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384 | def space(
box: Tensor | None = None,
*,
normalized: bool = True,
parallelepiped: bool = True,
remapped: bool = True,
) -> (Callable, Callable):
r"""Define a simulation space.
This function is fundamental in constructing simulation spaces derived from
subsets of $\mathbb{R}^{D}$ (where $D = 1$, $2$, or $3$) and is
instrumental in setting up simulation environments with specific
characteristics (e.g., periodic boundary conditions). The function returns
a displacement function and a shift function to compute particle
interactions and movements in space.
This function supports deformation of the simulation cell, crucial for
certain types of simulations, such as those involving finite deformations
or the computation of elastic constants.
Parameters
----------
box : Tensor | None, default=None
Interpretation varies based on the value of `parallelepiped`. If
`parallelepiped` is `True`, must be an affine transformation, $T$,
specified in one of three ways:
1. a cube, $L$;
2. an orthorhombic unit cell, $[L_{x}, L_{y}, L_{z}]$; or
3. a triclinic cell, upper triangular matrix.
If `parallelepiped` is `False`, must be the edge lengths.
If `box` is `None`, the simulation space has free boundary conditions.
normalized : bool, default=True
If `normalized` is `True`, positions are stored in the unit cube.
Displacements and shifts are computed in a normalized simulation space
and can be transformed back to real simulation space using the affine
transformation. If `normalized` is `False`, positions are expressed and
computations performed directly in the real simulation space.
parallelepiped : bool, default=True
If `True`, the simulation space is defined as a ${1, 2, 3}$-dimensional
parallelepiped with periodic boundary conditions. If `False`, the space
is defined on a ${1, 2, 3}$-dimensional hypercube.
remapped : bool, default=True
If `True`, positions and displacements are remapped to stay in the
bounds of the defined simulation space. A rempapped simulation space is
topologically equivalent to a torus, ensuring that particles exiting
one boundary re-enter from the opposite side.
Returns
-------
(Callable[[Tensor, Tensor], Tensor], Callable[[Tensor, Tensor], Tensor])
A pair of functions:
1. The displacement function, $\vec{d}$, measures the
difference between two points in the simulation space, factoring in
the geometry and boundary conditions. This function is used to
calculate particle interactions and dynamics.
2. The shift function, $u$, applies a displacement vector to a point
in the space, effectively moving it. This function is used to
update simulated particle positions.
Examples
--------
transformation = torch.tensor([10.0])
displacement_fn, shift_fn = space(
transformation,
normalized=False,
)
normalized_displacement_fn, normalized_shift_fn = space(
transformation,
normalized=True,
)
normalized_position = torch.rand([4, 3])
position = transformation * normalized_position
displacement = torch.randn([4, 3])
torch.testing.assert_close(
displacement_fn(position[0], position[1]),
normalized_displacement_fn(
normalized_position[0],
normalized_position[1],
),
)
"""
if isinstance(box, (int, float)):
box = torch.tensor([box])
if box is None:
def displacement_fn(
input: Tensor,
other: Tensor,
*,
perturbation: Tensor | None = None,
**_,
) -> Tensor:
if len(input.shape) != 1:
raise ValueError
if input.shape != other.shape:
raise ValueError
if perturbation is not None:
transform = input - other
match transform.ndim:
case 0:
return perturbation * transform
case 1:
return torch.einsum(
"i,...i->...i",
transform,
perturbation,
)
case 2:
return torch.einsum(
"ij,...j->...i",
transform,
perturbation,
)
case _:
raise ValueError
return input - other
def shift_fn(input: Tensor, other: Tensor, **_) -> Tensor:
return input + other
return displacement_fn, shift_fn
if parallelepiped:
inverted_transform = beignet.invert_transform(box)
if normalized:
def displacement_fn(
input: Tensor,
other: Tensor,
*,
perturbation: Tensor | None = None,
**kwargs,
) -> Tensor:
_transform = box
_inverted_transform = inverted_transform
