Turbulence and the Navier-Stokes equations
Turbulence and the Navier-Stokes Equations
R. M. Kiehn
Emeritus Professor of Physics, University of Houston
Retired to Mazan, France
http://www.cartan.pair.com
Abstract: The concept of continuous topological evolution, based upon Car-
tan’s methods of exterior differential systems, is used to develop a topological
theory of non-equilibrium thermodynamics, within which there exist processes
that exhibit continuous topological change and thermodynamic irreversibility.
The technique furnishes a universal, topological foundation for the partial differ-
ential equations of hydrodynamics and electrodynamics; the technique does not
depend upon a metric, connection or a variational principle. Certain topological
classes of solutions to the Navier-Stokes equations are shown to be equivalent to
thermodynamically irreversible processes.
Prologue
THE POINT OF DEPARTURE
This presentation summarizes a portion of 40 years of research interests1 in applied
physics from a perspective of continuous topological evolution. The motivation for the
past and present effort continues to be based on the recognition that topological evolution
(not geometrical evolution) is required if non-equilibrium thermodynamic systems and irre-
versible turbulent processes are to be understood without the use of statistics. This essay
is written (by an applied physicist) as an alternative response to the (more mathematical)
challenge of the Clay Institute regarding the properties of the Navier-Stokes equations and
their relationship to hydrodynamic turbulence. To replicate a statement made by the Clay
Institute:
”The challenge is to make substantial progress toward a mathematical theory
which will unlock the secrets hidden in the Navier-Stokes equations.”
1This work is summarized in a series of reference monographs [40], [41], [42], [43], [44] which have been
constructed and updated from numerous publications. These volumes contain many examples and proofs of
the basic concepts.
http://arxiv.org/abs/0704.1596v1
http://www.cartan.pair.com
The point of departure starts with a topological (not statistical) formulation of Thermo-
dynamics, which furnishes a universal foundation for the Partial Differential Equations of
classical hydrodynamics and electrodynamics [40]. The topology that is of significance is
defined in terms of Cartan’s topological structure [34], which can be constructed from an ex-
terior differential 1-form, A, defined on a pre-geometric domain of base variables. The topo-
logical method extends the classical geometrical approach to the study of non-equilibrium
thermodynamic systems.
Claim 1 The topological method permits the conclusion that among the solutions to the
Navier-Stokes equations there are C2 smooth, thermodynamically irreversible processes which
permit description of topological change and the decay of turbulence.
In addition, the method permits examples to be constructed showing the difference be-
tween certain piecewise-linear processes which are reversible, but which are different from
certain smooth processes which are irreversible. Such concepts (of smoothness) seem to be
of direct interest to the challenge of the Clay Institute, and are to be associated with the
fact that there are differences between piecewise linear, smooth, and topological manifolds
(see p. 106, [38]).
However, the methods of topological thermodynamics go well beyond these types of
questions. In particular, the methods permit non-statistical engineering design criteria to
be developed for non-equilibrium systems. The theory of Topological Thermodynamics,
based upon Continuous Topological Evolution [35] of Cartan’s topological structure, can
explain why topologically coherent, compact structures, far from equilibrium, will emerge
as long-lived artifacts of thermodynamically irreversible, turbulent, continuous processes. I
want to present the idea that:
Theorem 2 The Pfaff Topological Dimension (PTD) of a Thermodynamic System can
change dynamically and continuously via irreversible dissipative processes from a non-equilibrium
turbulent state of PTD = 4 to an excited “topologically stationary, but excited,” state of PTD
= 3, which is still far from equilibrium! The PTD=3 state admits an extremal Hamiltonian
evolutionary process which, if dominant, produces a relatively long lifetime.
There exist C2 smooth processes that can describe the topological evolution from an
Open non-equilibrium turbulent domain of Pfaff Topological Dimension 4 to Closed, but
non-equilibrium, domains of Pfaff Topological Dimension 3, and ultimately to equilibrium
domains of Pfaff dimension 2 or less. The Topological domains of Pfaff Topological Di-
mension 3 emerge via thermodynamically irreversible, dissipative processes as topologically
coherent, deformable defects, embedded in the turbulent environment of Pfaff Topological
Dimension 4.
Now I am well aware of the fact that Thermodynamics (much less Topological Thermo-
dynamics) is a topic often treated with apprehension. In addition, I must confess, that as
undergraduates at MIT we used to call the required physics course in Thermodynamics, The
Hour of Mystery! Let me present a few quotations (taken from Uffink, [39]) that describe
the apprehensive views of several very famous scientists:
Any mathematician knows it is impossible to understand an elementary course
in thermodynamics ....... V. Arnold 1990.
It is always emphasized that thermodynamics is concerned with reversible pro-
cesses and equilibrium states, and that it can have nothing to do with irreversible
processes or systems out of equilibrium ......Bridgman 1941
No one knows what entropy really is, so in a debate (if you use the term entropy)
you will always have an advantage ...... Von Neumann (1971)
On the other hand Uffink states:
Einstein, ..., remained convinced throughout his life that thermodynamics is the
only universal physical theory that will never be overthrown.
I wish to demonstrate that from the point of view of Continuous Topological Evolution
(which is based upon Cartan’s theory of exterior differential forms) many of the mysteries of
non-equilibrium thermodynamics, irreversible processes, and turbulent flows, can be resolved.
In addition, the non-equilibrium methods can lead to many new processes and patentable
devices and concepts.
There are many intuitive, yet disputed, definitions of what is meant by turbulence, but
the one property of turbulence that everyone agrees upon is that turbulent evolution in a
fluid is a thermodynamic irreversible process. Isolated or equilibrium thermodynamics can
be defined on a 4D space-time variety in terms of a connected Cartan topology of Pfaff
Topological Dimension 2 or less. Non-equilibrium thermodynamics can be constructed in
terms of disconnected Cartan topology of Pfaff Topological Dimension of 3 or more. As
irreversibility requires a change in topology, the point of departure for this article will be to
use the thermodynamic theory of continuous topological evolution in 4D space-time. It will
be demonstrated, by example, that the non-equilibrium component of the Cartan topology
can support topological change, thermodynamic irreversible processes and turbulent solu-
tions to the Navier-Stokes equations, while the equilibrium topological component cannot.
In addition, it will be demonstrated that complex isotropic macroscopic Spinors are the
source of topological fluctuations and irreversible processes in the topological dynamics of
non-equilibrium systems. This, perhaps surprising, fact has been ignored by almost all re-
searchers in classical hydrodynamics who use classic real vector analysis and symmetries to
produce conservation laws, which do not require Spinor components. The flaw in such sym-
metrical based theories is that they describe evolutionary processes that are time reversible.
Time irreversibility requires topological change.
EXTERIOR DIFFERENTIAL FORMS
overcomes the
LIMITATIONS of REAL VECTOR ANALYSIS
During the period 1965-1992 it became apparent that new theoretical foundations were
needed to describe non-equilibrium systems and continuous irreversible processes - which
require topological (not geometrical) evolution. I selected Cartan’s methods of exterior
differential topology to encode Continuous Topological Evolution. The reason for this
choice is that many years of teaching experience indicated that such methods were rapidly
learned by all research scientists and engineers. In short:
1. Vector and Tensor analysis is not adequate to study the evolution of topology. The
tensor constraint of diffeomorphic equivalences implies that the topology of the initial
state must be equal to the topology of the final state. Turbulence is a thermodynamic,
irreversible process which can not be described by tensor fields alone.
2. However, Cartan’s methods of exterior differential systems and the topological perspec-
tive of Continuous Topological Evolution (not geometrical evolution) CAN be used to
construct a theory of non-equilibrium thermodynamic systems and irreversible pro-
cesses.
3. Bottom Line: Exterior differential forms carry topological information and can be
used to describe topological change induced by processes. Real ”Vector” direction-
fields alone cannot describe processes that cause topological change; but Spinor direc-
tionfields can.
A cornerstone of classic Vector (tensor) analysis is the constraint of functional equivalence
with respect to diffeomorphisms. However, diffeomorphisms are a subset of a homeomor-
phisms, and homeomorphisms preserve topology. Hence to study topological change, Vector
(tensor) analysis is not adequate. In topological thermodynamics, processes are defined in
terms of directionfields which may or may not be tensors. The ubiquitous concepts of 1-1 dif-
feomorphic equivalence, and non-zero congruences, for the eigen directionfields of symmetric
matrices do not apply to the eigen directionfields of antisymmetric matrices. The eigen di-
rection fields of antisymmetric matrices (which are equivalent to Cartan’s isotropic Spinors)
may be used to define components of a thermodynamic process, but such Spinors have a null
congruence (zero valued quadratic form), admit chirality, and are not 1-1. Where classic
geometric evolution is described in terms of symmetries and conservation laws, topological
evolution is described in terms of antisymmetries.
Cartan’s theory of exterior differential forms is built over completely antisymmetric struc-
tures, and therefore is the method of choice for studying topological evolution. The exterior
differential defines limit sets; the Lie differential defines continuous topological evolution.
The concept of Spinors arise naturally in theories using Cartan’s methods of exterior dif-
ferential forms; i.e., Spinors are not added to the theory ad hoc. The Cartan theory of
extended differential forms can be used to study topological change. The word extended is
used to emphasize the fact that differential forms are functionally well defined with respect a
larger class of transformations than those used to define tensors. Extended differential forms
behave as scalars with respect to C1 maps which do not have an inverse, much less an inverse
Jacobian. Both the inverse map and the inverse Jacobian are required by a diffeomorphism.
The exterior differential form on the final state of such C1 non-invertible maps permits the
functional form of the differential form on the initial state to be functionally well defined in
a retrodictive, pullback sense - not just at a point, but over a neighborhood.
Theorem 3 Tensor fields can be neither retrodicted nor predicted in functional form by
maps that are not diffeomorphisms [14].
CONTINUOUS TOPOLOGICAL EVOLUTION
Objectives of CTE The objectives of the theory of Continuous Topological Evolution
are to:
1. Establish the long sought for connection between irreversible thermodynamic processes
and dynamical systems – without statistics!
2. Demonstrate the connection between thermodynamic irreversibility and Pfaff Topolog-
ical Dimension equal to 4. The result suggests that “2-D Turbulence is a myth” for it
is a thermodynamic system of Pfaff Topological Dimension equal to 3 [21].
3. Demonstrate that topological thermodynamics leads to universal topological equiva-
lences between Electromagnetism, Hydrodynamics, Cosmology, and Topological Quan-
tum Mechanics.
4. Demonstrate that Cartan’s methods of exterior differential forms permits important
topological concepts to be displayed in a useful, engineering format.
New Concepts deduced from CTE The theory of Continuous Topological Evolution
introduces several new important concepts that are not apparent in a geometric equilibrium
analysis.
1. Continuous Topological Evolution is the dynamical equivalent of the FIRST LAW OF
THERMODYNAMICS.
2. The Pfaff Topological Dimension, PTD, is a topological property associated with any
Cartan exterior differential 1-form, A. The PTD can change via topologically contin-
uous processes.
3. Topological Torsion is a 3-form (on any 4D geometrical domain) that can be used to
describe irreversible processes. As a 4D non-equilibrium direction field it is completely
determined by the coefficient functions that encode the thermodynamic system. Other
process directionfields are determined by the system topology based upon the 1-form
of Action, A, and the refinement based on the topology of the 1-form of work, W .
