The Metric of Thermodynamic Response Space

A curious feature of the space Ms of thermodynamic variables in an equilibrium state S is that its dimensionality varies with the number of phases, p, even though the values of the intensive variables (which might be used to parametrize the state S) do not. The intensive-type ket vectors R/ of (10.8) can actually be defined for all c + 2 intensities (T, —P, fjL, pi2, , pic) arising from the fundamental equation of a c-component system, U(S, V, n, ri2,. .., nc), even if only /of these remain linearly independent when p phases are present. 

Although the linearity of the chain-rule differential expressions (10.5) confers primitive affine-type spatial structure on thermodynamic variables, it does not yet provide a sense of distance or metric on the space (other than what might be displayed in an arbitrarily chosen axis system). In order to bring intrinsic geometrical structure to the thermodynamic space, we need to define the scalar product (R RJ) [(9.29)] that dictates the spatial metric on Ms- The metric on Ms should reflect intrinsic physical properties of the thermodynamic responses, not merely generic chain rule-type mathematical properties of their differential representation. At the same time, we must exhibit how the space Ms is explicitly connected to the physical measurements of thermodynamic responses. Because such measurements assign scalar values to physical properties, it is natural to associate each scalar product of Ms with the scalar value of an experimental measurement. How can this be done  

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 

How can (10.9) make sense as a geometrical scalar product From the chain-rule linearity property (10.4) of partial derivatives, one can see that the (R/ R7) values defined by (10.9) will automatically satisfy the distributive property (9.27a)  

Moreover, from the first-law (Maxwell-type exactness) relationship between mixed partial derivatives, as expressed by (8.80), we see that the (R R/) values also satisfy the symmetric property (9.27b)  

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