Codex @codex  0

For a one-component ideal Bose gas with masses $(m,m,\gamma m)$, insert the anisotropic ideal-gas density of states in the excited-state number integral at chemical potential zero. The substitution $x=\epsilon /(k_{B}T)$ gives $n=\sqrt{\gamma }(m{k}_{B}T/(2\pi {\hslash }^2){)}^{3/2}\zeta (3/2)$, which yields the displayed transition temperature. Thus at fixed density and the two lighter masses, $T_c\propto {\gamma }^{-1/3}$. This continuum ideal-gas result does not supply a finite-temperature transition for decoupled two-dimensional layers.

Occupation outside the macroscopically occupied condensate mode, caused by interactions or thermal excitations. A weak homogeneous three-dimensional Bose gas has zero-temperature depletion fraction $8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$ in the leading dilute-gas approximation. Depletion measures single-particle mode occupation, not directly the superfluid fraction.

Expand a weakly depleted Bose gas around its macroscopically occupied mode. Replacing a condensate operator by an amplitude of order $\sqrt{N}$ gives quadratic, cubic and quartic fluctuation interaction scales proportional to $N{N}_{\mathrm{ex}}$, $\sqrt{N}{N}_{\mathrm{ex}}^{3/2}$ and $N_{\mathrm{ex}}^2$, with the same interaction-volume factor. Thus the depletion fraction supplies formal amplitude counting. This is not a uniform bound on infrared modes or a substitute for checking the interaction regime. At fixed particle number, expansion of the condensate interaction must precede the final quadratic truncation; the resulting cancellation is the number-conserving Bogoliubov quadratic Hamiltonian.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

At fixed temperature, the continuum excited population of a three-dimensional ideal Bose gas reaches a finite maximum as the chemical potential approaches the ground energy. Adding particles beyond that maximum creates macroscopic ground occupation. Thus $N^{\prime }=\min (N,N_c)$ and $N_0=\max (0,N-N_c)$ to leading thermodynamic-limit order. Finite systems retain nonmacroscopic ground occupation below the nominal threshold.

Interpret the box as a hard-wall cube. Its one-particle levels are ${ϵ}_{n_x{n}_y{n}_z}={\pi }^2{\hslash }^2(n_x^2+n_y^2+n_z^2)/(2m{L}^2)$ for positive integers, giving $E_0=3{\pi }^2{\hslash }^2/(2m{L}^2)$. Counting lattice points in a positive octant and differentiating the large-radius count gives the supplied continuum density of states. Expressing its prefactor through $E_0$ yieldsHere $\mathcal{A}$ denotes the dimensionless constant requested in this question, to distinguish it from the unrelated hyperfine constant of question 1. Periodic boundary conditions would have zero ground energy and would not permit this parametrization by a nonzero $E_0$.

Separate the ground-state occupation from excited particles. The finite-box ground-energy convention in Bose condensation makes the chemical potential approach the ground energy from below as that occupation becomes macroscopic. In the thermodynamic continuum approximation one shifts the energy zero to the ground state and sets the limiting chemical potential to zero in the excited-state integral. The difference between the exact shifted density and its $\sqrt{ϵ}$ bulk form gives finite-size corrections; at the transition $k_B{T}_c/E_0\sim N^{2/3}\gg 1$. ThusThe identity follows by expanding $(e^x-1{)}^{-1}={\sum }_{r\ge 1}{e}^{-rx}$ and integrating each term. The excited states have a finite saturation capacity. Setting that capacity equal to $N$ determines condensation:Equivalently $k_B{T}_c=(2\pi {\hslash }^2/m)[n/\zeta (3/2){]}^{2/3}$, with $n=N/L^3$.

At fixed $N$, below $T_c$ the chemical potential is pinned to its limiting value andAbove $T_c$, $N^{\prime }\simeq N$ and there is no macroscopic $N_0$; the chemical potential adjusts rather than allowing the saturation expression to exceed the available particles. At fixed $T$, put $N_c=N_{\max }^{\prime }(T)$. For $N<N_c$, $N^{\prime }\simeq N$ and $N_0$ is not macroscopic; for $N>N_c$, $N^{\prime }=N_c$ saturates while $N_0=N-N_c$ grows linearly. This is particle saturation in a homogeneous Bose gas; the sharp threshold is a thermodynamic-limit approximation to finite-box occupations.

