In this paper, we discuss the regularities of average energies with a fixed angular momentum I (denoted as
EI′s) in many-body systems interacting via a two-body random ensemble. It is found that
EI′s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ (minimum of
I) or $I\ensuremath{\sim}{I}_{\mathrm{max}}$ (maximum of
I) have large probabilities [denoted as
P(I)] to be the smallest in energy, and
P(I) is close to zero elsewhere. A simple argument assuming the randomness of the two-particle coefficients of fractional parentage is given to explain these observations. A compact trajectory of the energy
EI vs
I(I+1) is found to be robust. Other regularities, such that there are two or three sizable
P(I)′s with $I\ensuremath{\sim}{I}_{\mathrm{min}}$ but $\mathcal{P}(I)\ensuremath{\ll}\mathcal{P}{(I}_{\mathrm{max}})'\mathrm{s}$ with $I\ensuremath{\sim}{I}_{\mathrm{max}},$ and that the coefficients C defined by
〈EI〉min=CI(I+1) are sensitive to the orbits and not sensitive to particle number, etc., are discovered and studied for the first time.
The distribution of nuclear ground-state spin in the two-body random ensemble (TBRE) is studied by using a general classification neural network (NN) model with the two-body interaction matrix elements as input features and corresponding ground-state spins as labels or output predictions. It seems that quantum many-body system problem exceeds the capability of our optimized neural networks when it comes to accurately predicting the ground-state spin of each sample within the TBRE. However, our neural network model effectively captures the statistical properties of the ground-state spin. This may be attributed to the fact that the neural network (NN) model has learned the empirical regularity of the ground-state spin distribution in TBRE, as discovered by human physicists.