Abstract
The behavior of the classical degenerate parametric oscillator ( \omega_{i} = \omega_{s} ) with small linear dissipation is considered and an expression for the steady-state probability distribution for the subharmonic amplitude is obtained. The treatment is limited to the case 1 \ll Q_{p} \ll Q_{s} where Q p and Q s are the Q factors at respectively the pump and signal frequencies. The behavior is analogous to that of the Brownian motion of a particle in a bistable potential well. This leads to a tractable equation for the relaxation towards the steady-state distribution by thermally activated jumps over the barrier. Near threshold, the behavior is similar to that of a system undergoing a second order phase transition in the mean field approximation. Analogies between first-order phase transitions and transitions in oscillating systems are also pointed out.
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