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| {{Condensed matter physics|expanded=States of matter}}
| | == Ka Wang Zhen == |
| An ideal '''Bose gas''' is a quantum-mechanical version of a classical [[ideal gas]]. It is composed of [[bosons]], which have an integer value of spin, and obey [[Bose–Einstein statistics]]. The statistical mechanics of bosons were developed by [[Satyendra Nath Bose]] for [[photon gas|photons]], and extended to massive particles by [[Albert Einstein]] who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a [[Bose–Einstein condensate]].
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| == Thomas–Fermi approximation ==
| | Ka Wang Zhen, Yan Qiu ancestral gods nodded in agreement.<br><br>'I walked deal of that Luo Feng, come back soon.' Hao Lei Xing Zhu finished then ride a palace treasure, quickly turned into a streamer speed away.<br><br>Ka Wang Zhen, Yan Qiu ancestral god phase with each other,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_72.htm オークリー サングラス レンズ], then coincidentally, are connected to the master of the family ordered universe.<br><br>'boat in the universe, the universe boat to leave as soon as possible.'<br><br>'If unable to leave short time,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_0.htm オークリー サングラス], then hidden in the universe boat of some hidden place repeatedly,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_62.htm オークリー サングラス 交換レンズ], that Luo Feng never be met.' shock Gad the king in his small universe of divine incarnation,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_11.htm オークリー サングラス ゴルフ], it is immediately ordered,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_58.htm ロードバイク サングラス オークリー], Qiu Yan ancestral God same.<br><br>......<br><br>the passage of time.<br><br>spin Sawaumi,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_39.htm サングラス オークリー 人気], reef.<br><br>stars tower stands in the middle silently waiting, but they have been around 13 main universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_51.htm 激安オークリーサングラス], apparently about to spin Sawaumi born pinnacle treasure ...... is known to have many strong and Feng Luo without killing spree, plus Treasure the temptation,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm オークリー サングラスケース], there are still two |
| {{seealso|Thomas–Fermi model}}
| | 相关的主题文章： |
| The thermodynamics of an ideal Bose gas is best calculated using the [[grand partition function]]. The grand partition function for a Bose gas is given by:
| | <ul> |
| | | |
| :<math>\mathcal{Z}(z,\beta,V) = \prod_i \left(1-ze^{-\beta\epsilon_i}\right)^{-g_i}</math>
| | <li>[http://www.iotp.co.uk/cgi-bin/mr.cgi http://www.iotp.co.uk/cgi-bin/mr.cgi]</li> |
| | | |
| where each term in the product corresponds to a particular energy ε<sub>i </sub>, g<sub>i </sub> is the number of states with energy ε<sub>i </sub>, ''z '' is the absolute activity (or "fugacity"), which may also be expressed in terms of the [[chemical potential]] μ by defining:
| | <li>[http://web-entry.admissions.keio.ac.jp/ryugaku1/entrance.cgi http://web-entry.admissions.keio.ac.jp/ryugaku1/entrance.cgi]</li> |
| | | |
| :<math>z(\beta,\mu)= e^{\beta \mu}</math>
| | <li>[http://forum.radio4.ru/cgi-bin/search.cgi http://forum.radio4.ru/cgi-bin/search.cgi]</li> |
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| and β defined as:
| | </ul> |
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| :<math>\beta = \frac{1}{kT}</math>
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| where ''k '' is [[Boltzmann's constant]] and ''T '' is the [[temperature]]. All thermodynamic quantities may be derived from the grand partition function and we will consider all thermodynamic quantities to be functions of only the three variables ''z '', β (or ''T ''), and ''V ''. All partial derivatives are taken with respect to one of these three variables while the other two are held constant. It is more convenient to deal with the dimensionless [[grand potential]] defined as:
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| :<math>\Omega=-\ln(\mathcal{Z}) = \sum_i g_i \ln\left(1-ze^{-\beta\epsilon_i}\right).</math>
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| Following the procedure described in the [[gas in a box]] article, we can apply the [[Thomas-Fermi model|Thomas-Fermi approximation]] which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral:
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| :<math>\Omega\approx \int_0^\infty \ln\left(1-ze^{-\beta E}\right)\,dg.</math>
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| The degeneracy ''dg '' may be expressed for many different situations by the general formula:
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| :<math>dg = \frac{1}{\Gamma(\alpha)}\,\frac{E^{\,\alpha-1}}{ E_c^{\alpha}} ~dE</math>
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| where α is a constant, <math>E_c</math> is a "critical energy", and Γ is the [[Gamma function]]. For example, for a massive Bose [[gas in a box]], α=3/2 and the critical energy is given by:
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| :<math>\frac{1}{(\beta E_c)^\alpha}=\frac{Vf}{\Lambda^3}</math>
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| where Λ is the [[thermal wavelength]]. For a massive Bose [[gas in a harmonic trap]] we will have α=3 and the critical energy is given by:
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| :<math>\frac{1}{(\beta E_c)^\alpha}=\frac{f}{(\hbar\omega\beta)^3}</math>
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| where ''V(r)=mω<sup>2</sup>r<sup>2</sup>/2 '' is the harmonic potential. It is seen that ''E<sub>c </sub>'' is a function of volume only.
