= Solution
A complex <order parameter> captures both the amplitude and phase rigidity of a <superfluid>. A local isotropic <Ginzburg-Landau theory> writes its free-energy functional, to fourth order in amplitude and second order in gradients, as
$$
\mathcal F[\Psi]=\int d^3r\left[\alpha|\Psi|^2+\frac\beta2|\Psi|^4+K|\nabla\Psi|^2\right],\qquad \beta,K>0.
$$
The uniform minimum has $\Psi=0$ for $\alpha>0$ and $|\Psi_0|^2=-\alpha/\beta$ for $\alpha<0$. Its phase remains arbitrary under a global $U(1)$ transformation. Minimizing with respect to $\Psi^*$ gives $-K\nabla^2\Psi+\alpha\Psi+\beta|\Psi|^2\Psi=0$. The phenomenological description specifies the allowed structures and coefficients; microscopic input is needed to identify the normalization of $\Psi$ and calculate those coefficients. A static functional by itself does not fix the dynamical equation or mode frequencies.
For a dilute three-dimensional Bose gas with short-range repulsion and particle <mass> $m_B$, the microscopic low-energy Hamiltonian is
$$
\widehat H=\int d^3r\left[
\frac{\hbar^2}{2m_B}\nabla\widehat\psi^\dagger\cdot\nabla\widehat\psi
+V_{\rm ext}\widehat\psi^\dagger\widehat\psi
+\frac{g_B}{2}\widehat\psi^\dagger\widehat\psi^\dagger\widehat\psi\widehat\psi\right],
\qquad g_B=\frac{4\pi\hbar^2a_s}{m_B}>0.
$$
The interaction parameter is expressed through the low-energy <s-wave scattering length> $a_s$, rather than an unrenormalized bare delta potential. In a symmetry-selected description, $\Psi=\langle\widehat\psi\rangle$ and $|\Psi|^2=n_0$ is condensate density. Equivalently, fixed-number <off-diagonal long-range order> identifies the macroscopically occupied natural orbital and its amplitude without a nonzero one-point expectation.
Evaluating $\widehat H-\mu\widehat N$ in a <coherent-field condensate variational state> gives the <Ginzburg-Landau functional of a dilute Bose gas>, with the mean-field microscopic identifications
$$
\boxed{K=\frac{\hbar^2}{2m_B},\qquad
\alpha=V_{\rm ext}-\mu,\qquad\beta=g_B.}
$$
Its stationary variational equation is the <Gross–Pitaevskii equation>. The time-dependent variational action additionally contains $i\hbar\Psi^*\partial_t\Psi$, giving
$$
i\hbar\partial_t\Psi=\left[-\frac{\hbar^2\nabla^2}{2m_B}+V_{\rm ext}+g_B|\Psi|^2\right]\Psi.
$$
This term expresses <density-phase conjugacy>. It is the missing microscopic dynamical input; a purely dissipative relaxation law would describe a different approximation.
<Bogoliubov theory> writes $\widehat\psi=\Psi+\delta\widehat\psi$ and retains quadratic fluctuations. In a uniform condensate, $\mu=g_Bn_0$ and the fluctuation matrix is
$$
\begin{pmatrix}
\epsilon_k+g_Bn_0&g_Bn_0\\
-g_Bn_0&-\epsilon_k-g_Bn_0
\end{pmatrix}.
$$
Its eigenvalues are $\pm\sqrt{\epsilon_k(\epsilon_k+2g_Bn_0)}$. The positive branch is the gapless sound mode, continuously crossing over to particle-like excitations at larger $k$. The same result follows by linearizing the time-dependent <Gross–Pitaevskii equation> about its uniform solution. Thus the microscopic condensate amplitude, scattering interaction and <mass> determine the mean-field static coefficients and the collective mode. Quadratic quantum fluctuations also produce <quantum depletion of a Bose condensate>, which a pure coherent-state product omits.
These identifications are controlled for a dilute weakly interacting gas away from a fluctuation-dominated critical region. They are not a universal statement that the exact finite-temperature GL coefficient is always $-\mu$ or that its quartic coefficient never receives corrections. Integrating out noncondensed and short-wavelength modes produces effective temperature-dependent coefficients and can change their near-critical behavior. The phase rigidity described by $K$ and the thermodynamic order-parameter expansion remain the shared organizing structure.
For <BCS theory>, the microscopic <order parameter> is instead an anomalous pair amplitude, commonly normalized as an <energy> gap $\Delta\propto\langle\widehat\psi_\downarrow\widehat\psi_\uparrow\rangle$. It is not an ordinary single-electron field expectation or automatically the square root of a pair number density. Integrating out <fermions> and expanding in small $\Delta$ and slow gradients near $T_c$ gives the <microscopic derivation of Ginzburg-Landau theory>. The <BCS pairing-field effective action> supplies its coefficients through normal-state pair-response functions and thermal <fermion> sums.
For example, a clean isotropic three-dimensional weak-coupling s-wave model, with $N(0)$ the <density of states> per spin at the <Fermi energy> and $\Delta$ measured in <energy> units, gives
$$
\mathcal F_s-\mathcal F_n=\int d^3r
\left[\alpha_\Delta|\Delta|^2+\frac{\beta_\Delta}{2}|\Delta|^4
+K_\Delta\left|\left(\nabla-\frac{iq\mathbf A}{\hbar}\right)\Delta\right|^2
+\frac{|\mathbf B|^2}{2\mu_0}\right],\qquad q=-2e,
$$
with <microscopic Ginzburg-Landau coefficients from BCS theory>
$$
\boxed{\alpha_\Delta\simeq N(0)\frac{T-T_c}{T_c},\qquad
\beta_\Delta=\frac{7\zeta(3)N(0)}{8\pi^2(k_BT_c)^2},\qquad
K_\Delta=\frac{7\zeta(3)N(0)\hbar^2v_F^2}{48\pi^2(k_BT_c)^2}.}
$$
The numerical factors arise from fermionic <Matsubara sums> and the isotropic velocity average; they depend on the stated gap and density-of-states conventions. Rescaling the <order parameter> rescales all three coefficients. The expansion applies near the transition with sufficiently slow spatial variation; disorder, anisotropic pairing and strong coupling alter the coefficients or require additional components. Microscopic derivation also fixes the pair charge in the covariant gradient, producing the superconducting screening current and the <Meissner effect>.<Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Superconductivity by Lev Gor'kov>[1959, near-transition gap expansion and twice-electron charge]{ref}
The neutral Bose and charged Fermi theories therefore share a complex <order parameter> and a phase-stiffness functional, while deriving that field from different microscopic objects. The <Bogoliubov transformation> mixes particle and hole fluctuations of the bosonic condensate in one case and paired fermionic excitations in the other. The static GL limit organizes long-distance behavior; the full microscopic theories retain excitation spectra and <temperature> dependence beyond that local near-transition expansion.
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