Mean Field Solver
Choose a method, enter system parameters, and compute the self-consistent order parameter. Advanced controls improve stability near critical temperature.
Formula Used
This solver uses a standard mean-field self-consistency condition for an Ising-like order parameter:
- m is the dimensionless magnetization (order parameter) in [-1, 1].
- z is the coordination number (nearest neighbors per site).
- J is the coupling energy per neighbor interaction.
- h is an external field energy bias (set h=0 for symmetry).
- T is temperature in Kelvin, and kB is Boltzmann’s constant.
The mean-field critical temperature is T_c = zJ/kB. Below T_c, nonzero solutions can appear when h ≈ 0.
How to Use This Calculator
- Enter the coordination number z for your lattice model.
- Provide coupling J and choose a unit (J, eV, meV, or kJ/mol).
- Set field energy h to study bias or spontaneous order.
- Choose temperature T in Kelvin and select a method.
- Adjust damping and tolerance for stable convergence.
- Click Solve to see results above the form.
- Use Download CSV or Download PDF for exports.
Example Data Table
These sample inputs are typical for demonstrating ordering below and above the critical temperature.
| Case | z | J | h | T (K) | Expected behavior |
|---|---|---|---|---|---|
| Below Tc | 4 | 1.0e-21 J | 0 | 200 | Nonzero m can appear with small m₀. |
| Near Tc | 4 | 1.0e-21 J | 1.0e-24 J | 290 | Slow convergence; damping helps stability. |
| Above Tc | 4 | 1.0e-21 J | 0 | 500 | m approaches zero for symmetric conditions. |
Mean Field Solver Guide
This short reference explains how the solver connects physical inputs to a stable numerical solution. It highlights where mean-field assumptions apply, how the critical point is identified, and which controls improve convergence for parameter sweeps and classroom demonstrations.
1) Mean-field modeling in statistical physics
Mean-field theory replaces fluctuations with an average field, turning many-body interactions into an effective single-site problem. This calculator applies it to an Ising-like order parameter, computing a self-consistent magnetization m from user inputs and stability controls.
2) Self-consistency equation as a fixed-point problem
The solver finds m from m = tanh((zJm + h)/(kB T)). Treat the right side as f(m); a solution is a fixed point where m = f(m). At low temperatures and tiny h, more than one fixed point can coexist.
3) Coordination number and interaction amplification
Coordination z multiplies the interaction term zJm and sets Tc = zJ/kB. Larger z increases the mean field felt at each site, so ordering persists to higher temperatures. Choose z that matches your lattice geometry and neighbor count.
4) Coupling strength and unit choices
J is the interaction energy scale. Inputs can be given in Joules, eV, meV, or kJ/mol and are converted internally to Joules. Pick magnitudes so Tc falls in a sensible range for the materials or model you are emulating.
5) Temperature and critical behavior
Temperature controls beta = 1/(kB T). For h = 0 above Tc the stable solution is m ≈ 0. Near Tc the mapping steepens, so convergence can slow or oscillate. Tc in the results helps you label ordered, critical, and disordered regimes.
6) External field and symmetry breaking
The field term h breaks symmetry and selects the magnetization sign. Near Tc, even small h can produce noticeable m because the susceptibility grows. Set h = 0 to explore spontaneous symmetry breaking and use m0 to target different branches.
7) Numerical strategy, damping, and stability
Fixed-point updates are simple and robust, while Newton updates can converge faster when derivatives behave well. Damping mixes old and new values to prevent overshoot; lower damping is helpful close to Tc. Tight tolerance improves accuracy but may require more iterations.
8) Interpreting outputs for research workflows
Outputs include m, Tc, an estimated dm/dh, and an iteration snapshot showing stability. Use CSV export to log parameter sweeps and PDF export for quick reporting. Common workflows scan T across Tc, compare methods, and record how damping affects convergence.
FAQs
1) What does the solver compute?
It finds a self-consistent magnetization m that satisfies m = tanh((zJm + h)/(kB T)). The result is a dimensionless order parameter between −1 and 1, along with convergence details and a susceptibility estimate.
2) Why can there be multiple solutions when h is zero?
Below the critical temperature, the equation can admit symmetric positive and negative magnetization fixed points. Which one you reach depends on the initial guess and numerical path, reflecting spontaneous symmetry breaking in mean-field theory.
3) When should I use damping?
Use damping when iterations oscillate, diverge, or converge slowly, especially near Tc. Reducing damping mixes updates more gently, improving stability. Typical values between 0.2 and 0.8 work well for most parameter sets.
4) Fixed-point or Newton update: which is better?
Newton often converges in fewer iterations when the derivative is well-behaved. Fixed-point updates are more conservative and may be more reliable near sensitive regions. If Newton behaves erratically, lower damping or switch to fixed-point.
5) What does “susceptibility” mean here?
The solver estimates dm/dh in mean-field form, describing how strongly magnetization changes with field. It can grow large near Tc and may appear undefined if the denominator approaches zero, indicating critical-like amplification.
6) How do I pick an initial guess m0?
For h = 0 near Tc, choose a small nonzero m0 such as 0.01 to encourage symmetry breaking. For biased cases, the sign of m0 can match the sign of h. Multiple runs can probe different branches.
7) Can I use this for parameter sweeps?
Yes. Solve across a grid of temperatures or couplings, then export CSV for each run. For smooth sweeps, reuse the previous solution as the next m0 to improve stability and reduce iterations near critical regions.