Thermodynamics, responses

The calculus-based symbolism (10.1), (10.2) provides dependable, if unwieldy, descriptions of thermodynamic responses. The differential aspect (10.2) often transcends the functional aspect (10.1) for thermodynamic purposes, and, indeed, special criteria may be invoked to test whether an underlying function (10.1) exists for measured differential... 

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 ... 

As shown by these values, the pressure vector dominates the thermodynamic responsiveness of the ideal gas, conferring near one-dimensional character on its thermodynamic geometry. 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

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