hw08soln - 1 Homework 8 (due December 16) Problem 8-1 (5...

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1 Homework 8 (due December 16) Problem 8-1 (5 pts.) Starting with Langevin equation for a 1D Brownian particle, f nd the mean square displacement as a function of time τ : D ( x ( t + τ ) x ( t )) 2 E t . Neglect the inertial e f ects, and take the particle mobility to be b . Consider the following cases: a) Free particle, U ( x )=0 . b) Particle subjected to a uniform force, U ( x )= Fx . c) Particle in a Harmonic potential, U ( x )= kx 2 / 2 . d) By comparing the result of (a) with the solution of the di f usion equation, express di f usion coe cient D in terms of mobility b . Solution: a) ˙ x = u ( t ) , h u ( t + t 0 ) u ( t ) i = 2 T b δ ( t 0 ) D ( x ( t + τ ) x ( t )) 2 E = * τ Z 0 τ Z 0 u ( t + t 00 ) u ( t + t 0 ) dt 0 dt 00 + = 2 T b τ Z 0 τ Z 0 δ ( t 0 t 00 ) dt 0 dt 00 = 2 T b τ b) ˙ x = F b + u ( t ) , D ( x ( t + τ ) x ( t )) 2 E = μ b 2 + * τ Z 0 τ Z 0 u ( t + t 00 ) u ( t + t 0 ) dt 0 dt 00 + = 2 T b τ + μ b 2 c) ˙ x k b x = u
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hw08soln - 1 Homework 8 (due December 16) Problem 8-1 (5...

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