"""A module for the crack model in the isometric ensemble.
This module consist of the class :class:`CrackIsometric` which
contains methods for computing quantities in the
isometric (constant displacement) thermodynamic ensemble.
"""
import numpy as np
import numpy.linalg as la
from .monte_carlo import CrackMonteCarlo
[docs]
class CrackIsometric(CrackMonteCarlo):
"""The crack model class for the isometric ensemble.
"""
def __init__(self, **kwargs):
"""Initializes the :class:`CrackIsometric` class.
Initialize and inherit all attributes and methods
from a :class:`.CrackMonteCarlo` class instance.
"""
CrackMonteCarlo.__init__(self, **kwargs)
rescaled_A = self.H_U_00()/self.kappa
inv_rescaled_A = la.inv(rescaled_A)
self.a_U_00 = np.array([
inv_rescaled_A[0, 0], inv_rescaled_A[0, 1],
inv_rescaled_A[1, 0], inv_rescaled_A[1, 1],
inv_rescaled_A[0, -1], inv_rescaled_A[0, -2],
inv_rescaled_A[1, -1], inv_rescaled_A[1, -2],
inv_rescaled_A[-1, 0], inv_rescaled_A[-1, 1],
inv_rescaled_A[-2, 0], inv_rescaled_A[-2, 1],
inv_rescaled_A[-1, -1], inv_rescaled_A[-1, -2],
inv_rescaled_A[-2, -1], inv_rescaled_A[-2, -2],
])
self.det_H_U_00 = la.det(self.H_U_00())
[docs]
def Q_isometric(self, v, transition_state=False):
r"""The nondimensional isometric partition function
as a function of the nondimensional end separation,
.. math::
Q(v) = \int d\lambda \
Q_0(v,\boldsymbol{\lambda})
e^{-\beta U_1(\boldsymbol{\lambda})},
approximated using the asymptotic relation
.. math::
Q(v) \sim
Q_0(v, \hat{\boldsymbol{\lambda}})
\prod_{j=1}^M \sqrt{\frac{2\pi}{\beta u''(\hat{\lambda}_j)}}
\, e^{-\beta u(\hat{\lambda}_j)},
which is valid for :math:`\varepsilon\gg 1`,
where :math:`\hat{\boldsymbol{\lambda}}`
is from minimizing :math:`\beta U`.
Args:
v (array_like): The nondimensional end separation.
approach (str, optional, default='asymptotic'):
The calculation approach.
transition_state (bool, optional, default=False):
Whether or not to calculate in the transition state.
Returns:
numpy.ndarray: The nondimensional isometric partition function.
Example:
Plot the rescaled nondimensional equilibrium radial distribution
as a function of the nondimensional end separation
in the isometric ensemble
for an increasing nondimensional bond energy
and compare to the limit given by the reference system:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 250)
>>> _ = plt.figure()
>>> for varepsilon in [10, 25, 100, 1000]:
... model = CrackIsometric(varepsilon=varepsilon)
... r_Q = (model.Q_isometric(v)/model.Q_isometric(1)
... )**(model.N**3/3/model.kappa)
... _ = plt.plot(v - 1, (v - 1)**2*r_Q,
... label=r'$\varepsilon=$'+str(varepsilon))
>>> r_Q_0 = (model.Q_0_isometric(v, [1, 1])
... / model.Q_0_isometric(1, [1, 1])
... )**(model.N**3/3/model.kappa)
>>> _ = plt.plot(v - 1, (v - 1)**2*r_Q_0,
... 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(\Delta v)^2$' +
... r'$\left[\frac{Q(v)}{Q(1)}\right]^{N^3/3\kappa}$')
>>> _ = plt.legend()
>>> plt.show()
"""
lambda_hat = self.minimize_beta_U(
v, transition_state=transition_state
)[2][-self.M:]
Q_isometric = self.Q_0_isometric(v, lambda_hat)
Q_isometric *= np.prod(
np.sqrt(
2*np.pi/self.beta_u_pp(lambda_hat[transition_state:])
), axis=0
)
Q_isometric *= np.exp(-np.sum(self.beta_u(lambda_hat), axis=0))
return Q_isometric
[docs]
def beta_A_abs_isometric(self, v, approach='asymptotic',
transition_state=False):
r"""The absolute nondimensional Helmholtz free energy
as a function of the nondimensional end separation,
.. math::
\beta A(v) = -\ln Q(v).
