The decay to equilibrium - A Symplectic to Contact phase transition

When physical systems admit description in terms of exterior differential forms, and when physical processes admit description in terms of singly parameterized vector fields, then it becomes apparent that whether or not the domain of indepedent variables is of even or odd dimension is crucial to the resulting solution possibilities. Given a 1-form of Action describing a physical system, the Pfaff dimension is the irreducible number of functions required to describe the system over the domain. The original discription may be in the form of a large number (mole numbers) of variables, but the Pfaff dimension may be only 2n+1 or 2n+2 where n is a small integer. If the Pfaff dimension is odd the manifold upon which it is built will support a contact structure. If the Pfaff dimension is even, the manifold will support a symplectic structre.

On the odd dimensional contact structure (of maxial rank) there exists a unique direction field completely determined by the components of the system. This direction field forms the unique eigen vector with null eigen value for the anti-symmetric matrix of coefficients that define the 2-form F=dA. This extremal direction forms the "extremal" vector of the calculus of variations, and evolution in the direction of this extremal field is conservative and admits a Hamiltonian representation. When the 1-form of virtual work W is defined as W=i(v)F, it is apparent that the virtual work is zero for extremal Hamiltonian evolution. Such is the case for the "force free" plasma.

When the Pfaff dimension of a physical system implies an even dimensional symplectic structure, there do not exist any null eigen vectors of F=dA. Hence the unique extremal evolutionary field does not exist on maximal rank domains of even dimension. There still exist "Hamiltonian" flows, but these processes are generated from a Bernoulli function, such that the Work 1-form is a perfect gradient. Such is the case for Eulerian hydrodynamics.
The Bernoulli function will generate the flow lines following a "Hamiltonian" recipe, but the Bernouilli function dow not necessarily have the same value from flow line to flow line. For both the extremal and the Bernoulli types of processes, the "flux of topological vorticity" ( or electromagnetic flux) is a local (open integration domain) evolutionary invariant.

For the even dimensional case, there does exist a direction field that is uniquely determined from the functions that define the physical system, A. That vector field is defined as the "Torsion vector". Evolution of the Action A in the direction of the torsion field is thermodynamically irreversible!!! In fact, evolution of the Action in the direction of the Torsion vector is proportional to itself. The function of proportionality is not constant and can be attractive to or repelling from a zero set.
Under certain circumstances, the attractor can be stabile and zero. Suppose the top Pfaffian is of the form F^F. Then F^F non-zero would imply a symplectic system, and F^F=0, A^F non-zero would imply a contact system. The form F^F decays in the direction of the Torsion vector to a zero value, which is now must be an element of a contact domain. The contact domain is defined on the original even dimensional space, but the 2-form F no longer is of maximal rank. This latter result implies that there can exist two null eigen vectors, one of which is precisely the torsion vector, in spaces of dimension 2n+2.

The sympectic (even dimensional) system decays by an irreversible (not symplectic) process in the direction of the Torsion vector, A^F with a decay rate determined by F^F. Once the system has decayed to the value 0, The Pfaff dimension of the domain is odd, and subsequent evolution can be made by a uniquely defined Hamiltonian process.

image As I have time I will replace this under construction
symbol with a link to a more detailed pdf or tex file download.

Copyright © CSDC Inc. All rights reserved.
Last update 07/02/2001
to HomePage