The Gaussian orthogonal ensemble (GOE) of random matrices has been widely employed to describe diverse phenomena in strongly coupled quantum systems. In particular, it has often been invoked to explain the fluctuations in decay rates that follow the $\chi$-squared distribution for one degree of freedom, as originally proposed by Brink and by Porter and Thomas. However, we find that the coupling to the decay channels can change the effective number of degrees of freedom from one to two. Our conclusions are based on a configuration-interaction Hamiltonian originally constructed to test the validity of transition-state theory, also known as the Rice-Ramsperger-Kassel-Marcus theory in chemistry. The internal Hamiltonian consists of two sets of GOE reservoirs connected by an internal channel. We find that the effective number of degrees of freedom depends on the control parameter $\rho \mathrm{\Gamma }$, where $\rho$ is the level density in the first reservoir and $\mathrm{\Gamma }$ is the level decay width. The distribution for two degrees of freedom is a well-known property of the Gaussian unitary ensemble (GUE); our model demonstrates that the GUE fluctuations can be present under much milder conditions. Our treatment of the model permits an analytic derivation for $\rho \mathrm{\Gamma }\gtrsim 1$.

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• Quantum chaotic systems
• Random matrix theory