Bose-Einstein Condensation

Derivation

Bose-Einstein Condensation

D-244 Home PU-302 Threads matter · chance · symmetry · energy Depends on Bose-Einstein and Fermi-Dirac Occupation Numbers

Statement

For an ideal gas of \(N\) non-interacting spin-0 bosons of mass \(m\) in a volume \(V\), the number of particles that the excited states can hold is bounded. Below a critical temperature \(T_c\) this bound falls below \(N\), forcing a macroscopic fraction \(N_0/N = 1-(T/T_c)^{3/2}\) into the single-particle ground state, with \(k_BT_c = \dfrac{2\pi\hbar^2}{m}\left(\dfrac{n}{\zeta(3/2)}\right)^{2/3}\) where \(n=N/V\).

Why it matters

Bose-Einstein condensation is the paradigm of a phase transition driven by quantum statistics alone, with no interaction potential responsible for the ordering. It is the mechanism behind superfluid \(^4\)He and the dilute-gas condensates of alkali atoms first realised in 1995, and it exhibits macroscopic occupation of a single quantum state — the seed of off-diagonal long-range order and coherence.

The derivation isolates exactly which piece of the physics is essential: the saturation of the excited-state population when the chemical potential \(\mu\) is pinned to the ground-state energy. That single kinematic fact fixes both \(T_c\) and the temperature dependence of the condensate, and it explains why the effect exists in three dimensions but not in two.

Assumptions

Ideal (non-interacting) gas.With interactions the ground state is a many-body state, the transition acquires a finite-interaction shift and critical fluctuations, and \(T_c\) is renormalised; the free-particle spectrum \(\varepsilon=\hbar^2k^2/2m\) no longer holds.

Fixed particle number \(N\) (canonical/grand-canonical with \(\langle N\rangle\) fixed).If \(N\) is not conserved (e.g. photons, phonons) then \(\mu=0\) identically and there is no condensation threshold in this sense.

Large system, \(\varepsilon_0\to 0\), continuum density of states for excited states.If dropped, the discrete sum must be kept; the transition sharpens only in the thermodynamic limit and the ground state must be separated by hand.

Three spatial dimensions with \(\varepsilon\propto k^2\).The convergence of \(\zeta(3/2)\) is what makes the excited-state capacity finite. In 2D the corresponding integral diverges and no condensation occurs at \(T>0\); a different dispersion changes the exponent.

Thermal equilibrium and thermodynamic limit \(N,V\to\infty\) at fixed \(n\).Without it the sharp non-analytic kink at \(T_c\) is rounded into a smooth crossover set by finite-size level spacing.

Derivation

\[ \bar n(\varepsilon)=\frac{1}{e^{\beta(\varepsilon-\mu)}-1},\qquad \beta=\frac{1}{k_BT},\qquad \mu<\varepsilon_0=0 \]

Bose-Einstein occupation from quantum-occupation statistics; \(\mu\) must lie below the lowest level or an occupancy diverges negative. A

\[ N=\sum_{\text{states}}\bar n(\varepsilon)=\underbrace{\frac{z}{1-z}}_{N_0}+\underbrace{\sum_{\varepsilon>0}\frac{1}{z^{-1}e^{\beta\varepsilon}-1}}_{N_{\text{ex}}},\qquad z\equiv e^{\beta\mu} \]

Separate the ground state (\(\varepsilon_0=0\)) from the rest before taking any continuum limit, because \(N_0=z/(1-z)\) can be macroscopic while its density of states is zero. B

\[ g(\varepsilon)\,d\varepsilon=\frac{V}{(2\pi)^3}4\pi k^2\,dk=\frac{V}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\varepsilon^{1/2}\,d\varepsilon \]

Free-particle density of states in 3D from \(\varepsilon=\hbar^2k^2/2m\), counting one spin state; legal because level spacing \(\to 0\) in the thermodynamic limit for \(\varepsilon>0\). B

\[ N_{\text{ex}}=\int_0^\infty \frac{g(\varepsilon)\,d\varepsilon}{z^{-1}e^{\beta\varepsilon}-1}=\frac{V}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sum_{\ell=1}^{\infty}z^\ell\int_0^\infty \varepsilon^{1/2}e^{-\ell\beta\varepsilon}\,d\varepsilon \]

