en>Tudor987 at 16:24, 31 August 2014

2014-08-31T16:24:33Z

en>Karho.Yau: /* References */

2014-01-22T03:06:53Z

References

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	The ~~writer's name~~ is ~~Christin~~ and ~~she feels comfortable when people use~~ the ~~full name~~. ~~Her husband~~ and ~~her live~~ in ~~American Samoa but her husband wants them~~ to ~~act~~. ~~To garden precisely what love arranging~~. ~~Filing~~ is ~~a few things i do for a living~~ and ~~Do not think think I'll change it anytime~~ inside the. ~~I've been working~~ with ~~my website for~~ a ~~short time now~~. ~~Get it done here: http:~~//~~www~~.~~ristouir~~.it/~~nike~~-~~free~~-3-0-v4-~~prezzo~~-~~basso~~.~~html~~<br><br>~~My blog -~~ [~~http:~~/~~/www~~.~~ristouir~~.it/~~nike~~-~~free~~-3-0-v4-~~prezzo~~-~~basso~~.~~html Nike Free 3~~.~~0 V4~~]		{{Statistical mechanics\|cTopic=[[Statistical ensemble (mathematical physics)\|Ensembles]]}}

			'''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 open statistical ensemble is similar to the [[grand canonical ensemble]] (GCE) with the key difference being that the OSE is not spatially fixed (it may expand and contract naturally).

			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 the average number of particles in a given volume 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

			<math> 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}, </math>

			where <math> p^v_m </math> is the probability to find <math> ~m </math> particles in volume <math> ~v </math>, <math> ~m \geq 1 </math>, <math> ~z </math> is activity, <math> ~\Upsilon_v </math> - OSE partition function. <math> ~\psi^v_i </math> and <math> ~\chi^v_j = 1 - \psi^v_j </math> - characteristic functions, equal to unity inside and outside the system, respectively, and <math> ~{\mathcal B}^{(m,t)}_{1...m+t} </math> - 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

			<math> 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, </math>

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

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

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

			<math>\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 }, </math>

			unlike the partition function of GCE

			<math> \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 }, </math>

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

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

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

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

			is determined from thermodynamic considerations

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

			where <math> ~R_{min} </math> - minimum work of 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 <math> ~p=\varrho v </math>.

			* Separation of fluctuations. When volume is much greater than the size of the molecule [[squared deviation]] of the number of particles is divided into bulk and surface terms.

			== References ==
			* Zaskulnikov V. M., Open statistical ensemble and surface phenomena: [http://arxiv.org/abs/0911.3106 arXiv:0911.3106]
			* Zaskulnikov V. M., Open statistical ensemble: new properties (scale invariance, application to small systems, meaning of surface particles, etc.): [http://arxiv.org/abs/1004.0896 arXiv:1004.0896v1]

			{{Statistical mechanics topics}}

			{{DEFAULTSORT:Open Statistical Ensemble}}
			[[Category:Statistical ensembles]]

en>Hajatvrc: clean up, typos fixed: , → , using AWB

2012-08-08T23:00:10Z

clean up, typos fixed: , → , using AWB

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The writer's name is Christin and she feels comfortable when people use the full name. Her husband and her live in American Samoa but her husband wants them to act. To garden precisely what love arranging. Filing is a few things i do for a living and Do not think think I'll change it anytime inside the. I've been working with my website for a short time now. Get it done here: http://www.ristouir.it/nike-free-3-0-v4-prezzo-basso.html<br><br>My blog - [http://www.ristouir.it/nike-free-3-0-v4-prezzo-basso.html Nike Free 3.0 V4]

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en>Tudor987 at 16:24, 31 August 2014

en>Karho.Yau: /* References */

en>Hajatvrc: clean up, typos fixed: , → , using AWB