Args:
v (array_like): The nondimensional end separation.
approach (str, optional, default='asymptotic'):
The calculation approach.
transition_state (bool, optional, default=False):
Whether or not to calculate in the transition state.
Returns:
numpy.ndarray: The absolute nondimensional Helmholtz free energy.
"""
if approach == 'asymptotic':
lambda_hat = self.minimize_beta_U(
v, transition_state=transition_state
)[2][-self.M:]
beta_A_abs_isometric = \
self.beta_A_0_abs_isometric(v, lambda_hat)
beta_A_abs_isometric += 0.5*np.sum(
np.log(
self.beta_u_pp(lambda_hat[transition_state:])/2/np.pi
), axis=0
)
beta_A_abs_isometric += np.sum(self.beta_u(lambda_hat), axis=0)
elif approach == 'monte carlo':
beta_A_abs_isometric = np.nan*v
return beta_A_abs_isometric
[docs]
def beta_A_isometric(self, v, approach='asymptotic', **kwargs):
r"""The relative nondimensional Helmholtz free energy
as a function of the nondimensional end separation,
.. math::
\beta\Delta A(v) = \beta A(v) - \beta A(1).
Args:
v (array_like): The nondimensional end separation.
approach (str, optional, default='asymptotic'):
The calculation approach.
**kwargs: Arbitrary keyword arguments.
Passed to :meth:`~.beta_A_isometric_monte_carlo`.
Returns:
numpy.ndarray: The relative nondimensional Helmholtz free energy.
Example:
Plot the rescaled nondimensional relative Helmholtz free energy
as a function of the nondimensional end separation
in the isometric ensemble
for an increasing nondimensional bond energy
and compare to the limit given by the reference system:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for varepsilon in [10, 25, 100, 1000]:
... model = CrackIsometric(varepsilon=varepsilon)
... beta_A = \
... model.beta_A_isometric(v, approach='asymptotic')
... _ = plt.plot(v - 1, model.N**3/3/model.kappa*beta_A,
... label=r'$\varepsilon=$'+str(varepsilon))
>>> beta_A_0 = model.beta_A_0_isometric(v, [1, 1])
>>> _ = plt.plot(v - 1, model.N**3/3/model.kappa*beta_A_0,
... 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)\Delta\beta A$')
>>> _ = plt.legend()
>>> plt.show()
"""
if approach == 'asymptotic':
beta_A_isometric = \
self.beta_A_abs_isometric(v, approach=approach) - \
self.beta_A_abs_isometric(1, approach=approach)
elif approach == 'monte carlo':
beta_A_isometric = self.beta_A_isometric_monte_carlo(v, **kwargs)
return beta_A_isometric
[docs]
def p_isometric(self, v, approach='asymptotic', **kwargs):
r"""The nondimensional end force
as a function of the nondimensional end separation
in the isometric ensemble,
.. math::
p(v) = \frac{\partial}{\partial v}\,\beta A(v).
Args:
v (array_like): The nondimensional end separation.
approach (str, optional, default='asymptotic'):
The calculation approach.
**kwargs: Arbitrary keyword arguments.
Passed to :meth:`~.p_isometric_monte_carlo`.
Returns:
numpy.ndarray: The nondimensional end force.
Example:
Plot the rescaled nondimensional end force
as a function of the nondimensional end separation
in the isometric ensemble
for an increasing nondimensional bond energy
and compare to the limit given by the reference system:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for varepsilon in [10, 25, 100, 1000]:
... model = CrackIsometric(varepsilon=varepsilon)
... p = model.p_isometric(v, approach='asymptotic')
... _ = plt.plot(v - 1, model.N**3/3/model.kappa*p,
... label=r'$\varepsilon=$'+str(varepsilon))
>>> p_0 = model.p_0_isometric(v, [1, 1])
>>> _ = plt.plot(v - 1, model.N**3/3/model.kappa*p_0,
... 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)p$')
>>> _ = plt.legend()
>>> plt.show()
"""
if approach == 'asymptotic':
s_hat = self.minimize_beta_U(v)[2]
lambda_hat = s_hat[-self.M:]
p_isometric = self.p_0_isometric(v, lambda_hat)
elif approach == 'monte carlo':
p_isometric = self.p_isometric_monte_carlo(v, **kwargs)
return p_isometric
[docs]
def k_isometric(self, v, approach='asymptotic', **kwargs):
r"""The nondimensional forward reaction rate coefficient
as a function of the nondimensional end separation
in the isometric ensemble.
