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Extra resources for An introduction to mechanics Solutions Manual
18 1 Physics of Langevin Equation However, the microscopic origin of the free Brownian motion is the deterministic mechanics. Therefore, we seek to formulate the free Brownian motion from the viewpoint of Newtonian mechanics while we keep the timescale of interest much larger than τm . To this end, we take into account all the mechanical forces that the Brownian particle receives from the thermal environment. 14) with the average 0, ξˆ (t) = 0. Another force is the viscous friction force, −γ v(t), where v(t) is the velocity of the Brownian particle at time t with respect to the thermal environment.
In fact, the momentum conservation of the particle plus environment adds a memory effect to ξ (t), called (hydrodynamic) longtime tail [10, 11]. 16) we assumed that the effect of inertia of the Brownian particle is more important than that of the surrounding fluid. When the mass density of the bead is comparable to that of the surrounding fluid, the neglect of the total momentum conservation is very crude approximation and the memory in ξ (t) is not negligible at short time. 16), therefore, requires attentions when we compare with experiments.
16). 17 Hereafter, we will often omit the hat, “ˆ”, which signifies the stochastic process. 16) are stochastic processes. d. Gaussian random variables with zero average. 1 and in (b) Γ = 2. 1 Random Events or Fluctuations 19 Remark About the Effect of Inertia The above equations are not invariant under the Galilee transformation, p → p − mV and x → x − V t, with a constant velocity, V . It is understandable because Galilean invariance applies to the whole system consisting of the Brownian particle plus the thermal environment.