  Spectral statistics of quantum systems have been studied in detail using the
nearest neighbour level spacings, which for generic chaotic systems follows
random matrix theory predictions. In this work, the probability density of the
closest neighbour and farther neighbour spacings from a given level are
introduced. Analytical predictions are derived using a $3 \times 3$ matrix
model. The closest neighbour density is generalized to the $k-$th closest
neighbour spacing density, which allows for investigating long-range
correlations. For larger $k$ the probability density of $k-$th closest
neighbour spacings is well described by a Gaussian. Using these $k-$th closest
neighbour spacings we propose the ratio of the closest neighbour to the second
closest neighbour as an alternative to the ratio of successive spacings. For a
Poissonian spectrum the density of the ratio is flat, whereas for the three
Gaussian ensembles repulsion at small values is found. The ordered spacing
statistics and their ratio are numerically studied for the integrable circle
billiard, the chaotic cardioid billiard, the standard map and the zeroes of the
Riemann zeta function. Very good agreement with the predictions is found.

Chaotic Dynamics;Quantum Physics