By Robert Kiehn

Adventures in utilized Topology...Continuous topological evolution can be utilized to encode non equilibrium thermodynamic platforms and irreversible tactics with out using facts or chance. Such equipment are common and follow to structures from the microscale to the cosmological scale. Thermodynamic irreversibility and the arrow of time are artifacts of topological switch in 4 dimensions or extra.

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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 that Cartan’s magic formula is another way to state that the first law of thermodynamics makes practical sense. It is remarkable that although the symbols are diﬀerent, the same basic constructions and conclusions apply to most classical physical systems. Examples given in Part II will demonstrate this feature of topological universality.

2) The topological constraint in eﬀect defines (or presumes) a limit process. The constraint on the more general Lie diﬀerential leads to the concept of the Lie derivative of the 1-form A. The evolution is represented by the infinitesimal propagation of the 50 Topological Thermodynamics 1-form, A, down the flow lines generated by the 1-parameter group. 6. In this series, the Lie diﬀerential (not Lie derivative) is the fundamental generator of continuous topological evolution. " It is important to remember that the concept of a diﬀerential form is diﬀerent from the concept of a derivative, where a (topological) limit has been defined.

For example, the 3-form of charge-current density is exact, J = dG. 3). 17) z1d z1d z1d Z = i(ρV k )F + 0 ⇒ 0. 18) z1d It follows that i(ρV k )dA must be zero on the cycle z1d for any deformation parameter ρ. 5). 4 Holder Norms, Period Integrals and Topological Quantization Besides the invariant structures considered above, the Cartan methods may be used to generate other sets of topological invariants. Realize that over a domain of Pfaﬀ dimension n less than or equal to N, the Cartan criteria admits a submersive map to be made from N to a space of minimal dimension n.