Effective potential for classical field theories subject to stochastic noise

Abstract

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.

5 pages, LaTeX 209, ReV_TeX 3.2. V2: Cosmetic changes to improve handling of Fadeev--Popov ghosts. V3: References updated. This paper has now been superseded and the contents of this paper have now been incorporated into cond-mat/9904215 [Phys. Rev. E60 (1999) 6343-6360]; cond-mat/9904391 [Physica A280 (2000) 437-455]; and cond-mat/9904413 [Phys. Lett. A, in press]