Expand \((z^{-1}e^{\beta\varepsilon}-1)^{-1}=\sum_{\ell\ge1}z^\ell e^{-\ell\beta\varepsilon}\), valid for \(0<z\le1\); term-by-term integration justified by uniform convergence on \(\varepsilon>0\). C

\[ \int_0^\infty \varepsilon^{1/2}e^{-\ell\beta\varepsilon}\,d\varepsilon=\frac{\Gamma(3/2)}{(\ell\beta)^{3/2}}=\frac{\sqrt\pi}{2}\,\frac{(k_BT)^{3/2}}{\ell^{3/2}} \]

Standard Gamma integral \(\int_0^\infty x^{s-1}e^{-ax}dx=\Gamma(s)a^{-s}\) with \(s=3/2\), \(\Gamma(3/2)=\tfrac{\sqrt\pi}{2}\). A

\[ N_{\text{ex}}=\frac{V}{\lambda_T^3}\,g_{3/2}(z),\qquad \lambda_T=\frac{h}{\sqrt{2\pi m k_BT}},\qquad g_{3/2}(z)=\sum_{\ell=1}^\infty\frac{z^\ell}{\ell^{3/2}} \]

Collect the prefactor; the constants reassemble exactly into the thermal de Broglie wavelength \(\lambda_T\) and the polylogarithm \(g_{3/2}=\mathrm{Li}_{3/2}\). Pure algebra of the previous line. B

\[ g_{3/2}(z)\ \text{increases monotonically on }(0,1],\qquad g_{3/2}(1)=\zeta\!\left(\tfrac32\right)\approx 2.612 \]

Because \(\mu\le0\Rightarrow z\le1\), the excited-state occupancy is bounded above by its value at \(z=1\); the series is finite there since \(3/2>1\). This saturation is the crux. C

\[ N_{\text{ex}}^{\max}(T)=\frac{V}{\lambda_T^3}\,\zeta\!\left(\tfrac32\right)\ \propto\ T^{3/2} \]

Set \(z=1\) in Step 6. As \(T\) falls this ceiling drops; when it drops below \(N\) the excess particles have nowhere to go but the ground state. A

\[ N=\frac{V}{\lambda_{T_c}^3}\,\zeta\!\left(\tfrac32\right)\ \Longrightarrow\ n\,\lambda_{T_c}^3=\zeta\!\left(\tfrac32\right) \]

Define \(T_c\) as the temperature where the excited states are exactly filled with \(z\to1\); equivalently the phase-space density reaches \(\zeta(3/2)\). B

\[ n=\frac{(2\pi m k_BT_c)^{3/2}}{h^3}\zeta\!\left(\tfrac32\right)\ \Longrightarrow\ k_BT_c=\frac{2\pi\hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3} \]

Insert \(\lambda_{T_c}=h/\sqrt{2\pi m k_BT_c}\) and solve for \(T_c\), using \(h=2\pi\hbar\Rightarrow h^2/2\pi=2\pi\hbar^2\). A

\[ T<T_c:\ \mu\to0^-,\ z\to1\ \Rightarrow\ N_{\text{ex}}(T)=\frac{V}{\lambda_T^3}\zeta\!\left(\tfrac32\right)=N\left(\frac{T}{T_c}\right)^{3/2} \]

Below \(T_c\) the chemical potential is pinned at \(0\) (it cannot exceed \(\varepsilon_0\)); divide Step 8 by Step 9 so \(V\zeta(3/2)\) cancels, leaving the pure temperature ratio. B

\[ N_0=N-N_{\text{ex}}\ \Longrightarrow\ \boxed{\ \frac{N_0}{N}=1-\left(\frac{T}{T_c}\right)^{3/2}\ }\quad(T\le T_c) \]

Particle-number conservation (Step 2) with \(N_{\text{ex}}\) from Step 11; the remainder is macroscopic and sits in one state — the condensate. A

Result

\[ k_BT_c=\frac{2\pi\hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3},\qquad \frac{N_0}{N}=1-\left(\frac{T}{T_c}\right)^{3/2} \]

Reading. Condensation sets in when the phase-space density \(n\lambda_T^3\) reaches \(\zeta(3/2)\approx2.612\) — physically, when the thermal wavelength grows to the interparticle spacing so wavepackets overlap. Below \(T_c\) the excited states are "full" at their maximum thermal capacity \(\propto T^{3/2}\), and every particle beyond that capacity condenses into the ground state, giving a condensate fraction that rises from 0 at \(T_c\) to 1 at \(T=0\).