Args:
v (array_like): The nondimensional end separation.
approach (str, optional, default='asymptotic'):
The calculation approach.
**kwargs: Arbitrary keyword arguments.
Passed to :meth:`~.k_isometric_monte_carlo`.
Returns:
numpy.ndarray: The nondimensional forward reaction rate.
Example:
Plot the nondimensional forward reaction rate coefficient
as a function of the nondimensional end separation
in the isometric ensemble
for an increasing nondimensional bond energy
and compare to the limit given by the reference system:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for varepsilon in [10, 25, 100, 1000]:
... model = CrackIsometric(varepsilon=varepsilon)
... _ = plt.semilogy(
... v - 1, model.k_isometric(v),
... label=r'$\varepsilon=$'+str(varepsilon))
>>> _ = plt.semilogy(v - 1, model.k_0_isometric(v, [1, 1]),
... 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$k$')
>>> _ = plt.legend()
>>> plt.show()
"""
if approach == 'asymptotic':
k_isometric = (
self.Q_isometric(v, transition_state=True)/self.Q_isometric(v)
) / (
self.Q_isometric(1, transition_state=True)/self.Q_isometric(1)
)
elif approach == 'monte carlo':
k_isometric = self.k_isometric_monte_carlo(v, **kwargs)
return k_isometric
[docs]
def k_rev_isometric(self, v):
r"""The nondimensional reverse reaction rate coefficient
as a function of the nondimensional end separation
in the isometric ensemble.
Args:
v (array_like): The nondimensional end separation.
Returns:
numpy.ndarray: The nondimensional reverse reaction rate.
"""
model_2 = CrackIsometric(
N=self.N + 1, M=self.M - 1, kappa=self.kappa,
alpha=self.alpha, varepsilon=self.varepsilon
)
return (
self.Q_isometric(v, transition_state=True)/model_2.Q_isometric(v)
) / (
self.Q_isometric(1, transition_state=True)/model_2.Q_isometric(1)
)
[docs]
def k_net_isometric(self, v):
r"""The nondimensional net reaction rate coefficient
as a function of the nondimensional end separation
in the isometric ensemble.
Args:
v (array_like): The nondimensional end separation.
Returns:
numpy.ndarray: The nondimensional net reaction rate.
Example:
Plot the nondimensional net reaction rate coefficient
as a function of the nondimensional end separation
in the isometric ensemble
for an increasing system size:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import Crack
>>> Np = np.logspace(-1, np.log(6)/np.log(10), 33)
>>> _ = plt.figure()
>>> for N in [4, 8, 16, 64]:
... model = Crack(N=N)
... v = model.v(Np/N, ensemble='isometric')
... _ = plt.semilogy(
... 3*model.kappa*(v - 1)/N**2,
... model.k_net(v, ensemble='isometric'),
... label=r'$N=$'+str(N))
>>> _ = plt.xlabel(r'$3\kappa\Delta v/N^2$')
>>> _ = plt.ylabel(r'$k^\mathrm{net}/k_\mathrm{ref}$')
>>> _ = plt.legend()
>>> plt.show()
"""
return self.k_isometric(v, approach='asymptotic') \
- self.k_rev_isometric(v)
[docs]
def Q_0_isometric(self, v, lambda_):
r"""The nondimensional isometric partition function
as a function of the nondimensional end separation
for the reference system,
.. math::
Q_0(v,\boldsymbol{\lambda}) =
Q_b(v,\lambda_1,\lambda_2)
e^{-\beta U_{01}(\boldsymbol{\lambda})}.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional isometric partition function.
