H-stable potential

In statistical mechanics of continuous systems, a potential for a many-body system is called H-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 ; the interaction or potential between a particle in position and a particle in position is
where is a real, even function. Then is H-stable if there exists such that, for any and any,

Applications

If and, for every and every, it holds
If the potential is stable, then, for any bounded domain, any and, the series
The grand canonical partition function, in finite volume, is
H-stability doesn't necessary imply the existence of the infinite volume pressure. For example, in a Coulomb system the potential is
If the potential is not bounded, H-stability is not a necessary condition for the existence of the grand canonical partition function in finite volume. For example, in the case of Yukawa interaction in two dimensions,

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.
The definition of stability is :
where E₀ 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.