4. Closed thermodynamic topological defects of Pfaff Topological Dimension 3 can emerge
from Open thermodynamic systems of Pfaff Topological Dimension 4 by means of irre-
versible dissipative processes that represent topological evolution and change. When
the topologically coherent defect structures emerge, their evolution can be dominated
by a Hamiltonian component (modulo topological fluctuations), which maintains the
topological deformation invariance, and yields hydrodynamic wakes [20] and other Soli-
ton structures. These objects are of Pfaff Topological Dimension 3 and are far from
equilibrium. They behave as if they were ”stationary excited” states above the equi-
librium ground state. Falaco Solitons are an easily reproduced hydrodynamic example
that came to my attention in 1986 [41] [33] .
PRESENTATION OUTLINE
The essay is constructed in several sections:
Section 1. Topological Thermodynamics In Section 1, the concepts of topological
thermodynamics in a space-time variety are reviewed (briefly) in terms of Cartan’s method
of exterior differential forms. A thermodynamic system is encoded in terms of a 1-form
of Action, A. Thermodynamic processes are encoded in terms of the Lie differential with
respect to a directionfield, V , acting on the 1-form, A, to produce a 1-form, Q. The process
directionfield can have Vector and Spinor components. The definition of the Lie differential
is a statement of cohomology and defines Q as the composite of a 1-form, W , and a perfect
differential, dU . The formula abstractly represents a dynamical version of the First Law of
Thermodynamics.
The existence of a 1-form on a 4D space-time variety generates a Cartan topology. If
the Pfaff Topological (not geometrical) Dimension of the 1-form of Action, A, is 2 or less,
then the thermodynamic system is an isolated or equilibrium system on the 4D variety. If
the Pfaff Topological Dimension of A is greater than 3, then the system is a non-equilibrium
system on the 4D variety. Examples of systems of Pfaff Topological Dimension 4 which admit
processes which are thermodynamically irreversible are given in the reference monographs
(see footnote page 1).
Section 2. Applications In Section 2, the abstract formalism will be given a specific real-
ization appropriate for fluids in general. First, an electromagnetic format will be described
because my teaching experience has demonstrated that the concepts of non-equilibrium
phenomena are more readily recognized in an electromagnetic format. Then it will be
demonstrated how the PDE’s representing the Hamiltonian version of the hydrodynamic
Lagrange-Euler equations arise from the constraint that the work 1-form, W , should vanish
(Pfaff Topological Dimension of W = 0). The Bernoulli flow will be obtained by constrain-
ing the thermodynamic Work 1-form to be exact, W = dΘ (Pfaff Topological Dimension
1), and the Helmholtz flow will follow from the constraint that the thermodynamic Work
1-form be closed, but not necessarily exact, dW = 0. Such reversible dynamical processes
belong to the connected component of the Work 1-form; irreversible processes belong to the
disconnected component of the Work 1-form.
Section 3. The Navier-Stokes system In Section 3, the topological constraints of
isolated equilibrium systems will be relaxed to produce more general PDE’s defining the
topological evolution of the system relative to an applied process. These relaxed topolog-
ical constraints will include the partial differential equations known as the Navier-Stokes
equations. The method used will be to augment the topology induced by the 1-form of
Action, A, by studying the topological refinements induced by the 1-form of Work, W . It
will be demonstrated that when the Pfaff Topological Dimension of A and W and Q are 4,
there exist C2 solutions (processes) to the Navier-Stokes equations which are thermodynam-
ically irreversible (the most significant property of turbulent flow). An interesting result is
the set of conditions on solutions of the Navier-Stokes equations that produce an adiabatic
irreversible flow.
Those topological refinements of the Work 1-form, required to include the Navier-Stokes
equations, can be related directly to the concept of macroscopic Spinors. Macroscopic,
complex Spinor solutions occur naturally in terms of the eigendirection fields of (real) anti-
symmetric matrices with non-zero eigenvalues, whenever the thermodynamic Work 1-form
is not zero. Spinors can also be associated with topological fluctuations of position and
velocity about kinematic perfection generated by 1-parameter groups. These topological
fluctuations are presumed to be representations of pressure and temperature.
Section 4. Closed States of Topological Coherence embedded as deformable
defects in Turbulent Domains One of the key interests of the Clay problem has to
do with the smoothness of the solutions to the Navier-Stokes equations. In Section 4, the
problem will be attacked from the point of view of thermodynamics. First, the properties
of the different species of topological defects will be discussed. These defects are non-
equilibrium closed domains (of PTD = 3) which can emerge by C2 smooth irreversible
process in open domains (of PTD = 4), as excited states far from equilibrium, yet with long
relative lifetimes. Falaco Solitons are an easily reproduced experimental example of such
topological defects, and are discussed in detail in [41].
The properties of two different species of PTD = 3 defect domains will be given in detail.
In addition, an analytic solution of a thermodynamically irreversible process that causes the
defect domain to emerge will be displayed.
Finally, an example will be given where by combinations of Spinor solutions produce piece-
wise linear processes. These piecewise linear processes are thermodynamically reversible,
while the Spinor solutions of which they are composed are not.
Section 5. Topological Fluctuations and Spinors In Section 5, a few concluding
remarks will be made about the ongoing research concerning topological fluctuations, as
generated by Spinors. The methods of fiber bundle theory are used extend the 4D thermo-
dynamic domain. Such topological fluctuations can be associated with fluid pressure and
temperature.
1 Topological Thermodynamics
1.1 The Axioms of Topological Thermodynamics
The topological methods used herein are based upon Cartan’s theory of exterior differential
forms. The thermodynamic view assumes that the physical systems to be studied can
be encoded in terms of a 1-form of Action Potentials (per unit source, or, per mole), A,
on a four-dimensional variety of ordered independent variables, {ξ1, ξ2, ξ3, ξ4}. The variety
supports a differential volume element Ω4 = dξ
1ˆdξ2ˆdξ3ˆdξ4. This statement implies that
the differentials of the µ = 4 base variables are functionally independent. No metric, no
connection, no constraint of gauge symmetry is imposed upon the four-dimensional pre-
geometric variety. Topological constraints can be expressed in terms of exterior differential
systems placed upon this set of base variables [1].
In order to make the equations more suggestive to the reader, the symbolism for the
variety of independent variables will be changed to the format {x, y, z, t}, but be aware that
no constraints of metric or connection are imposed upon this variety, at this, thermodynamic,
level. For instance, it is NOT assumed that the variety is spatially Euclidean.
With this notation, the Axioms of Topological Thermodynamics can be summarized as:
Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1-
form of covariant Action Potentials, Aµ(x, y, z, t...), on a four-dimensional ab-
stract variety of ordered independent variables, {x, y, z, t}. The variety supports
differential volume element Ω4 = dxˆdyˆdzˆdt.
Axiom 2. Thermodynamic processes are assumed to be encoded, to within a
factor, ρ(x, y, z, t...), in terms of a contravariant Vector and/or complex Spinor
directionfields, symbolized as V4(x, y, z, t).
Axiom 3. Continuous Topological Evolution of the thermodynamic system can
be encoded in terms of Cartan’s magic formula (see p. 122 in [10]). The Lie
differential with respect to the process, ρV4, when applied to an exterior differen-
tial 1-form of Action, A = Aµdx
µ, is equivalent, abstractly, to the first law of
thermodynamics.
Cartan’s Magic Formula L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (1)
First Law : W + dU = Q, (2)
Inexact Heat 1-form Q = W + dU = L(ρV4)A, (3)
Inexact Work 1-form W = i(ρV4)dA, (4)
Internal Energy U = i(ρV4)A. (5)
Axiom 4. Equivalence classes of systems and continuous processes can be de-
fined in terms of the Pfaff Topological Dimension and topological structure gen-
erated by of the 1-forms of Action, A, Work, W , and Heat, Q.
Axiom 5. If QˆdQ 6= 0, then the thermodynamic process is irreversible.
1.2 Cartan’s Magic Formula ≈ First Law of Thermodynamics
The Lie differential (not Lie derivative) is the fundamental generator of Continuous Topo-
logical Evolution. When acting on an exterior differential 1-form of Action, A = Aµdx
Cartan’s magic (algebraic) formula is equivalent abstractly to the first law of thermodynam-
L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (6)
= W + dU = Q. (7)
In effect, Cartan’s magic formula leads to a topological basis of thermodynamics, where
the thermodynamic Work, W , thermodynamic Heat, Q, and the thermodynamic internal
energy, U , are defined dynamically in terms of Continuous Topological Evolution. In effect,
the First Law is a statement of Continuous Topological Evolution in terms of deRham
cohomology theory; the difference between two non-exact differential forms is equal to an
exact differential, Q−W = dU .
My recognition (some 30 years ago) of this correspondence between the Lie differential
and the First Law of thermodynamics has been the corner stone of my research efforts in
applied topology.
It is important to realize that the Cartan formula is to be interpreted algebraically. Many
textbook presentations of the Cartan-Lie differential formula presume a dynamic constraint,
such that the vector field V4(x, y, z, t) be the generator of a single parameter group. If
true, then the topological constraint of Kinematic Perfection cn be established as an exterior
differential system of the format:
Kinematic Perfection : dxk −Vkdt⇒ 0. (8)
The topological constraint of Kinematic Perfection, in effect, defines (or presumes) a limit
process. This constraint leads to the concept of the Lie derivative2 of the 1-form A. The
evolution then is represented by the infinitesimal propagation of the 1-form, A, down the
2Professor Zbigniew Oziewicz told me that Slebodzinsky was the first to formulate the idea of the Lie
derivative in his thesis (in Polish).
flow lines generated by the 1-parameter group. Cartan called this set of flow lines ”the tube
of trajectories”.
However, such a topological, kinematic constraint is not imposed in the presentation found
in this essay; the directionfield, V4, may have multiple parameters. This observation leads
to the important concept of topological fluctuations (about Kinematic Perfection), such as
given by the expressions:
Topological : Fluctuations
(dxk −Vkdt) = ∆xk 6= 0, ( ∼ Pressure) (9)
(dV k −Akdt) = (∆Vk) 6= 0, ( ∼ Temperature) (10)
d(∆xk) = −(dVk −Akdt)ˆdt = −(∆Vk)ˆdt, (11)
In this context it is interesting to note that in Felix Klein’s discussions [4] of the development
of calculus, he says
”The primary thing for him (Leibniz) was not the differential quotient (the deriva-
tive) thought of as a limit. The differential, dx, of the variable x had for him
(Leibniz) actual existence...”
The Leibniz concept is followed throughout this presentation. It is important for the
reader to remember that the concept of a differential form is different from the concept of a
derivative, where a (topological) limit has been defined, thereby constraining the topological
evolution.
The topological methods to be described below go beyond the notion of processes which
are confined to equilibrium systems of kinematic perfection. Non-equilibrium systems and
processes which are thermodynamically irreversible, as well as many other classical ther-
modynamic ideas, can be formulated in precise mathematical terms using the topological
structure and refinements generated by the three thermodynamic 1-forms, A, W, and Q.
1.3 The Pfaff Sequence and the Pfaff Topological Dimension
1.3.1 The Pfaff Topological Dimension of the System 1-form, A
It is important to realize that the Pfaff Topological Dimension of the system 1-form of Action,
A, determines whether the thermodynamic system is Open, Closed, Isolated or Equilibrium.
