Cross-section fluctuations in chaotic scattering systems

Exact analytical expressions for the cross-section correlation functions of chaotic scattering systems have hitherto been derived only under special conditions. The objective of the present article is to provide expressions that are applicable beyond these restrictions. The derivation is based on a statistical model of Breit-Wigner type for chaotic scattering amplitudes which has been shown to describe the exact analytical results for the scattering ($S$)-matrix correlation functions accurately. Our results are given in the energy and in the time representations and apply in the whole range from isolated to overlapping resonances. The $S$-matrix contributions to the cross-section correlations are obtained in terms of explicit irreducible and reducible correlation functions. Consequently, the model can be used for a detailed exploration of the key features of the cross-section correlations and the underlying physical mechanisms. In the region of isolated resonances, the cross-section correlations contain a dominant contribution from the self-correlation term. For narrow states the self-correlations originate predominantly from widely spaced states with exceptionally large partial width. In the asymptotic region of well-overlapping resonances, the cross-section autocorrelation functions are given in terms of the $S$-matrix autocorrelation functions. For inelastic correlations, in particular, the Ericson fluctuations rapidly dominate in that region. Agreement with known analytical and experimental results is excellent.

Figure 1

$S$-matrix autocorrelation function in the time representation. The full black lines correspond to the analytical results, Eqs. (24) and (25) of Ref. [31], blue circles to those derived on the basis of the VWZ model [20], and red diamonds to the SWB model Eqs. (19) and (23). The $\mathrm{\Lambda }=52$ transmission coefficients corresponding to the values of ${\mathrm{\Gamma }}_W/d$ in the insets are listed in Table 1. All correlation functions were divided by $(1+{\delta }_{ab})T_a{T}_b$, predicted at $\tau =0$ for the inelastic case ($b\ne a$) in the upper panels and the elastic one ($b=a$) in the lower panels. The cross-correlation functions ($b\ne a$) in the middle panels vanish there at $\tau =0$.

Figure 2

Relative contributions of the individual terms entering Eqs. (36) and (39) to the cross-section correlation functions ${\tilde{C}}_{ab}(\tau =0)$ vs ${\mathrm{\Gamma }}_W/d$ for 52 equal transmission coefficients. Black dots correspond to $\mathrm{\Phi }={\tilde{\mathcal{F}}}_4$, red squares to $\mathrm{\Phi }={\tilde{\mathcal{G}}}_4$, green diamonds to $\mathrm{\Phi }={\tilde{\mathcal{H}}}_4$, orange triangles-up to $\mathrm{\Phi }={\tilde{\mathcal{F}}}_3$. Blue triangles-down display the Fourier transform of the Ericson fluctuation term in Eq. (37), maroon crosses that of the second term in Eq. (40). Upper panel: the inelastic case $a=1,b=2$. Lower panel: the elastic case $a=b=1$.

Figure 3

Same as Fig. 2, for the specific combinations $\mathrm{\Sigma }$ of the individual contributions given in (40) (black dots), (41) (blue squares), and (42) (red diamonds), respectively. It is clearly visible that the cross-section correlation function is well approximated by the sum Eq. (40) of the two terms which dominate the regions of isolated and overlapping resonances, respectively. With increasing ${\mathrm{\Gamma }}_W/d$, ${\tilde{C}}_{aa}(\tau =0)$ approaches ${\tilde{C}}_{aa}^{\left(as\right)}(\tau =0)$ given in (48), plotted as maroon crosses.

Figure 4

Ratios of $C_{ab}(\epsilon =0)$ to the corresponding $C_{ab}^{\left(as\right)}\left(\epsilon \right)$ defined in Eqs. (49) and (50), respectively. The transmission coefficients $T_1,T_2$, and ${\tau }_{abs}$ corresponding to the 20 values of ${\mathrm{\Gamma }}_W/d$ considered in the figure are given in Table 1. Black squares were obtained using the inverse Fourier transforms of (36) and (39) and of (24) and (25) in Ref. [31]. Blue crosses show the exact analytical results given in Ref. [24] and green diamonds the experimental ones. Upper panel: the inelastic case $a=1,b=2$. Lower panel: the elastic case for $a=b=1$. The scale has been chosen logarithmic for a better illustration of the good agreement between the different results.

Figure 5

Ratios of $C_{ab}(\epsilon =0)$ to the functions $C_{ab}^{\left(as\right)}\left(\epsilon \right)$ defined in Eqs. (49) and (50), respectively, for 52 equal transmission coefficients. Black squares were obtained using the inverse Fourier transforms of (36) and (39) and of (24) and (25) in Ref. [31], red circles and the blue line show the RMT simulations and the exact analytical results given in Ref. [24], respectively. Upper panel: the inelastic case $a=1,b=2$. Lower panel: the elastic case for $a=b=1$. The scale has been chosen logarithmic for a better illustration of the good agreement between the different results, especially in the inelastic case.

Figure 6

Cross-section correlation functions $C_{ab}\left(\epsilon \right)/C_{ab}\left(0\right)$. The 52 transmission coefficients $T_1,T_2$, and ${\tau }_{abs}$ corresponding to the values of ${\mathrm{\Gamma }}_W/d$ in the insets are given in Table 1. The full black lines were obtained using the inverse Fourier transforms of (36) and (39) and of (24) and (25) in Ref. [31]. The red circles correspond to the RMT results, the experimental ones are plotted as green diamonds. Upper panel: the inelastic case $a=1,b=2$. Lower panel: the elastic case $a=b=1$.

Figure 7

Cross-section correlation functions $C_{ab}\left(\epsilon \right)/C_{ab}\left(0\right)$ for 52 equal transmission coefficients. The corresponding values of ${\mathrm{\Gamma }}_W/d$ are given in the insets. The black full lines were obtained using the inverse Fourier transforms of (36) and (39), and of (24) and (25) in Ref. [31]. Red circles illustrate the RMT results. Upper panel: the inelastic case $a=1,b=2$. Lower panel: the elastic case $a=b=1$.

Figure 8

Ratios of $C_{ab}(\epsilon =0)$ to $C_{ab}^{\left(as\right)}(\epsilon =0)$ defined Eqs. (49) and (50). Here, 202 open channels with equal transmission coefficients were taken, corresponding to a small $\rho =1/202$. Red squares show the approximations (56) and (59) obtained by taking into account only the self-correlations for the elastic case (lower panel) and the inelastic case (upper panel), respectively. The blue line shows the exact analytical results obtained from Ref. [24]. A logarithmic scale was chosen to better illustrate their good agreement up to ${\mathrm{\Gamma }}_W/d\simeq 0.2$.