Open statistical ensemble

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 fluid and 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

p^v_m = \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^v_i\chi^v_j \right ] {\mathcal B}^{(m,t)}_{1...m+t} d\boldsymbol{r}_1...d\boldsymbol{r}_{m+t},

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

The first term of the series corresponds to the distribution of the GCE with the accuracy up to normalizing factors - partition functions.

Summing the series we obtain the expression

p^v_m = \frac{1}{m! \Upsilon_v} \int \left [ \prod_{i = 1}^m \psi^v_i \right ] \varrho^{(m)}_{G,1...m}(\chi^v) d\boldsymbol{r}_1...d\boldsymbol{r}_m,

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

\varrho^{(m)}_{G,1...m} \approx z^m \exp(-\beta U^{m}_{1...m})

and corresponding renormalization, where ~U^{m}_{1...m} - ~m -particle interaction potential, ~1/\beta = k_B T , ~k_B - Boltzmann constant, ~T - 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

\Upsilon_v = \exp { \sum_{t=1}^\infty \frac{z^t}{t!} \int \left [1 - \prod_{i = 1}^{t} \chi^v_i \right ] {\mathcal U}^{(t)}_{1...t} d\boldsymbol{r}_1...d\boldsymbol{r}_t },

unlike the partition function of GCE

\Xi_v = \exp { \sum_{t=1}^\infty \frac{z^t}{t!} \int \left [ \prod_{i = 1}^{t} \psi^v_i \right ] {\mathcal U}^{(t)}_{1...t} d\boldsymbol{r}_1...d\boldsymbol{r}_t },

where ~{\mathcal U}^{(t)}_{1...t} - Ursell factors. Collapsed series of activities for ~\Upsilon_v we obtain the alternative representation

~\Upsilon_v = \exp{\beta [ vP(z,T) + a\sigma (z,T) ]},

where ~P(z,T) is the pressure, ~\sigma (z,T) - coefficient of surface tension on the interface of the fluid and hard solid, ~a - 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 ~v in the fluid

~p^v_0 = \exp-{\beta [ vP(z,T) + a\sigma (z,T) ]},

is determined from thermodynamic considerations

~p^v_0 \propto \exp{( -\beta R_{min})},

where ~R_{min} - minimum work of formation of such fluctuations.

Some properties of the OSE

References