Statistical mechanics |
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Thermodynamics · Kinetic theory |
Microcanonical · Canonical
Grand canonical Isothermal–isobaric Isoenthalpic–isobaric Open statistical |
Open statistical ensemble (OSE) corresponds to a physical system, which exchanges energy with the environment, being with it in thermal equilibrium, exchanges particles with the environment for a given chemical potential and correctly takes into account the surface terms.
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 correlation 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 the strict respect of probability and potential limitations.
Unlike the GCE, for OSE average number of particles in a given volume is strictly coincides with the volume term. Also in contrast to the GCE, the correlation functions of the OSE strictly satisfy the requirement of translational invariance.
The expression for the general term of the distribution has the form
where is the probability to find particles in volume , , is activity, - OSE partition function. and - characteristic functions, equal to unity inside and outside the system, respectively, and - 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
where - correlation function, depending on , and expressed through a series of . The last expression is equivalent to the GCE distribution with the replacement
and corresponding renormalization, where - -particle interaction potential, , - Boltzmann constant, - 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
unlike the partition function of GCE
where - Ursell factors. Collapsed series of activities for we obtain the alternative representation
where is the pressure, - coefficient of surface tension on the interface of the fluid and hard solid, - 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 in the fluid
is determined from thermodynamic considerations
where - minimum work of formation of such fluctuations.
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