8.13: Appendix IV- Correlations in the Langevin formalism

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As shown above, integrating the Langevin equation \(\Dp+\gamma p = F + \eta(t)\) yields \[p(t)=p(0)\,e^{-\gamma t} +{F\over\gamma}\>\big(1-e^{-\gamma t}\big) + \int\limits_0^t\!\!ds\>\eta(s)\,e^{\gamma (s-t)}\ .\]. Thus, the momentum autocorrelator is \[\begin{split} \blangle p(t)\,p(t')\brangle -\blangle p(t)\brangle \blangle p(t')\brangle &= \int\limits_0^t\!\!ds\!\int\limits_0^{t'}\!\!ds'\> e^{\gamma(s-t)}\,e^{\gamma(s'-t')}\,\blangle\eta(s)\,\eta(s')\brangle\\ &=\Gamma\,e^{-\gamma(t+t')}\!\! \int\limits_0^{t\ns_{min}}\!\!\!ds\>e^{2\gamma s} =M\kT\,\Big( e^{-\gamma |t-t'|} - e^{-\gamma(t+t')}\Big)\ , \end{split}\] where \[t\ns_{min}={min}(t,t')= \begin{cases} t&{if}\ t<t' \\ t'&{if}\ t'<t \end{cases}\] is the lesser of \(t\) and \(t'\). Here we have used the result \[\begin{split} \int\limits_0^t\!\!ds\!\int\limits_0^{t'}\!\!ds'\> e^{\gamma(s+s')}\,\delta(s-s')&= \int\limits_0^{t\ns_{min}}\!\!\!ds\!\int\limits_0^{t\ns_{min}}\!\!\!ds'\>e^{\gamma(s+s')}\,\delta(s-s')\\ &=\int\limits_0^{t\ns_{min}}\!\!\!ds\>e^{2\gamma s}={1\over 2\gamma}\Big(e^{2\gamma t\ns_{min}}-1\Big)\ . \end{split}\] One way to intuitively understand this result is as follows. The double integral over \(s\) and \(s'\) is over a rectangle of dimensions \(t\times t'\). Since the \(\delta\)-function can only be satisfied when \(s=s'\), there can be no contribution to the integral from regions where \(s>t'\) or \(s'>t\). Thus, the only contributions can arise from integration over the square of dimensions \(t\ns_{min}\times t\ns_{min}\). Note also \[t+t'-2\,{min}(t,t')=|t-t'|\ .\]

[Fssprime] Regions for some of the double integrals encountered in the text.

Let’s now compute the position \(x(t)\). We have \[\begin{split} x(t)&=x(0)+{1\over M}\!\int\limits_0^t\!\!ds\>p(s)\\ &=x(0) + \int\limits_0^t\!\!ds\, \Bigg[\bigg(v(0)-{F\over\gamma M}\bigg)\,e^{-\gamma s} + {F\over \gamma M}\Bigg] +{1\over M}\!\int\limits_0^t\!\!ds\!\int\limits_0^{s}\!\!ds\ns_1\>\eta(s\ns_1)\,e^{\gamma(s\ns_1-s)}\\ &=\blangle x(t)\brangle + {1\over M}\!\int\limits_0^t\!\!ds\!\int\limits_0^{s}\!\!ds\ns_1\>\eta(s\ns_1)\,e^{\gamma(s\ns_1-s)}\ , \end{split}\] with \(v=p/M\). Since \(\big\langle\eta(t)\big\rangle=0\), we have \[\begin{split} \blangle x(t)\brangle&=x(0)+\int\limits_0^t\!\!ds\, \Bigg[\bigg(v(0)-{F\over\gamma M}\bigg)\,e^{-\gamma s} + {F\over \gamma M}\Bigg]\\ &=x(0)+{Ft\over \gamma M} + {1\over\gamma}\bigg(v(0)-{F\over\gamma M}\bigg)\,\big(1-e^{-\gamma t}\big)\ . \end{split}\] Note that for \(\gamma t\ll 1\) we have \(\big\langle x(t)\big\rangle = x(0)+v(0)\, t + \half M^{-1} F t^2 + \CO(t^3)\), as is appropriate for ballistic particles moving under the influence of a constant force. This long time limit of course agrees with our earlier evaluation for the terminal velocity, \(v\ns_\infty=\big\langle p(\infty)\big\rangle/M = F/\gamma M\).

We next compute the position autocorrelation: \[\begin{aligned} \blangle x(t)\,x(t')\brangle - \blangle x(t)\brangle \blangle x(t')\brangle & = {1\over M^2}\!\int\limits_0^t\!\!ds\! \int\limits_0^{t'}\!\!ds'\>e^{-\gamma(s+s')}\!\int\limits_0^s\!\!ds\ns_1\!\int\limits_0^{s'}\!\!ds'_1\> e^{\gamma(s\ns_1+s\ns_2)}\,\blangle\eta(s\ns_1)\,\eta(s\ns_2)\brangle\nonumber\\ &={\Gamma\over 2\gamma M^2}\int\limits_0^t\!\!ds\!\int\limits_0^{t'}\!\!ds'\> \Big(e^{-\gamma|s-s'|}-e^{-\gamma(s+s')}\Big)\end{aligned}\] We have to be careful in computing the double integral of the first term in brackets on the RHS. We can assume, without loss of generality, that \(t\ge t'\). Then \[\begin{split} \int\limits_0^t\!\!ds\!\int\limits_0^{t'}\!\!ds'\>e^{-\gamma|s-s'|}&=\int\limits_0^{t'}\!\!ds'\,e^{\gamma s'}\!\int\limits_{s'}^t\!\!ds\> \>e^{-\gamma s}+ \int\limits_0^{t'}\!\!ds'\,e^{-\gamma s'}\!\int\limits_0^{s'}\!\!ds\>e^{\gamma s}\\ &=2\gamma^{-1}t' + \gamma^{-2}\big(e^{-\gamma t} + e^{-\gamma t'} - 1 - e^{-\gamma (t-t')} \big)\ . \end{split}\] We then find, for \(t>t'\), \[\blangle x(t)\,x(t')\brangle - \blangle x(t)\brangle \blangle x(t')\brangle = {2\kT\over \gamma M}\> t' + {\kT\over \gamma^2 M}\,\big(2e^{-\gamma t} + 2 e^{-\gamma t'} -2 -e^{-\gamma(t-t')} - e^{-\gamma(t+t')} \big)\ .\] In particular, the equal time autocorrelator is \[\blangle x^2(t)\brangle - \blangle x(t)\brangle^{\!2} = {2\kT\over \gamma M}\> t + {\kT\over \gamma^2 M}\,\big(4 e^{-\gamma t} - 3 - e^{-2\gamma t}\big)\ .\] We see that for long times \[\blangle x^2(t)\brangle - \blangle x(t)\brangle^{\!2} \sim 2Dt\ ,\] where \(D=\kT/\gamma M\) is the diffusion constant.