Units check. \([\hbar^2/m]=\mathrm{J^2 s^2\,kg^{-1}}\); \([n^{2/3}]=\mathrm{m^{-2}}\). Product: \(\mathrm{J^2 s^2\,kg^{-1}m^{-2}}=\mathrm{J\cdot(J\,s^2\,kg^{-1}m^{-2})}=\mathrm{J\cdot(kg\,m^2 s^{-2}\cdot s^2\,kg^{-1}m^{-2})}=\mathrm{J}\), matching \(k_BT_c\). The fraction \(N_0/N\) is dimensionless, as \((T/T_c)^{3/2}\) is a pure ratio.

Limiting cases

\(T\to T_c^-\): \(N_0/N\to0^+\) continuously — the order parameter turns on at the transition with infinite slope in \(dN_0/dT\).
\(T\to0\): \(N_0/N\to1\), all particles in the ground state; the excited-state thermal cloud vanishes as \(T^{3/2}\).
\(T>T_c\): \(z<1\), \(N_0/N=O(1/N)\to0\) in the thermodynamic limit — no macroscopic condensate.
Classical limit \(n\lambda_T^3\ll1\): \(g_{3/2}(z)\approx z\), recovering \(z\approx n\lambda_T^3\) and the Maxwell-Boltzmann ideal gas, far from any condensation.
Heavy mass or high density: \(T_c\propto n^{2/3}/m\) rises, so lighter, denser gases condense at higher temperature (why \(^4\)He condenses near a few K but alkali gases only at nK).

Breaks when

Two dimensions (or \(d\le2\) with \(\varepsilon\propto k^2\)). The excited-state integral becomes \(\int_0^\infty d\varepsilon/(z^{-1}e^{\beta\varepsilon}-1)\propto-\ln(1-z)\), which diverges as \(z\to1\). The states are never saturated, \(T_c=0\), and there is no condensation at finite temperature.
Interactions become important. For a dense system such as real \(^4\)He, the ideal-gas \(T_c\approx3.1\,\mathrm{K}\) only roughly locates the observed \(\lambda\)-transition at \(2.17\,\mathrm{K}\); strong interactions deplete the true condensate to \(\sim10\%\) even at \(T=0\), and the \(T^{3/2}\) law and mean-field picture fail near \(T_c\).
Finite systems / strong confinement. When level spacing is not negligible (few atoms, tight traps) the sum cannot be replaced by an integral; the sharp kink at \(T_c\) is smeared into a crossover and \(N_0\) grows smoothly.
Non-parabolic dispersion. For \(\varepsilon\propto k^s\) in \(d\) dimensions the relevant exponent is \(d/s\); if \(d/s\le1\) the polylog diverges and BEC is destroyed (e.g. \(\varepsilon\propto k\) photonic-like dispersion in low \(d\)).

