Abstract:
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\left(t,x,x_1,\cdots ,x_N\right) 
where
x_n 
(
n =
1,2,\cdots 
,
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:
\displaystyle\sum _n = 0^N\left(-\dfrac\mathrmd\mathrmdt\right)^n\dfrac\partial L\partial x_n-\mu x_1+\xi \left(t\right) = 0 ,
where
\mu 
represents effective constant of viscosity and
\xi \left(t\right) 
is Gaussian noise． Note that we set
x_0 = x 
in the above equation．
Define
p_N-1 = \dfrac\partial L\left(t,x,x_1,\cdots ,x_N\right)\partial x_N 
． From this equation, we can solve
x_N 
and express it as a function
x_N = \varphi (t,x,x_1,\cdots , p_N-1) 
． The Fokker-Planck equation corresponding to generalized Langevin equation is derived, which may be expressed as
\dfrac\partial \rho \partial t = -\displaystyle\sum _n = 0^N-1\left\\dfrac\partial \partial x_n\left(x_n+1\rho \right)+\dfrac\partial \partial p_n\left\left(\dfrac\partial L\partial x_n-p_n-1\right)\rho \right\right\+\mu k_\mathrmBT\dfrac\partial ^2\rho \partial p_0^2 ,
where
\rho = \rho (t,x,x_1,\cdots,x_N-1,p_0,p_1,\cdots,p_N-1) 
is the distribution function in phase space．
T is temperature of the bath． Note that we set
p_-1 = \mu x_1 
and replace
x_N with a function of
t,x,x_1,\cdots ,p_N-1 
in the above equation．
As an example, we consider Pais-Uhlenbeck oscillator whose Lagrangian is
L = \dfracY2x_2^2-\left(\omega _1^2+\omega _2^2\right)x_1^2+\omega _1^2\omega _2^2x^2, 
where
Y is a constant, and frequencies
\omega _1,\omega _2 
are independent of time． The corresponding Langevin equation and Fokker-Planck equation are
Y\left\dfrac\mathrmd^4x\mathrmdt^4+\left(\omega _1^2+\omega _2^2\right)\dfrac\mathrmd^2x\mathrmdt^2+\omega _1^2\omega _2^2x\right-\mu \dfrac\mathrmdx\mathrmdt+\xi \left(t\right) = 0,
and
\dfrac\partial \rho \partial t = -x_1\dfrac\partial \rho \partial x-\dfracp_1Y\dfrac\partial \rho \partial x_1-\left(Y\omega _1^2\omega _2^2x-\mu x_1\right)\dfrac\partial \rho \partial p_0+\leftY\left(\omega _1^2+\omega _2^2\right)x_1+p_0\right\dfrac\partial \rho \partial p_1+\mu k_\mathrmBT\dfrac\partial ^2\rho \partial p_0^2 ,
respectively．
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