Effective action for stochastic partial differential equations

Abstract

Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern-formation, and the structural development of the universe itself. It is reasonably well-known that certain SPDEs can be manipulated to be equivalent to (non-quantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these non-quantum field theories are fully interacting. The limitation to one-loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the BRST formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this "direct approach" is the SPDE analog of canonical quantization using physical fields.) After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action, and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from QFT to the case of SPDEs. An important result is that the amplitude of the two-point function governing the noise acts as the loop-counting parameter and is the analog of Planck's constanth in this SPDE context. We derive a general expression for the one-loop effective potential of an arbitrary SPDE subject to translation-invariant Gaussian noise, and compare this with the one-loop potential for QFT.

Об Уравнении Кардара-Паризи-Чжана И Его Матричном Обобщении

“…Для получения отличного от нуля значения p 0 в непертурбатив-ной области далее мы рассмотрим матричное обобщение уравнения КПЧ. В завершение этого раздела дадим несколько комментариев по поводу серии ра-бот [14]- [16]. В этих работах было получено выражение для эффективного действия в однопетлевом приближении в случае достаточно общего стохастического диффе-ренциального уравнения в частных производных и корреляционной функции шу-ма достаточно общего вида.…”

“…К настоящему моменту для уравнения КПЧ в крити-ческой размерности D = 2 известны критические экспоненты с двухпетлевой точ-ностью [10]- [12], а также высказана гипотеза о виде бета-функции во всех порядках теории возмущений (ТВ) [13]. Помимо этого, в литературе рассматривалось приме-нение формализма эффективного действия к стохастическим дифференциальным уравнениям в частных производных [14], в частности к уравнению КПЧ. Явное вы-ражение для однопетлевого эффективного потенциала получено в работах [15], [16].…”

Heat kernel regularization of the effective action for stochastic reaction-diffusion equations

“…As argued in Ref. [4] the effective potential is a superb tool for studying the onset of pattern formation (i.e., symmetry breaking) about the static and spatially homogeneous solutions of the SPDE. This quantity takes into account both interactions and fluctuations to a given number of loops.…”

Variational and potential formulation for stochastic partial differential equations

“…In Table 1 some particular cases of operators and Φ i functions are presented (a p are the coefficients of the polynomial of order p while κ and ω are real scalars; for details see [6] and references therein). Hereafter, Einstein summation convention and Euclidian metric tensor are assumed.…”

“…Effective actions can be found by means of the Martin-Siggia-Rose formalism [5], a perturbative procedure that makes use of both physical and "conjugate" (auxiliary) fields. Also, Hochberg et al have proposed in a series of papers [6,7,8] a "direct" approach, finding effective actions and potentials for SPDEs using a functional integral formalism similar in structure to those of quantum field theory. Last year Ao [9] has reported a worth consulting novel approach for constructing potentials associated with first order SPDEs.…”