Failure modes

Including the ground state in the integral. The density of states \(g(\varepsilon)\propto\varepsilon^{1/2}\) vanishes at \(\varepsilon=0\), so the continuum integral silently discards \(N_0\). One must split it off (Step 2) before integrating, or the condensate is invisible.
Letting \(\mu>0\). Setting \(\mu\) above the ground-state energy makes an occupation number negative or divergent; \(\mu\) is bounded above by \(\varepsilon_0\) and pins to it below \(T_c\).
Using \(g_{3/2}(z)\) for \(z>1\). The series \(\sum z^\ell/\ell^{3/2}\) diverges for \(z>1\); \(z\) never exceeds 1, and the "extra" particles go into \(N_0\), not into a larger \(g_{3/2}\).
Confusing \(\lambda_T\) definitions. Some texts write \(\lambda_T=h/\sqrt{2\pi mk_BT}\), others use \(\hbar\); mixing conventions puts stray factors of \(2\pi\) into \(T_c\). Keep \(h=2\pi\hbar\) explicit.
Reading \(\zeta(3/2)=2.612\) as a temperature or energy. It is a dimensionless number (the value of the Riemann zeta function), the maximum phase-space density, not a physical scale on its own.
Claiming the ideal-gas \(T_c\) equals the \(^4\)He \(\lambda\) point exactly. The numerical closeness (3.1 K vs 2.17 K) is suggestive but interactions matter; treating them as equal is a category error.

Discussion

The essential physics is a competition between two quantities that both scale as \(T^{3/2}\): the actual particle number \(N\) (fixed) and the maximum thermal capacity of the excited states \(N_{\text{ex}}^{\max}(T)=V\zeta(3/2)/\lambda_T^3\). Above \(T_c\) the capacity exceeds \(N\), so \(z<1\) adjusts to hold exactly \(N\) particles in the excited states. Below \(T_c\) the capacity has shrunk below \(N\); the chemical potential can no longer rise to compensate because it is capped at the ground-state energy, and the deficit \(N-N_{\text{ex}}\) is dumped into a single quantum state. The transition is thus a saturation phenomenon, not driven by any energetic preference — it is entropy and quantum indistinguishability that force the accumulation.

This is a genuine thermodynamic phase transition even in the ideal gas: the specific heat has a cusp at \(T_c\) (the famous \(\lambda\)-shape in the interacting case), and the condensate density serves as an order parameter. The connection to symmetry breaking is subtle — the condensate wavefunction acquires a well-defined phase, breaking the global \(U(1)\) gauge symmetry associated with particle-number conservation, which is why BEC underlies superfluidity and the appearance of a macroscopic coherent matter wave.

The dimensional sensitivity is a lesson in how phase transitions depend on the density of states near the band bottom. In 3D, \(g(\varepsilon)\sim\varepsilon^{1/2}\) suppresses low-energy states enough that their total capacity converges; in 2D, \(g(\varepsilon)\sim\varepsilon^0\) is constant and the low-energy states can absorb arbitrarily many particles as \(\mu\to0\), so saturation — and hence condensation — never occurs at finite \(T\). This is a special case of the Mermin-Wagner-Hohenberg circle of ideas about the absence of long-range order in low dimensions.

More rigorously, the sharp transition is an artefact of the thermodynamic limit and the interchange of \(\lim_{N\to\infty}\) with the continuum integral. For finite \(N\) the ground-state occupation \(N_0=z/(1-z)\) is analytic in \(T\); the non-analyticity of \(N_0/N\) at \(T_c\) emerges only as \(N\to\infty\) with \(n\) fixed. Careful treatments (e.g. via the grand potential and a saddle-point or Poisson-summation analysis of the discrete sum) show the condensate fraction acquires finite-size corrections of order \(N^{-1/3}\) and the "kink" is rounded over a width set by the trap level spacing — precisely what is measured in dilute-atom experiments, where the harmonic-trap density of states changes the exponent to \(N_0/N=1-(T/T_c)^3\).

Common misconceptions. BEC is not "all the atoms freezing into one place" — the condensate is a delocalised single-particle mode, not a spatial clump (though in a trap the ground-state wavefunction is spatially compact). Nor does condensation require attractive interactions or cooling below a chemical binding energy; the ideal gas condenses with no interactions at all. And the condensate does not have zero energy or zero momentum spread in a real trap — it has the zero-point energy and width of the ground state.