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| We can solve the equation for the grand potential by integrating the Taylor series of the integrand term by term, or by realizing that it is proportional to the [[Mellin transform]] of the Li<sub>1</sub>(z exp(-β E)) where Li<sub>s</sub>(x) is the [[polylogarithm]] function. The solution is:
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| :<math>\Omega\approx-\frac{\textrm{Li}_{\alpha+1}(z)}{\left(\beta E_c\right)^\alpha}.</math>
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| The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the [[Bose–Einstein condensate]] and will be dealt with in the next section.
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| == Inclusion of the ground state ==
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| The total [[Particle number|number of particles]] is found from the grand potential by
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| :<math>N = -z\frac{\partial\Omega}{\partial z} \approx\frac{\textrm{Li}_\alpha(z)}{(\beta E_c)^\alpha}</math>
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| The polylogarithm term must remain real and positive, and the maximum value it can possibly have is at z=1 where it is equal to ζ(α) where ζ is the [[Riemann zeta function]]. For a fixed ''N '', the largest possible value that β can have is a critical value β<sub>c </sub> where
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| :<math>N = \frac{\zeta(\alpha)}{(\beta_c E_c)^\alpha}</math> | |
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| This corresponds to a critical temperature T<sub>c</sub>=1/kβ<sub>c</sub> below which the Thomas-Fermi approximation breaks down. The above equation can be solved for the critical temperature:
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| :<math>T_c=\left(\frac{N}{\zeta(\alpha)}\right)^{1/\alpha}\frac{E_c}{k}</math>
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| For example, for <math>\alpha=3/2</math> and using the above noted value of <math>E_c</math> yields
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| :<math>T_c=\left(\frac{N}{Vf\zeta(3/2)}\right)^{2/3}\frac{h^2}{2\pi m k}</math>
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| Again, we are presently unable to calculate results below the critical temperature, because the particle numbers using the above equation become negative. The problem here is that the Thomas-Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so the equation breaks down. It turns out, however, that the above equation gives a rather accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term:
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| :<math>N = N_0+\frac{\textrm{Li}_\alpha(z)}{(\beta E_c)^\alpha}</math>
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| where ''N<sub>0 </sub>'' is the number of particles in the ground state condensate:
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| :<math>N_0 = \frac{g_0\,z}{1-z}</math>
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| [[Image:BoseEinsteinGas1.png|thumb|400px|right|Figure 1: Various Bose gas parameters as a function of normalized temperature τ. The value of α is 3/2. Solid lines are for N=10,000, dotted lines are for N=1000. Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential μ is shown in red, and green lines are the values of z. It has been assumed that k =ε<sub>c</sub>=1.]]
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| This equation can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for α=3/2, with k=ε<sub>c</sub>=1 which corresponds to a [[gas in a box|gas of bosons in a box]]. The solid black line is the fraction of excited states ''1-N<sub>0</sub>/N '' for ''N ''=10,000 and the dotted black line is the solution for ''N ''=1000. The blue lines are the fraction of condensed particles ''N<sub>0</sub>/N '' The red lines plot values of the
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| negative of the chemical potential μ and the green lines plot the corresponding values of ''z ''. The horizontal axis is the normalized temperature τ defined by
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| :<math>\tau=\frac{T}{T_c}</math>
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| It can be seen that each of these parameters become linear in τ<sup>α</sup> in the limit of low temperature and, except for the chemical potential, linear in 1/τ<sup>α</sup> in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.
