Wave-number dependent current correlation for a harmonic oscillator

The wave-number $k$ dependent current-correlation function is considered for a harmonic oscillator model. An explicit analytic expression for the Laplace transformed correlation function is derived. It is compared with numerical solutions and results obtained by the recurrence relation method. Several limiting cases such as the long-wavelength limit $k\to 0$ and the deep inelastic limit $k\to \infty$ are discussed in detail. In particular, we show that the deep inelastic limit allows for an explicit summation of the continued fraction. An approximation scheme for the recurrants at intermediate values of $k$ is also considered.

Figure 1

(Color online) The normalized current-correlation function as a function of the time $t$ in units of the inverse eigenfrequency ${\Omega }^{-1}$. Several values of the parameter $A$ are considered.

Figure 2

(Color online) The normalized current-correlation function as a function of the parameter $A$ and the frequency $z$. Besides the surface plot of the correlation function, also a contour plot is shown.

Figure 3

(Color online) The slope of the current-correlation function as given by Eq. (19). The expansions for small $A$ [Eq. (20)] and large $A$ [Eq. (21)] are also shown.

Figure 4

(Color online) Normalized current-correlation function as a function of the frequency $z$ for small values of the parameter $A$. We compare the exact expression Eq. (18) with the $A$ expansion Eq. (25) and the $A\to 0$ limit [Eq. (22)].

Figure 5

(Color online) The recurrants ${\Delta }_1,\dots ,{\Delta }_5$ as a function of the parameter $A$. Also shown are the approximate expressions for ${\Delta }_1{\Delta }_1,\dots ,{\Delta }_5$ in the limits $k\to 0$ and $k\to \infty$ [see Eqs. (34, 38), respectively].

Figure 6

(Color online) Laplace transform as given by Eq. (52). Various values of $A$ are considered. We also show the continued fraction for the deep inelastic limit [see Eqs. (39, 40)].

Figure 7

(Color online) Laplace transform as given by Eq. (52) for large values of the frequency $z$. The parameters are $A=0.5,1,2$. The leading order is indeed given by $-\left(3A+1\right)z^3/{\Omega }^3$ [cf. Eq. (47)].

Figure 8

(Color online) Laplace transform as given by Eq. (52) for large values of the frequency $z$. The parameters are $A=0.5,1.0,2.0$. Compared is the approximate expression [Eq. (55)] with the exact expression [Eq. (18)] and a simple truncation of the continued fraction beyond the third order.