Worked examples

Example 1 — Critical temperature of ideal \(^4\)He at liquid density. Take \(m=6.65\times10^{-27}\,\mathrm{kg}\), number density \(n=2.18\times10^{28}\,\mathrm{m^{-3}}\) (from liquid density \(145\,\mathrm{kg\,m^{-3}}\)). Find \(T_c\). A

\[ k_BT_c=\frac{2\pi\hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3} \]

\[ \frac{2\pi\hbar^2}{m}=\frac{2\pi(1.055\times10^{-34})^2}{6.65\times10^{-27}}=1.051\times10^{-41}\ \mathrm{J\,m^2} \]

\[ \frac{n}{\zeta(3/2)}=\frac{2.18\times10^{28}}{2.612}=8.35\times10^{27}\ \mathrm{m^{-3}},\quad \left(8.35\times10^{27}\right)^{2/3}=4.11\times10^{18}\ \mathrm{m^{-2}} \]

\[ k_BT_c=(1.051\times10^{-41})(4.11\times10^{18})=4.32\times10^{-23}\ \mathrm{J} \]

\[ T_c=\frac{4.32\times10^{-23}}{1.381\times10^{-23}}\approx 3.1\ \mathrm{K} \]

Reading. The ideal-gas estimate lands remarkably close to the observed \(^4\)He \(\lambda\)-transition at \(2.17\,\mathrm{K}\); the \(\sim40\%\) overshoot is the fingerprint of the interactions this model ignores.

Example 2 — \(T_c\) and condensate fraction of a dilute \(^{87}\)Rb gas. Take \(m=1.44\times10^{-25}\,\mathrm{kg}\) and \(n=1.0\times10^{19}\,\mathrm{m^{-3}}\). Find \(T_c\), then the condensate fraction at \(T=T_c/2\). B

\[ \frac{2\pi\hbar^2}{m}=\frac{2\pi(1.055\times10^{-34})^2}{1.44\times10^{-25}}=4.85\times10^{-43}\ \mathrm{J\,m^2} \]

\[ \frac{n}{\zeta(3/2)}=\frac{1.0\times10^{19}}{2.612}=3.83\times10^{18}\ \mathrm{m^{-3}},\quad \left(3.83\times10^{18}\right)^{2/3}=2.45\times10^{12}\ \mathrm{m^{-2}} \]

\[ k_BT_c=(4.85\times10^{-43})(2.45\times10^{12})=1.19\times10^{-30}\ \mathrm{J}\ \Rightarrow\ T_c=\frac{1.19\times10^{-30}}{1.381\times10^{-23}}\approx 86\ \mathrm{nK} \]

\[ \frac{N_0}{N}\Big|_{T=T_c/2}=1-\left(\tfrac12\right)^{3/2}=1-0.354=0.646 \]

\[ T_c\approx 86\ \mathrm{nK},\qquad \frac{N_0}{N}\approx 0.65\ \text{at}\ T=T_c/2 \]

Reading. Dilute alkali gases condense only in the nanokelvin regime — five orders of magnitude colder than helium — precisely because their far lower density gives a tiny \(n^{2/3}\). At half the critical temperature already about two-thirds of the atoms occupy the ground state.

