The second order extension of the Gibbs state

Abstract

We present the second order thermodynamics (in the sense of higher order thermodynamics of the first author) on the base of gauge invariant (forH →H + const. transformation) parameterization of second order states, i.e.,

$$\rho = Z^{ - 1} \exp \left[ { - \beta H - \beta '\left( {H - U} \right)^2 } \right]$$

whereU is determined self-consistently fromU = TrρH. By this formulation, we can give definite physical interpretation to temperature parametersβ andβ′. In particular, the first order temperatureβ is identified with the conventional concept of temperature, and for large systems the second order state gives the same results as the conventional Gibbs state. We show these facts by some general considerations and by concrete calculations for three simple models, ideal gas, a semiharmonic oscillatory chain of hard-core particles, and a 3-dimensional solid with harmonic oscillations. The results are weekly model-dependent.