4.4.6 pde.pdes.kpz_interface module

The Kardar–Parisi–Zhang (KPZ) equation describing the evolution of an interface.

class KPZInterfacePDE(nu=0.5, lmbda=1, *, bc=None, noise=0, rng=None)[source]

Bases: SDEBase

The Kardar–Parisi–Zhang (KPZ) equation.

The mathematical definition is

\[\partial_t h = \nu \nabla^2 h + \frac{\lambda}{2} \left(\nabla h\right)^2 + \eta(\boldsymbol r, t)\]

where \(h\) is the height of the interface in Monge parameterization. The dynamics are governed by the two parameters \(\nu\) and \(\lambda\), while \(\eta\) is Gaussian white noise, whose strength is controlled by the noise argument.

Parameters:

  • nu (float) – Parameter \(\nu\) for the strength of the diffusive term

  • lmbda (float) – Parameter \(\lambda\) for the strength of the gradient term

  • bc (BoundariesData | None) – The boundary conditions applied to the field. Boundary conditions are generally given as a dictionary with one condition for each axis side. For periodic axes, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using specific identifiers, like x- and y+). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘auto_periodic_neumann’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

  • noise (float) – Variance of the (additive) noise term

  • rng (Generator) – Random number generator (default: default_rng()) used for stochastic simulations. Note that this random number generator is only used for numpy functions, while compiled numba code uses the random number generator of numba. Moreover, in simulations using multiprocessing, setting the same generator in all processes might yield unintended correlations in the simulation results.

default_bc = 'auto_periodic_neumann'

Default boundary condition used when no specific conditions are chosen.

evolution_rate(state, t=0)[source]

Evaluate the right hand side of the PDE.

Parameters:

  • state (ScalarField) – The scalar field describing the concentration distribution

  • t (float) – The current time point

Returns:

Scalar field describing the evolution rate of the PDE

Return type:

ScalarField

explicit_time_dependence: bool | None = False

Flag indicating whether the right hand side of the PDE has an explicit time dependence.

Type:

bool

property expression: str

the expression of the right hand side of this PDE

Type:

str

make_evolution_rate(state, backend)[source]

Create a compiled function evaluating the right hand side of the PDE.

Parameters:

  • state (ScalarField) – An example for the state defining the grid and data types

  • backend (str or BackendBase) – The backend used for numerical operations

Returns:

A function with signature (state_data, t), which can be called with an instance of the state data and time to obtain the associated evolution rate.

Return type:

Callable[[TNativeArray, float], TNativeArray]