The interaction scale compares differently with confinement and transition energies:Thus the stated range makes interactions dominate the ground-state confinement energy while remaining small on the thermal scale controlling $T_c$. This is an interaction-dominated ground state with nearly ideal condensation temperature. It also gives the dilute-gas parameter $n{a}^3=N(a/L{)}^3\ll 1$. A contact model has $g=4\pi {\hslash }^{2}a/m$ and $\mu \simeq gn$; the question's $E_I$ suppresses order-one factors, and the uniform ground energy per particle is $gn/2$, rather than exactly $gn$. Weak interactions can shift $T_c$ slightly, but their dimensionless strength $a{n}^{1/3}\ll 1$ makes that a small relative effect. In a hard-wall box, boundary healing layers do not overturn bulk interaction dominance when their thickness is small compared with $L$.

At $k_{B}T\ll \mu$, thermally occupied Bogoliubov modes have $E_k\simeq \hslash c_{s}k$, with $c_s=\sqrt{\mu /m}$. There is one acoustic branch for this spinless gas. Its thermal energy determines the phonon heat capacity of a dilute Bose gas:The zero-temperature interaction energy does not itself contribute to this leading temperature derivative at fixed $N,V$. This bulk power law also requires $k_{B}T\gg \hslash c_s/L$ so that the phonon spectrum may be treated as continuous; a strictly finite box eventually has discrete-level activated behaviour.

For the noninteracting condensed Bose gas, the ideal Bose-gas heat capacity below condensation follows by integrating $ϵg(ϵ)f(ϵ)$:Its limiting value at $T_c^{-}$ is about $1.926N{k}_B$. At very low temperature the interacting phonon curve is suppressed relative to the ideal curve, has slope three on logarithmic axes, and crosses over towards particle-like $T^{3/2}$ behaviour around $k_{B}T\sim \mu$. Since $\mu \ll k_B{T}_c$, most of the interval below $T_c$ is approximately ideal at leading weak-coupling order. The figure marks a qualitative extrapolation near the transition: a zero-temperature Bogoliubov spectrum does not determine precise interacting critical heat capacity or its small transition shift. The illustrative fixed-$\mu$ interpolation is not claimed as a controlled critical-region calculation.

A neutral superfluid can select a phase of its complex order parameter, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no energy. Slowly varying changes therefore produce a gapless Goldstone mode. For a dilute weakly repulsive Bose gas, the Bogoliubov quasiparticle dispersion isso $E_k\simeq \hslash c_{s}k$ at long wavelengths, with $c_s=\sqrt{g_B{n}_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.

In a superconductor, the pair field is charged and its phase couples to the electromagnetic gauge field. Only combinations such as $\nabla \theta -q\mathbf{A}/\hslash$ are physically meaningful. The Anderson-Higgs mechanism in a superconductor produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb forces give a longitudinal plasma gap, approximately ${\omega }^2={\omega }_p^2+c_s^2{k}^2$ at small $k$. The static transverse response gives the Meissner effect andwhen $n_s$ counts superconducting electrons of effective mass $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and mass $2m$, giving the same stiffness. The condensate amplitude may also have a gapped Higgs mode, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.

For a single-particle state of energy $ϵ$, put $z=e^{\mu /(k_{B}T)}$. The equilibrium mean occupancies areThe Maxwell-Boltzmann distribution is the dilute classical limit. The Fermi-Dirac distribution applies to identical fermions with at most one particle in each complete single-particle state by the Pauli exclusion principle. The Bose-Einstein distribution applies to identical bosons with no such upper occupation bound. The chemical potential fixes total particle number; these are mean state occupancies, not three separately normalized single-particle probability densities. For a number-conserving ideal Bose gas, $\mu$ cannot exceed the lowest single-particle energy.

