Bose gas

An idealBose gasis a quantum-mechanicalphase of matter,analogous to a classicalideal gas.It is composed ofbosons,which have an integer value of spin and abide byBose–Einstein statistics.The statistical mechanics of bosons were developed bySatyendra Nath Bosefor aphoton gasand extended to massive particles byAlbert 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 aBose–Einstein condensate.

Introduction and examples

Bosonsarequantum mechanicalparticles that followBose–Einstein statistics,or equivalently, that possess integerspin.These particles can be classified as elementary: these are theHiggs boson,thephoton,thegluon,theW/Zand the hypotheticalgraviton;or composite like the atom ofhydrogen,the atom of¹⁶O,the nucleus ofdeuterium,mesonsetc. Additionally, somequasiparticlesin more complex systems can also be considered bosons like theplasmons(quanta ofcharge density waves).

The first model that treated a gas with several bosons, was thephoton gas,a gas of photons, developed byBose.This model leads to a better understanding ofPlanck's lawand theblack-body radiation.The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. Thephonongas,also known asDebye model,is an example where thenormal modesof vibration of the crystal lattice of a metal, can be treated as effective massless bosons.Peter Debyeused the phonon gas model to explain the behaviour ofheat capacityof metals at low temperature.

An interesting example of a Bose gas is an ensemble ofhelium-4atoms. When a system of⁴He atoms is cooled down to temperature nearabsolute zero,many quantum mechanical effects are present. Below 2.17K,the ensemble starts to behave as asuperfluid,a fluid with almost zeroviscosity.The Bose gas is the most simple quantitative model that explains thisphase transition.Mainly when a gas of bosons is cooled down, it forms aBose–Einstein condensate,a state where a large number of bosons occupy the lowest energy, theground state,and quantum effects are macroscopically visible likewave interference.

The theory of Bose-Einstein condensates and Bose gases can also explain some features ofsuperconductivitywherecharge carrierscouple in pairs (Cooper pairs) and behave like bosons. As a result, superconductors behave like having noelectrical resistivityat low temperatures.

The equivalent model for half-integer particles (likeelectronsorhelium-3atoms), that followFermi–Dirac statistics,is called theFermi gas(an ensemble of non-interactingfermions). At low enough particlenumber densityand high temperature, both the Fermi gas and the Bose gas behave like a classicalideal gas.

Macroscopic limit

The thermodynamics of an ideal Bose gas is best calculated using thegrand canonical ensemble.Thegrand potentialfor a Bose gas is given by:

$\Omega =-\ln({\mathcal {Z}})=\sum _{i}g_{i}\ln \left(1-ze^{-\beta \epsilon _{i}}\right).$

where each term in the sum corresponds to a particular single-particle energy levelε_i;g_iis the number of states with energyε_i;zis the absolute activity (or "fugacity" ), which may also be expressed in terms of thechemical potentialμby defining:

$z(\beta,\mu )=e^{\beta \mu }$

andβdefined as:

$\beta ={\frac {1}{k_{\rm {B}}T}}$

wherek_Bis theBoltzmann constantandTis thetemperature.All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variablesz,β(orT), andV.All partial derivatives are taken with respect to one of these three variables while the other two are held constant.

The permissible range ofzis from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0).

Macroscopic limit, result for uncondensed fraction

Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas,Bose gas) in three dimensions. The Bose gas pressure is lower than an equivalent classical gas, especially below the critical temperature (marked with ★) where particles begin moving en masse into the zero-pressure condensed phase.

Following the procedure described in thegas in a boxarticle, we can apply theThomas–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. This replacement gives the macroscopic grand potential function$\Omega _{m}$,which is close to$\Omega$:

$\Omega _{\rm {m}}=\int _{0}^{\infty }\ln \left(1-ze^{-\beta E}\right)\,dg\approx \Omega.$

The degeneracydgmay be expressed for many different situations by the general formula:

$dg={\frac {1}{\Gamma (\alpha )}}\,{\frac {E^{\,\alpha -1}}{E_{\rm {c}}^{\alpha }}}~dE$

whereαis a constant,E_cis acriticalenergy, and Γ is thegamma function.For example, for a massive Bose gas in a box,α= 3/2and the critical energy is given by:

${\frac {1}{(\beta E_{\rm {c}})^{\alpha }}}={\frac {Vf}{\Lambda ^{3}}}$

where Λ is thethermal wavelength,andfis a degeneracy factor (f= 1for simple spinless bosons). For a massive Bosegas in a harmonic trapwe will haveα= 3and the critical energy is given by:

${\frac {1}{(\beta E_{c})^{\alpha }}}={\frac {f}{(\hbar \omega \beta )^{3}}}$

whereV(r) =mω²r²/2 is the harmonic potential. It is seen thatE_cis a function of volume only.

