SpecificHeat - crowlogic/arb4j GitHub Wiki

The Riemann zeta function is related to Debye's theory of specific heat through the Debye model of solids.

The specific heat of solids at low temperatures is a long-standing problem in the field of condensed matter physics. Both the Einstein and Debye models were developed to address this issue, but it was the Debye model that accurately predicted the behavior of specific heat at low temperatures.

The Debye model treats the vibrations of the atomic lattice (phonons) as phonon gases in a box, considering the sound waves that propagate through a solid, and treats them as a kind of quantum mechanical wave, much like photons of light.

The specific heat predicted by Debye's model is proportional to $T^3$ at low temperatures, which matches experimental observations much better than the $T$ prediction of the Einstein model.

The connection to the Riemann zeta function comes into play when we calculate the Debye function, which gives the average energy of the phonon gas. The specific heat can then be found by taking the derivative of this average energy with respect to temperature.

The Debye function involves an integral over all the modes of vibration, and that integral takes the form of the Riemann zeta function, specifically the zeta function at 3, ζ(3). So, the calculation of the Debye function and thus the specific heat in the Debye model of solids involves the use of the Riemann zeta function. This is a prime example of the unexpected ways in which different areas of mathematics and physics can interact.