Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

Abstract

A point-like particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the 1D Fokker-Planck (Kramers) equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m→0, with a series of corrections expanded in powers of m/γ, γ denotes the friction coefficient. The corrections are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.

Appendices

Appendix A: Exact Solution

The Green’s function (3.2) solving the FPK equation with zero potential, Eq. (3.1), is calculated here. First we introduce the scaled coordinates ξ, u, τ according to Eqs. (3.3) and define the function Γ(ξ,u,τ;ξ′,u′,τ′),

$$ G\bigl(x,v,t;x',v',t' \bigr)=e^{-u^2/2}\varGamma\bigl(\xi,u,\tau;\xi',u', \tau'\bigr) e^{u'^2/2}, $$

(A.1)

satisfying the transformed equation (3.1),

(A.2)

the exponential factor becomes unity due to δ(u−u′).

After the Fourier transform in ξ and τ,

$$ \varGamma\bigl(\xi,u,\tau;\xi',u', \tau'\bigr)=\int\frac{dk\,d\nu}{4\pi^2} e^{ik(\xi-\xi')-i\nu(\tau-\tau')} \varGamma_{k,\nu}\bigl(u;u'\bigr), $$

(A.3)

and shifting the velocities by ik/2, w=u+ik/2 and w′=u′+ik/2, the equation

$$ \bigl[-i\nu-\partial_w^2+w^2-1+k^2/4 \bigr]\varGamma_{k,\nu}\bigl(w;w'\bigr)= \frac{\gamma\beta}{4} \delta\bigl(u-u'\bigr) $$

(A.4)

becomes solvable if Γ _k,ν (w;w′) is expressed in the basis set of the linear harmonic oscillator ψ _n (w),

$$ \varGamma_{k,\nu}\bigl(w;w'\bigr)=\sum _{n=0}^{\infty}\varGamma_n(k,\nu) \psi_n(w) \psi_n^*\bigl(w'\bigr). $$

(A.5)

The eigenfunctions ψ _n (w) satisfy

$$ \bigl(-\partial_w^2+w^2 \bigr) \psi_n(w)=\lambda_n\psi_n(w) =(2n+1) \psi_n(w) $$

(A.6)

and we use the integral representation of the Hermite polynomials H _n (w) [37]

$$ \psi_n(w)=\frac{1}{\sqrt[4]{\pi}\sqrt{2^n n!}}H_n(w)e^{-w^2/2} =\sqrt{\frac{2^n}{n!\pi^{3/2}}}e^{-w^2/2} \int_{-\infty}^{\infty}(w+ir)^ne^{-r^2}\,dr $$

(A.7)

in the next calculation.

Using the transformations above, we find

$$ \varGamma_n(k,\nu)=\frac{\gamma\beta/4}{-i\nu+\lambda_n-1+k^2/4}. $$

(A.8)

Applying it in the formulas (A.5) and (A.3), we integrate the last one over ν in the complex plane,

(A.9)

Θ(x) denotes the Heaviside unit step function and D ₀=1/γβ is the diffusion constant. Now the integral relation (A.7) is used for ψ _n (w) and $\psi_{n}^{*}(w')$ and the summation over n can be readily completed. Finally, the straightforward triple integration over k, r, r′ is performed and using the transformation (A.1) results in the formula (3.2).

Appendix B: Quadratic Approximation

Derivation of the formula (4.29) for the effective diffusion coefficient D(x) with all the derivatives higher than U″(x) neglected is presented here. This approximation corresponds to local replacing of the potential by a parabola, U(x)≃κ(x−x ₀)²/2+U ₀, where κ, x ₀ and U ₀ are fitting parameters.

First we simplify Eq. (4.27). For quadratic potential, the right hand side can be rewritten as

$t_{0}\hat{Z}=\sum_{n=1}^{\infty}t_{0}\hat{Z}_{n}$; the difference contains only the higher derivatives of U(x), which are zero. Hence

$$ D(x)/D_0=1+e^{-\beta U(x)}\sum _{n=1}^{\infty}t_0^n \hat{Z}_ne^{\beta U(x)}. $$

(B.1)

The formulas for D(x) have been derived considering stationary flow; j(x,t)=j is constant. It simplifies the relation (4.7); ∂ _t j=0. Thus the right hand side represents stationary flux, which can be directly compared with Eq. (4.8), giving a much simpler relation between $\hat{Z}_{n}$ and $\hat{I}_{n}$ than Eq. (4.10),

$$ e^{-\beta U(x)}\hat{Z}_n(x)\partial_xe^{\beta U(x)}p(x)= \partial_x\hat{I}_n(x)p(x) $$

(B.2)

for any stationary solution p(x). Calculation of the coefficients of D(x) according to Eq. (B.1) requires us to take ∂ _x exp[βU(x)]p(x)=exp[βU(x)], hence finally

$$ e^{-\beta U(x)}\hat{Z}_ne^{\beta U(x)}= \frac{2}{\sqrt{\pi}} \partial_x\int_{-\infty}^{\infty}u^2 \,du\,e^{-u^2} \hat{\omega}_n(x,u)e^{-\beta U(x)}\int dx\, e^{\beta U(x)} $$

(B.3)

after application of Eq. (4.6).

Before writing the explicit formulas for $\hat{\omega}_{n}$ for the quadratic potential, we define the polynomials

(B.4)

n=1,2,… , coming from the expansions of $\hat{\omega}_{n}$ and $\hat{\eta}_{n}$ for zero potential in t ₀, Eqs. (3.15). The first few polynomials are visible in the round brackets of Eq. (3.19). One can check by direct integration that

(B.5)

corresponding to the normalization of $\hat{\eta}_{n}$, Eq. (4.5), and the relations (3.16), (3.17), proving no correction to the Smoluchowski equation in the case U(x)=0.

The operators $\hat{\omega}_{n}$ and $\hat{\eta}_{n}$ for the quadratic potential have the form

(B.6)

with the coefficients

$$ c_{n,k}=D_0^n\frac{2k\ (2n-1)!}{(n-k)!(n+k)!}. $$

(B.7)

Due to the integrals, Eq. (B.5), only the first terms with P ₁(u) in Eq. (B.6) contribute to the expansion of D(x), Eq. (B.3). Then the functions become

(B.8)

taking U ⁽³⁾(x)=0 into account. Applied in Eq. (B.1) it results in the expansion of D(x), Eq. (4.29).

Finally, one has to verify that the formulas (B.6) satisfy the recurrence relations (4.13) and (4.17), acting on the function p(x)=exp[−βU(x)]∫dxexp[βU(x)]. Although the equations simplify notably due to neglecting the derivatives higher than U″(x), we omit the details of this tedious but straightforward calculation.