Also, it is important to realize that the Pfaff Topological Dimension of the thermodynamic
Work 1-form, W , determines a specific category of reversible and/or irreversible processes.
It is therefore of some importance to understand the meaning of the Pfaff Topological Di-
mension of a 1-form. Given the functional format of a general 1-form, A, on a 4D variety
it is an easy step to compute the Pfaff Sequence, using one exterior differential operation,
and several algebraic exterior products. For a differential 1-form, A, defined on a geometric
domain of 4 base variables, the Pfaff Sequence is defined as:
Pfaff Sequence {A, dA,AˆdA, dAˆdA...} (12)
It is possible that over some domains, as the elements of the sequence are computed, one of
the elements (and subsequent elements) of the Pfaff Sequence will vanish. The number of
non-zero elements in the Pfaff Sequence (PS) defines the Pfaff Topological Dimension (PTD)
of the specified 1-form3. Modulo singularities, the Pfaff Topological Dimension determines
the minimum number M of N functions of base variables (N ≥ M) required to define the
topological properties of the connected component of the 1-form A.
The Pfaff Topological Dimension of the 1-form of Action, A, can be put into correspon-
dence with the four classic topological structures of thermodynamics. Equilibrium, Isolated,
Closed, and Open systems. The classic thermodynamic interpretation is that the first two
structures do not exchange mass (mole numbers) or radiation with their environment. The
Closed structure can exchange radiation with its environment but not mass (mole numbers).
The Open structure can exchange both mass and radiation with its environment. The fol-
lowing table summarizes these properties. For reference purposes, I have given the various
elements of the Pfaff sequence specific names:
Topological
p-form name
element
Nulls PTD
Thermodynamic
system
Action A dA = 0 1 Equilibrium
Vorticity dA AˆdA = 0 2 Isolated
Torsion AˆdA dAˆdA = 0 3 Closed
Parity dAˆdA − 4 Open
Table 1 Applications of the Pfaff Topological Dimension.
The four thermodynamic systems can be placed into two disconnected topological cate-
gories. If the Pfaff Topological Dimension of A is 2 or less, the first category is determined
by the closure (or differential ideal) of the 1-form of Action, A∪ dA. This Cartan topology
is a connected topology. In the case that the Pfaff Topological Dimension is greater than
2, the Cartan topology is based on the union of two closures, {A ∪ dA ∪ AˆdA ∪ dAˆdA},
and is a disconnected topology.
3The Pfaff Topological dimension has been called the ”class” of a 1-form in the old literature. I prefer
the more suggestive name.
It is a topological fact that there exists a (topologically) continuous C2 process from a
disconnected topology to a connected topology, but there does not exist a C2 continuous
process from a connected topology to a disconnected topology. This fact implies that
topological change can occur continuously by a ”pasting” processes representing the decay
of turbulence by ”condensations” from non-equilibrium to equilibrium systems. On the other
hand, the creation of Turbulence involves a discontinuous (non C2) process of ”cutting” into
parts. This warning was given long ago [19] to prove that computer analyses that smoothly
match value and slope will not replicate the creation of turbulence, but can faithfully replicate
the decay of turbulence.
1.3.2 The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W
The topological structure of the thermodynamic Work 1-form, W , can be used to refine the
topology of the physical system; recall that the physical system is encoded by the Action
1-form, A.
Claim 4 The PDE’s that represent the system dynamics are determined by the Pfaff Topo-
logical Dimension of the 1-form of Work, W , and the 1-form of Action, A, that encodes the
physical system.
The Pfaff Topological Dimension of the thermodynamic Work 1-form depends upon both
the physical system, A, and the process, V4. In particular if the Pfaff Dimension of the
thermodynamic Work 1-form is zero, (W = 0), then system dynamics is generated by an
extremal vector field which admits a Hamiltonian realization. However, such extremal
direction fields can occur only when the Pfaff Topological Dimension of the system encoded
by A is odd, and equal or less than the geometric dimension of the base variables.
For example, if the geometric dimension is 3, and the Pfaff Topological Dimension of
A is 3, then there exists a unique extremal field on the Contact manifold defined by dA.
This unique directionfield is the unique eigen directionfield of the 3x3 antisymmetric matrix
(created by the 2-form F = dA) with eigenvalue equal to zero.
If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 3, then
there exists a two extremal fields on the geometric manifold. These directionfields are those
generated as the eigen directionfields of the 4x4 antisymmetric matrix (created by the 2-form
F = dA) with eigenvalue equal to zero.
If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 4, then there
do not exist extremal fields on the Symplectic manifold defined by dA. All of the eigen
directionfields of the 4x4 antisymmetric matrix (created by the 2-form F = dA) are complex
isotropic spinors with pure imaginary eigenvalues not equal to zero.
1.4 Topological Torsion and other Continuous Processes.
1.4.1 Reversible Processes
Physical Processes are determined by directionfields4 with the symbol, V4, to within an
arbitrary function, ρ. There are several classes of direction fields that are defined as follows
Associated Class:i(ρV4)A = 0, (13)
Extremal Class:i(ρV4)dA = 0, (14)
Characteristic Class:i(ρV4)A = 0, (15)
and : i(ρV4)dA = 0, (16)
Helmholtz Class: d(i(ρV4)dA) = 0, (17)
Extremal Vectors (relative to the 1-form of Action, A) produce zero thermodynamic work,
W = i(ρV4)dA = 0, and admit a Hamiltonian representation. Associated Vectors (relative
to the 1-form of Action, A) can be adiabatic if the process remains orthogonal to the 1-
form, A. Helmholtz processes (which include Hamiltonian processes, Bernoulli processes
and Stokes flow) conserve the 2-form of Topological vorticity, dA. All such processes are
thermodynamically reversible. Many examples of these systems are detailed in the reference
monographs (see footnote on page 1).
1.4.2 Irreversible Processes
There is one directionfield that is uniquely defined by the coefficient functions of the 1-form,
A, that encodes the thermodynamic system on a 4D geometric variety. This vector exists
only in non-equilibrium systems, for which the Pfaff Topological Dimension of A is 3 or 4.
This 4 vector is defined herein as the topological Torsion vector, T4. To within a factor, this
directionfield5 has the four coefficients of the 3-form AˆdA, with the following properties:
4Which include both vector and spinor fields.
5A direction field is defined by the components of a vector field which establish the ”line of action” of
the vector in a projective sense. An arbitrary factor times the direction field defines the same projective
line of action, just reparameterized. In metric based situations, the arbitrary factor can be interpreted as a
renormalization factor.
Properties of :Topological Torsion T4 on Ω4 (18)
i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt = AˆdA, (19)
W = i(T4)dA = σ A, (20)
dW = dσˆA+ σdA = dQ (21)
U = i(T4)A = 0, T4 is associative (22)
i(T4)dU = 0 (23)
i(T4)Q = 0 T4 is adiabatic (24)
L(T4)A = σ A, T4 is homogeneous (25)
L(T4)dA = dσˆA+ σdA = dQ, (26)
QˆdQ = L(T4)AˆL(T4)dA = σ
2AˆdA 6= 0, (27)
dAˆdA = d(AˆdA) = d{(i(T4)Ω4} = (div4T4)Ω4, (28)
L(T4)Ω4 = d{(i(T4)Ω4} = (2σ)Ω4, (29)
If the Pfaff Topological Dimension of A is 4 (an Open thermodynamic system), then T4
has a non-zero 4 divergence, (2σ), representing an expansion or a contraction of the 4D
volume element Ω4. The Heat 1-form, Q, generated by the process, T4, is NOT integrable.
Q is of Pfaff Topological Dimension greater that 2, when σ 6= 0. Furthermore the T4 process
is locally adiabatic as the change of internal energy in the direction of the process path is
zero. Therefore, in the Pfaff Topological Dimension 4 case, where dAˆdA 6= 0, the T4
direction field represents an irreversible, adiabatic process.
When σ is zero and dσ = 0, but AˆdA 6= 0, the Pfaff Topological Dimension of the
system is 3 (a Closed thermodynamic system). In this case, the T4 direction field becomes
a characteristic vector field which is both extremal and associative, and induces a Hamilton-
Jacobi representation (the ground state of the system for which dQ = 0).
For any process and any system, equation ( 27) can be used as a test for irreversibility.
It seems a pity, that the concept of the Topological Torsion vector and its association
with non-equilibrium systems, where it can be used to establish design criteria to minimize
energy dissipation, has been ignored by the engineering community.
1.4.3 The Spinor class
It is rather remarkable (and only fully appreciated by me in February, 2005) that there is a
large class of direction fields useful to the topological dynamics of thermodynamic systems
(given herein the symbol ρS4) that do not behave as vectors (with respect to rotations).
They are isotropic complex vectors of zero length, defined as Spinors by E. Cartan [2], but
which are most easily recognized as the eigen directionfields relative to the antisymmetric
matrix, [F ], generated by the component of the 2-form F = dA:
The Spinor Class [F ] ◦ |ρS4〉 = λ |ρS4〉 6= 0, (30)
〈ρS4| ◦ |ρS4〉 = 0, λ 6= 0 (31)
In the language of exterior differential forms, if the Work 1-form is not zero, the process
must contain Spinor components:
W = i(ρS4)dA 6= 0 (32)
As mentioned above, Spinors have metric properties, behave as vectors with respect to
transitive maps, but do not behave as vectors with respect to rotations (see p. 3, [2]).
Spinors generate harmonic forms and are related to conjugate pairs of minimal surfaces.
The notation that a Spinor is a complex isotropic directionfield is preferred over the names
”complex isotropic vector”, or ”null vector” that appear in the literature. As shown below,
the familiar formats of Hamiltonian mechanical systems exclude the concept of Spinor process
directionfields, for the processes permitted are restricted to be represented by direction fields
of the extremal class, which have zero eigenvalues.
Remark 5 Spinors are normal consequences of antisymmetric matrices, and, as topological
artifacts, they are not restricted to physical microscopic or quantum constraints. According
to the topological thermodynamic arguments, Spinors are implicitly involved in all processes
for which the 1-form of thermodynamic Work is not zero. Spinors play a role in topological
fluctuations and irreversible processes.
The thermodynamic Work 1-form, W , is generated by a completely antisymmetric 2-
form, F , and therefore, if not zero, must have Spinor components. In the odd dimensional
Contact manifold case there is one eigen Vector, with eigenvalue zero, which generates the
extremal processes that can be associated with a Hamiltonian representation. The other
two eigendirection fields are Spinors. In the even dimensional Symplectic manifold case,
any non-zero component of work requires that the evolutionary directionfields must contain
Spinor components. All eigen directionfields on symplectic spaces are Spinors.
The fundamental problem of Spinor components is that there can be more than one Spinor
direction field that generates the same geometric path. For example, there can be Spinors
of left or right handed polarizations and Spinors of expansion or contraction that produce
the same optical (null congruence) path. This result does not fit with the classic arguments
of mechanics, which require unique initial data to yield unique paths. Furthermore, the
concept of Spinor processes can annihilate the concept of time reversal symmetry, inherent
in classical hydrodynamics. The requirement of uniqueness is not a requirement of non-
equilibrium thermodynamics, where Spinor ”entanglement” has to be taken into account.
1.5 Emergent Topological Defects
Suppose an evolutionary process starts in a domain of Pfaff Topological Dimension 4, for
which a process in the direction of the Topological Torsion vector, T4 , is known to represent
an irreversible process. Examples can demonstrate that the irreversible process can proceed
to a domain of the geometric variety for which the dissipation coefficient, σ, becomes zero.
