Editing Langevin equation (section)

=== Harmonic oscillator in a fluid ===
{{See also|Phase space}}
<math display="block"> m\frac{dv}{dt} = -\lambda v + \eta (t)-k x </math>

A particle in a fluid is described by a Langevin equation with a potential energy function, a damping force, and thermal fluctuations given by the [[fluctuation dissipation theorem]]. If the potential is quadratic then the constant energy curves are ellipses, as shown in the figure. If there is dissipation but no thermal noise, a particle continually loses energy to the environment, and its time-dependent phase portrait (velocity vs position) corresponds to an inward spiral toward 0 velocity. By contrast, thermal fluctuations continually add energy to the particle and prevent it from reaching exactly 0 velocity. Rather, the initial ensemble of stochastic oscillators approaches a steady state in which the velocity and position are distributed according to the [[Maxwell–Boltzmann distribution]]. In the plot below (figure 2), the long time velocity distribution (blue) and position distributions (orange) in a harmonic potential (<math display="inline"> U = \frac{1}{2} k x^2 </math>) is plotted with the Boltzmann probabilities for velocity (green) and position (red). In particular, the late time behavior depicts thermal equilibrium. 

[[File:Squared_displacements_of_simulated_free_Brownian_particles.png|thumb|right|upright=1.75|	
Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3''k''<sub>B</sub>''T''/''m'', and 6''k''<sub>B</sub>''T''/''m'' respectively, with 3''k''<sub>B</sub>''T''/''m'' being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.]]