In between the national security lectures of past few days, I got the following impressionistic sketchy thoughts on a simplified rule of thumb conjecture for how to deal with Frohlich coherence for pumped open dissipative structures including living matter and high temperature operating superconductors as well as other applications.
Toy Model 1
Zero rest mass (or zero energy gap) bosons of frequency f, with zero chemical potential so total number of particles are not conserved e.g. photons, acoustic phonon branch in crystals et-al
I. Ansatz 1: effective non-equilibrium "temperature" T(f)' is
T(f)' = T[1 - P(f)/hf^2]^-1
P(f) is the external pump power spectrum at frequencies f for the gapless bosons.
T is the equilibrium temperature when P(f) = 0.
Therefore,
T(f)' > 0 —> infinity when P(f)/hf^2 —> 1- = critical threshold for non-equilibrium analog of spontaneous symmetry
Furthermore
T(f)' < 0 for P(f)/hf^2 > 1, i.e. population inversion for qubit sources of the bosons of energy levels E1 & E2
hf = E1 - E2
Toy Model 2 effective room temperature and higher non-equilibrium super-conductors and also Frohlich pumped membranes, microtubules et-al.
zero chemical potential
T(f)' = T[1 + P(f)/hf^2]^-1
T(f)' —> 0 monotonically as P(f)/hf^2 —> infinity
Toy model 3 If there is a chemical potential u (Lagrangian multiplier constraint conserving total number of particles N)
T(f)' = T[1 + P(f)/hf^2 + uN/kT]^-1
II. For (3D + 1) bosons with zero dissipation in the non-equilibrium regime
n(f) = [e^hf/kT(f') - 1]^-1
N = integral of n(f) x density of oscillators df
III. For (3D + 1) bosons with non-zero dissipation r(f) in the non-equilibrium regime
n(f) = [e^hf/kT(f') - 1+ r(f)]^-1 ?
IV Fluctuation-Dissipation around T(f)'
"The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermal fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.
The fluctuation–dissipation theorem relies on the assumption that the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation. Therefore, the theorem connects the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium.[1] Often the linear response takes the form of one or more exponential decays.
The fluctuation–dissipation theorem was originally formulated by Harry Nyquist in 1928,[2] and later proven by Herbert Callen and Theodore A. Welton in 1951.[3]" https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem
Phase transitions happen when the small fluctuations become unstable.