Physical examples [41] such as the skidding bowling ball proceed with irreversible dissipation
(PTD = 6) until the ”no-slip” condition is reached (PTD = 5). In fluid systems the
topological defects can emerge as long lived states far from equilibrium. The process is
most simply visualized as a ”condensation” from a turbulent gas, such as the creation of
a star in the model which presumes the universe is a very dilute, turbulent van der Waals
gas near its critical point. The red spot of Jupiter, a hurricane, the ionized plasma ring
in a nuclear explosion, Falaco Solitons, the wake behind an aircraft are all exhibitions of
the emergence process to long lived topological structures far from equilibrium. It is most
remarkable that the emergence of these experimental defect structures occurs in finite time.
The idea is that a subdomain of the original system of Pfaff Topological Dimension 4 can
evolve continuously with a change of topology to a region of Pfaff Topological Dimension
3. The emergent subdomain of Pfaff Topological Dimension 3 is a topological defect, with
topological coherence, and often with an extended lifetime (as a soliton structure with a
dominant Hamiltonian evolutionary path), embedded in the Pfaff dimension 4 turbulent
background.
The Topological Torsion vector in a region of Pfaff Topological Dimension 3 is an extremal
vector direction field in systems of Pfaff Topological Dimension 3; it then has a zero 4D
divergence, and leaves the volume element invariant. Moreover the existence of an extremal
direction field implies that the 1-form of Action can be given a Hamiltonian representation,
k + H(P, q, t)dt. In the domain of Pfaff dimension 3 for the Action, A, the subse-
quent continuous evolution of the system, A, relative to the process T4, can proceed in an
energy conserving, Hamiltonian manner, representing a ”stationary” or ”excited” state far
from equilibrium (the ground state). This argument is based on the assumption that the
Hamiltonian component of the direction field is dominant, and any Spinor components in
the PTD = 3 domain, representing topological fluctuations, can be ignored. These excited
states, far from equilibrium, can be interpreted as the evolutionary topological defects that
emerge and self-organize due to irreversible processes in the turbulent dissipative system of
Pfaff dimension 4.
The descriptive words of self-organized states far from equilibrium have been abstracted
from the intuition and conjectures of I. Prigogine [9]. The methods of Continuous Topolog-
ical Evolution correct the Prigogine conjecture that ”dissipative structures” can be caused
by dissipative processes and fluctuations. The long-lived excited state structures created
by irreversible processes are non-equilibrium, deformable topological defects almost void
of irreversible dissipation. The topological theory presented herein presents for the first
time a solid, formal, mathematical justification (with examples) for the Prigogine conjec-
tures. Precise definitions of equilibrium and non-equilibrium systems, as well as reversible
and irreversible processes can be made in terms of the topological features of Cartan’s ex-
terior calculus. Using Cartan’s methods of exterior differential systems, thermodynamic
irreversibility and the arrow of time can be well defined in a topological sense, a technique
that goes beyond (and without) statistical analysis [23]. Thermodynamic irreversibility and
the arrow of time requires that the evolutionary process produce topological change.
2 Applications
2.1 An Electromagnetic format
The thermodynamic identification of the terms in Cartan’s magic formula are not whimsical.
To establish an initial level of credence in the terminology, consider the 1-form of Action, A,
where the component functions are the symbols representing the familiar vector and scalar
potentials in electromagnetic theory. The coefficient functions have arguments over the four
independent variables {x, y, z, t},
A = Aµ(x, y, z, t)dx
µ = A ◦ dr− φ dt. (33)
Construct the 2-form of field intensities as the exterior differential of the 1-form of Action,
F = dA = (∂Ak/∂x
j − ∂Aj/∂xk)dxjˆdxk (34)
= Fjkdx
jˆdxk = +Bzdxˆdy...+ Exdxˆdt... . (35)
The engineering variables are defined as electric and magnetic field intensities:
E = −∂A/∂t− grad φ, B = curl A. (36)
Relative to the ordered set of base variables, {x, y, z, t}, define a process directionfield,
ρV4, as a 4-vector with components, [V, 1], with a scaling factor, ρ.
ρ[V4] = ρ[V, 1]. (37)
Note that this direction field can be used to construct a useful 3-form of (matter) current,
C, in terms of the 4-volume element, Ω4 = dxˆdyˆdzˆdt :
C = i(ρV4)dxˆdyˆdzˆdt = i(C4)Ω4. (38)
The process 3-form, C, is not necessarily the same as electromagnetic charge current density
3-form of electromagnetic theory, J . The 4-divergence of C, need not be zero: dC 6= 0.
Using the above expressions, the evaluation of the thermodynamic work 1-form in terms
of 3-vector engineering components becomes:
The thermodynamic Work 1-form: W = i(ρV4)dA = i(ρV4)F, (39)
⇒ −ρ{E+V ×B} ◦ dr+ ρ{V ◦ E}dt (40)
= −ρ{fLorentz} ◦ dr+ ρ{V ◦ E}dt. (41)
The Lorentz force = −{fLorentz} ◦ dr (spatial component) (42)
The dissipative power = +{V ◦ E}dt (time component). (43)
For those with experience in electromagnetism, note that the construction yields the
format, automatically and naturally, for the ”Lorentz force” as a derivation consequence,
without further ad hoc assumptions. The dot product of a 3 component force, fLorentz, and
a differential spatial displacement, dr, defines the elementary classic concept of ”differential
work”. The 4-component thermodynamic Work 1-form,W , includes the spatial component
and a differential time component, Pdt, with a coefficient which is recognized to be the
”dissipative power”, P = {V ◦ E}. The thermodynamic Work 1-form, W , is not necessarily
a perfect differential, and therefore can be path dependent. Closed cycles of Work need not
be zero.
Next compute the Internal Energy term, relative to the process defined as ρV4:
Internal Energy: U = i(ρV4)A = ρ(V ◦A− φ). (44)
The result is to be recognized as the ”interaction” energy density in electromagnetic
plasma systems. It is apparent that the internal energy, U , corresponds to the interaction
energy of the physical system; that is, U is the internal stress energy of system deformation.
Therefore, the electromagnetic terminology can be used to demonstrate the premise that
Cartan’s magic formula is not just another way to state that the first law of thermodynamics
makes practical sense. The topological methods permit the long sought for integration of
mechanical and thermodynamic concepts, without the constraints of equilibrium systems,
and/or statistical analysis.
It is remarkable that although the symbols are different, the same basic constructions and
conclusions apply to many classical physical systems. The correspondence so established
between the Cartan magic formula acting on a 1-form of Action, and the first law of ther-
modynamics is taken both literally and seriously in this essay. The methods yield explicit
constructions for testing when a process acting on a physical system is irreversible. The
methods permit irreversible adiabatic processes to be distinguished from reversible adiabatic
processes, analytically. Adiabatic processes need not be ”slow” or quasi-static.
Given any 1-form, A, W, and/or Q, the concept of Pfaff Topological Dimension (for each
of the three 1-forms, A, W , Q) permits separation of processes and systems into equiva-
lence classes. For example, dynamical processes can be classified in terms of the topological
Pfaff dimension of the thermodynamic Work 1-form, W . All extremal Hamiltonian systems
have a thermodynamic Work 1-form, W , of topological Pfaff dimension of 1, (dW = 0).
Hamiltonian systems can describe reversible processes in non-equilibrium systems for which
the topological Pfaff dimension is 3. Such systems are topological defects whose topology is
preserved by the Hamiltonian dynamics, but all processes which preserve topology are re-
versible. In non-equilibrium systems, topological fluctuations can be associated with Spinors
of the 2-form, F = dA. Even if the dominant component of the process is Hamiltonian,
Spinor fluctuations can cause the system (ultimately) to decay.
2.1.1 Topological 3-forms and 4-forms in EM format
Construct the elements of the Pfaff Sequence for the EM notation,
{A, F = dA,AˆF, FˆF}, (45)
and note that the algebraic expressions of Topological Torsion, AˆF , can be evaluated in
terms of 4-component engineering variables T4 as:
Topological Torsion vector (46)
AˆF = i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt (47)
T4 = [T, h] = −[E×A+Bφ,A ◦B]. (48)
The exterior 3-form, AˆF , with physical units of (~/unit mole)2, is not found (usually) in
classical discussions of electromagnetism6.
If T4 is used as to define the direction field of a process, then
L(T4)A = σA, i(T4)A = 0. (49)
where 2σ = {div4(T4)} = 2(E ◦B). (50)
The important (universal) result is that if the acceleration associated with the direction
field, E, is parallel to the vorticity associated with the direction field, B, then according to
the equations starting with eq. (18) et. seq. the process is dissipatively irreversible. This
result establishes the design criteria for engineering applications to minimize dissipation from
turbulent processes.
The Topological Torsion vector has had almost no utilization in applications of classical
electromagnetic theory.
2.1.2 Topological Torsion quanta
The 4-form of Topological Parity, FˆF , can be evaluated in terms of 4-component engineering
variables as:
Topological Parity
d(AˆF ) = FˆF = {div4(T4)}Ω4 = {2E ◦B} Ω4. (51)
6The unit mole number is charge, e, in EM theory.
This 4-form is also known as the second Poincare Invariant of Electromagnetic Theory.
The fact that FˆF need not be zero implies that the Pfaff Topological Dimension of the
1-form of Action, A, must be 4, and therefore A represents a non-equilibrium Open thermo-
dynamic system. Similarly, if FˆF = 0, but AˆF 6= 0, then the Pfaff Topological Dimension
of the 1-form of Action, A, must be 3, and the physical system is a non-equilibrium Closed
thermodynamic system. When FˆF = 0, the corresponding three-dimensional integral of
the closed 3-form, AˆF , when integrated over a closed 3D-cycle, becomes a deRham period
integral, defined as the Torsion quantum. In other words, the closed integral of the (closed)
3-form of Topological Torsion becomes a deformation (Hopf) invariant with integral values
proportional to the integers.
Torsion quantum =
3D cycle
AˆF. (52)
On the other hand, topological evolution and transitions between ”quantized” states of
Torsion require that the respective Parity 4-form is are not zero. As,
L(T4)Ω4 = d{i(T4)Ω4} = (2σ) Ω4 = 2(E ◦B) Ω4 6= 0, (53)
it is apparent that the evolution of the differential volume element, Ω4, depends upon the
existence and colinearity of both the electric field, E, and the magnetic field, B. It is here
that contact is made with the phenomenological concept of ”4D bulk” viscosity = 2σ. It is
tempting to identify σ2 with the concept of entropy production. Note that the Topological
Torsion directionfield appears only in non-equilibrium systems. These results are universal
and can be used in hydrodynamic systems discussed in that which follows.
2.2 A Hydrodynamic format
2.2.1 The Topological Continuum vs. the Geometrical Continuum
In many treatments of fluid mechanics the (geometrical) continuum hypothesis is invoked
from the start. The idea is ”matter” occupies all points of the space of interest, and that
properties of the fluid can be represented by piecewise continuous functions of space and
time, as long as length and time scales are not too small. The problem is that at very small
scales, one has been led to believe the molecular or atomic structure of particles will become
evident, and the ”macroscopic” theory will breakdown. However, these problems of scale
are geometric issues, important to many applications, but not pertinent to a topological
perspective, where shape and size are unimportant. Suppose that the dynamics can be
formulated in terms of topological concepts which are independent from sizes and shapes.
