Euler integration of thermodynamic systems in the presence of fields

Abstract

The significance and validity of Euler integration and the Gibbs–Duhem equation for thermodynamic systems in the presence of fields is considered. It is shown that the presence of fields can change the fundamental property concerning Euler integrability and applicability of the Gibbs–Duhem equation that underlines classical thermodynamics. In this context, an assembly of charges, current elements or a combination thereof, is not expected to be a first-order homogeneous function of its extensive variables. Consequently, Euler integration of systems that include charges, current elements, or generally involve interaction with electromagnetic fields can be meaningless. This means that, in general, such systems are not expected to satisfy the Gibbs–Duhem equation. Specific examples of capacitors, charged spheres, magnetic circuit of variable gap and a sphere in a uniform field, are used to illustrate the effect of the field on the energy and thermodynamic properties of polarizable systems. It is shown that, apart from limited configurations, these specific systems do not behave as first-order homogeneous functions of their extensive variables and their Euler integration can lead to results of no physical significance. Finally, in the presence of fields, the field-independent part of the energy differential can be Euler integrated, but the field-dependent part must be integrated as an exact differential. Consequently, the sum of the field-independent and field-dependent exact differentials is generally not Euler integrable.