H-stable potential

Consider a system of particles in positions ${\displaystyle x_{1},x_{2},\ldots \in R^{\nu }}$; the interaction or potential between a particle in position ${\displaystyle x_{i}}$ and a particle in position ${\displaystyle x_{j}}$ is

${\displaystyle \phi (x_{i}-x_{j})\,}$

where ${\displaystyle \phi (x)}$ is a real, even (possibly unbounded) function. Then ${\displaystyle \phi (x)}$ is H-stable if there exists ${\displaystyle B>0}$ such that, for any ${\displaystyle n\geq 1}$ and any ${\displaystyle x_{1},x_{2},\ldots ,x_{n}\in R^{\nu }}$,

${\displaystyle V_{n}(x_{1},x_{2},\ldots x_{n}):=\sum _{i<j=1}^{n}\phi (x_{i}-x_{j})\geq -Bn\,}$

• If ${\displaystyle \phi (0)<\infty }$ and, for every ${\displaystyle n\geq 1}$ and every ${\displaystyle x_{1},x_{2},\ldots x_{n}\in R^{\nu }}$, it holds

${\displaystyle \sum _{i,j=1}^{n}\phi (x_{i}-x_{j})\geq 0}$

then the potential ${\displaystyle \phi (x)}$ is stable (with the constant ${\displaystyle B}$ given by ${\displaystyle {\frac {\phi (0)}{2}}}$). This condition applies for example to potentials that are: a) positive functions; b) positive-definite functions.

• If the potential ${\displaystyle \phi (x)}$ is stable, then, for any bounded domain ${\displaystyle \Lambda }$, any ${\displaystyle \beta >0}$ and ${\displaystyle z>0}$, the series

${\displaystyle \sum _{n\geq 1}{\frac {z^{n}}{n!}}\int _{\Lambda ^{n}}\!dx_{1}\cdots dx_{n}\;\exp[-\beta V_{n}(x_{1},x_{2},\ldots x_{n})]}$

is convergent. In fact, for bounded, upper-semi-continuous potentials the hypothesis is not only sufficient, but also necessary!

• The grand canonical partition function, in finite volume, is

${\displaystyle \Xi (\beta ,z,\Lambda ):=1+\sum _{n\geq 1}{\frac {z^{n}}{n!}}\int _{\Lambda ^{n}}\!dx_{1}\cdots dx_{n}\;\exp[-\beta V_{n}(x_{1},x_{2},\ldots x_{n})]}$

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

${\displaystyle \phi (x)={\frac {1}{4\pi |x|}}}$

and, if the charges of all the particles are equal, then the potential energy is

${\displaystyle V_{n}(x_{1},\ldots ,x_{n})=\sum _{i<j}\phi (x_{i}-x_{j})}$

and the system is H-stable with ${\displaystyle B=0}$; 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,

${\displaystyle \phi (x)\sim -{\frac {1}{2\pi }}\ln {m|x|}\qquad {\rm {for}}\quad x\sim 0}$

if the particles can have charges with different signs, the potential energy is

${\displaystyle H_{n}({\underline {q}},{\underline {x}})=\sum _{i<j}q_{i}q_{j}\phi (x_{i}-x_{j})}$

where ${\displaystyle q_{j}}$ is the charge of the particle ${\displaystyle j}$. ${\displaystyle H_{n}({\underline {q}},{\underline {x}})}$ in not bounded from below: for example, when ${\displaystyle n=2}$ and ${\displaystyle q_{1}q_{2}=1}$, the two body potential has infimum

${\displaystyle \inf _{x_{1},x_{2}}\phi (x_{1}-x_{2})=-\infty }$

Yet, Frohlich[1] proved the existence of the thermodynamics limit for ${\displaystyle \beta <4\pi }$.