Example:
Plot the rescaled nondimensional equilibrium radial distribution
as a function of the nondimensional end separation
for the reference system in the isometric ensemble
for an increasing number of broken bonds :math:`N`
and compare to the thermodynamic limit:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 250)
>>> _ = plt.figure()
>>> for N in [5, 10, 25, 100]:
... model = CrackIsometric(N=N)
... r_Q_0 = (model.Q_0_isometric(v, [1, 1])
... / model.Q_0_isometric(1, [1, 1])
... )**(model.N**3/3/model.kappa)
... _ = plt.plot(v - 1, (v - 1)**2*r_Q_0,
... label='$N=$'+str(N))
>>> _ = plt.plot(v - 1, (v - 1)**2*np.exp(-(v - 1)**2/2),
... 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(\Delta v)^2$' +
... r'$\left[\frac{Q_0(v)}{Q_0(1)}\right]^{N^3/3\kappa}$')
>>> _ = plt.legend()
>>> plt.show()
"""
return self.Q_b_isometric(v, lambda_) * \
np.exp(-self.beta_U_01(lambda_))
[docs]
def beta_A_0_abs_isometric(self, v, lambda_):
r"""The absolute nondimensional Helmholtz free energy
as a function of the nondimensional end separation
for the reference system,
.. math::
\beta A_0(v,\boldsymbol{\lambda}) =
-\ln Q_0(v,\boldsymbol{\lambda}) =
\beta A_b(v,\lambda_1,\lambda_2)
+ \beta U_{01}(\boldsymbol{\lambda}).
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The absolute nondimensional Helmholtz free energy.
"""
return self.beta_A_b_isometric_abs(v, lambda_) \
+ self.beta_U_01(lambda_)
[docs]
def beta_A_0_isometric(self, v, lambda_):
r"""The relative nondimensional Helmholtz free energy
as a function of the nondimensional end separation
for the reference system,
.. math::
\beta\Delta A_0(v,\boldsymbol{\lambda}) =
\beta A_0(v,\boldsymbol{\lambda}) -
\beta A_0(v_0,\boldsymbol{\lambda}).
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The relative nondimensional Helmholtz free energy.
Example:
Plot the rescaled nondimensional relative Helmholtz free energy
as a function of the nondimensional end separation
for the reference system in the isometric ensemble
for an increasing number of broken bonds :math:`N`
and compare to the thermodynamic limit:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25, 100]:
... model = CrackIsometric(N=N)
... beta_A_0 = model.beta_A_0_isometric(v, [1, 1])
... _ = plt.plot(v - 1, N**3/3/model.kappa*beta_A_0,
... label='$N=$'+str(N))
>>> _ = plt.plot(v - 1, (v - 1)**2/2, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)\beta\Delta A_0$')
>>> _ = plt.legend()
>>> plt.show()
"""
return self.beta_A_0_abs_isometric(v, lambda_) \
- self.beta_A_0_abs_isometric(1, lambda_)
[docs]
def p_0_isometric(self, v, lambda_):
r"""The nondimensional end force
as a function of the nondimensional end separation
for the reference system in the isometric ensemble,
.. math::
p_0(v,\lambda_1,\lambda_2) =
\frac{\partial}{\partial v}\,\beta A_0(v,\lambda_1,\lambda_2) =
p_b(v,\boldsymbol{\lambda}).
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional end force.
Example:
Plot the rescaled nondimensional end force
as a function of the nondimensional end separation
for the reference system in the isometric ensemble
for an increasing number of broken bonds :math:`N`
and compare to the thermodynamic limit:
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackIsometric
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25, 100]:
... model = CrackIsometric(N=N)
... p_0 = model.p_0_isometric(v, [1, 1])
... _ = plt.plot(v - 1, N**3/3/model.kappa*p_0,
... label='$N=$'+str(N))
>>> _ = plt.plot(v - 1, v - 1, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)p_0$')
>>> _ = plt.legend()
>>> plt.show()
"""
return self.p_b_isometric(v, lambda_)
[docs]
def k_0_isometric(self, v, lambda_):
"""The nondimensional forward reaction rate coefficient
as a function of the nondimensional end separation
for the reference system in the isometric ensemble.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional forward reaction rate.
"""
return self.k_b_isometric(v, lambda_)
[docs]
def Q_b_isometric(self, v, lambda_):
r"""The nondimensional isometric partition function
as a function of the nondimensional end separation
for the isolated bending system,
.. math::
Q_b(v,\lambda_1,\lambda_2) =
\sqrt{\frac{(2\pi)^N}{\det\mathbf{H}}}
\,e^{\frac{1}{2}\mathbf{g}^T\cdot\mathbf{H}^{-1}\cdot\mathbf{g}-f},
where :math:`\mathbf{H}`,
the nondimensional Hessian of the potential energy
for the isolated bending system, and
:math:`\mathbf{g}` and :math:`f` are
.. math::
\mathbf{g}(v,\lambda_1,\lambda_2) = \kappa\left(
2v, -v, 0, \ldots, 0, -\lambda_1, 4\lambda_1 - \lambda_2
\right)^T,
.. math::
f(v,\lambda_1,\lambda_2) = \frac{\kappa}{2}\left[v^2 + \lambda_1^2
+ \left(2\lambda_1 - \lambda_2\right)^2\right]
.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional isometric partition function.
