高阶动力学粒子的布朗运动

Brownian motion with high-derivative dynamics

讨论了与谐振子热浴耦合的高阶动力学粒子的运动行为,导出了含高阶动力学的朗之万方程及其对应的福克-普朗克方程.

Brownian motion with higher-derivative dynamics is investigated in this work.As a model,we consider a particle coupling with a heat bath consisting of harmonic oscillators.Assume that motion of particl without bath is determined by a LagrangianL=L(t,x,x_(1),···,x_(N))where x_(n)(n=1,2,···,N)is the n-th order derivativ of x with respect to time t.After integrating variables of bath,we derived a generalized Langevin equation fo Brownian motion as follows:∑_(n=0)(gd/dt)^(n)δL/δx_(n)-μx_(1)+ξ(t)=0,whereμrepresents effective constant of viscosity andξ(t)is Gaussian noise.Note that we setx_(0)=x in the above equation.Define p_(N-a)=δL(t,x,x_(1),...,x_(N))/δX_(N).From this equation,we can solve x_(N)and express it as a function x_(N)=φ(t,x,x_(1),···,p_(N-1)).The Fokker-Planck equation corresponding to generalized Langevin equation is derived,which may be expressed as■whereρ=ρ(t,x,x_(1),···,x_(N-1),p_0,p_1,···,p_(N-1))is the distribution function in phase space.T is temperature of the bath Note that we setp_(-1)=μx_(1)and replace x_(N)with a function oft,x,x_(1),···,p_(N-1)in the above equation.As an example,we consider Pais-Uhlenbeck oscillator whose Lagrangian is■where Y is a constant,and frequenciesω_1,ω_2 are independent of time.The corresponding Langevin equation and Fokker-Planck equation are■and■respectively.