Open statistical ensemble

The Open statistical ensemble (OSE) corresponds to a physical system that exchanges energy and particles with a reservoir at a given temperature and chemical potential, thus being at thermodynamic equilibrium with it, and in addition correctly takes into account the surface terms.

The open statistical ensemble is similar to the grand canonical ensemble (GCE) with the key difference being that the OSE has no fictitious surface on its boundaries.

The expression for the partition function of the ensemble is similar to the partition function of the grand canonical ensemble (GCE), with the replacement of the Boltzmann factor in terms of the series on the distribution functions of special form.

The coefficient of surface tension, included in the partition function, corresponds to the interface of a fluid and a hard solid, due to respecting probability and potential limitations.

Unlike the GCE, for OSE the average number of particles in a given volume coincides with the volume term. Also in contrast to the GCE, the correlation and distribution functions of the OSE strictly satisfy the requirement of translational invariance.

The expression for the general term of the distribution has the form

${\displaystyle p_{m}^{v}={\frac {z^{m}}{m!\Upsilon _{v}}}\sum _{t=0}^{\infty }{\frac {z^{t}}{t!}}\int \left[\prod _{i=1}^{m}\prod _{j=m+1}^{m+t}\psi _{i}^{v}\chi _{j}^{v}\right]{\mathcal {B}}_{1...m+t}^{(m,t)}d{\boldsymbol {r}}_{1}...d{\boldsymbol {r}}_{m+t},}$

where ${\displaystyle p_{m}^{v}}$ is the probability to find ${\displaystyle ~m}$ particles in volume ${\displaystyle ~v}$, ${\displaystyle ~m\geq 1}$, ${\displaystyle ~z}$ is activity and ${\displaystyle ~\Upsilon _{v}}$ is the OSE partition function. ${\displaystyle ~\psi _{i}^{v}}$ and ${\displaystyle ~\chi _{j}^{v}=1-\psi _{j}^{v}}$ are characteristic functions, equal to unity inside and outside the system, respectively, and ${\displaystyle ~{\mathcal {B}}_{1...m+t}^{(m,t)}}$ are partial localization factors which generalize the notions of the Boltzmann and Ursell factors and contain them as extreme cases.

The first term of the series corresponds, up to normalization, to the distribution of GCE partition functions.

Summing the series we obtain the expression

${\displaystyle p_{m}^{v}={\frac {1}{m!\Upsilon _{v}}}\int \left[\prod _{i=1}^{m}\psi _{i}^{v}\right]\varrho _{G,1...m}^{(m)}(\chi ^{v})d{\boldsymbol {r}}_{1}...d{\boldsymbol {r}}_{m},}$

where ${\displaystyle ~\varrho _{G,1...m}^{(m)}}$ is a distribution function, depending on ${\displaystyle ~z}$ and ${\displaystyle ~\chi ^{v}}$ and is expressed through a series of ${\displaystyle ~{\mathcal {B}}_{1...m+t}^{(m,t)}}$. The last expression is equivalent to the GCE distribution with the replacement

${\displaystyle \varrho _{G,1...m}^{(m)}\approx z^{m}e^{-\beta U_{1...m}^{m}}}$

and corresponding renormalization, where ${\displaystyle ~U_{1...m}^{m}}$ is an ${\displaystyle ~m}$-particle interaction potential, ${\displaystyle ~1/\beta =k_{B}T}$, with ${\displaystyle ~k_{B}}$ being the Boltzmann constant and ${\displaystyle ~T}$ being the temperature. This expression shows that the GCE is a low-density approximation of the OSE.

For the partition function of the OSE, we have the expression

${\displaystyle \Upsilon _{v}=\exp {\sum _{t=1}^{\infty }{\frac {z^{t}}{t!}}\int \left[1-\prod _{i=1}^{t}\chi _{i}^{v}\right]{\mathcal {U}}_{1...t}^{(t)}d{\boldsymbol {r}}_{1}...d{\boldsymbol {r}}_{t}},}$

unlike the partition function of the GCE

${\displaystyle \Xi _{v}=\exp {\sum _{t=1}^{\infty }{\frac {z^{t}}{t!}}\int \left[\prod _{i=1}^{t}\psi _{i}^{v}\right]{\mathcal {U}}_{1...t}^{(t)}d{\boldsymbol {r}}_{1}...d{\boldsymbol {r}}_{t}},}$

where ${\displaystyle ~{\mathcal {U}}_{1...t}^{(t)}}$ are the Ursell factors. Collapsing the series of activities for ${\displaystyle ~\Upsilon _{v}}$ we obtain the alternative representation

${\displaystyle ~\Upsilon _{v}=\exp {\beta [vP(z,T)+a\sigma (z,T)]},}$

where ${\displaystyle ~P(z,T)}$ is the pressure, ${\displaystyle ~\sigma (z,T)}$ the coefficient of surface tension on the interface of the fluid and hard solid and ${\displaystyle ~a}$ is the surface bounding the system.

It should be stressed that an open system does not singled out, and the surface tension is created due to the fluctuation component of the partition function.

The last expression is exactly consistent with the probability of the formation of the hole volume ${\displaystyle ~v}$ in the fluid

${\displaystyle ~p_{0}^{v}=e^{-\beta \left(vP{\left(z,T\right)}+a\sigma {\left(z,T\right)}\right)}}$

and is determined from thermodynamic considerations

${\displaystyle ~p_{0}^{v}\propto e^{-\beta R_{min}},}$

where ${\displaystyle ~R_{min}}$ is the minimum work required for the formation of such fluctuations.

Some properties of the OSE

• Scale invariance. In contrast to grand canonical ensemble, an open statistical ensemble satisfies the scale invariance requirement: general term of the included subsystem distribution corresponds to that of the original system.
• Application to small systems. This distribution may be applied to however small volumes including those less than the size of a molecule. In this case it degenerates into a Bernoulli distribution with ${\displaystyle ~p=\varrho v}$.
• Separation of fluctuations. When the volume is much greater than the size of the molecule, the squared deviation of the number of particles is divided into bulk and surface terms.

References

• Zaskulnikov V. M., Open statistical ensemble and surface phenomena: arXiv:0911.3106
• Zaskulnikov V. M., Open statistical ensemble: new properties (scale invariance, application to small systems, meaning of surface particles, etc.): arXiv:1004.0896