Year-end Sale % OFF 
Abstract
Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of . Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.
Highlights
It has been known since the work of Gibbs [1,2] that thermodynamics can be formulated in the language of contact geometry
Because the fundamental thermodynamic relation takes the form of a contact structure on an odd-dimensional manifold, closed systems in thermal equilibrium correspond to Legendrian submanifolds with respect to this contact structure
We find a generating set of differential invariants and invariant derivations with respect to Aff(V ), and we notice that the heat capacity is one of the fundamental scalar differential invariants
Summary
It has been known since the work of Gibbs [1,2] that thermodynamics can be formulated in the language of contact geometry. For each integer k ≥ 2, the kth central moment gives a symmetric k-form on Legendrian submanifolds [6]. In this framework, the group of affine transformations on V plays an important role, as it acts on. If we treat Legendrian submanifolds that are related by a transformation in this group as equivalent, it is clear that the important quantities are those that are invariant under this group action Such quantities will be the main focus of this paper. We compute differential invariants with respect to two different subgroups of Aff(V ) before we finish with a discussion of the significance of differential invariants
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
