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 x 1 , x 2 , … ∈ R ν {\displaystyle x_{1},x_{2},\ldots \in R^{\nu }} ; the interaction or potential between a particle in position x i {\displaystyle x_{i}} and a particle in position x j {\displaystyle x_{j}} is
ϕ ( x i − x j ) {\displaystyle \phi (x_{i}-x_{j})\,}
where ϕ ( x ) {\displaystyle \phi (x)} is a real, even (possibly unbounded) function. Then ϕ ( x ) {\displaystyle \phi (x)} is H-stable if there exists B > 0 {\displaystyle B>0} such that, for any n ≥ 1 {\displaystyle n\geq 1} and any x 1 , x 2 , … , x n ∈ R ν {\displaystyle x_{1},x_{2},\ldots ,x_{n}\in R^{\nu }} ,
V n ( x 1 , x 2 , … x n ) := ∑ i < j = 1 n ϕ ( x i − x j ) ≥ − B n {\displaystyle V_{n}(x_{1},x_{2},\ldots x_{n}):=\sum _{i<j=1}^{n}\phi (x_{i}-x_{j})\geq -Bn\,}
Applications If ϕ ( 0 ) < ∞ {\displaystyle \phi (0)<\infty } and, for every n ≥ 1 {\displaystyle n\geq 1} and every x 1 , x 2 , … x n ∈ R ν {\displaystyle x_{1},x_{2},\ldots x_{n}\in R^{\nu }} , it holds
∑ i , j = 1 n ϕ ( x i − x j ) ≥ 0 {\displaystyle \sum _{i,j=1}^{n}\phi (x_{i}-x_{j})\geq 0}
then the potential ϕ ( x ) {\displaystyle \phi (x)} is stable (with the constant B {\displaystyle B} given by ϕ ( 0 ) 2 {\displaystyle {\frac {\phi (0)}{2}}} ). This condition applies for example to potentials that are: a) positive functions; b) positive-definite functions. If the potential ϕ ( x ) {\displaystyle \phi (x)} is stable, then, for any bounded domain Λ {\displaystyle \Lambda } , any β > 0 {\displaystyle \beta >0} and z > 0 {\displaystyle z>0} , the series
∑ n ≥ 1 z n n ! ∫ Λ n d x 1 ⋯ d x n exp [ − β V n ( x 1 , x 2 , … x n ) ] {\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})]}
… excerpt ends here. Continue reading the full article.
