{'fraction_non_alphanumeric': 0.04215116279069767, 'fraction_numerical': 0.0, 'mean_word_length': 4.838983050847458, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Bounds on the energy of the N-body problem with linear combinations of power-law potentials and the logarithmic potential.

We study the ground-state energy of a system of N identical particles interacting via pair potentials such as f(r) = -..cap alpha../r+..beta..r+delta ln(r). Kinetic potentials are used in a geometrical approximation theory which yields simple analytical formulas for upper and lower bounds to the energy. General energy-bound formulas are provided for arbitrary linear combinations of power laws and the logarithmic potential. Specific results are presented with parameter values appropriate for few-quark systems interacting via the linear-plus-Coulomb potential.