Then such a theory of a Topological Continuum would be valid at all scales. Such is the
goal of this monograph.
Remark 6 However, one instance where ”scale” many have topological importance is as-
sociated with the example of a surface with a ”teeny” hole. If the hole, no matter what
its size, has a twisted ear (Moebius band) then the whole surface is non-orientable, no mat-
ter how ”small” the hole. Could it be that the world of the quantum is, in effect, that of
non-orientable defects embedded in an otherwise orientable manifold that originally had no
such defects. Note further the strong correspondence with Fermions with non-oriented (half-
integer) multiplet ribbons, and Bosons with oriented (integer) multiplet ribbons of both right
and left twists [44].
As will be developed below, the fundamental equations of exterior differential systems can
lead to field equations in terms of systems of Partial Differential Equations (PDE’s). The
format of the fundamental theory will be in terms of objects (exterior differential forms)
which, although composed of algebraic constructions of tensorial7 things, are in a sense
scalars (or pseudo scalars) that are homogeneous with respect to concepts of scale. The
theory then developed is applicable to hydrodynamics at all scales, from the microworld
to the cosmological arena. The ”breakdown” of the continuum model is not relevant.
The topological system may consist of many disconnected parts when the system is not in
thermodynamic equilibrium or isolation, and the parts can have topological obstructions
or defects, some of which can be used to construct period integrals that are topologically
”quantized”. Hence the ”quantization” of the micro-scaled geometric systems can have it
genesis in the non-equilibrium theory of thermodynamics. However, from the topological
perspective, the rational topological quantum values can also occur at all scales.
2.2.2 Topological Hydrodynamics
The axioms of Topological Thermodynamics are summarized in Section 1.1. For hydrody-
namics (or electrodynamics the axioms are essentially the same. Just exchange the word,
hydrodynamics (or electrodynamics) for the word, thermodynamics, in the formats of Section
By 1969 it had become evident to me that electromagnetism (without geometric con-
straints), when written in terms of differential forms, was a topological theory, and that the
concept of dissipation and irreversible processes required more than that offered by Hamil-
tonian mechanics. At that time I was interested in possible interactions of the gravitational
field and the polarizations of an electromagnetic signal. One of the first ideas discovered
7Relative to diffeomorphisms.
about topological electrodynamics was that there existed an intrinsic transport theorem [11]
that introduced the concept of what is now called Topological Spin, AˆG, into electromag-
netic theory [43]. As a transport theorem not recognized by classical electromagnetism,
the first publication was as a letter to Physics of Fluids. That started my interest in a
topological formulation of fluids.
It was not until 1974 that the Lie differential acting on exterior differential forms was es-
tablished as the key to the problem of intrinsically describing dissipation and the production
of topological defects in physical systems; but methods of visualization of such topological
defects in classical electrodynamics were not known [12]. It was hoped that something in
the more visible fluid mechanics arena would lend credence to the concepts of topological
defects. The first formulations of the PDE’s of fluid dynamics in terms of differential forms
and Cartan’s Magic formula followed quickly [13].
In 1976 it was argued that topological evolution was at the cause of turbulence in fluid
dynamics, and the notion of what is now called Topological Torsion, AˆF, became recognized
as an important concept. It was apparent that streamline flow imposed the constraint that
AˆF = 0 on the equations of hydrodynamics. Turbulent flow, being the antithesis of
streamline flow, must admit AˆF 6= 0. In 1977 it was recognized that topological defect
structures could become ”quantized” in terms of deRham period integrals [15], forming a
possible link between topology and both macroscopic and microscopic quantum physics.
The research effort then turned back to a study of topological electrodynamics in terms of
the dual polarized ring laser, where it was experimentally determined that the speed of an
electromagnetic signal outbound could be different from the speed of an electromagnetic
signal inbound: a topological result not within the realm of classical theory.
Then in 1986 the long sought for creation and visualization of topological defects in
fluids [16] became evident. The creation of Falaco Solitons in a swimming pool was the
experiment that established credence in the ideas of what had, by that time, become a
theory of continuous topological evolution. It was at the Cambridge conference in 1989
[17] that the notions of topological evolution, hydrodynamics and thermodynamics were put
together in a rudimentary form, but it was a year later at the Permb conference in 1990
[18] that the ideas were well established. The Permb presentation also suggested that the
ambiguous (at that time) notion of coherent structures in fluids could be made precise in
terms of topological coherence. A number of conference presentations followed in which
the ideas of continuous thermodynamic irreversible topological evolution in hydrodynamics
were described [19], but the idea that the topological methods of thermodynamics could be
used to distinguish non-equilibrium processes and non-equilibrium systems and irreversible
processes with out the use of statistics slowly came into being in the period 1985-2005 [22].
These efforts have been summarized in [40], and a collection of the old publications appears
in [45].
2.3 Classical Hydrodynamic Theory
There are two classical techniques for describing the evolutionary motion of a fluid: the
Lagrangian method and the Eulerian method. Both methods treat the fluid relative to a
Euclidean 3D manifold, with time as a parameter. The first (Lagrangian) technique treats
a fluid as a collection of ”particles, or parcels” and the flow is computed in terms of ”initial”
data {a, b, c, τ} imposed upon solutions to Newtonian equations of motion for ”particles,
or parcels”. The solution functions describe a map from an initial state {a, b, c; τ} to a
final state {x, y, z, t}. This method is related to solutions of kinematic equations, and is
contravariant in the sense of an immersion to velocities (the tangent space). The kinematic
basis for the Lagrangian motion draws heavily from the Frenet-Serret analysis of a point
moving along a space curve.
The second technique treats a fluid as ”field”, and is representative of a ”wave” point of
view of a Hamiltonian system. The functions that define the field (the covariant momenta)
depend on ”final data” {x, y, z, t}, and are covariant in the sense of a submersion. Each
method has its preimage in the form of an exterior differential 1-form of Action. The
primitive classical Lagrangian Action concept is written in the form
AL = L(x
k, V m, t)dt, (54)
and the primitive Eulerian Action is written as
AE = pkdx
k + Hdt. (55)
As both Actions supposedly describe the same fluid, are they equivalent? That is,
does AE ⇔ AL ? (56)
Note that AL is composed from only two functions (L and t) such that at most AL is of
Pfaff Topological Dimension 2. On the other hand, the Pfaff Topological Dimension of
AE (as written) could be as high as 8, if all functions and differentials are presumed to be
independent. So the two formulations are NOT equivalent, unless constraints reduce the
topological dimension of AE to 2, or if additions are made to the 1-form AL to increase its
Pfaff dimension.
2.3.1 The Lagrange-Hilbert Action
The classic addition to AL is of the form, pk(dx
k − V kdt), where the pk are presumed to be
Lagrange multipliers of the fluctuations in kinematic velocity. The result is defined as the
Cartan - Hilbert 1-form of Action:
ACH = L(x
k, V k, t)dt+ pk(dx
k − V kdt). (57)
Note that at first glance it appears that there are 10=3N+1 independent geometric variables
{xk, V k, pk, t} in the formula for ACH , but if the Pfaff Sequence is constructed, the Pfaff
Topological Dimension turns out to be 8. So with this addition of Lagrange multipliers
to AL, the topological dimensions of the two actions are the same. However, note that by
rearranging variables,
ACH = pkdx
k + (L(xk, V m, t)− pkV k)dt, (58)
= pkdx
k + Hdt = AE . (59)
For a fluid the Eulerian ”momenta” per unit parcel of mass is usually defined as
pk/m = vk, (60)
such that the Eulerian Action per unit mass becomes
AE ⇒ vkdxk + Hdt. (61)
The bottom line is that the Lagrangian and Eulerian point of view can be made compatible
if fluctuations in Kinematic Perfection are allowed.
Recall that the development of elasticity theory (and its emphasis on symmetrical tensors)
also spawned the development of hydrodynamics. Much of the theory of classical hydro-
dynamics was phrased geometrically in the language of vector analysis. The development
followed the phenomenological concepts of an extended Newtonian theory of elasticity. The
classical theory was developed from ”balance” equations for a bounded sample, or parcel, of
matter (mass), which express assumptions (defined as the conservation of mass, momenta
and energy) in terms of integrals over the bounded sample, or parcel, of matter. The
classical integrals are performed usually over three-dimensional volumes (and not over 4D
space-time). The classical Cauchy result (for the momentum equations) is
ρ{∂v/∂t + v ◦ ∇v} = ∇◦T+ ρf , (62)
ρ{∂v/∂t + grad(v ◦ v/2)− v×curlv} = ∇◦T+ ρf , (63)
Constitutive assumptions are then made for the 3D stress tensor T , such that (in matrix
format)
[T] = (−P + λ(∇ ◦ v) [I] + ν{[∇v] + [∇v]T }, (64)
where P is the Pressure, ν the affine ”shear” viscosity and λ the ”expansion” viscosity. The
antisymmetric components {[∇v]− [∇v]T } have been ignored.
It will be demonstrated how the Axioms of Hydrodynamics yield topological information
about the classic Cauchy development, and goes beyond the symmetrical formulations by
recognizing that antisymmetries can introduce complex spinor contributions to the dynamics.
2.4 Euler flows and Hamiltonian fluids
Consider the Action 1-form per unit source (in thermodynamics, the unit source is mole
number, or sometimes mass), constructed from a covariant 3D velocity field, v = vk(x, y, z, t),
and a scalar potential function, φ:
A = v ◦ dr−φdt = vk(x, y, z, t)dxk − φdt. (65)
Compute the exterior differential dA and define the following (3D vector) functions as,
ω = curl v and a = +{∂v/∂t + grad(φ)}, (66)
such that,
F = dA = {∂Ak/∂xj − ∂Aj/∂xk}dxjˆdxk = Fjkdxjˆdxk (67)
= ωzdxˆdy + ωxdyˆdz + ωydzˆdx− axdxˆdt− aydyˆdt− azdzˆdt.
These vector fields always satisfy the Poincare-Faraday induction equations, dF = ddA =
0, or,
curl (−a) + ∂ω/∂t = 0, div ω = 0. (68)
The Eulerian Fluid Consider a process created by the contravariant vector directionfield,
V4 = [V
x,Vy,Vz, 1] and use Cartan’s magic formula,
L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A) =W + dU = Q, (69)
to compute the thermodynamic Work 1-form,W . The expressions for Work,W, and internal
energy, U, become:
W = i(ρV4)dA = −ρ{−∂v/∂t − grad(φ) +V × ω} ◦ dr
− ρV ◦ {∂v/∂t + grad(φ)}dt, (70)
= ρ{a−V × ω} ◦ dr−ρV ◦ {a)}dt (71)
U = i(V4)A = ρ(V · v−φ). (72)
At first, topologically constrain the thermodynamic Work 1-form to be of the Bernoulli
class in terms of the exterior differential system:
W = −dP (73)
ρ{+a−V × ω} ◦ dr−ρV ◦ {a)}dt = −grad P ◦ dr−∂P/∂t dt (74)
Assume that formally V = v, and the potential is equal to φ = v · v/2. Compare the
coefficients of dr to deduce the classic equations of motion for the Eulerian fluid.
