WallGo.freeEnergy

Class that does phase tracing, computes the effective potential in the minimum and interpolate it.

Classes

FreeEnergy(effectivePotential, ...[, ...])	Describes properties of a local effective potential minimum.
FreeEnergyValueType(veffValue, fieldsAtMinimum)	Data class containing the field value that minimizes the potential and the value of the potential in the minimum.

class FreeEnergy(effectivePotential, startingTemperature, startingPhaseLocationGuess, initialInterpolationPointCount=1000)

Describes properties of a local effective potential minimum. This is used to keep track of a minimum with respect to the temperature. By definition, the free energy density of a phase equals the value of the effective potential in its local minimum.

Initialize a FreeEnergy object

Parameters:

• effectivePotential (EffectivePotential) – EffectivePotential object used to compute the free energy.

• startingTemperature (float) – Temperature at which the interpolation of the effective potential starts.

• startingPhaseLocationGuess (Fields) – Approximate position of the phase at startingTemperature.

• initialInterpolationPointCount (int, optional) – Initial number of points sampled for the interpolation. The default is 1000.

constructInterpolationFromArray(freeEnergyArrays, dT)

Constructs the interpolation table directly from arrays of temperatures, field values, and potential values, bypassing the phase tracing process.

Parameters:

• freeEnergyArrays (FreeEnergyArrays) – Object containing arrays of the temperature, minimum and value of the free energy.

• dT (float) – Small step in temperature used in derivatives, used here to ensure endpoints of temperature range not exceeded when taking derivatives later.

Return type:

None

derivative(x, order=1, bUseInterpolation=True)

Override of InterpolatableFunction.derivative() function. Specifies accuracy based on our internal variables and puts the results in FreeEnergyValueType format. Otherwise similar to the parent function.

Parameters:

• x (list[float] or np.ndarray) – Points where the free energy will be evaluated.

• bUseInterpolatedValues (bool, optional) – Whether or not to use interpolation to evaluate the function. The default is True.

• order (int)

• bUseInterpolation (bool)

Returns:

FreeEnergyValueType object containing the value of the free energy’s derivative and the field value that minimizes the potential.

Return type:

FreeEnergyValueType

evaluate(x, bUseInterpolatedValues=True)

Evaluate the free energy.

Parameters:

• x (list[float] or np.ndarray) – Points where the free energy will be evaluated.

• bUseInterpolatedValues (bool, optional) – Whether or not to use interpolation to evaluate the function. The default is True.

Returns:

FreeEnergyValueType object containing the value of the free energy and the field value that minimizes the potential.

Return type:

FreeEnergyValueType

tracePhase(TMin, TMax, dT, rTol=1e-06, spinodal=True, paranoid=True, phaseTracerFirstStep=None, interpolationDegree=1)

Traces minimum of potential

Finds the potential minimum for the range over which it exists. Takes a temperature derivative of the minimsation condition, and solves for \(\phi_i^\text{min}(T)\) as an initial value problem

\[\frac{\partial^2 V^\text{eff}}{\partial \phi_i \partial \phi_j}\bigg|_{\phi=\phi^\text{min}} \frac{\partial \phi^\text{min}_j}{\partial T} + \frac{\partial^2 V^\text{eff}}{\partial \phi_i \partial T}\bigg|_{\phi=\phi^\text{min}} = 0,\]

starting from a solution at the starting temperature. It uses scipy.integrate.solve_ivp() to solve the problem. Stops if a mass squared goes through zero.

Parameters:

• TMin (float) – Minimal temperature at which the phase tracing will be done.

• TMax (float) – Maximal temperature at which the phase tracing will be done.

• dT (float) – Maximal temperature step size used by the phase tracer.

• rTol (float, optional) – Relative tolerance of the phase tracing. The default is 1e-6.

• spinodal (bool, optional) – If True, stop tracing if a mass squared turns negative. The default is True.

• paranoid (bool, optional) – If True, re-solve minimum after every step. The default is True.

• phaseTracerFirstStep (float or None, optional) – If a float, this gives the starting step size in units of the maximum step size dT. If None then uses the initial step size algorithm of scipy.integrate.solve_ivp. Default is None

• interpolationDegree (int, optional) – Degree of the splines used in FreeEnergy to interpolate the potential and its derivatives. Default is 1.

Return type:

None

class FreeEnergyValueType(veffValue, fieldsAtMinimum)

Data class containing the field value that minimizes the potential and the value of the potential in the minimum.

Parameters:

• veffValue (ndarray)

• fieldsAtMinimum (Fields)

static fromArray(arr)

ASSUMES that the last column is Veff value.

Parameters:

arr (ndarray)

Return type:

FreeEnergyValueType