The mass renormalization for the translational Brownian motion

Abstract

The damped harmonic oscillator is modeled as a local mode X with mass m and frequency $\omega _{0}$ immersed in a phonon bath with spectral density function $j_{0}(\omega$). This function behaves as $\omega ^{s}\, (s= 1,2,3,\ldots )$ when $\omega \rightarrow 0$. The limit $\omega _{0} = 0$ represents translational (free) Brownian motion. The earlier work (Hakim and Ambegaokar in Phys Rev A 32:423, 1985) concluded that the so defined limit transition is prohibited for spectral densities with $s<2$. In the present study we demonstrate that a specially constructed preliminary excitation changing the original bath spectrum as $j_{0}(\omega ) \rightarrow j(\omega )$ allows for treating the free damped motion of X with no restriction for the initial spectrum dimensionality. This procedure validates the finite mass renormalization (i.e. $m\rightarrow M$ when $\omega _{0}\rightarrow 0)$ for the conventional bath spectra with $s=1,2$. We show that the new spectral density $j(\omega )$ represents the momentum bilinear interaction between mode X and the environmental modes, whereas the conventional function $j_{0}(\omega )$ is inherent to the case of bilinear coordinate interaction in terms of the same variables. The translational damping kernel is derived based on the new spectral density.

Appendix

The full expression for the function $f(\omega )$ (23) is

$$\begin{aligned} f(\omega )=\omega ^{2}\left\{ {\left( {px^{s-2}e^{-x}} \right) ^{2}+\left( {\frac{2}{\pi }pI_s (x)-1} \right) ^{2}} \right\} \end{aligned}$$

where

$$\begin{aligned} I_s (x)=\int \limits _0^\infty {\frac{{x}'^{(s-1)}e^{-{x}'}d{x}'}{x^{2}-{x}'^{2}}} \quad (s=1,2,3) \end{aligned}$$

In particular [11]:

$$\begin{aligned} I_1 (x)=\frac{1}{2x}\left[ {e^{-x}\bar{{E}}i(x)+e^{x}Ei(-x)} \right] \end{aligned}$$

whereas $I_2 (x)=xI_1 (x)$ and $I_3 (x)=x^{2}I_1 (x)-1$.

Expressions (33) for $z_{s}(x)$ arise based on these prescriptions.