if "transform" in kwargs:
_transform = kwargs["transform"]
if "updated_transform" in kwargs:
_transform = kwargs["updated_transform"]
if len(input.shape) != 1:
raise ValueError
if input.shape != other.shape:
raise ValueError
displacement = beignet.apply_transform(
torch.remainder(input - other + 1.0 * 0.5, 1.0) - 1.0 * 0.5,
_transform,
)
if perturbation is not None:
match displacement.ndim:
case 0:
return perturbation * displacement
case 1:
return torch.einsum(
"i,...i->...i",
displacement,
perturbation,
)
case 2:
return torch.einsum(
"ij,...j->...i",
displacement,
perturbation,
)
case _:
raise ValueError
return displacement
if remapped:
def u(input: Tensor, other: Tensor) -> Tensor:
return torch.remainder(input + other, 1.0)
def shift_fn(input: Tensor, other: Tensor, **kwargs) -> Tensor:
_transform = box
_inverted_transform = inverted_transform
if "transform" in kwargs:
_transform = kwargs["transform"]
_inverted_transform = beignet.invert_transform(_transform)
if "updated_transform" in kwargs:
_transform = kwargs["updated_transform"]
return u(input, beignet.apply_transform(other, _inverted_transform))
return displacement_fn, shift_fn
def shift_fn(input: Tensor, other: Tensor, **kwargs) -> Tensor:
_transform = box
_inverted_transform = inverted_transform
if "transform" in kwargs:
_transform = kwargs["transform"]
_inverted_transform = beignet.invert_transform(_transform)
if "updated_transform" in kwargs:
_transform = kwargs["updated_transform"]
return input + beignet.apply_transform(other, _inverted_transform)
return displacement_fn, shift_fn
def displacement_fn(
input: Tensor,
other: Tensor,
*,
perturbation: Tensor | None = None,
**kwargs,
) -> Tensor:
_transform = box
_inverted_transform = inverted_transform
if "transform" in kwargs:
_transform = kwargs["transform"]
_inverted_transform = beignet.invert_transform(_transform)
if "updated_transform" in kwargs:
_transform = kwargs["updated_transform"]
input = beignet.apply_transform(input, _inverted_transform)
other = beignet.apply_transform(other, _inverted_transform)
if len(input.shape) != 1:
raise ValueError
if input.shape != other.shape:
raise ValueError
displacement = beignet.apply_transform(
torch.remainder(input - other + 1.0 * 0.5, 1.0) - 1.0 * 0.5,
_transform,
)
if perturbation is not None:
match displacement.ndim:
case 0:
return perturbation * displacement
case 1:
return torch.einsum(
"i,...i->...i",
displacement,
perturbation,
)
case 2:
return torch.einsum(
"ij,...j->...i",
displacement,
perturbation,
)
case _:
raise ValueError
return displacement
if remapped:
def u(input: Tensor, other: Tensor) -> Tensor:
return torch.remainder(input + other, 1.0)
def shift_fn(input: Tensor, other: Tensor, **kwargs) -> Tensor:
_transform = box
_inverted_transform = inverted_transform
if "transform" in kwargs:
_transform = kwargs["transform"]
_inverted_transform = beignet.invert_transform(
_transform,
)
if "updated_transform" in kwargs:
_transform = kwargs["updated_transform"]
return beignet.apply_transform(
u(
beignet.apply_transform(_inverted_transform, input),
beignet.apply_transform(_inverted_transform, other),
),
_transform,
)
return displacement_fn, shift_fn
def shift_fn(input: Tensor, other: Tensor, **_) -> Tensor:
return input + other
return displacement_fn, shift_fn
def displacement_fn(
input: Tensor,
other: Tensor,
*,
perturbation: Tensor | None = None,
**_,
) -> Tensor:
if len(input.shape) != 1:
raise ValueError
if input.shape != other.shape:
raise ValueError
displacement = torch.remainder(
input - other + box * 0.5,
box,
)
if perturbation is not None:
transform = displacement - box * 0.5
match transform.ndim:
case 0:
return perturbation * transform
case 1:
return torch.einsum(
"i,...i->...i",
transform,
perturbation,
)
case 2:
return torch.einsum(
"ij,...j->...i",
transform,
perturbation,
)
case _:
raise ValueError
return displacement - box * 0.5
if remapped:
def shift_fn(input: Tensor, other: Tensor, **_) -> Tensor:
return torch.remainder(input + other, box)
else:
def shift_fn(input: Tensor, other: Tensor, **_) -> Tensor:
return input + other
return displacement_fn, shift_fn
|