Problems

Thermal wavelength. Compute the thermal de Broglie wavelength \(\lambda_T=h/\sqrt{2\pi m k_BT}\) for \(^{87}\)Rb (\(m=1.44\times10^{-25}\,\mathrm{kg}\)) at \(T=100\,\mathrm{nK}\), and compare to the mean spacing \(n^{-1/3}\) at \(n=1.0\times10^{19}\,\mathrm{m^{-3}}\).
Solution
\(2\pi m k_BT=2\pi(1.44\times10^{-25})(1.381\times10^{-23})(1.0\times10^{-7})=1.249\times10^{-54}\). \(\sqrt{\cdot}=1.118\times10^{-27}\). \(\lambda_T=6.626\times10^{-34}/1.118\times10^{-27}=5.9\times10^{-7}\,\mathrm{m}=0.59\,\mu\mathrm{m}\). Spacing \(n^{-1/3}=(1.0\times10^{19})^{-1/3}=4.6\times10^{-7}\,\mathrm{m}=0.46\,\mu\mathrm{m}\). Since \(\lambda_T\gtrsim n^{-1/3}\), i.e. \(n\lambda_T^3\approx(0.59/0.46)^3\approx2.1\gtrsim\zeta(3/2)\), the gas is essentially at/below its condensation threshold — consistent with \(T_c\approx86\,\mathrm{nK}\) from Example 2.
Deriving the \(T^{3/2}\) law. Starting from \(N_{\text{ex}}(T)=V\zeta(3/2)/\lambda_T^3\) for \(T\le T_c\) and \(N=V\zeta(3/2)/\lambda_{T_c}^3\), show \(N_{\text{ex}}/N=(T/T_c)^{3/2}\).
Solution
Divide: \(N_{\text{ex}}/N=\lambda_{T_c}^3/\lambda_T^3\). Since \(\lambda_T\propto T^{-1/2}\), we have \(\lambda_T^3\propto T^{-3/2}\), so \(\lambda_{T_c}^3/\lambda_T^3=(T_c^{-1/2}/T^{-1/2})^3=(T/T_c)^{3/2}\). Hence \(N_{\text{ex}}/N=(T/T_c)^{3/2}\) and \(N_0/N=1-(T/T_c)^{3/2}\). \(V\) and \(\zeta(3/2)\) cancel exactly because \(\mu=0\) below \(T_c\).
Critical temperature. An ideal Bose gas of atoms with \(m=3.32\times10^{-27}\,\mathrm{kg}\) (\(^2\)H-like) has \(n=5.0\times10^{20}\,\mathrm{m^{-3}}\). Find \(T_c\).
Solution
\(2\pi\hbar^2/m=2\pi(1.055\times10^{-34})^2/3.32\times10^{-27}=2.105\times10^{-41}\,\mathrm{J\,m^2}\). \(n/\zeta(3/2)=5.0\times10^{20}/2.612=1.914\times10^{20}\). \((1.914\times10^{20})^{2/3}=(1.914)^{2/3}\times10^{40/3}=1.541\times3.32\times10^{13}\)... compute: \((1.914\times10^{20})^{1/3}=5.77\times10^{6}\); square \(=3.33\times10^{13}\,\mathrm{m^{-2}}\). \(k_BT_c=(2.105\times10^{-41})(3.33\times10^{13})=7.01\times10^{-28}\,\mathrm{J}\). \(T_c=7.01\times10^{-28}/1.381\times10^{-23}=5.1\times10^{-5}\,\mathrm{K}=51\,\mu\mathrm{K}\).
Condensate fraction. At what temperature (as a fraction of \(T_c\)) is exactly \(90\%\) of the gas condensed? At what fraction is \(50\%\) condensed?
Solution
\(N_0/N=1-(T/T_c)^{3/2}=0.90\Rightarrow(T/T_c)^{3/2}=0.10\Rightarrow T/T_c=0.10^{2/3}=0.215\). For \(50\%\): \((T/T_c)^{3/2}=0.50\Rightarrow T/T_c=0.50^{2/3}=0.630\). So \(90\%\) condensed at \(T\approx0.22\,T_c\) and \(50\%\) at \(T\approx0.63\,T_c\).
No BEC in 2D. For a 2D ideal Bose gas the density of states is constant, \(g(\varepsilon)=mA/(2\pi\hbar^2)\). Show that \(N_{\text{ex}}\) diverges as \(z\to1\), so \(T_c=0\).
Solution
\(N_{\text{ex}}=\dfrac{mA}{2\pi\hbar^2}\displaystyle\int_0^\infty\dfrac{d\varepsilon}{z^{-1}e^{\beta\varepsilon}-1}\). Substitute \(x=\beta\varepsilon\): integral \(=k_BT\int_0^\infty dx/(z^{-1}e^{x}-1)=-k_BT\ln(1-z)\). Thus \(N_{\text{ex}}=\dfrac{mA k_BT}{2\pi\hbar^2}\big[-\ln(1-z)\big]\), which \(\to+\infty\) as \(z\to1^-\) for any \(T>0\). The excited states can therefore accommodate any \(N\) with \(z<1\); no macroscopic ground-state occupation is forced, so condensation occurs only at \(T=0\). Equivalently, the "\(g_1(z)\)" polylog is \(-\ln(1-z)\), which lacks a finite value at \(z=1\) — the divergence of \(\zeta(1)\) — unlike \(\zeta(3/2)\) in 3D.