---
tag: 713MRSZ9XA
kind: concept
name: Ginzburg–Landau equation
msc: [35Q56, 82D55]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Ginzburg–Landau_theory
---

## Summary

The Ginzburg–Landau equations describe a superconductor through a complex order parameter $\psi$ and a magnetic vector potential $A$.
They are the Euler–Lagrange equations of the Ginzburg–Landau energy, which penalises variations of $\psi$, deviations of $\abs{\psi}$ from $1$, and magnetic energy.
The quantity $\abs{\psi}^2$ measures the density of superconducting electrons.
Their solutions include vortices, points or lines around which $\psi$ winds and where $\abs{\psi}$ vanishes.

## Explanations

### L1 human

Below a critical temperature some metals conduct electricity with no resistance and push out magnetic fields.
Ginzburg and Landau described this state by a complex number $\psi$ at each point.
Where $\abs{\psi}$ is near $1$ the metal is superconducting, and where it is $0$ the metal is normal.
The energy prefers $\abs{\psi} = 1$, prefers $\psi$ to vary slowly, and charges for magnetic field.
The two simplest solutions are the fully superconducting state $\psi = 1$ with no field and the normal state $\psi = 0$ with a uniform field.
In between, a magnetic field can enter through thin vortex tubes.