{∂v/∂t + grad(v · v/2)− v× ω} = −grad(P )/ρ. (75)
This formula should be compared to the derivation of the Lorentz force term for Work
in electromagnetic systems. The functional format of the hydrodynamic 1-form of Action,
A, is the same as that specified above for the electromagnetic system. All that is changed
is the notation. In essence, the two topological theories are equivalent to the extent that
there is a correspondence between functions:
A ⇔ v, φ ⇔ v · v/2, (76)
E ⇔ −a, B ⇔ ω. (77)
All the results of the preceding section using an electromagnetic format can be translated
into the hydrodynamic format.
Note that the Bernoulli ”pressure”, P , is an evolutionary invariant along a trajectory,
L(ρV4)P = i(V4)dP = i(V4)i(V4)dA = 0. (78)
If the Pressure is barotropic, then the Bernoulli function becomes, dΘ = dP/ρ. The function
Θ can be amalgamated with the potential, φ, such that the thermodynamic Work 1-form
becomes equal to zero. The system then admits an extremal Hamiltonian direction field
such that the thermodynamic Work 1-form is zero.
W = i(VH)dA = 0. (79)
For a process defined in terms of an extremal directionfield, the First Law indicates that the
1-form of Heat, Q, is exact, dQ = 0, and equal to the change in internal energy, Q = dU .
Any Hamiltonian process is reversible, as QˆdQ = 0.
The time-like component of the exterior differential system W + dP = 0 leads to the
equation,
∂P/∂t = −ρv ◦ {∂v/∂t + grad(v · v/2) = ρ(v ◦ a). (80)
It is apparent that if the velocity, v, and the acceleration, a, are orthogonal, then the time
rate of change of the Bernoulli pressure is zero.
It also follows that the ”Master” equation is valid, with the only difference being that
curl v is defined as ω, the vorticity of the hydrodynamic flow. The master equation becomes,
curl(v× ω) = ∂ω/∂t, (81)
and this equation is to be recognized as the equivalent of Helmholtz’ equation for the con-
servation of vorticity. When the Pfaff Topological Dimension of the Work 1-form is 1, it is
possible to show that the ”Master” leads to a diffusion equation.
In the hydrodynamic sense, conservation of vorticity implies uniform continuity. In other
words, the Eulerian flow is not only Hamiltonian, it is also uniformly continuous, and satisfies
both the master equation and the conservation of vorticity constraints. In addition, it may
be demonstrated that such systems are at most of Pfaff dimension 3, and admit a relative
integral invariant which generalizes the hydrodynamic concept of invariant helicity. In the
electromagnetic topology, the Hamiltonian constraint is equivalent to the statement that
the Lorentz force vanishes, a condition that has been used to define the ”ideal” plasma or
”force-free” plasma state [46].
3 The Navier-Stokes fluid
3.1 The classic Navier-Stokes equations
It can be demonstrated that the ”ideal fluid” has a Hamiltonian representation, for which the
dynamics preserves a ”Hamiltonian” energy. This result is in disagreement with experiment
in that it is observed that motions of ”non-ideal” fluids exhibit decay to a stationary state.
The Lagrange Euler equations must be modified to accommodate dissipation of kinetic energy
and angular momentum. In fact, the ideal fluid constraint of zero affine shear stresses
should be replaced by dissipative terms related to both affine shears and a new phenomena
of rotational and expansion shears which have a fixed point. The classical phenomenological
outcome is the Navier-Stokes PDE’s,
∂v/∂t + grad(v · v/2)− v× curl v = −gradϕ− (1/ρ) gradP
+ ν∆v
− (µB + ν) grad div v, (82)
where µB is the ”bulk” viscosity coefficient and ν is the ”shear” viscosity coefficient. If the
fluid is ”incompressible” then the last term, which includes corrections due to bulk viscosity,
vanishes; the incompressible constraint requires that div v ⇒ 0.
In that which follows, the basic momentum equation (82) will be deduced from the per-
spective of Continuous Topological Evolution. Different topological equivalence classes of
thermodynamic processes depend upon the Pfaff Topological Dimension of the Work 1-form.
The different classes of thermodynamic processes are related to the velocity field in a Hy-
drodynamic system. The phenomenological (geometrical) derivation of the equations of
hydrodynamics will be replaced by determining the format of the PDE’s that agree with the
constraint required to satisfy the various PTD equivalence classes the Work 1-form. The
1-form of Work (for barotropic flows as in eq. (75)) will be of Pfaff Topological Dimension 1.
The Pfaff Topological Dimension of the 1-form of Work for the Navier-Stokes fluid can be as
high as 4, and is required to be 4 if the flow is fully turbulent. Various intermediate classes
of the work 1-form are of interest, as well. In particular, the Pfaff Topological Dimension
of the work 1-form must be 3 for a baroclinic system, a result that admits frontal systems
with propagating tangential discontinuities as found in weather systems.
3.2 The Navier-Stokes equations embedded in a non-equilibrium
thermodynamic system
In this subsection, the topological refinement due to the Pfaff Topological Dimension of the
Work 1-form will be employed to demonstrate that processes in a non-equilibrium thermody-
namic system can be put into correspondence with solutions of the Navier-Stokes equations.
In order to go beyond extremal (Hamiltonian) processes, it is necessary that the Pfaff
Topological Dimension of the Work 1-form must be greater than 1. Recall that for any
process, the Work done is transverse to the process trajectory,
(i(ρV4)W = (i(ρV4)(i(ρV4)dA = 0. (83)
Hence, if the PTD of the Work 1-form, W , is to be greater than 1, it must have the format,
W = i(ρV4)dA = −dP +̟j(dxj − vjdt) = −dP +̟j∆xj , (84)
where the ”Bernoulli function”, P , if it exists, must be a first integral (a process invariant),
L(ρV4)P = (i(ρV4)dP = 0. (85)
It is also important to remember that such non-zero contributions to the work 1-form
are due to the complex, isotropic Cartan Spinors, which are the eigen direction fields of the
2-form, F .
The coefficients, ̟j, of the topological fluctuations, ∆x
j , act in the manner of Lagrange
multipliers, and mimic the concept of system forces. If ̟j/ρ is defined (arbitrarily
8) as
υ curl curl v then the spatial components of the thermodynamic Work 1-form, W , are con-
strained to yield the partial differential equations for a constant density Navier-Stokes fluid:
{∂v/∂t + grad(v · v/2)− v × ω} = −grad(P )/ρ+ υ curl curl v. (86)
Density variations can be included by adding a term λdiv(V) to the potential {v · v/2} to
yield:
∂v/∂t + grad{v.v/2} − v × curl v = −gradP/ρ (87)
+ λ{grad(div v)} (88)
+ υ{curl curl v}. (89)
Classically, v can be identified with the geometric kinematic shear viscosity, and λ = µB −υ.
The coefficient µB can be identified with the topological (space-time) bulk viscosity.
8This is one of many formal choices, but the choice demonstrates that the Navier-Stokes equations reside
within the domain of non-equilibrium thermodynamics. QED
3.3 The Topological Torsion process for the Navier-Stokes fluid
The Navier-Stokes constraint implies that the thermodynamic Work 1-form need not be
closed. Then there are thermodynamic processes represented by solutions to the Navier-
Stokes equations that are thermodynamically irreversible. In this subsection, the Topological
Torsion vector will be expressed in terms of the solutions of the Navier-Stokes equations.
From the work in section 2, the 1-form of Action will generate a 3-form of Topological
Torsion, AˆdA = i(T4)Ω4, and leads to the 4 Vector of Topological Torsion (written in
hydrodynamic notation):
T4 = [−a × v + {v.v/2} curl v, (v ◦ curl v)], (90)
= [−a × v + {v.v/2} ω,(v ◦ ω)] = [T, h]. (91)
Use the Navier-Stokes equations (82) to solve for a,
a = [grad{v.v/2}+ ∂v/∂t] (92)
= v × curl v − gradP/ρ
+ λ{grad(div v)}+ υ{curl curl v}, (93)
and then substitute this result into the expression for T4, to yield:
T = [hv − (v ◦ v/2)curl v− v × (gradP lρ)
+ λ{v× grad(div v)} − υ{v× (curl curl v)}], (94)
h = v · curl v, (95)
Note that T4 exists even for Euler flows, where υ = 0, if the flow is baroclinic. The
measurement of the components of the Torsion vector, T4, have been completely ignored by
experimentalists in hydrodynamics.
By a similar substitution, the topological parity 4-form, FˆF, becomes expressible in
terms of engineering quantities as,
K = {2(−a ◦ ω)}Ω4 = {2(σ)} ⇔ 2(E ◦B)
σ = {gradP/ρ ◦ curl v (96)
− λ{ grad(div v) ◦ curl v}
− υ{curl v ◦ (curl curl v)}}Ω4. (97)
The coefficient σ is a measure of the space-time bulk dissipation coefficient (not λ), and it is
the square of this number which must not be zero if the process is irreversible (see eq (27)).
Recall that a turbulent dissipative irreversible flow is defined when the Pfaff dimension of
the Action 1-form is equal to 4, which implies that K 6= 0.
From the expression for σ, it is apparent that if the 3D vector of vorticity is of Pfaff
dimension 2, such that ω◦curl ω = 0, then the last term vanishes, and there is no irreversible
dissipation due to shear viscosity, υ (a result useful in the theory of wakes).
Other useful situations and design criteria for dissipation, or the lack thereof, can be
gleaned from the formula. If the vector field is harmonic, then an irreversible process
requires that,
σ = (−a ◦ ω) = {(gradP/ρ− µBgrad(div v)) ◦ curl v} 6= 0. (98)
(Recall that harmonic vector fields are generators of minimal surfaces.) For fluids where
(µB) ⇒ 0, if the pressure gradient is orthogonal to the vorticity and the flow field is harmonic,
then there is no irreversible dissipation as σ = 0, and the flow is not turbulent. Note that
for many fluids the bulk viscosity is much greater than the shear viscosity. When σ = 0,
no topological torsion defects are created; the acceleration, a, and the vorticity, ω, of the
Navier-Stokes fluid are colinear.
Theorem 7 It is thereby demonstrated that solutions to the Navier-Stokes equations corre-
spond to processes of a non-equilibrium thermodynamic system of PTD(A) = 4, and Work
1-forms of PTD(W ) > 2. Such processes include Spinor direction fields generated by the
Topological Torsion vector. The Topological Torsion vector generates processes, and hence
solutions to the Navier-Stokes equations, that are thermodynamically irreversible..
These results should be compared to those generated by Lamb and Eckart [3] for the
fluid dissipation function, which is defined by the requirement that the dissipative flow has
a (geometric) entropy production rate greater than or equal to zero. More examples can be
found in, ”Wakes, Coherent Structures, and Turbulence” [42].
4 Closed States of Topological Coherence embedded as
deformable defects in Turbulent Domains
In this section 4, the problem of C2 smoothness will be attacked from the point of view of
topological thermodynamics. First, two distinct examples will be given demonstrating two
different emergent PTD = 3 states, that emerge from different 4D rotations (see p. 108,
[38]). Then, an example demonstrating the decay of a PTD = 4 state into a PTD = 3
state will be given in detail.
