H-stable potential

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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, 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[edit]


Consider a system of particles in positions x_1, x_2,\ldots \in R^\nu; the interaction or potential between a particle in position x_i and a particle in position x_j is

\phi(x_i-x_j) \,

where \phi(x) is a real, even (possibly unbounded) function. Then \phi(x) is H-stable if there exists B>0 such that, for any n\ge1 and any x_1, x_2,\ldots, x_n\in R^\nu,

V_n(x_1, x_2,\ldots x_n):=\sum_{i<j=1}^n  \phi(x_i-x_j)\ge-Bn\,


  • If \phi(0)<\infty and, for every n\ge 1 and every x_1, x_2,\ldots x_n\in R^\nu, it holds
\sum_{i,j=1}^n  \phi(x_i-x_j)\ge0
then the potential \phi(x) is stable (with the constant B given by \frac{\phi(0)}{2}). This condition applies for example to potentials that are: a) positive functions; b) positive-definite functions.
  • If the potential \phi(x) is stable, then, for any bounded domain \Lambda, any \beta>0 and z>0, the series

\sum_{n\ge 1}\frac{z^n}{n!} 
\int_{\Lambda^n}\!dx_1\cdots dx_n\; 
\exp[-\beta V_n(x_1, x_2,\ldots x_n)]
is convergent. In fact, for bounded, upper-semi-continuous potentials the hypothesis is not only sufficient, but also necessary!
 \Xi(\beta, z, \Lambda):=1+\sum_{n\ge 1}\frac{z^n}{n!} 
\int_{\Lambda^n}\!dx_1\cdots dx_n\; 
\exp[-\beta V_n(x_1, x_2,\ldots x_n)]
hence the H-stability is a sufficient condition for the partition function to exists in finite volume.
  • H-stability doesn't necessary imply the existence of the infinite volume pressure. For example, in a Coulomb system (in dimension three) the potential is
and, if the charges of all the particles are equal, then the potential energy is
V_n(x_1,\ldots, x_n)=\sum_{i<j} \phi(x_i-x_j)
and the system is H-stable with B=0; but the thermodynamic limit doesn't exist, because the potential is not tempered.
  • 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,
\phi(x)\sim-\frac{1}{2\pi}\ln{m|x|} \qquad
{\rm for}\quad x\sim 0
if the particles can have charges with different signs, the potential energy is
H_n(\underline q, \underline x)=\sum_{i<j} q_i q_j\phi(x_i-x_j)
where q_j is the charge of the particle j. H_n(\underline q, \underline x) in not bounded from below: for example, when n=2 and q_1q_2=1, the two body potential has infimum

\inf_{x_1, x_2}  \phi(x_1-x_2)=-\infty
Yet, Frohlich[1] proved the existence of the thermodynamics limit for \beta< 4\pi.

Quantum statistical mechanics[edit]

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 :

\exists B: \frac{E_0}{N} > -B, \,

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.


  1. ^ Frohlich, J. (1976). "Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa and Coulomb systems". Comm.Math.Phys. 47.