This integral expression for the grand potential evaluates to:

$\Omega _{\rm {m}}=-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\text{c}}\right)^{\alpha }}},$

where Li_s(x) is thepolylogarithmfunction.

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 sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the uncondensed portion of the gas.

Limit on number of particles in uncondensed phase, critical temperature

The totalnumber of particlesis found from the grand potential by

$N_{\rm {m}}=-z{\frac {\partial \Omega _{m}}{\partial z}}={\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{c})^{\alpha }}}.$

This increases monotonically withz(up to the maximumz= +1). The behaviour when approachingz= 1 is however crucially dependent on the value ofα(i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well).

Forα> 1,the number of particles only increases up to a finite maximum value, i.e.,N_mis finite atz= 1:

$N_{\rm {m,max}}={\frac {\zeta (\alpha )}{(\beta E_{\rm {c}})^{\alpha }}},$

whereζ(α) is theRiemann zeta function(usingLi_α(1) =ζ(α)). Thus, for a fixed number of particlesN_m,the largest possible value thatβcan have is a critical valueβ_c.This corresponds to a critical temperatureT_c= 1/k_Bβ_c,below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature:

$T_{\rm {c}}=\left({\frac {N}{\zeta (\alpha )}}\right)^{1/\alpha }{\frac {E_{\rm {c}}}{k_{\rm {B}}}}$

For example, for the three-dimensional Bose gas in a box ($\alpha =3/2$and using the above noted value ofE_c) we get:

$T_{\rm {c}}=\left({\frac {N}{Vf\zeta (3/2)}}\right)^{2/3}{\frac {h^{2}}{2\pi mk_{\rm {B}}}}$

Forα≤ 1,there is no upper limit on the number of particles (N_mdiverges aszapproaches 1), and thus for example for a gas in a one- or two-dimensional box (α= 1/2andα= 1respectively) there is no critical temperature.

Inclusion of the ground state

The above problem raises the question forα> 1:if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? 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 particles simply 'disappear' from the continuum of states. It turns out, however, that the macroscopic equation gives an 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 to accept the particles that fall out of the continuum:

$N=N_{0}+N_{\rm {m}}=N_{0}+{\frac {{\textrm {Li}}_{\alpha }(z)}{(\beta E_{\rm {c}})^{\alpha }}}$

whereN₀is the number of particles in the ground state condensate.

Thus in the macroscopic limit, whenT<T_c,the value ofzis pinned to 1 andN₀takes up the remainder of particles. ForT>T_cthere is the normal behaviour, withN₀= 0.This approach gives the fraction of condensed particles in the macroscopic limit:

${\frac {N_{0}}{N}}={\begin{cases}1-\left({\frac {T}{T_{\rm {c}}}}\right)^{\alpha }&{\mbox{if }}\alpha >1{\mbox{ and }}T<T_{\rm {c}},\\0&{\mbox{otherwise}}.\end{cases}}$

Limitations of the macroscopic Bose gas model

The above standard treatment of a macroscopic Bose gas is straightforward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow ageometric distribution,meaning that when condensation happens atT<T_cand most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that thecompressibilitybecomes unbounded forT<T_c.Calculations can instead be performed in thecanonical ensemble,which fixes the total particle number, however the calculations are not as easy.

Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite).See the article Bose–Einstein condensate.

Approximate behaviour in small gases

Figure 1: Various Bose gas parameters as a function of normalized temperatureτ.The value ofαis 3/2. Solid lines are forN=10000,dotted lines are forN= 1000.Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potentialμis shown in red, and green lines are the values ofz.It has been assumed thatk=ε_c= 1.

For smaller,mesoscopic,systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energyε=0 in the grand potential:

$\Omega =g_{0}\ln(1-z)+\Omega _{\rm {m}}$

which gives insteadN₀=⁠g₀z/1 −z⁠.Now, the behaviour is smooth when crossing the critical temperature, andzapproaches 1 very closely but does not reach it.

