For constrained physical systems that can be described by a 1-form of Action, the topological Pfaff dimension or class of the 1-form is either even or odd.
Action 1-forms of odd Pfaff class generate contact manifolds of dimension 2n+1,
and support an evolutionary path defined by the unique null eigenvector of the
matrix of coefficients of the 2-form, dA. Such evolutionary extremal paths are
conservative in a Hamiltonian sense, and describe processes which are
Action 1-forms of even Pfaff class generate symplectic manifolds of dimension
2n+2, for which null eigenvectors (extremal fields) of the 2-form dA do not
exist. Such symplectic manifolds do support a unique evolutionary path defined
but the Torsion Current, a 2n+1 form constructed as A^dA^..^dA.
Evolution in the unique direction of the components of the Torsion Current is,
in general, thermodynamically irreversible.
In the previous display, the notion of Pfaff dimension and Frobenious
integrability was discussed. Topological Torsion of a 1-form of Action required
that the Pfaff dimension of A be 3 or more. Topological Torsion has been
defined as the 3-form A^dA. The Torsion current is defined as the 2n+1 form,
A^dA^dA....^dA. In a 4 dimensional domain, the concepts are equivalent; e.g.,
in four dimensions the Torsion Current is the same as the Topological Torsion
In classical thermodynamics, the concept of irreversibility is associated with the idea that the 1-form of Heat, Q, does not admit an integrating factor. That is, for irreversible processes it is impossible to write Q in the form Q = TdS.
Such a criteria implies that Q^dQ is not zero for irreversible processes. The
Pfaff dimension of Q must be 3 or greater.
Now consider a physical system that can be described by a 1-form of Action, A,
on space time. Consider a process described by a vector field of flow (or a
current in space time). Then from Cartan's magic formula
Consider those V that have components proportional to the Torsion current, T, which is constructed from the 3-form of Topological Torsion, A^dA
It can be shown that
For such processes in the direction of the Torsion current,
Hence, such processes are thermodynamically irreversible.