• dS (?L/?vk-pk)dvk
• dS will be defined (tentatively) as an
• Entropy production rate

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Lagrangian Example

• Evolution starts on the 2k2 symplectic manifold
  with orbits being attracted to 2k1 domains,
  where the momenta become canonical p - ?L/?v ?
  0.
• Topological evolution can either continue to
  reduce the Pfaff topological dimension, or
• the process on the Contact 2k1 manifold can
  become extremal, and the topological change
  stops.
• The resulting contact manifold becomes a
  stationary non-dissipating Hamiltonian state, n
  gt 2.
• Far from Equilibrium.

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Fluctuations in Work W

• Consider the Lagrangian Work 1-form on a 2n2
  manifold,
• W i(V)dA (? L/?v-p) ?v ?p?x
• A Lagrangian symplectic system requires that
• (? L/?v-p) ?v ? 0, and ?p?x ? 0.
• The Work done on a symplectic system is never
  zero, and the momenta are NOT canonical. (?
  L/?v-p) ? 0
• The Heisenberg Fluctuation cannot be Zero. ?p?x
  ? 0
• The process must have fluctuations that produce
• Temperature and Pressure.

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Fluctuations in Work W

• Consider the Work 1-form on a 2n2 manifold, (?
  L/?v-p) ? 0.
• Work i(V)dA (? L/?v-p) ?v ?p?x
• Processes on a 2n2 symplectic manifold require W
  ? 0.
• To be a symplectic manifold requires that the
  first term in the expression for work, W, is not
  zero. The momenta cannot be canonical, and the
  Velocity fluctuations must be non-zero. This
  implies the existence of a non-zero temperature,
  and leads to the analogue of the Planck concept
  of a zero point energy on the symplectic 2n2
  topological manifold.

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Fluctuations in Work W

• Conclusions
• Spinor Fluctuations are at the foundations of
• Symplectic Lagrangian systems
• Non-canonical momentum (Entropy production)
• Pressure, ?x ? 0
• Temperature, ?v ? 0
• Heisenberg Uncertainty, ?p?x ? 0