Ginzburg–Landau Solver Calculator

This tool solves the stationary one-dimensional Ginzburg–Landau equation for a real order parameter ψ(x):

A finite-difference grid approximates the second derivative, and an under/over-relaxed fixed-point iteration updates ψ until the maximum change drops below the chosen tolerance.

• Bulk amplitude (when α<0): ψ₀ = √(−α/β)
• Condensation energy magnitude: |Δf| = α²/(2|β|)
• Thermodynamic critical field (approx.): Hc = |α|/√(|β| μ₀)
• GL parameter: κ = λ/ξ (when λ is provided)

1. Select α mode: temperature model or direct α.
2. Enter β, ξ, λ (optional), L, and N for the spatial grid.
3. Set boundary values ψ(0) and ψ(L) to reflect your setup.
4. Adjust ω and tolerance if convergence is slow or unstable.
5. Press Solve; results and the first profile points appear above.
6. Use Download CSV/PDF to export the full solution and summary.

Article

1) What the solver computes

This calculator computes a stationary one-dimensional Ginzburg–Landau order-parameter profile ψ(x) on [0,L]. A finite-difference grid and relaxed fixed-point updates satisfy your Dirichlet boundaries. Results include the full profile, convergence metrics, and derived diagnostics for quick reporting and downstream plotting.

2) Physical meaning of α and β

α determines whether ψ tends toward zero (α>0) or a finite amplitude (α<0). β sets nonlinear saturation and stabilizes the solution. Using α(T)=α₀(T/Tc−1) lets you scan temperature near Tc and observe the transition in a controlled way. When α is near zero, the interior amplitude is small and can converge more slowly.

3) Role of coherence length ξ

ξ weights the gradient term and governs how quickly ψ can change. Larger ξ enforces smoother variations and thicker boundary layers; smaller ξ allows sharper features. Numerically, extreme ξ relative to Δx can stiffen the update, so refine N or lower ω when needed.

4) Boundary conditions and interfaces

Dirichlet boundaries model contacts, pair-breaking surfaces, or imposed proximity strength. ψ=0 at both ends often yields interior recovery toward ψ₀ when α<0, while nonzero boundaries can seed superconductivity. Choose boundaries to match your setup, then interpret the interior profile rather than the endpoints alone.

5) Numerical stability and relaxation ω

Relaxation ω controls stability. Values near 1 are conservative; larger values accelerate but can oscillate. If “Converged” is No, reduce ω, increase max iterations, and check that Δx is not too coarse. The max-update value should fall well below your tolerance.

6) Free-energy interpretation and checks

The discrete free energy approximates ∫[ξ²(∂ψ/∂x)² + αψ² + (β/2)ψ⁴]dx. Absolute values depend on scaling, but trends are useful: stable superconducting solutions typically lower the functional when α<0. Comparing energies between parameter sweeps helps flag overconstrained boundaries or unstable parameter choices. Inspect smoothness and boundedness as basic sanity checks.

7) Using κ and critical fields

Providing λ enables κ=λ/ξ, an important GL classifier in standard theory. The reported Hc≈|α|/√(|β|μ₀) is a quick diagnostic within your chosen normalization, not a full electrodynamic prediction. For field penetration, vortices, currents, or phase winding, extended coupled equations are required.

8) Practical workflow for experiments

A practical workflow is: pick geometry (L) and material scales (ξ, λ), set boundaries, then sweep T or α. Increase N until the profile and key outputs stop changing appreciably. Export CSV for plotting and keep the PDF report with parameters, convergence metrics, and date for reproducibility.

FAQs

1) Why does the solution stay near zero even when α<0?

If both boundaries force ψ=0 and the domain is short compared with ξ, the gradient penalty can keep ψ suppressed. Increase L, decrease ξ, or seed a small nonzero boundary value to test interior recovery.

2) What does “Converged: No” mean here?

The maximum pointwise update did not drop below your tolerance within the iteration limit. Reduce ω, increase max iterations, or adjust N so Δx is not too large relative to ξ.

3) How should I choose N for accurate results?

Increase N until key outputs like max ψ, free energy, and the overall profile change negligibly. A practical check is doubling N and confirming differences fall within your acceptable error band.

4) Can I use this for complex ψ or magnetic fields?

This implementation solves a real ψ in 1D without electromagnetic vector potential. For vortices, phase winding, or field penetration, you need the complex GL equation coupled to Maxwell terms.

5) Why is β required to be nonzero?

With β=0, the model loses nonlinear saturation and can become ill-posed for superconducting states. A nonzero β stabilizes finite ψ and enables the bulk amplitude relation ψ₀=√(−α/β) when α<0.

6) What units should I use for α and β?

Units depend on your chosen normalization of ψ and free energy. Keep α, β, and ψ consistent so the equation is dimensionally coherent. Use the tool for relative studies unless you have a calibrated convention.

7) How do I interpret the discrete free energy value?

It is a numerical approximation to the GL functional on your grid. Compare energies between runs with the same normalization to rank configurations or parameter sweeps; absolute magnitudes may not be meaningful across different scalings.