Thermodynamic Vectors and Geometry

In the abstract space Ais associated with a given equilibrium state S, the fundamental geometry for thermodynamic vectors R,-), R7) is fixed by the scalar product 

Classical and Geometrical Theory of Chemical and Phase Thermodynamics. By Frank Weinhold Copyright 2009 John Wiley Sons, Inc. 

We shall say that two vectors R ) and R7) are equal if and only if their separation R, — Ry vanishes, 

In short, the theorems, terminology, and working methods of Euclidean geometry can be carried over intact into this abstract thermodynamic domain. 

Equation (11.8) expresses the geometrical necessity of linear dependence among any /+ 1 vectors in an /-dimensional space. Such linear dependence (corresponding to a null 

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It is important to recognize that the small subset of matrix equations introduced in the main text (typically, restricted to real matrix elements) will be found sufficient to exploit the geometrical simplicity that underlies equilibrium thermodynamics. Nevertheless, it is useful to introduce the thermodynamic vector geometry in the broader framework of matrix theory and Dirac notation that is broadly applicable to the advanced thermodynamic topics of Chapters 11-13, as well as to many other areas of modem physical chemistry research. 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

Having made this long detour into vector geometry and metric spaces, the student of thermodynamics will naturally be impatient to learn the missing link that connects these disparate domains, i.e., that associates the scalar products of the geometry domain... 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

Algebra geometry trigonometry calculus (including vector calculus) differential equations statistics numerical analysis algorithmic science computational methods circuits statics dynamics fluids materials thermodynamics continuum mechanics stability theory wave propagation diffusion heat and mass transfer fluid mechanics atmospheric engineering solid mechanics. 

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