Researchers try to get the temperature-independent energy spectrum of some materials for obtaining all their thermodynamic properties from their experimental heat capacities alone.
However, measurements are made at finite temperatures in practice, such as with phonon spectra obtained by inelastic neutron scattering, so it is very difficult to obtain a T-independent energy spectrum for any real material. One promising method to obtain such a spectrum is to solve the so-called specific heat-phonon spectrum inversion (SPI) problem.
In the past two decades, a research group at Fudan has proved the unique existence theorem of SPI, obtained the exact solution formula with a parameter s for avoiding divergence, proposed the asymptotic behavior control (ABC) theory, and developed the universal function set method based on Hermite functions (UFS-H) for practical calculations.
The SPI equation is a Fredholm integral equation of the first kind, which is notoriously ill-conditioned: a small deviation in input can lead to a large deviation in output. The instabilities are classified into three classes: Hadamard instability, data incompleteness instability, and method-dependent instability. Hadamard instability is always present but can be reduced by limiting the solution to a function class without high frequencies. ABC theory has proved to be crucial in reducing the data incompleteness instability. A new DM function (DMF) set method was developed and the T-independent phonon spectrum of the negative thermal expansion material ZrW2O8 was obtained. Compared with the inversion result from the previous UFS-H method, the T-independent phonon spectrum obtained for ZrW2O8 is almost method-independent. The third class of instability is well controlled.
According to this T-independent phonon spectrum, all the related thermodynamic quantities for ZrW2O8 lattice vibrations, such as the thermodynamic potential, entropy, internal energy, Helmholtz free energy, grand partition function, etc., can then be obtained by integration to avoid the derivative.
It is well known that thermodynamics attributes the concrete properties of a macroscopic system to its state equation and specific heat. However, for some special systems the authors can obtain all the thermodynamic functions based only on their specific heat. In this, the first realization of a holographic study in Gibbs statistics, all the results are consistent with the laws of thermodynamics. This work tests the foundations of Gibbs statistics and finds results consistent with those foundations.
Holographic study ensures that all the data obtained are exactly of the same sample under a particular physical condition. This cannot be achieved by multi-measurement in different experiments. These thermodynamic functions, obtained from the T-independent phonon spectrum, can be tested by future experiments. The entropy obtained by this procedure has a derivative that tends to be zero, so it can be regarded as a check for the third law of thermodynamics. Holographic studies of materials have opened a door to studying the properties of materials at ultra-low temperatures. It is expected that future studies to address the properties of materials at ultra-low temperatures via SPI using T-independent spectra like those researchers have found for ZrW2O8.
See the article: Ji F M, Dai X X, Stevens R, et al, Thermodynamic functions of ZrW2O8 from its heat capacity. Sci China-Phys Mech Astron, 2012, 55(4): 563