``Anti-Equilibrium'': The limiting frozen state of kappa distributions

The kappa distribution of particle velocities provides an unambiguous replacement of the Maxwell distribution for systems out of thermal equilibrium. The kappa index is a measure of how far the system of particles is from thermal equilibrium. This ``thermodynamic distance'' is inversely proportional to the kappa index; it becomes zero at thermal equilibrium where the kappa index is infinite, while it obtains its maximum at the furthest state from thermal equilibrium, where the kappa index is zero, a state called ``anti-equilibrium.'' By keeping fixed the temperature and decreasing the kappa index, the particles approach this peculiar state of anti-equilibrium and are characterized by a power-law distribution density with spectral index $\sim$ 1.5; this constitutes a universal behavior, independent of the system's number of particles or degrees of freedom. As the kappa decreases and the system approaches the anti-equilibrium state, the particles lose their kinetic energy. This procedure of ``kappa-freezing'' by decreasing the kappa index at a fixed temperature is similar to the more familiar freezing procedure of decreasing temperature and approaching the ``absolute zero'' for a fixed kappa index.