4.1 Examples of PTD = 3 domains and their Emergence
In section 3, it was demonstrated that there are solutions (thermodynamic processes) to
the Navier Stokes equations in non-equilibrium thermodynamic domains. The properties
of those PTD = 3 domains which emerge by C2 irreversible solutions from domains of
PTD = 4 are of particular interest. From section 2, it is apparent that the key feature of
PTD = 3 domains is that the electric E field (acceleration field a in hydrodynamics) must
be orthogonal to the magnetic B field (vorticity field ω in hydrodynamics). There are 8
cases to consider (including chirality),
Pfaff Topological Dimension 3
E = 0, ±B 6= 0, (99)
B = 0, ±E 6= 0, (100)
E ◦B = 0, with chirality choices, ±E = ±B 6= 0, (101)
of which two will be discussed in detail.
The Finite Helicity case (both E and B finite) PTD = 3 Start with the 4D ther-
modynamic domain, and first consider the 1-form of Action, A, with the format9:
A = Ax(z)dx+ Ay(z)dy − φ(z)dt, (102)
and its induced 2-form, F = dA,
F = dA = (∂Ax(z)/∂z)dzˆdx + (∂Ay(z)/∂z)dzˆdy − (∂φ(z)/∂z)dzˆdt, (103)
= Bx(z)dzˆdx−By(z)dzˆdy + Ez(z)dzˆdt. (104)
9The +E, +B chirality has been selected.
The 3-form of Topological Torsion 3-form becomes
i(T4)Ω4 = AˆF where (105)
T4(z) = [EzAy + φBx, −EzAx + φBy, 0, AxBx + AyBy] (106)
with div4(T4(z)) = 2(E ◦B) = 0, A ◦B 6= 0. (107)
The Zero Helicity case (both E and B finite) PTD = 3 Start with the 4D thermo-
dynamic domain, and consider the 1-form of Action, A, with the format:
A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y)dt, (108)
and its induced 2-form, F = dA,
F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy}
− (∂φ(x, y)/∂x)dxˆdt− (∂φ(x, y)/∂y)dyˆdt, (109)
= Bz(x, y)dxˆdy + Ex(x, y)dxˆdt+ Ey(x, y)dyˆdt. (110)
The 3-form of Topological Torsion 3-form becomes,
i(T4)Ω4 = AˆF where (111)
T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (112)
with div4(T4(x, y)) = 2(E ◦B) = 0, A ◦B = 0. (113)
This case of zero helicity (A ◦B = 0), has the Topological Torsion vector, T4(x, y), colinear
with the B field.
Zero Helicity case: PTD = 4 decays to PTD = 3 The two distinct cases, modulo
chirality, are suggestive of the idea (see p. 108 [38]) that the rotation group of a 4D domain
is not simple. The example,immediately above, is particularly useful because the algebra of
the decay from Pfaff dimension 4 to 3 is transparent.
Start with the 4D thermodynamic domain, and consider the 1-form of Action, A, with
the format:
A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y, z, t)dt, (114)
and its induced 2-form, F = dA,
F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy}
− (∂φ(x, y, z)/∂x)dxˆdt− (∂φ(x, y, z)/∂y)dyˆdt− (∂φ(x, y, z)/∂z)dzˆdt, (115)
= Bz(x, y)dxˆdy (116)
+ Ex(x, y, z, t)dxˆdt+ Ey(x, y, z, t)dyˆdt+ Ez(x, y, z, t)dyˆdt. (117)
The 3-form of Topological Torsion 3-form becomes,
i(T4)Ω4 = AˆF with PTD(A) = 4 (118)
T4(x, y, z, t) = [−EzAy,+EzAx, (ExAy − EyAx) + φBz, 0] (119)
with div4(T4(x, y, z, t)) = 2{Ez(x, y, z, t)Bz(x, y)} 6= 0. (120)
In this case, the helicity (A ◦B = 0) is still zero, but now the Topological Torsion vector,
T4(x, y, z, t), has three spatial components. Moreover, the Process generated by T4(x, y, z)
is thermodynamically irreversible, as (E ◦B) 6= 0. The example 1-form is of PTD = 4.
To demonstrate the emergence of the PTD = 3 state, suppose the potential function in
this example has the format,
φ = ψ(x, y) + ϕ(z)e−αt (121)
Ez(z, t) = −(∂ϕ(z)/∂z)e−αt = Ez(z)e−αt. (122)
Then the irreversible dissipation function decays as {Ez(z)Bz}e−αt. By addition of Spinor
fluctuation terms to represent the very small components of irreversible dissipation at late
times, the PTD = 3 solution,
T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (123)
becomes dominant, and represents a long lived ”stationary” state far from equilibrium,
modulo the small Spinor decay terms10.
4.2 Piecewise Linear Vector Processes vs. C2 Spinor processes
It will be demonstrated on thermodynamic spaces of Pfaff Topological Dimension 3, that
there exist piecewise continuous processes (solutions to the Navier-Stokes equations) which
are thermodynamically reversible. These Vector processes can be fabricated by combinations
of Spinor processes, each of which is irreversible. This topological result demonstrates, by
example, the difference between piecewise linear 3-manifolds and smooth complex manifolds.
It appears that the key feature of the irreversible processes is that they have a fixed point
of ”rotation or expansion”.
Consider those abstract physical systems that are represented by 1-forms, A, of Pfaff
Topological Dimension 3. The concept implies that the topological features can be described
10The experimental fact that the defect structures emerge in finite time is still an open topological problem,
although some geomtric success has been achieved through Ricci flows.
in terms of 3 functions (of perhaps many geometrical coordinates and parameters) and their
differentials. For example, if one presumes the fundamental independent base variables
are the set {x, y, z}, with an exterior differential oriented volume element consisting of a
product11 of exact 1-forms Ω3 = +dxˆdyˆdz, (then a local) Darboux representation for a
physical system could have the appearance,
A = xdy + dz. (124)
The objective is to use the features of Cartan’s magic formula to compute the possible
evolutionary features of such a system. The evolutionary dynamics is essentially the first
law of thermodynamics:
LρVA = i(ρV)dA+ di(ρV)A) = W + dU = Q. (125)
The elements of the Pfaff sequence for this Action become,
A = xdy + dz., (126)
dA = dxˆdy, (127)
AˆdA = dxˆdyˆdz, (128)
dAˆdA = 0. (129)
Note that for this example the coefficient of the 3-form of Topological Torsion is not zero,
and depends upon the Enstrophy (square of the Vorticity) of the fluid flow.
4.3 The Vector Processes
Relative to the position vector R = [x, y, z] of ordered topological coordinates {x, y, z},
consider the 3 abstract, linearly independent, orthogonal (supposedly) vector direction fields:
, (130)
, (131)
. (132)
11More abstract systems could be constructed from differential forms which are not exact.
These direction fields can be used to define a class of (real) Vector processes, but these real
vectors do not exhibit the complex Spinor class of eigendirection fields for the 2-form, dA.
The Spinor eigendirection fields are missing from this basis frame. The important fact is
that thermodynamic processes defined in terms of a real basis frame (and its connection) are
incomplete, as such processes ignore the complex spinor direction fields.
For each of the real direction fields, deform the (assumed) process by an arbitrary function,
ρ. Then construct the terms that make up the First Law of topological thermodynamics.
First construct the contractions to form the internal energy for each process,
UVx = i(ρVx)A = 0, dUVx = 0, (133)
UVy = i(ρVy)A = ρx, dUVy = d(ρx), (134)
UE = i(ρE)A = ρ, dUE = dρ. (135)
The extremal vector E is the unique eigenvector with eigenvalue zero relative to the maximal
rank antisymmetric matrix generated by the 2-form, dA. The associated vector Vx (relative
to the 1-form of Action, A, is orthogonal to the y, z plane. Recall that any associated vector
represents a local adiabatic process, as the Heat flow is transverse to the process. The
linearly independent thermodynamic Work 1-forms for evolution in the direction of the 3
basis vectors are determined to be,
WVx = i(ρVx)dA = +ρdy, (136)
WVy = i(ρVy)dA = −ρdx, (137)
WE = i(ρE)dA = 0. (138)
From Cartan’s Magic Formula representing the First Law as a description of topological
evolution,
L(V)A = i(ρV)dA+ d(i(ρV)A) ≡ Q, (139)
it becomes apparent that,
QVx = −ρdy, dQVx = −dρˆdy, (140)
QVy = +xdρ, dQVy = −dρˆdx, (141)
QE = dρ dQE = 0, (142)
All processes in the extremal direction satisfy the conditions that QEˆdQE = 0. Hence, all
extremal processes are reversible. It is also true that evolutionary processes in the direction
of the other basis vectors, separately, are reversible, as the 3-form QˆdQ vanishes forVx, Vy,
or E. Hence all such piecewise continuous, transitive, processes are thermodynamically
reversible.
Note further that the ”rotation” induced by the antisymmetric matrix [dA] acting on Vx
yields Vy and the 4th power of the matrix yields the identity rotation,
[dA] ◦ |Vx〉 = |Vy〉 , (143)
2 ◦ |Vx〉 = − |Vx〉 , (144)
4 ◦ |Vx〉 = + |Vx〉 . (145)
This concept is a signature of Spinor phenomena.
4.4 The Spinor Processes
Now consider processes defined in terms of the Spinors. The eigendirection fields of the
antisymmetric matrix representation of F = dA,
[F ] =
0 1 0
−1 0 0
0 0 0
, (146)
are given by the equations:
EigenSpinor1 |Sp1〉 =
Eigenvalue = +
-1, (147)
EigenSpinor2 |Sp2〉 =
Eigenvalue = -
-1 (148)
EigenVector1 |E〉 =
Eigenvalue = 0 (149)
Now consider the processes defined by ρ times the Spinor eigendirection fields. Compute
the change in internal energy, dU , the Work, W and the Heat, Q, for each Spinor eigendi-
rection field:
= i(ρSp1)A =
−1ρx d(UρSp
−1d(ρx), (150)
= i(ρSp2)A = −
−1ρx d(UρSp
) = −
−1d(ρx), (151)
UρE = i(ρE)A = ρ, d(UρE) = dρ, . (152)
= i(ρSp1)dA = ρ(dy −
-1dx), (153)
= i(ρSp2)dA = +ρ(dy +
-1dx) (154)
WρV1 = i(ρV1)dA = 0, . (155)
= Li(ρSp
)A = ρ(dy −
-1dx) +
-1d(ρx), (156)
= Li(ρSp
)A = ρ(dy +
-1dx)−
-1d(ρx), (157)
QρV1 = Li(ρV1)A = dρ. (158)
4.5 Irreversible Spinor processes
Next compute the 3-forms of QˆdQ for each direction field, including the spinors:
QρV1ˆdQρV1 = 0, (159)
ˆdQρSp
-1ρdρˆdxˆdy, (160)
ˆdQρSp
-1ρdρˆdxˆdy. (161)
It is apparent that evolution in the direction of the Spinor fields can be irreversible in a
thermodynamic sense, if dρˆdxˆdy is not zero. This is not true for the ”piecewise linear”
combinations of the complex Spinors that produce the real vectors, V and V⊥.
Evolution in the direction of ”smooth” combinations of the base vectors may not satisfy
the reversibility conditions, QˆdQ = 0, when the combination involves a fixed point in the
x, y plane. For example, it is possible to consider smooth rotations (polarization chirality)
in the x, y plane:
Vrotation right = V⊥ +
-1V = Sp1, (162)
QˆdQ = −
-1ρdρˆdxˆdy. (163)
Vrotation left = V −
-1V⊥= Sp2, (164)
QˆdQ = +
-1ρdρˆdxˆdy. (165)
The non-zero value ofQˆdQ for the continuous rotations are related to the non-zero Godbillon-
Vey class [7]. A key feature of the rotations is that they have a fixed point in the plane;
the motions are not transitive. If the physical system admits an equation of state of the
form, θ = θ(x, y, ρ) = 0, then the rotation or expansion processes are not irreversible.
