No Problem. Of Multiple Inverse Problems In Statistical Physics Before Study

Investigation shows that studies of inverse problems may become a growth point in modern physics. Considering the needs in studies of mechanisms of high Tc superconductivity, one proposed the specific heat-phonon spectrum inversion (SPI) problem, and obtained an exact solution formula with a parameter s for eliminating divergence and avoiding Reimann hypothesis, called Dai's formula. Montroll's formula is its a special case. We obtain a new complete and orthogonal function set and establish a new solution method (DMF method) for this inverse problem, combining with Dai's formula and the asymptotic behavior control (ABC) condition. Because DMF method incorporates the ABC conditions, which is of crucial importance for actual inversions, so DMF is a method suitable for practical computations. Using this new DMF method we obtain the phonon spectra of high Tc superconductor YBCO and negative thermal expansion material ZrWO from their real heat capacity data.The black-body radiation inversion (BRI) is to determine the area-temperature distribu-tion on the surface of a black-body by measuring its radiated power spectrum. This problem received much attention for its potential applications in remote sensing. Our DMF method is also generalized and applied to BRI. It shows an extraordinarily high precision in numerical computations, then solves the famous:"low-temperature distribution difficulty" in BRI, which has puzzled us for a long time.The emissivity inversion (EMI) is stated as obtaining the emissivity of a "gray-body" from the measured total radiated power. How to reduce the emissivity is an important task in anti-remote sensing. Because of the character of high computation precision, the DMF method in ETI exhibits its own irreplaceable merits.These kinds of inverse problems are all concerned with solving a Fredholm integral equation of first kind, so we will inevitably face the notorious instability problems. People have used many physical and mathematical constrains to control the instability. However, so hard conditions like the normalization of phonon spectrum have not any contribution yet. In our work, the normalization problems of SPI and BRI are studied for the first time, then the normalization conditions are used in numerical computations successfully. It confirms the inversion results are normalized. The inversion quality will be improved.The energy spectra (phonon spectra) obtained via SPI are strictly T-independent. This allows us to carry out holographic studies of materials, in which the information encoded in the heat capacity only is transformed into the energy spectrum and hence into all the other related thermodynamic properties. The holographic studies of YBCO and ZrWO are realized. The information of these two materials are obtained by one experiment on a single sample. Our work also includes deriving Chen's formula from Dai's formula in BRI problem. When and only when Dai's ABC conditions are valid, the unique existence theorem of Chen's formula and the convergence of Chen's series are proved for the first time. Thus the two famous theories in BRI are unified. Two sets of exact solutions are also obtained by Chen's formula.At last, a new transmissivity inversion (TRI) problem is proposed. The transmissivity of a film is suggested to be determined by measuring only the transmitted total radiated power. It may become a new approach to transmissivity which is rather difficult to be measured directly as a function of frequency.