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| The equation for the number of particles can be written in terms of the normalized temperature as:
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| :<math>N = \frac{g_0\,z}{1-z}+N~\frac{\textrm{Li}_\alpha(z)}{\zeta(\alpha)}~\tau^\alpha</math>
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| For a given ''N '' and τ, this equation can be solved for τ<sup>α</sup> and then a series solution for ''z '' can be found by the method of [[inversion of series]], either in powers of τ<sup>α</sup> or as an asymptotic expansion in inverse powers of τ<sup>α</sup>. From these expansions, we can find the behavior of the gas near ''T =0'' and in the Maxwell-Boltzmann as ''T '' approaches infinity. In particular, we are interested in the limit as ''N '' approaches infinity, which can be easily determined from these expansions.
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| == Thermodynamics ==
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| Adding the ground state to the equation for the particle number corresponds to adding the equivalent ground state term to the grand potential:
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| :<math>\Omega = g_0\ln(1-z)-\frac{\textrm{Li}_{\alpha+1}(z)}{\left(\beta E_c\right)^\alpha}</math>
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| All thermodynamic properties may now be computed from the grand potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in <math>\tau^\alpha</math> is shown.
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| {| border="1" cellpadding="4" cellspacing="0" align="center" style="margin: 0 0 1em; border-color:#ccc"
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| |-
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| ! align="center" |Quantity
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| ! align="center" |General
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| ! align="center" |<math>T \ll T_c\,</math>
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| ! align="center" |<math>T \gg T_c\,</math>
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| |-
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| ! align="center" |z
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| ! align="center" |
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| ! align="center" |<math>=1\,</math>
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| ! align="center" |<math>\approx \frac{\zeta(\alpha)}{\tau^\alpha}
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| -\frac{\zeta^2(\alpha)}{2^\alpha\tau^{2\alpha}}</math>
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| |-
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| ! align="center" |Vapor fraction<br><math>1-\frac{N_0}{N}\,</math>
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| ! align="center" |<math>=\frac{\textrm{Li}_{\alpha}(z)}{\zeta(\alpha)}
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| \,\tau^\alpha</math>
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| ! align="center" |<math>=\tau^\alpha\,</math>
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| ! align="center" |<math>=1\,</math>
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| |-
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| ! align="center" |Equation of state<br><math>\frac{PV\beta}{N}=
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| -\frac{\Omega}{N}\,</math>
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| ! align="center" |<math>=
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| \frac{\textrm{Li}_{\alpha\!+\!1}(z)}{\zeta(\alpha)}\,\tau^\alpha</math>
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| ! align="center" |<math>=
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| \frac{\zeta(\alpha\!+\!1)}{\zeta(\alpha)}\,\tau^\alpha</math>
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| ! align="center" |<math>\approx
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| 1-\frac{\zeta(\alpha)}{2^{\alpha\!+\!1}\tau^\alpha}</math>
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| |-
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| ! align="center" |Gibbs Free Energy<br><math>G=\ln(z)\,</math>
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| ! align="center" |<math>=\ln(z)\,</math>
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| ! align="center" |<math>=0\,</math>
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| ! align="center" |<math>\approx
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| \ln\left(\frac{\zeta(\alpha)}{\tau^\alpha}\right)
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| -\frac{\zeta(\alpha)}{2^{\alpha}\tau^\alpha}</math> | |
| |}
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| It is seen that all quantities approach the values for a classical [[ideal gas]] in
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| the limit of large temperature. The above values can be used to calculate other
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| thermodynamic quantities. For example, the relationship between internal energy and
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| the product of pressure and volume is the same as that for a classical ideal gas over
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| all temperatures:
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| :<math>U=\frac{\partial \Omega}{\partial \beta}=\alpha PV</math>
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| A similar situation holds for the specific heat at constant volume
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| :<math>C_v=\frac{\partial U}{\partial T}=k(\alpha+1)\,U\beta</math>
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| The entropy is given by:
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| :<math>TS=U+PV-G\,</math> | |
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| Note that in the limit of high temperature, we have
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| :<math>TS=(\alpha+1)+\ln\left(\frac{\tau^\alpha}{\zeta(\alpha)}\right)</math>
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| which, for α=3/2 is simply a restatement of the [[Sackur-Tetrode equation]]. In one dimension bosons with delta interaction behave as fermions, they obey [[Pauli exclusion principle]]. In one dimension Bose gas with delta interaction can be solved exactly by [[Bethe ansatz]]. The bulk free energy and thermodynamic potentials were calculated by [[Chen Nin Yang]]. In one dimensional case [http://books.google.com/books?id=kaZ0pKIHhxAC&dq=quantum+inverse+scattering+method&printsec=frontcover&source=bl&ots=4AaoICh4Q3&sig=8RoPelCcAOLTG1vY21DDLCulNGY&hl=en&ei=I6zuScSBE5a-M9q1pOwP&sa=X&oi=book_result&ct=result&resnum=8#v=onepage&q&f=false correlation functions] also were evaluated. The In one dimension Bose gas is equivalent to quantum [[non-linear Schroedinger equation]].