Note that the (supposedly) Vector processes of the preceding subsection are combinations
of the Spinor processes,
Vx = (a · Sp1 + b · Sp2)/2 (166)
Vy = −
-1(a · Sp1− b · Sp2)/2. (167)
Almost always, a process defined in terms a linear combinations of the Spinor direction fields
will generate a Heat 1-form, Q, that does not satisfy the Frobenius integrability theorem,
and therefore all such processes are thermodynamically irreversible: QˆdQ 6= 0. However,
with the requirement that a2 is precisely the same as b2, then either piecewise linear process
is reversible, for QˆdQ = 0.
If the coefficients, and therefore the Spinor contributions, have slight fluctuations, the
cancellation of the complex terms is not precise. Then either of the (now approximately)
piecewise continuous process will NOT be reversible due to Spinor fluctuations.
Remark 8 The facts that piecewise (sequential) C1 transitive evolution along a set of di-
rection fields in odd (3) dimensions can be thermodynamically reversible, QˆdQ = 0, while
(smooth) C2 evolution processes composed from complex Spinors can be thermodynamically
irreversible, QˆdQ 6= 0, is a remarkable result which appears to have a relationship to Nash’s
theorem on C1 embedding. Physically, the results are related to tangential discontinuities
such as hydrodynamic wakes.
For systems of Pfaff dimension 4, all of the eigendirection fields are Spinors. The Spinors
occur as two conjugate pairs. If the conjugate variables are taken to be x,y and z,t then the
z,t spinor pair can be interpreted in terms of a chirality of expansion or contraction, where
the x,y pair can be interpreted as a chirality of polarization. In this sense it may be said
that thermodynamic time irreversibility is an artifact of dimension 4.
It is remarkable that a rotation and an expansion can be combined (eliminating the fixed
point) to produce a thermodynamically reversible process.
Ian Stewart points out that there are three types of manifold structure: piecewise linear,
smooth, topological. Theorems on piecewise-linear manifolds may not be true on smooth
manifolds. The work above seems to describe such an effect. Piecewise continuous processes
are reversible, where smooth continuous processes are not (see page 106, [38])!
5 Epilogue: Topological Fluctuations and Spinors
This Section 5 goes beyond the original objective of demonstrating that the Navier-Stokes
equations, based upon continuous topological evolution, can describe the irreversible decay
of turbulence, but not its creation. However, the key features of process irreversibility and
turbulence are entwined with the concept of Topological Torsion and Spinors. Hence this
epilogue calls attention to the fact that the Cartan topological methods permit the analysis
of Spinor entanglement, as well as the analysis of fluctuations about kinematic perfection.
This research area is in its infancy, and extends the thermodynamic approach to the realm
of fiber bundles. A few of the introductory ideas are presented below.
Remark 9 These concepts go beyond the scope of this essay which has the objective of
presenting the important topological ideas in a manner palatable (if not recognizable) to the
engineering community of hydrodynamics.
5.1 The Cartan-Hilbert Action 1-form
To start, consider those physical systems that can be described by a function L(q,v,t) and
a 1-form of Action given by Cartan-Hilbert format,
A = L(qk,vk,t)dt + pk·(dqk − vkdt). (168)
The classic Lagrange function, L(qk,vk,t)dt, is extended to include fluctuations in the
kinematic variables, (dqk−vkdt) 6= 0. It is no longer assumed that the equation of Kinematic
Perfection is satisfied. Fluctuations of the topological constraint of Kinematic Perfection
are permitted;
Topological Fluctuations in position: ∆q = (dqk − vkdt) 6= 0. (169)
As the fluctuations are 1-forms, it is some interest to compute their Pfaff Topological Di-
mension. The first step in the construction of the Pfaff Sequence is to compute the exterior
differential of the fluctuation 1-form:
Fluctuation 2-form: d(∆q) = −(dvk − akdt)ˆdt (170)
= −∆vˆdt, (171)
Topological Fluctuations in velocity:∆v =(dvk − akdt) 6= 0. (172)
It is apparent that the Pfaff Topological Dimension of the fluctuations is at most 3, as
∆qˆ∆vˆdt 6= 0, and has a Heisenberg component,
When dealing with fluctuations in this prologue, the geometric dimension of independent
base variables will not be constrained to the 4 independent base variables of the Thermody-
namic model. At first glance it appears that the domain of definition is a (3n+1)-dimensional
variety of independent base variables, {pk,qk,vk,t}. Do not make the assumption that the
pk are constrained to be canonically defined. Instead, consider pk to be a (set of) Lagrange
multiplier(s) to be determined later. Also, do not assume at this stage that v is a kinematic
velocity function, such that (dqk−vkdt) ⇒ 0. The classical idea is to assert that topological
fluctuations in position are related to pressure, and topological fluctuations in velocity are
related to temperature.
For the given Action, construct the Pfaff Sequence (12) in order to determine the Pfaff
dimension or class [8] of the Cartan-Hilbert 1-form of Action. The top Pfaffian is defined
as the non-zero p-form of largest degree p in the sequence. The top Pfaffian for the Cartan-
Hilbert Action is given by the formula,
Top Pfaffian is 2n+2
(dA)n+1 = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}ˆΩ2n+1, (173)
Ω2n+1 = dp1ˆ...dpnˆdq
1ˆ..dqnˆdt. (174)
The formula is a bit surprising in that it indicates that the Pfaff Topological Dimension of
the Cartan-Hilbert 1-form is 2n+2, and not the geometrical dimension 3n + 1. For n =
3 ”degrees of freedom”, the top Pfaffian indicates that the Pfaff Topological Dimension of
the 2-form, dA is 2n + 2 = 8. The value 3n + 1 = 10 might be expected as the 1-form
was defined initially on a space of 3n+ 1 ”independent” base variables. The implication is
that there exists an irreducible number of independent variables equal to 2n + 2 = 8 which
completely characterize the differential topology of the first order system described by the
Cartan-Hilbert Action. It follows that the exact 2-form, dA, satisfies the equations
(dA)n+1 6= 0, but Aˆ(dA)n+1 = 0. (175)
Remark 10 The idea that the 2-form, dA, is a symplectic generator of even maximal rank,
2n+2, implies that ALL eigendirection fields of the 2-form, F = dA, are complex isotropic
Spinors, and all processes on such domains have Spinor components.
The format of the top Pfaffian requires that the bracketed factor in the expression above,
{Σnk=1(∂L/∂vk − pk)dvk}, can be represented (to within a factor) by a perfect differential,
dS = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}. (176)
The result is also true for any closed addition γ added to A; e.g., the result is ”gauge
invariant”. Addition of a closed 1-form does not change the Pfaff dimension from even to
odd. On the other hand the result is not renormalizable, for multiplication of the Action
1-form by a function can change the algebraic Pfaff dimension from even to odd.
On the 2n+2 domain, the components of (2n+1)-form T = Aˆ(dA)n generate what
has been defined herein as the Topological Torsion vector, to within a factor equal to the
Torsion Current. The coefficients of the (2n+1)-form are components of a contravariant
vector density Tm defined as the Topological Torsion vector, the same concept as defined
previously on a 4D thermodynamic domain, but now extended to (2n+2)-dimensions. This
vector is orthogonal (transversal) to the 2n+2 components of the covector, Am. In other
words,
AˆT = Aˆ(Aˆ(dA)n) = 0 ⇒ i(T)(A) =
TmAm = 0. (177)
This result demonstrates that the extended Topological Torsion vector represents an adia-
batic process. This topological result does not depend upon geometric ideas such as metric.
It was demonstrated above that, on a space of 4 independent variables, evolution in the
direction of the Topological Torsion vector is irreversible in a thermodynamic sense, subject
to the symplectic condition of non-zero divergence, d(AˆdA) 6= 0. The same concept holds
on dimension 2n+2.
The 2n+2 symplectic domain so constructed can not be compact without boundary for it
has a volume element which is exact. By Stokes theorem, if the boundary is empty, then the
surface integral is zero, which would require that the volume element vanishes; but that is
in contradiction to the assumption that the volume element is finite. For the 2n+2 domain
to be symplectic, the top Pfaffian can never vanish. The domain is therefore orientable,
but has two components, of opposite orientation. Examination of the constraint that the
symplectic space be of dimension 2n+2 implies that the Lagrange multipliers, pk, cannot be
used to define momenta in the classical ”conjugate or canonical” manner.
Define the non-canonical components of the momentum, ℏkj, as,
non-canonical momentum: ℏkj = (pj − ∂L/∂vj), (178)
such that the top Pfaffian can be written as,
(dA)n+1 = (n+ 1)!{Σnj=1ℏkjdvj}ˆΩ2n+1, (179)
Ω2n+1 = dp1ˆ...dpnˆdq
1ˆ..dqnˆdt. (180)
For the Cartan-Hilbert Action to be of Pfaff Topological Dimension 2n+2, the factor
{Σnj=1ℏkjdvj} 6= 0. It is important to note, however, that as (dA)n+1 is a volume ele-
ment of geometric dimension 2n+2, the 1-form Σnj=1ℏkjdv
j is exact (to within a factor, say
T (qk, t, pk,Sv)); hence,
Σnj=1ℏkjdv
j = TdSv. (181)
Tentatively, this 1-form, dSv, will be defined as the Topological Entropy production relative
to topological fluctuations of momentum, kinematic differential position and velocity. If
ℏkj is defined as the deviation about the canonical definition of momentum, ℏkj = ∆pj , and
noting the the expression for the top Pfaffian can be written as (n+1)!{Σnj=1ℏkj∆vj}ˆΩ2n+1,
leads to an expression for the entropy production rate in the suggestive ”Heisenberg” format:
TdSv = ∆pj∆v
j . (182)
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	Topological Thermodynamics
	The Axioms of Topological Thermodynamics
	Cartan's Magic Formula  First Law of Thermodynamics
	The Pfaff Sequence and the Pfaff Topological Dimension
	The Pfaff Topological Dimension of the System 1-form, A
	The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W
	Topological Torsion and other Continuous Processes.
	Reversible Processes
	Irreversible Processes
	The Spinor class
	Emergent Topological Defects
	Applications
	An Electromagnetic format
	Topological 3-forms and 4-forms in EM format
	Topological Torsion quanta
	A Hydrodynamic format
	The Topological Continuum vs. the Geometrical Continuum
	Topological Hydrodynamics
	Classical Hydrodynamic Theory
	The Lagrange-Hilbert Action
	Euler flows and Hamiltonian fluids
	The Navier-Stokes fluid
	The classic Navier-Stokes equations
	The Navier-Stokes equations embedded in a non-equilibrium thermodynamic system
	The Topological Torsion process for the Navier-Stokes fluid
	Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains
	Examples of PTD = 3 domains and their Emergence
	Piecewise Linear Vector Processes vs. C2 Spinor processes
	The Vector Processes
	The Spinor Processes
	Irreversible Spinor processes
	Epilogue:  Topological Fluctuations and Spinors
	The Cartan-Hilbert Action 1-form