This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation forα= 3/2,withk=ε_c= 1,which corresponds to agas of bosons in a box.The solid black line is the fraction of excited states1 −N₀/NforN=10000and the dotted black line is the solution forN= 1000.The blue lines are the fraction of condensed particlesN₀/N.The red lines plot values of the negative of the chemical potentialμand the green lines plot the corresponding values ofz.The horizontal axis is the normalized temperatureτdefined by

$\tau ={\frac {T}{T_{\rm {c}}}}$

It can be seen that each of these parameters become linear inτ^αin the limit of low temperature and, except for the chemical potential, linear in 1/τ^αin the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.

The equation for the number of particles can be written in terms of the normalized temperature as:

$N={\frac {g_{0}\,z}{1-z}}+N~{\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}~\tau ^{\alpha }$

For a givenNandτ,this equation can be solved forτ^αand then a series solution forzcan be found by the method ofinversion of series,either in powers ofτ^αor as an asymptotic expansion in inverse powers ofτ^α.From these expansions, we can find the behavior of the gas nearT= 0and in the Maxwell–Boltzmann asTapproaches infinity. In particular, we are interested in the limit asNapproaches infinity, which can be easily determined from these expansions.

This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number, which is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.

Thermodynamics

Expanded out, the grand potential is:

$\Omega =g_{0}\ln(1-z)-{\frac {{\textrm {Li}}_{\alpha +1}(z)}{\left(\beta E_{\rm {c}}\right)^{\alpha }}}$

All thermodynamic properties can be computed from this 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$\tau ^{\alpha }$is shown.

Quantity	General	$T\ll T_{c}\,$	$T\gg T_{c}\,$
$z$		$\approx {\frac {\zeta (\alpha )}{\tau ^{\alpha }}}-{\frac {\zeta ^{2}(\alpha )}{2^{\alpha }\tau ^{2\alpha }}}$	$=1\,$
Vapor fraction $1-{\frac {N_{0}}{N}}\,$	$={\frac {{\textrm {Li}}_{\alpha }(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }$	$=\tau ^{\alpha }\,$	$=1\,$
Equation of state ${\frac {PV\beta }{N}}=-{\frac {\Omega }{N}}\,$	$={\frac {{\textrm {Li}}_{\alpha \!+\!1}(z)}{\zeta (\alpha )}}\,\tau ^{\alpha }$	$={\frac {\zeta (\alpha \!+\!1)}{\zeta (\alpha )}}\,\tau ^{\alpha }$	$\approx 1-{\frac {\zeta (\alpha )}{2^{\alpha \!+\!1}\tau ^{\alpha }}}$
Gibbs Free Energy $G=\ln(z)\,$	$=\ln(z)\,$	$=0\,$	$\approx \ln \left({\frac {\zeta (\alpha )}{\tau ^{\alpha }}}\right)-{\frac {\zeta (\alpha )}{2^{\alpha }\tau ^{\alpha }}}$

It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures:

$U={\frac {\partial \Omega }{\partial \beta }}=\alpha PV$

A similar situation holds for the specific heat at constant volume

$C_{V}={\frac {\partial U}{\partial T}}=k_{\rm {B}}(\alpha +1)\,U\beta$

The entropy is given by:

$TS=U+PV-G\,$

Note that in the limit of high temperature, we have

$TS=(\alpha +1)+\ln \left({\frac {\tau ^{\alpha }}{\zeta (\alpha )}}\right)$

which, forα= 3/2 is simply a restatement of theSackur–Tetrode equation.In one dimension bosons with delta interaction behave as fermions, they obeyPauli exclusion principle.In one dimension Bose gas with delta interaction can be solved exactly byBethe ansatz.The bulk free energy and thermodynamic potentials were calculated byChen-Ning Yang.In one dimensional case correlation functions also were evaluated.In one dimension Bose gas is equivalent to quantumnon-linear Schrödinger equation.

See also

• Tonks–Girardeau gas

References

General references

• Huang, Kerson (1967).Statistical Mechanics.New York: John Wiley and Sons.
• Isihara, A. (1971).Statistical Physics.New York: Academic Press.
• Landau, L. D.; E. M. Lifshitz (1996).Statistical Physics, 3rd Edition Part 1.Oxford: Butterworth-Heinemann.
• Pethick, C. J.; H. Smith (2004).Bose–Einstein Condensation in Dilute Gases.Cambridge: Cambridge University Press.
• Yan, Zijun (2000)."General Thermal Wavelength and its Applications"(PDF).Eur. J. Phys.21(6): 625–631.Bibcode:2000EJPh...21..625Y.doi:10.1088/0143-0807/21/6/314.S2CID250870934.