Condensation of Eigen Microstate in Statistical Ensemble and Phase Transition

In a statistical ensemble with $M$ microstates, we introduce an $M \times M$ correlation matrix with the correlations between microstates as its elements. Using eigenvectors of the correlation matrix, we can define eigen microstates of the ensemble. The normalized eigenvalue by $M$ represents the weight factor in the ensemble of the corresponding eigen microstate. In the limit $M \to \infty$, weight factors go to zero in the ensemble without localization of microstate. The finite limit of weight factor when $M \to \infty$ indicates a condensation of the corresponding eigen microstate. This indicates a phase transition with new phase characterized by the condensed eigen microstate. We propose a finite-size scaling relation of weight factors near critical point, which can be used to identify the phase transition and its universality class of general complex systems. The condensation of eigen microstate and the finite-size scaling relation of weight factors have been confirmed by the Monte Carlo data of one-dimensional and two-dimensional Ising models.Comment: 9 pages, 16 figures, accepted for publication in Sci. China-Phys. Mech. Astro

A Two-stage Polynomial Method for Spectrum Emissivity Modeling

Spectral emissivity is a key in the temperature measurement by radiation methods, but not easy to determine in a combustion environment, due to the interrelated influence of temperature and wave length of the radiation. In multi-wavelength radiation thermometry, knowing the spectral emissivity of the material is a prerequisite. However in many circumstances such a property is a complex function of temperature and wavelength and reliable models are yet to be sought. In this study, a two stages partition low order polynomial fitting is proposed for multi-wavelength radiation thermometry. In the first stage a spectral emissivity model is established as a function of temperature; in the second stage a mathematical model is established to describe the dependence of the coefficients corresponding to the wavelength of the radiation. The new model is tested against the spectral emissivity data of tungsten, and good agreement was found with a maximum error of 0.64