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8.13: Appendix IV- Correlations in the Langevin formalism

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As shown above, integrating the Langevin equation ˙p+γp=F+η(t) yields p(t)=p(0)eγt+Fγ(1eγt)+t0dsη(s)eγ(st) .. Thus, the momentum autocorrelator is p(t)p(t)p(t)p(t)=t0dst0dseγ(st)eγ(st)η(s)η(s)=Γeγ(t+t)tmin0dse2γs=MkBT(eγ|tt|eγ(t+t)) , where tmin=min(t,t)={tif t<ttif t<t is the lesser of t and t. Here we have used the result t0dst0dseγ(s+s)δ(ss)=tmin0dstmin0dseγ(s+s)δ(ss)=tmin0dse2γs=12γ(e2γtmin1) . One way to intuitively understand this result is as follows. The double integral over s and s is over a rectangle of dimensions t×t. Since the δ-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 tmin×tmin. Note also t+t2min(t,t)=|tt| .

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

Let’s now compute the position x(t). We have x(t)=x(0)+1Mt0dsp(s)=x(0)+t0ds[(v(0)FγM)eγs+FγM]+1Mt0dss0ds1η(s1)eγ(s1s)=x(t)+1Mt0dss0ds1η(s1)eγ(s1s) , with v=p/M. Since η(t)=0, we have x(t)=x(0)+t0ds[(v(0)FγM)eγs+FγM]=x(0)+FtγM+1γ(v(0)FγM)(1eγt) . Note that for γt1 we have x(t)=x(0)+v(0)t+12M1Ft2+O(t3), 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=p()/M=F/γM.

We next compute the position autocorrelation: x(t)x(t)x(t)x(t)=1M2t0dst0dseγ(s+s)s0ds1s0ds1eγ(s1+s2)η(s1)η(s2)=Γ2γM2t0dst0ds(eγ|ss|eγ(s+s)) 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 tt. Then t0dst0dseγ|ss|=t0dseγstsdseγs+t0dseγss0dseγs=2γ1t+γ2(eγt+eγt1eγ(tt)) . We then find, for t>t, x(t)x(t)x(t)x(t)=2kBTγMt+kBTγ2M(2eγt+2eγt2eγ(tt)eγ(t+t)) . In particular, the equal time autocorrelator is x2(t)x(t)2=2kBTγMt+kBTγ2M(4eγt3e2γt) . We see that for long times x2(t)x(t)22Dt , where D=kBT/γM is the diffusion constant.


This page titled 8.13: Appendix IV- Correlations in the Langevin formalism is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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