By Meghnad Saha
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Additional info for A treatise on heat: Including kinetic theory of gasses [sic], thermodynamics and recent advances in statistical thermodynamics
If solutions to a particular evolutionary problem exist, then the solutions are not unique. Envelope solutions, such as Huygen wavelets and propagating tangential discontinuities (called signals) are classic examples of solution non-uniqueness. 13. Cartan’s Magic formula, in terms of the Lie diﬀerential acting on exterior diﬀerential 1-forms establishes the long sought for combination of dynamics and thermodynamics, enabling non-equilibrium systems and many irreversible processes to be computed in terms of continuous topological evolution, without resort to probability theory and statistics.
In this monograph, those processes which can change Unique Continuous Evolutionary Processes 43 continuously the topology of a physical system are of major interest, for topological change is a necessary requirement of thermodynamic irreversibility. Intuitively, a process applied to a physical system can be arbitrary, which implies that the process V is not necessarily dictated by the topological structure of the physical system, A. This intuitive idea is not precise. In contrast, for a given physical system, A, there are two vector direction fields that are determined uniquely from the functions that defined the topological structure of the physical system.
Evolutionary processes on space-time will be determined in terms of four-dimensional contravariant direction fields, V4 (x, y, z, t) , to within a non-zero factor, ρ. Continuous topological evolution (see Chapter 7) will be defined in terms of Cartan’s magic formula for the Lie diﬀerential, Cartan-Lie diﬀerential : L(ρV4 ) A = i(ρV4 )dA + d(i(ρV4 )A). 1) The definition is to be interpreted algebraically, using the properties of the exterior diﬀerential and the inner product associated with exterior diﬀerential forms.