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

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})]}

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

Ξ

(

β

,

z

,

Λ

)

:=

1

+

∑

n

≥

1

z

n

n

!

∫

Λ

n

d

x

1

⋯

d

x

n

exp

⁡

[

−

β

V

n

(

x

1

,

x

2

,

…

x

n

)

]

{\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.