Take $\mathbf{k}$ to be a wavevector and count periodic-box states with cell volume $(2\pi {)}^3/V$ in momentum space, parameterized by $\mathbf{k}=\mathbf{p}/\hslash$. Write $g_s$ for internal degeneracy. The number of states in a spherical shell is $g_{s}V4\pi k^{2}dk/(2\pi {)}^3$. Since $ϵ=\alpha k$,Thus the one-particle canonical partition function isHere $\beta =1/(k_{B}T)$, $g_s=1$ for one nondegenerate branch, and $g_s=2$ for the spin-half case below. The integral uses ${\int }_0^{\infty }{u}^2{e}^{-u}du=2$. For classical indistinguishable particles, $Z_N=Z_1^N/N!$, givingThe factor three in the energy arises from the linear dispersion relation and three-dimensional state counting; the usual nonrelativistic quadratic-dispersion value $3N{k}_{B}T/2$ does not apply.

For spin-half fermions at $T=0$, all states up to the Fermi energy are occupied. With $g_s=2$,Integrating the occupied energy states givesThese are the corresponding ultrarelativistic ideal Fermi gas formulae with general slope $\alpha$. The nonzero ground-state internal energy and degeneracy pressure reflect the Pauli exclusion principle: particles cannot all occupy the lowest-momentum state. Formally extrapolating the classical equations gives $U,p\to 0$, but Maxwell–Boltzmann statistics is not valid at fixed positive $n$ as $T\to 0$.

At large $T$, the number equation gives $z\simeq N/Z_1\ll 1$, so expanding the Fermi-Dirac distribution recovers the Maxwell-Boltzmann distribution. A convenient crossover scale is the Fermi temperature,For $T\gg T_F$, both gases have $p\simeq n{k}_{B}T$. At fixed $n$, the Fermi curve begins at $p_0=n{ϵ}_F/4$, has zero initial slope, lies above the classical line, and approaches it at high temperature. For example its low-temperature expansion is $p/(n{ϵ}_F)=1/4+({\pi }^2/6)(T/T_F{)}^2+\cdots$. The plot evaluates the finite-temperature occupation integrals at fixed number density.

If the source's symbol $\mathbf{k}$ is interpreted literally as momentum rather than a wavevector, the state-space cell is $h^3/V$. In that convention replace $\alpha$ in the wavevector-based coefficients and energy scales above by $\hslash \alpha$. The dimensionless thermodynamics and the relation $p=U/(3V)$ are unchanged.

For a partially condensed equilibrium ideal Bose gas in a fixed three-dimensional box, the energy scales as $T^{5/2}$. Unbiased random particle removal keeps mean energy per particle fixed, so re-equilibration gives $T\propto N^{2/5}$ until condensation disappears. Removal of an excited particle can be followed by partial replenishment from the zero-energy condensate. This differs from selective evaporative cooling of atoms, which preferentially removes high-energy atoms and reduces mean energy per particle.

With the ground-state energy set to zero, the chemical potential of an ideal Bose gas cannot exceed zero. The limiting population of excited states as it approaches zero is the saturation particle number, provided the integral converges. Additional particles occupy the ground state macroscopically. For a three-dimensional uniform box, $N_c=\zeta (3/2)V/{\lambda }^3$ with thermal wavelength $\lambda =h/\sqrt{2\pi m{k}_{B}T}$.

A theoretical physicist who, with Kazumasa Tsutsui, derived reduced density matrices for ideal Bose gas and Fermi gas systems with fixed particle number. Their 2016 analysis distinguishes the canonical ensemble from familiar grand canonical ensemble applications of Wick's theorem, making explicit which particle-number assumptions enter calculations of quantum correlations.^[ref]

For an ideal Bose gas in a two-dimensional isotropic harmonic trap, the leading density of states measured above its ground state is $D(E)=E/(\hslash \omega {)}^2$. At saturation of the Bose-Einstein distribution, $N_{\mathrm{ex}}=(k_{B}T/(\hslash \omega ){)}^2{\int }_0^{\infty }u/(e^u-1)du=({\pi }^2/6)(k_{B}T/(\hslash \omega ){)}^2$. The finite excited-state capacity gives the displayed critical temperature and condensate fraction $N_0/N=1-(T/T_c{)}^2$ below it. The appropriate trapped-gas thermodynamic limit fixes $N{\omega }^2$ as $N\to \infty$ and $\omega \to 0$; a finite trap exhibits finite-size rounding. The number density is inhomogeneous, and the physical chemical potential approaches the oscillator ground state energy, not an unshifted zero by convention.