MartinSiggiaRose

Martin–Siggia–Rose (MSR) is a formalism in statistical physics for turning stochastic differential equations into a field-theoretic path integral. Introduced in 1973 by Martin, Siggia, and Rose, it provides a generating functional for correlation and response functions of systems governed by Langevin dynamics with noise. The core idea is to enforce the stochastic equation with a delta functional and to exponentiate this constraint using an auxiliary response field, often denoted tilde{x}. After averaging over the noise, one obtains an effective action S[x, tilde{x}] that plays the role of a classical field action; observables are computed by functional derivatives of the generating functional with respect to external sources.

The formalism makes it possible to apply diagrammatic and renormalization-group methods to non-equilibrium problems, including dynamic critical phenomena (such as Model A and related models), reaction–diffusion systems, turbulence, and glassy dynamics. It is common to use the MSRJD variant, named for Janssen and De Dominicis, which emphasizes causality and provides a convenient framework for dealing with multiplicative noise.

The MSR approach has both conceptual and practical subtleties, such as discretization choices (Ito vs Stratonovich) and Jacobians arising from changing variables, which can affect the form of the action and the interpretation of results. Nevertheless, it remains a foundational tool in non-equilibrium statistical mechanics and stochastic field theory.