Kernels

`lazy_xy_matrix`(x, y, L[, boundary_conditions])	XY matrix as LazyTensors.
`lazy_interaction_kernel`(x, y, Rx, L, ...[, ...])	Interaction kernel as LazyTensor.
`lazy_overlapping_kernel`(x, y, Rx, Ry, L, ...)	Overlapping kernel as LazyTensor.
`lazy_morse`(x, y, Ca, la, Cr, lr[, p, mx, my])	LazyTensor of the forces derived from the Morse potential.
`lazy_quadratic`(x, y, R, L[, boundary_conditions])	LazyTensor of the forces derived from of a quadratic potential.

sisyphe.kernels.squared_distances_tensor(x, y, L, boundary_conditions='periodic')

Squared distances matrix as torch Tensors.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) Tensor) – Columns.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".

Returns

The tensor whose \((i,j)\) coordinate is the squared distance between x[i,:] and y[j,:].

Return type

(M,N) Tensor

sisyphe.kernels.lazy_xy_matrix(x, y, L, boundary_conditions='periodic')

XY matrix as LazyTensors.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) – Columns.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".

Returns

The LazyTensor A such that for each coordinate \((i,j)\), A[i,j,:] is the LazyTensor LazyTensor(y[j,:]-x[i,:]).

Return type

(M,N,d) LazyTensor

sisyphe.kernels.squared_distances(x, y, L, boundary_conditions='periodic')

Squared distances LazyTensor.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) – Columns.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".

Returns

The LazyTensor whose \((i,j)\) coordinate is the squared distance between x[i,:] and y[j,:].

Return type

(M,N) LazyTensor

sisyphe.kernels.sqdist_angles(x, y, x_orientation, L, boundary_conditions='periodic')

Squared distances and angles LazyTensors.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) – Columns.
x_orientation ((M,d) Tensor) – Orientation of x.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".

Returns

The LazyTensor whose \((i,j)\) coordinate is the squared distance between x[i,:] and y[j,:] and the LazyTensor whose \((i,j)\) coordinate is the cosine of the angle between y[j,:]-x[i,:] and x_orientation[i,:].

Return type

(M,N) LazyTensor, (M,N) LazyTensor

sisyphe.kernels.lazy_interaction_kernel(x, y, Rx, L, boundary_conditions, vision_angle=6.283185307179586, axis=None, **kwargs)

Interaction kernel as LazyTensor.

Parameters

x ((M,d) Tensor) – Row.
y ((N,d) Tensor) – Columns.
Rx ((N,) Tensor or float) – Interaction radius of x.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".
vision_angle (float, optional) – Angle of vision of x. Default is \(2\pi\).
axis ((M,d) Tensor, optional) – Orientation of x. Default is None. Must be specified if vision_angle is not \(2\pi\).
**kwargs – Arbitrary keywords arguments.

Returns

The LazyTensor whose \((i,j)\) coordinate is 1 if y[j,:] is in the cone of vision of x[i,:] and 0 otherwise.

Return type

(M,N) LazyTensor

sisyphe.kernels.lazy_overlapping_kernel(x, y, Rx, Ry, L, boundary_conditions, **kwargs)

Overlapping kernel as LazyTensor.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) Tensor) – Columns.
Rx ((M,) Tensor or float) – Radius of x.
Ry ((N,) Tenosr or float) – Radius of y.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".
**kwargs – Arbitrary keywords arguments.

Returns

The LazyTensor whose \((i,j)\) coordinate is 1 if x[i,:] and y[j,:] are at a distance smaller than Rx[i]+Ry[j] (or smaller than Rx+Ry if all the x and y particles have the same radius).

Return type

(M,N) LazyTensor

Raises

NotImplementerError – if Rx and Ry do not have the same type, or when Rx and Ry are float but Rx is not equal to Ry.

sisyphe.kernels.lazy_morse(x, y, Ca, la, Cr, lr, p=2, mx=1.0, my=1.0, **kwargs)

LazyTensor of the forces derived from the Morse potential.

The Morse potential is defined by

\[U(z) = -C_a \exp(-|z|^p / \ell_a^p) + C_r \exp(-|z|^p / \ell_r^p)\]

The force exerted by a particle located in \(y_j\) on a particle located on \(x_i\) is

\[F = - m_x m_y \nabla U(x_i - y_j)\]

where \(m_x\) and \(m_y\) are the masses of the particles in \(x_i\) and \(y_j\) respectively.

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) Tensor) – Columns.
Ca (float) – Attraction coefficient.
la (float) – Attraction length.
Cr (float) – Repulsion coefficient.
lr (float) – Repulsion length.
p (int, optional) – Exponent. Default is 2.
mx ((M,) Tensor or float, optional) – Mass of x. Default is 1.
my ((N,) Tensor or float, optional) – Mass of y. Default is 1.
**kwargs – Arbitrary keywords arguments.

Returns

The LazyTensor whose \((i,j,:)\) coordinate is the force exerted by y[j,:] on x[i,:] derived from the Morse potential.

Return type

(M,N,d) LazyTensor

sisyphe.kernels.lazy_quadratic(x, y, R, L, boundary_conditions='periodic', **kwargs)

LazyTensor of the forces derived from of a quadratic potential.

The quadratic potential is defined by:

\[U(z) = \frac{1}{2R} |z|^2 - |z|\]

where \(R\) is a fixed parameter. The force exerted on a particle located on \(y_j\) on a particle located on \(x_i\) is

\[F = \left(\frac{1}{R}-\frac{1}{|y_j-x_i|}\right)(y_j-x_i).\]

Parameters

x ((M,d) Tensor) – Rows.
y ((N,d) Tensor) – Columns.
R ((M,) Tensor or float) – Radius of the x particles.
L ((d,) Tensor) – Box size.
boundary_conditions (list or str, optional) –
Boundary conditions. Can be one of the following:
- list of size \(d\) containing 0 (periodic) or 1 (wall with reflecting boundary conditions) for each dimension.
- "open" : no boundary conditions.
- "periodic" : periodic boundary conditions.
- "spherical" : reflecting boundary conditions on the sphere of radius L[0]/2 and center L/2.
Default is "periodic".
**kwargs – Arbitrary keywords arguments.

Returns

The LazyTensor whose \((i,j,:)\) coordinate is the force exerted by y[j,:] on x[i,:] derived from the quadratic potential.

Return type

(M,N,d) LazyTensor