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Gerhard Dickel
Time Dependence of Thermodynamic Potentials
Keywords: First Law, Hilbert’s Independence Theorem, Thermodynamic Potentials, Least Action, Time Dependence and Collapse of Thermodynamic States
Integrals being dependent on the limits only are denominated potentials. According to Hilbert’s Independence Theorem potentials are given by the limits of an action integral. Whereas the upper limit presents the actual action the lower limit concerns the least action. The action can be a constant, the least action can be zero or can be given by a finite value. As to thermodynamics the latter is true as found by Planck.
According to Hilbert the path of integration must be taken along the geodesic slope as resulting from the Euler equations. Changing from dynamics to thermodynamics we change from single mass points to a system of mass points. In this case according to Caratheodory the amount as well as the direction of the geodesic slope of the components must be taken into account. As to the latter condition we consider the integrand of the action integral presented by a chain of vectors of the coordinated canonical variables. Searching according to calculus of variation for conditions that furnish a minimum value of the action integral the choice of the elliptic transcendente yields a sufficient condition. Since the latter can be reduced as shown by Legendre to three kinds of irreducible elliptic integrals the transition from the Lagrange representation into the Hamilton representation yields an irreducible action. It can be taken as compatible with the least action of Planck.
Whereas the action integral presents the actual time the integrand presents the energy state. As recognized already by Gauss the reverse function must be considered to get the familiar representation.
Zeitschrift für Physikalische Chemie, Oldenbourg Wissenschaftsverlag
Print ISSN: 0942-9352
Volume: 219, 12/2005
Pages: 1639 - 1647