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| == See also ==
| |
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| * [[Gas in a box]]
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| * [[Debye model]]
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| * [[Bose–Einstein condensate]]
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| * [[Bose–Einstein condensation: a network theory approach]]
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| == References ==
| |
| {{reflist}}
| |
| * {{cite book |last=Huang |first=Kerson |title=Statistical Mechanics
| |
| |year=1967 |publisher=John Wiley and Sons |location=New York |id= }}
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| * {{cite book |last=Isihara |first=A. |title=Statistical Physics
| |
| |year=1971 |publisher=Academic Press |location=New York |id= }}
| |
| * {{cite book |last=Landau |first=L. D. |coauthors=E. M. Lifshitz
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| |title=Statistical Physics, 3rd Edition Part 1
| |
| |year=1996 |publisher=Butterworth-Heinemann |location=Oxford |id= }}
| |
| * {{cite book |last=Pethick |first=C. J.|coauthors=H. Smith
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| |title=Bose–Einstein Condensation in Dilute Gases |year=2004
| |
| |publisher=Cambridge University Press |location=Cambridge |id= }}
| |
| * {{cite journal | last = Yan | first = Zijun | year = 2000
| |
| | title = General Thermal Wavelength and its Applications | journal = Eur. J. Phys
| |
| | volume = 21 | pages = 625–631
| |
| | url = http://www.iop.org/EJ/article/0143-0807/21/6/314/ej0614.pdf | format = PDF
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| | doi = 10.1088/0143-0807/21/6/314|bibcode = 2000EJPh...21..625Y | issue = 6 }}
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| [[Category:Bose–Einstein statistics]]
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| [[Category:Ideal gas]]
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| [[Category:Quantum mechanics]]
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| [[Category:Thermodynamics]]
| |
Ka Wang Zhen
Ka Wang Zhen, Yan Qiu ancestral gods nodded in agreement.
'I walked deal of that Luo Feng, come back soon.' Hao Lei Xing Zhu finished then ride a palace treasure, quickly turned into a streamer speed away.
Ka Wang Zhen, Yan Qiu ancestral god phase with each other,オークリー サングラス レンズ, then coincidentally, are connected to the master of the family ordered universe.
'boat in the universe, the universe boat to leave as soon as possible.'
'If unable to leave short time,オークリー サングラス, then hidden in the universe boat of some hidden place repeatedly,オークリー サングラス 交換レンズ, that Luo Feng never be met.' shock Gad the king in his small universe of divine incarnation,オークリー サングラス ゴルフ, it is immediately ordered,ロードバイク サングラス オークリー, Qiu Yan ancestral God same.
......
the passage of time.
spin Sawaumi,サングラス オークリー 人気, reef.
stars tower stands in the middle silently waiting, but they have been around 13 main universe,激安オークリーサングラス, apparently about to spin Sawaumi born pinnacle treasure ...... is known to have many strong and Feng Luo without killing spree, plus Treasure the temptation,オークリー サングラスケース, there are still two
相关的主题文章：