Physical systems of finite size and limited total energy $E$ have limited entropy content $S$ (alternatively, limited information-storing capacity). We demonstrate the validity of our previously conjectured bound on the specific entropy $\frac{S}{E}$ in numerous examples taken from quantum mechanics (number of energy levels upto given energy), free-field systems (entropy of miscellaneous radiations for given energy), and strongly interacting particles (number of many-hadron states up to given energy). In the quantum-mechanical examples we have compared the bound directly with the logarithm of the number of levels for the harmonic oscillator, the rigid rotator, and a particle in an arbitrary potential well. For many-particle systems such as radiations, there is no closed formula for the number of configurations associated with a specified one-particle spectrum. To overcome this barrier we use an efficient numerical algorithm to calculate the number of configurations up to given energy from the spectrum. In all our examples of systems of scalar, electromagnetic, and neutrino quanta contained in spaces of various shapes, the numerical results are in harmony with the bound on $\frac{S}{E}$. This conclusion is buttressed by an approximate analytical estimate of the peak $\frac{S}{E}$ which leaves little doubt as to the general applicability of the bound for systems of free quanta. We consider a gas of hadrons confined to a cavity as an example of a system of strongly interacting particles. Our numerical algorithm applied to the Hagedorn mass spectrum for hadrons confirms that the number of many-hadron states up to a given energy is consistent with the bound. Finally, we show that a rather general one-channel communication system has an information-carrying capacity which cannot exceed a bound akin to that on $\frac{S}{E}$. It is argued that a complete many-channel system is similarly limited.

• Received 4 April 1984

DOI:https://doi.org/10.1103/PhysRevD.30.1669