Effective potential for classical field theories subject to stochastic noise

Classical field theories coupled to stochastic noise provide an extremely powerful tool for modeling phenomena as diverse as turbulence, pattern-formation, and the structural development of the universe itself. In this Letter we sketch a general formalism that maps such systems into a field theory language, and demonstrate how to extract the one-loop physics for an arbitrary classical field theory coupled to Gaussian noise. The amplitude of the noise two-point function serves as the loop-counting parameter and is the analog of Planck's constant hbar in quantum field theory. We define the effective action and the effective potential, and derive a general formula for the one-loop effective potential of a classical field theory coupled to translation-invariant Gaussian noise.