# 4.3.5. pde.pdes.kpz_interface module

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

class KPZInterfacePDE(nu=0.5, lmbda=1, *, noise=0, bc='auto_periodic_neumann')[source]

Bases: PDEBase

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 strenth of the gradient term

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

• bc (Dict[str, Dict | str | BCBase] | Dict | str | BCBase | Tuple[Dict | str | BCBase, Dict | str | BCBase] | BoundaryAxisBase | Sequence[Dict[str, Dict | str | BCBase] | Dict | str | BCBase | Tuple[Dict | str | BCBase, Dict | str | BCBase] | BoundaryAxisBase]) – The boundary conditions applied to the field. Boundary conditions are generally given as a list with one condition for each axis. 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 a tuple of two conditions). 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 ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

diagnostics: Dict[str, Any]

Diagnostic information (available after the PDE has been solved)

Type:

dict

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