Editing Langevin equation (section)

=== Path integral ===
A [[Path integral formulation|path integral]] equivalent to a Langevin equation can be obtained from the corresponding [[Fokker&ndash;Planck equation]] or by transforming the Gaussian probability distribution <math>P^{(\eta)}(\eta)\mathrm{d}\eta</math> of the fluctuating force <math>\eta</math> to a probability distribution of the slow variables, schematically <math>P(A)\mathrm{d}A = P^{(\eta)}(\eta(A))\det(\mathrm{d}\eta/\mathrm{d}A)\mathrm{d}A</math>.
The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where <math>A(t+\Delta t)-A(t)</math> depends on <math>A(t)</math> but not on <math>A(t+\Delta t)</math>.  It turns out to be convenient to introduce auxiliary ''response variables'' <math>\tilde A</math>. The path integral equivalent to the generic Langevin equation then reads<ref name="Janssen1976">{{cite journal | title = Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties | journal = Z. Phys. B | year = 1976 | first = H. K. | last = Janssen | volume = 23 | issue = 4 | pages = 377–380|bibcode = 1976ZPhyB..23..377J |doi = 10.1007/BF01316547 | s2cid = 121216943 }}</ref>
<math display="block">\int P(A,\tilde{A})\,\mathrm{d}A\,\mathrm{d}\tilde{A} = N\int \exp \left( L(A,\tilde{A})\right) \mathrm{d}A\,\mathrm{d}\tilde{A},</math>
where <math>N</math> is a normalization factor and
<math display="block">L(A,\tilde{A}) = \int \sum_{i,j} \left\{ \tilde{A}_{i}\lambda_{i,j}\tilde{A}_{j}-\widetilde{A}_{i} \left \{ \delta_{i,j} \frac{\mathrm{d}A_j}{\mathrm{d}t}-k_\text{B}T\left[A_i, A_j\right]\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}A_{j}}+\lambda_{i,j}\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}A_j} - \frac{\mathrm{d}\lambda_{i,j}}{\mathrm{d}A_j} \right \} \right\} \mathrm{d}t.</math>
The path integral formulation allows for the use of tools from [[quantum field theory]], such as perturbation and renormalization group methods. This formulation is typically referred to as either the Martin-Siggia-Rose formalism <ref name="Martin1973">{{cite journal | title = Statistical Dynamics of Classical Systems | author = Martin, P. C. and Siggia, E. D. and Rose, H. A. | journal = Phys. Rev. A | volume = 8 | issue = 1 | pages =423-437 | year = 1973 | doi = 10.1103/PhysRevA.8.423}}</ref> or the Janssen-De Dominicis <ref name="Janssen1976" /><ref name="DeDominicis">{{cite journal | title = Techniques de Renormalisation de la Théorie des Champs et Dynamique des Phénomènes Critiques | author = De Dominicis, C. | journal = J. Phys. Colloques | year = 1976 | volume = 37 | issue = C1 | pages = 247-253 | doi = 10.1051/jphyscol:1976138}}</ref> formalism after its developers. The mathematical formalism for this representation can be developed on [[abstract Wiener space]].