Editing Langevin equation (section)

== Generic Langevin equation ==
There is a formal derivation of a generic Langevin equation from classical mechanics.<ref name="Kawasaki1973">{{cite journal | last=Kawasaki|first=K.|year=1973|title=Simple derivations of generalized linear and nonlinear Langevin equations | journal=J. Phys. A: Math. Nucl. Gen.|volume=6|issue=9|pages=1289–1295|bibcode=1973JPhA....6.1289K|doi=10.1088/0305-4470/6/9/004}}<!--| accessdate = 2014-05-30 --></ref><ref name="den2015">{{cite arXiv |last=Dengler |first=R. |eprint=1506.02650v2 |title=Another derivation of generalized Langevin equations | year=2015 |class=physics.class-ph }}</ref> This generic equation plays a central role in the theory of [[Critical phenomena|critical dynamics]],<ref name="HH1977">{{cite journal |title=Theory of dynamic critical phenomena |journal=[[Reviews of Modern Physics]] |year=1977 |first1=P. C. |last1=Hohenberg |first2=B. I. |last2=Halperin |volume=49 |issue=3 |pages=435–479 |doi=10.1103/RevModPhys.49.435 |bibcode = 1977RvMP...49..435H |s2cid=122636335 }}</ref> and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.

An essential step in the derivation is the division of the degrees of freedom into the categories ''slow'' and ''fast''. For example, local thermodynamic equilibrium in a liquid is reached within a few collision times, but it takes much longer for densities of conserved quantities like mass and energy to relax to equilibrium. Thus, densities of conserved quantities, and in particular their long wavelength components, are slow variable candidates. This division can be expressed formally with the [[Zwanzig projection operator]].<ref>{{cite journal |title=Memory effects in irreversible thermodynamics |journal=[[Physical Review|Phys. Rev.]] |year=1961 |first=R. |last=Zwanzig |volume=124 |issue=4 |pages=983–992 |doi=10.1103/PhysRev.124.983 |bibcode = 1961PhRv..124..983Z }}</ref> Nevertheless, the derivation is not completely rigorous from a mathematical physics perspective because it relies on assumptions that lack rigorous proof, and instead are justified only as plausible approximations of physical systems.

Let <math>A=\{A_i\}</math> denote the slow variables. The generic Langevin equation then reads
<math display="block">\frac{\mathrm{d}A_{i}}{\mathrm{d}t}=k_\text{B}T\sum\limits_{j}{\left[ {A_{i},A_{j}}\right] \frac{{\mathrm{d}}\mathcal{H}}{{\mathrm{d}A_{j}}}}-\sum\limits_{j}{\lambda _{i,j}\left( A\right) \frac{\mathrm{d}\mathcal{H}}{{\mathrm{d}A_{j}}}+}\sum\limits_{j}{\frac{\mathrm{d}{\lambda _{i,j}\left(A\right) }}{{\mathrm{d}A_{j}}}}+\eta _{i}\left( t\right).</math>

The fluctuating force <math>\eta_i\left( t\right)</math> obeys a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
<math display="block">\left\langle {\eta _{i}\left( t\right) \eta _{j}\left( t'\right) }\right\rangle =2\lambda _{i,j}\left( A\right) \delta \left( t-t'\right).</math>

This implies the [[Onsager reciprocal relations|Onsager reciprocity relation]] <math>\lambda_{i,j}=\lambda_{j,i}</math> for the damping coefficients <math>\lambda</math>. The dependence <math>\mathrm{d}\lambda_{i,j}/\mathrm{d}A_j</math> of <math>\lambda</math> on <math>A</math> is negligible in most cases. The symbol <math>\mathcal{H}=-\ln\left(p_0\right)</math> denotes the [[Hamiltonian mechanics|Hamiltonian]] of the system, where <math>p_0\left(A\right)</math> is the equilibrium probability distribution of the variables <math>A</math>. Finally, <math>[A_i,A_j]</math> is the projection of the [[Poisson bracket]] of the slow variables <math>A_i</math> and <math>A_j</math> onto the space of slow variables.

In the Brownian motion case one would have <math>\mathcal{H}=\mathbf{p}^2/\left(2mk_\text{B}T\right)</math>, <math>A=\{\mathbf{p}\}</math> or <math>A=\{\mathbf{x}, \mathbf{p}\}</math> and <math>[x_i, p_j]=\delta_{i,j}</math>. The equation of motion <math>\mathrm{d}\mathbf{x}/\mathrm{d}t=\mathbf{p}/m</math> for <math>\mathbf{x}</math> is exact: there is no fluctuating force <math>\eta_x</math> and no damping coefficient <math>\lambda_{x,p}</math>.