Editing Langevin equation (section)

=== Thermal noise in an electrical resistor ===
[[File:ResistorCapacitance.png|right|250px|thumb|An electric circuit consisting of a resistor and a capacitor.]]
There is a close analogy between the paradigmatic Brownian particle discussed above and [[Johnson noise]], the electric voltage generated by thermal fluctuations in a resistor.<ref>{{cite journal | last1 = Johnson | first1 = J. | year = 1928 | title = Thermal Agitation of Electricity in Conductors | url = http://link.aps.org/abstract/PR/v32/p97 | journal = Phys. Rev. | volume = 32 | issue = 1| page = 97 | bibcode = 1928PhRv...32...97J | doi = 10.1103/PhysRev.32.97 }}</ref> The diagram at the right shows an electric circuit consisting of a [[Electrical resistance and conductance|resistance]] ''R'' and a [[capacitance]] ''C''. The slow variable is the voltage ''U'' between the ends of the resistor. The Hamiltonian reads <math>\mathcal{H} = E / k_\text{B}T = CU^2 / (2k_\text{B}T)</math>, and the Langevin equation becomes
<math display="block">\frac{\mathrm{d}U}{\mathrm{d}t} =-\frac{U}{RC} + \eta \left( t\right),\;\;\left\langle \eta \left( t\right) \eta \left( t'\right)\right\rangle = \frac{2k_\text{B}T}{RC^{2}}\delta \left(t-t'\right).</math>

This equation may be used to determine the correlation function
<math display="block">\left\langle U\left(t\right) U\left(t'\right) \right\rangle = \frac{k_\text{B}T}{C} \exp \left(-\frac{\left| t - t'\right| } {RC}\right) \approx 2Rk_\text{B}T \delta \left( t - t'\right),</math>
which becomes white noise (Johnson noise) when the capacitance {{math|''C''}} becomes negligibly small.