Dear Mr. Liley,

From the point of view of harmonic functions affecting stock markets which we have analysed in Econophysical Quantum Mechanics using string matching field theory developed by us (Mallick (1993, 2015), Mallick, Hamburger & Mallick (2016, 2017, 2018)), that energy is best conserved by the first moment of your wave function in entropy, for which a topological solution is to convert the phase space in terms of radian measure, for which your moment generating function will become spherical and the natural iterative partition using the law of the iterated logarithm will be the equipartitioned quadrants as the sphere will represent the potential energy in the differential phase space and physical laws of symmetry will apply. We have used this strategy to quantize information-energy in electronic stock markets to derive systems equilibrium in spacetime in the internetworked string matching field. We have derived the Internetworked Fine Structure Constant in this quantum mechanical field by applying methods of Higgs-Mallick Mechanics in the Dbranes String Theoretic Stock & Flow RHMHM Calculus (Mallick et. al. as above), also Mallick (2018).You should discuss this matter with your adviser.

S.K.Mallick

for S.K.Mallick, S.Raychaudhury, S.Mallick

of RHMHM School,

USA, Japan, India

Behnam, thank you for the response. I've ready through derivations both for the generalized equipartition theorem and the virial theorem (which is even more general), and you are right the results do not depend on the parameters of the Hamiltonian. That's why I'm confused. It is clearly seen that if a parameter is temperature dependent, then taking the derivative with respect to temperature is going to introduce an extra term into the average energy, seemingly disagreeing with equipartition.

Drew> Z = Exp[-H / (kb * T)], and the average energy can be calculated as proportional to the derivative of ln[Z].

The exact formula, easy to prove, is ⟨H⟩ = -∂ln(Z)∂β, where β = 1/(kBT). However, as you probably already have noted, that is mathematically correct only when H is independent of β (i.e. temperature T).

One may easily imagine situations where the parameters of the Hamiltonian actually depend on temperature, because one is dealing with a phenomenological "effective" description^*, not taking into account the physics which leads to this temperature dependence. However, if such a dependence is large enough to make any difference, the standard thermodynamic interpretation^** of ln Z breaks down, and thereby all sacred relations of thermodynamics. Which is the absolutely last thing we should consider violating in physics.

If you want to escape the usual equipartition principle, this is easily violated by non-quadratic terms in a classical Hamiltonian, or introduction of quantum mechanics (without which even Hell would freeze over, due to its infinite heat capacity).

^*) Which in practise is always the case, since we don't even know what is going on at extremely small scales, and (mostly) don't have to worry about sub-atomic scales.

^**) ln Z = -β F = -β(U-TS), where F is the Helmholtz free energy.

PS. The very first answer to this question should be viewed as an attempt to repeat the notorious Sokal hoax, https://en.wikipedia.org/wiki/Sokal_affair (often perpetrated on RG).