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 particle without bath is determined by a Lagrangian L=L(t,x,x1,…,xN)where xn(n=1,2,…,N)is the n-th order derivative of x with respect to time t.After integrating variables of bath,we derived a generalized Langevin equation for Brownian motion as follows:N∑n=0(-d/dt)n∂L/∂xn-μx1+ζ(t)=0,where μ represents effective constant of viscosity and ξ(t)is Gaussian noise.Note that we set x0=x in the above equation. Define pN-1=∂L(t,x,x1…,xN)/∂xN.From this equation,we can so1ve xN and express it as a function xN=φ(t,x,x1,…,pN-1).The Fokker-Planck equation corresponding to generalized Langevin equation is derived,which may be expressed as ∂ρ/∂t=-N-1∑n=0{∂/∂xn(xn+1ρ)+∂+∂pn[(∂L/∂xn-pn-1)ρ]}+μkBT∂2ρ/∂p02,where ρ=ρ(t,x,x1,…,xN-1,p0,p1,…,pN-1)is the distribution function in phase space.T is temperature of the bath.Note that we set p-1=μx1 and replace xN with a function of t,x,x1,…,pN-1 in the above equation.As an example,we consider Pais-Uhlenbeck oscillator whose Lagrangian is L=Y/2[x22-(ω21+ω22)x21+ω21ω22x2],where Y is a constant,and frequencies ω1,ω2 are independent of time.The corresponding Langevin equation and Fokker-Planck equation are Y[d4x+dt4(ω21+ω22)d2x+dt2+ω21ω22x]-μdx/dt+ζ(t)0,and ∂ρ/∂t=-x1∂ρ/∂x-p1/Y∂ρ/∂x1-(Yω21ω22x-μx1)∂ρ/∂p0+[Y(ω21+ω22)x1+p0]∂ρ/∂p1+μkBT∂2ρ/∂p02,respectively.
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