Variational Principles in Conformation Dynamics - FU Berlin, FB MI

A.1. Diffusion in a quadratic potential 47Lemma A.3: With ν := (1−e −2θτ ),thetransitionfunctionp(x, y; τ) is given by a Gaussianp(x, y; τ) =1√2πανexp− (y − e−θτ x) 22αν. (A.14)Proof. We show that p satisfies the Smoluchowski Equation III.6 with initial conditionp(x, · ;0) = δ x . Since the initial condition is clearly fulfilled, we check that the differentialequation holds for all τ>0. In order to do so, we note that θ = b2 = D α mγk B,whereasT α 2= θ. Ontheonehandwefind:D= bα mγ ∂p 2(y − e −θτ∂τ = − x)θe −θτ x2αν − (y − e −θτ x) 2 4αθe −2θτp − 1 4α 2 ν 22 p2πα2θe−2θτ (A.15)2πανb 2=mγk B T p − yxe−θτ − e −2θτ x 2 − yxe −3θτ + x 2 e −4θτ − y 2 e −2θτ +2yxe −3θτ − x 2 e −4θτν 2− p θe−2θτνb 2=mγk B T p − yxe−θτ − e −2θτ x 2 + yxe −3θτ − y 2 e −2θτν 2On the other hand we have:1 ∂∂2(byp)+Dmγ ∂y ∂y p = θpν2 ν − b2 − yxe −θτ )νmγk B T p(y2 ν 2+b 2 − 2yxe −θτ + x 2 e −2θτmγk B T py2 ν 2= −p θe−2θτν++− p θe−2θτν− pθν(A.16)(A.17). (A.18)(A.19)(A.20)b 2 + yxe −θτ + y 2 e −2θτ − yxe −3θτmγk B T p−y2 ν(A.21)b 2 − 2yxe −θτ + x 2 e −2θτmγk B T py2 ν 2= −p θe−2θτνSince all terms cancel, the equation holds.+(A.22)b 2mγk B T p−yxe−θτ + y 2 e −2θτ − yxe −3θτ + x 2 e −2θτν 2 .(A.23)Knowing this, we can prove the statement about the eigenfunctions of the propagator: