potentials module
- class potentials.PotClass(dim, pot_id, stiff_eps)
Bases:
objectThis class implements several potential functions and their 1st-order derivatives.
- Parameters
pot_id (int) – index of the potential function.
- V(x)
This function computes the corresponding potential for given x, depending on the value of pot_id.
- grad_V(x)
This function computes the gradient of the corresponding potential for given x, depending on the value of pot_id.
- Returns
1st-order derivatives (gradients) at each point of the input.
- grad_modified_dw1d(xvec, a=1.0, b=1.0, c=0.7)
1st-order derivative of the 1d double-well potential.
- grad_v_1d_quadratic(x)
1st-order derivative of the 1d quadratic potential.
- grad_v_2d_3well(x)
Compute the gradient of potential
v_2d_3well().
- grad_v_2d_curved(x)
Compute the gradient of
v_2d_curved().
- grad_v_2d_dw_quadratic(x)
Compute the gradient of
v_2d_dw_quadratic().
- grad_v_2d_quaratic(x)
Compute the gradient of
v_2d_quadratic().
- grad_v_nd_3well_in_x01(x)
Compute the gradient of
v_nd_3well_in_x01().
- modified_dw1d(xvec, a=1.0, b=1.0, c=0.7)
1d double-well potential, defined as
\[\begin{split}f(x) = \begin{cases} \frac{b\pi^2}{4a^2} (x+a)^2 - cx \,, & x \in (-\infty, -a)\,; \\ \frac{b}{2}(\cos(\frac{x}{a}\pi) + 1.0)- cx\,, & x \in [-a,a)\,; \\ \frac{b\pi^2}{4a^2} (x-a)^2- cx \,, & x \in [a,+\infty)\,. \end{cases}\end{split}\]- Parameters
xvec (1d numpy array) – points for which to compute potential.
a,b,c (positive, double) – parameters used in defining \(f(x)\).
- Returns
potential values at each point of xvec, same length as xvec.
- Return type
1d numpy array.
- output_potential(Param)
- v_1d_dw(x)
1d double well potential. This function simply call
modified_dw1d().pot_id=2.
- v_1d_quadratic(x)
1d quadratic potential \(f(x)=\frac{x^2}{2}.\)
pot_id=1.
- v_2d_3well(x)
This is a 2d potential. It has three wells (low-potential regions) along radius.
pot_id=5.
- v_2d_curved(x)
a 2d potential pot_id=6
- v_2d_dw_quadratic(x)
2d potential, the sum of 1d double-well potential in \(x_1\) and 1d quadratic potential in \(x_2\).
pot_id=4.
- v_2d_quadratic(x)
2d quadratic potential \(f(x) = 0.5 x_1^2 + 2.0 x_2^2\,, \mbox{for}~ x=(x_1,x_2) \in \mathbb{R}^2\,\).
pot_id=3.
- v_nd_3well_in_x01(x)
This is a high-dimensional potential, a sum of
v_2d_3well()in (x[0], x[1]) and quadratic in other dimensions.pot_id=7.