Variational Principles in Conformation Dynamics - FU Berlin, FB MI

III.2. Diffusion process 23time ∆t is given by∆X(t) =∆tF (X(t),t)+σW t ,(III.1)meaning that the change is composed of a deterministic function F depending on the quantityX(t) and the time plus a random fluctuation, modelled by Brownian motion. Here, σ>0is simply a control parameter governing the strength of the perturbation. Upon dividing by∆t and passing to the limit ∆t → 0, oneistemptedtowritedownadifferentialequationdX(t)dt= F (X(t),t)+σdW t , (III.2)where we would like to have dW t denote the time derivative of Brownian motion. We havejust seen that such a derivative does not exist. However, Equation III.2 does make sense ifit is transformed into an integral form. This involves the definition of a stochastic integraland the Ito formula, [Behrends, 2011, ch.6,7].Inparticular,Brownianmotionitselfcanbeshown to solve Equation III.2 in this sense, if F =0.III.2. Diffusion processConsider a classical particle of mass m moving under the influence of friction γ>0 and ofaforcefieldF, generatedbyapotentialenergyfunctionV ,i.e. F = −∇V . By Newton’sequation, the position vector x(t) ∈ R 3 satisfies the relationm d2 x(t)dt 2= −mγ dx(t)dt−∇V (x(t)).(III.3)Let us now suppose that there are random perturbations, caused, for instance, by collisionswith small particles, which continuously alter the particle’s motion. We therefore makeEquation III.3 astochasticdifferentialequationbyaddingσdW t ,m d2 x(t)dt 2= −mγ dx(t)dt−∇V (x(t)) + σdW t ,(III.4)keeping in mind that it is meant to be solved in an integral sense. It is known thatσ = √ 2mγk B T ,wherek B =1.3806488J/K is Boltzmann’s constant and T is the systemtemperature. This equation is typically called the Langevin equation of our system,the solution x(t) =:X t is a Markov process, called a diffusion process. Equation III.4 holdsin the same way if we study a system of n particles, in this case, X t is a stochastic process in3n dimensions. If the friction parameter γ is large enough, it is justified to drop the secondderivative on the left-hand side of Equation III.4, [Zwanzig, 2001, sec. 2.2].Rearrangingtheterms modifies the equation todx(t)dt= − 1mγ ∇V (x(t)) + √ 2DdW t ,(III.5)