Introduction to Kibble–Zurek freeze-out and defect density
A Kibble–Zurek defect-density estimate starts with a simple physical problem: a system is driven through a continuous phase transition too quickly to remain equilibrated at every instant. Near the critical point, relaxation becomes slow while the equilibrium correlation length wants to grow. Eventually the changing control parameter outruns the system’s ability to coordinate itself. The size of the correlated regions at that point sets the likely spacing of domains and, therefore, the scaling of defects left behind.
This calculator performs the power-law bookkeeping for that picture. Enter the correlation-length exponent ν, the dynamical exponent z, a microscopic time τ₀, a quench time τQ, and the spatial dimension d. It reports the freeze-out time t̂, a normalized freeze-out length ξ̂, and a defect-density scaling estimate. Use it to compare ramps consistently, not as a replacement for a measurement or a full dynamical simulation.
What Kibble–Zurek defect-density question does this calculator answer?
The Kibble–Zurek question answered here is how a finite ramp duration changes the characteristic domain scale after a critical crossing. Topological defects can be vortices, kinks, domain walls, or other mismatches in the emerging order, depending on the material and transition. Their detailed type is not supplied by this tool; the result instead gives the broad scaling expected when domains choose their order independently beyond the freeze-out length.
That distinction matters. A density prediction is most meaningful when two scenarios use the same material, geometry, transition, and defect-counting convention. Then changing τQ tells a clean story about quench speed. Changing ν or z is also useful, but only when those exponents belong to the same universality class and dynamical model as the experiment or simulation being compared.
How to use the Kibble–Zurek defect density calculator
Enter positive numerical values for ν, z, τ₀, and τQ, then choose the dimension in which the density is being counted. The exponents are dimensionless. Both time fields must use seconds, so convert milliseconds, microseconds, or other source units before calculating. The spatial dimension d is normally 1, 2, or 3: use 1 for a line-like count, 2 for an areal density, and 3 for a volumetric density.
- Identify the published or modeled ν and z for the transition of interest.
- Enter τ₀, the short microscopic relaxation-time reference, in seconds.
- Enter τQ, the duration or characteristic time of the control-parameter ramp, in seconds.
- Select d for the space in which the resulting defects are counted.
- Choose Estimate Defect Density, then compare the three displayed freeze-out quantities with another run.
A useful first check is to hold everything else fixed and increase τQ. For ordinary positive exponents, the reported ξ̂ should increase and the predicted defect density should decrease. If that direction is surprising, inspect the time units before drawing a physical conclusion.
Inputs for Kibble–Zurek scaling and sensible parameter choices
The inputs describe complementary parts of one scaling model. The correlation-length exponent ν describes how the equilibrium correlation length diverges near criticality. The dynamical exponent z connects the growth of relaxation time to that length scale. Together, the product νz determines how strongly critical slowing down modifies the response to a quench.
τ₀ is the microscopic time scale that anchors the estimate. It is not necessarily the full duration of the experiment; it is the short-time reference appropriate to the model. τQ is the quench time, often related to the inverse slope of a linear ramp near the critical point. A large ratio τQ/τ₀ represents a relatively slow crossing on microscopic time scales. Because this calculator uses that ratio in a power law, both values must be stated in the same unit.
The dimension d controls the conversion from a characteristic separation to a density. For example, if defects are point-like and counted over a two-dimensional sample, d = 2 gives a result that scales as inverse area. It does not automatically determine defect codimension or distinguish line defects from point defects in a three-dimensional medium. If your observable follows a different geometric rule, treat the displayed density as a baseline scaling comparison and adapt the interpretation accordingly.
When values are uncertain, run a low, central, and high scenario rather than treating a single exponent or time scale as exact. This is especially helpful because a modest exponent change can produce a noticeable density shift when τQ/τ₀ spans many orders of magnitude.
Formulas for Kibble–Zurek freeze-out time, length, and density
The calculator first estimates the point at which the evolving system can no longer keep up with the ramp. The following Kibble–Zurek relation is the exact power-law form implemented for the freeze-out time and normalized correlation length:
The defect-density step treats one independent domain as occupying a region set by the freeze-out length. In proportional form, the conversion is:
The length formula shown above is normalized: a full dimensional correlation length would include a microscopic length prefactor ξ₀, which this calculator does not ask for. The results panel presents ξ̂ with metres and ndef with m−d under the conventional unit-length normalization ξ₀ = 1 m. For another microscopic length, multiply ξ̂ by ξ₀ before assigning physical length units, and use the corresponding inverse-length power for density. Nonuniversal prefactors are also omitted, so the strongest conclusion is generally the trend between comparable runs.
Worked example: a two-dimensional Kibble–Zurek quench
Consider a two-dimensional comparison with ν = 0.67, z = 2, τ₀ = 0.001 s, τQ = 1 s, and d = 2. The ratio τQ/τ₀ is 1,000. Using the implemented relations gives a freeze-out time of about 5.24 × 10−2 s, a normalized freeze-out length of about 7.22, and a normalized density of about 1.92 × 10−2 m−2 under the stated ξ₀ = 1 m convention.
Now slow only the quench while keeping the same exponents and microscopic reference. The length grows as a positive power of τQ, so the inverse-area density falls. The effect is not usually a one-for-one change: critical exponents soften the response. This is the practical value of calculating the powers rather than assuming that ten times more quench time always means ten times fewer defects.
Comparing Kibble–Zurek quench scenarios one variable at a time
A clean sensitivity study changes one physical assumption at a time. First compare two τQ values with ν, z, τ₀, and d fixed. Next, if needed, compare exponent sets only when they represent credible alternative models for the same transition. Record the inputs beside each output; otherwise an apparent improvement in density may actually come from a changed geometry or microscopic reference rather than a slower ramp.
For planning, focus on relative changes in t̂, ξ̂, and ndef. A larger t̂ marks a later loss of adiabatic tracking, while a larger ξ̂ indicates more extensive correlated regions. The density result condenses that length into the chosen dimensional counting convention. These quantities are related, but each highlights a different part of the freeze-out story.
How to interpret a Kibble–Zurek defect-density result
Read the result as a scaling estimate, beginning with its direction and units. A lower density means the model predicts fewer independently formed domains in the chosen dimension, not necessarily a guaranteed lower count after every later process. Defects can annihilate, coarsen, pin, or be created by mechanisms that lie outside the idealized crossing. A measured final density can therefore differ from the freeze-out estimate even when the initial Kibble–Zurek trend is correct.
The most defensible use is comparison: hold the system definition fixed, calculate several ramp times, and see how the expected initial spacing changes. If the normalized length needs to become an absolute experimental length, supply the appropriate ξ₀ from the material model before making that conversion.
Limitations and assumptions of this Kibble–Zurek defect-density estimate
This Kibble–Zurek calculator deliberately models a homogeneous, continuous transition with a simple scaling ramp. It does not simulate finite-size boundaries, spatially inhomogeneous fronts, noise, disorder, post-quench coarsening, defect annihilation, or a system-specific prefactor. It also assumes positive critical exponents and a physically meaningful common choice of ν and z.
Use extra care if τQ is not a linear-ramp time near the critical point, if the system crosses multiple transitions, or if the observed defect type has a geometry different from the d-dimensional inverse-volume rule used here. In those cases, the tool can still be useful as a transparent first estimate, but the interpretation should be checked against the relevant theory, numerical model, or experiment. Its best role is to expose the freeze-out assumptions and show which input most strongly influences the predicted scaling.