= Ginzburg-Landau functional of a dilute Bose gas
{c}
{title2=$K=\hbar^2/(2m_B),\quad\alpha=V_{\rm ext}-\mu,\quad\beta=4\pi\hbar^2a_s/m_B$}
Take $\Psi=\langle\widehat\psi\rangle$ normalized so that $|\Psi|^2$ is condensate density. The coherent-field expectation of a dilute contact-interacting Bose Hamiltonian gives the grand-potential functional $\int[\hbar^2|\nabla\Psi|^2/(2m_B)+(V_{\rm ext}-\mu)|\Psi|^2+g_B|\Psi|^4/2]d^3r$. Its leading microscopic coefficients are therefore those displayed, with positive <s-wave scattering length> $a_s$ for repulsion. Varying with respect to $\Psi^*$ gives the stationary <Gross–Pitaevskii equation>. Thermal and quantum fluctuations can renormalize the coefficients; these mean-field identities are not exact universal critical coefficients.
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