H-stable potential
In statistical mechanics of continuous systems, a potential for a many-body system is called H-stable (or simply stable) if the potential energy per particle is bounded below by a constant that is independent of the total number of particles. In many circumstances, if a potential is not H-stable, it is not possible to define a grand canonical partition function in finite volume, because of "catastrophic configurations" with infinite particles located in a finite space. Classical statistical mechanics. Definition. Consider a system of particles in positions formula_1; the "interaction" or "potential" between a particle in  position formula_2 and a particle in position formula_3 is where formula_5 is a real, even (possibly unbounded) function. Then formula_5 is H-stable if there exists formula_7 such that, for any formula_8 and any formula_9, Quantum statistical mechanics. The notion of H-stability in quantum mechanics is more subtle. While in the classical case the kinetic part of the Hamiltonian is not important as it can be zero independently of the position of the particles, in the quantum case the kinetic term plays an important role in the lower bound for the total energy because of the uncertainty principle. (In  fact, stability of matter was the historical reason for introducing such a principle in mechanics.) The definition of stability is : where "E" is the ground state energy. Classical H-stability implies quantum H-stability, but the converse is false. The criterion is especially useful in statistical mechanics, where H-stability is necessary to the existence of thermodynamics, i.e. if a system is not H-stable, the thermodynamic limit does not exist.