Surfaces of Constant Temperature for Glauber Dynamics

The wavefunction of a single spin system in a prepared initial state evolves to equilibrium with a heat bath. The average spin $$q(t) = p_{\uparrow}(t) - p_{\downarrow}(t)$$ exhibits a characteristic time for this evolution. With the proper choice of spin flip rates, a dynamical Ising model (Glauber) can be constructed with the same characteristic time for transition of the average spin to equilibrium. The Glauber dynamics are expressed as a Markoff process that possesses many of the same physical characteristics as its quantum mechanical counterpart. In addition, since the classical trajectories are those of an ergodic process (the time averages of a single trajectory are equivalent to the ensemble averages), the surfaces of constant temperature, in terms of the model parameters, may be derived for the single spin system.