Geometry in classical statistical thermodynamics

A Euclidean geometry for classical thermodynamics is discussed. The central physical idea is that it is useful to characterize the system in terms of a number of mean values of "relevant" observables. These mean values are written, as usual, as an expectation Σ_i,A_i,p ⁱ over a (classical) probability distribution. The expectation value is then interpreted as a scalar product between vectors belonging to dual spaces. A metric is introduced via the transformation from one space to another. In terms of the metric, the scalar product of two vectors belonging to the same space (e.g., two probability distributions or two observables) can be defined. In the space of all states the metric does not depend on the state of the system and the curvature tensor vanishes, i.e., the space is Euclidean.