"""
return np.exp(-self.beta_A_b_isometric_abs(v, lambda_))
[docs]
def beta_A_b_isometric_abs(self, v, lambda_):
r"""The absolute nondimensional Helmholtz free energy
as a function of the nondimensional end separation
for the isolated bending system,
.. math::
\beta A_b(v,\lambda_1,\lambda_2) =
-\ln Q_b(v,\lambda_1,\lambda_2).
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The absolute nondimensional Helmholtz free energy.
"""
rescaled_b_inv_A_b = self.np_array(v)**2*(
4*self.a_U_00[0]
- 2*(self.a_U_00[1] + self.a_U_00[2]) + self.a_U_00[3]
) + self.np_array(v)*(
(4*lambda_[0] - lambda_[1])*(
2*(self.a_U_00[4] + self.a_U_00[8])
- (self.a_U_00[6] + self.a_U_00[9])
) + lambda_[0]*(
self.a_U_00[7] + self.a_U_00[11]
- 2*(self.a_U_00[5] + self.a_U_00[10])
)
) + lambda_[0]**2*(
self.a_U_00[15]
) + (4*lambda_[0] - lambda_[1])**2*(
self.a_U_00[12]
) - lambda_[0]*(4*lambda_[0] - lambda_[1])*(
self.a_U_00[13] + self.a_U_00[14]
)
return self.kappa/2*(
self.np_array(v)**2
+ lambda_[0]**2 + (2*lambda_[0] - lambda_[1])**2
- rescaled_b_inv_A_b
) + np.log(self.det_H_U_00) - self.N*np.log(2*np.pi)/2
[docs]
def beta_A_b_isometric(self, v, lambda_):
r"""The relative nondimensional Helmholtz free energy
as a function of the nondimensional end separation
for the isolated bending system,
.. math::
\beta\Delta A_b(v,\lambda_1,\lambda_2) =
\beta A_b(v,\lambda_1,\lambda_2) -
\beta A_b(v_0,\lambda_1,\lambda_2).
In the thermodynamic limit of a large number of broken bonds
:math:`N`, this function has the asymptotic relation
.. math::
\beta\Delta A_b(v,\lambda_1,\lambda_2) \sim
\frac{3\kappa}{2N^3}(v - v_0)^2 \quad\text{for }N\gg 1.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The relative nondimensional Helmholtz free energy.
"""
return self.beta_A_b_isometric_abs(v, lambda_) \
- self.beta_A_b_isometric_abs(1, lambda_)
[docs]
def p_b_isometric(self, v, lambda_):
r"""The nondimensional end force
as a function of the nondimensional end separation
for the isolated bending system in the isometric ensemble,
.. math::
p_b(v,\lambda_1,\lambda_2) =
\frac{\partial}{\partial v}\,\beta A_b(v,\lambda_1,\lambda_2).
In the thermodynamic limit of a large number of broken bonds
:math:`N`, this function has the asymptotic relation
.. math::
p_b(v,\lambda_1,\lambda_2) \sim
\frac{3\kappa}{N^3}(v - v_0) \quad\text{for }N\gg 1.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional end force.
"""
rescaled_dbdv_inv_A_b = self.np_array(v)*(
4*self.a_U_00[0]
- 2*(self.a_U_00[1] + self.a_U_00[2])
+ self.a_U_00[3]
) + lambda_[0]*(self.a_U_00[7] - 2*self.a_U_00[5]) \
+ (4*lambda_[0] - lambda_[1])*(2*self.a_U_00[4] - self.a_U_00[6])
return self.kappa*(v - rescaled_dbdv_inv_A_b)
[docs]
def k_b_isometric(self, v, lambda_):
"""The nondimensional forward reaction rate coefficient
as a function of the nondimensional end separation
for the isolated bending system in the isometric ensemble.
Args:
v (array_like): The nondimensional end separation.
lambda_ (array_like): The intact bond stretches.
Returns:
numpy.ndarray: The nondimensional forward reaction rate.
"""
return (
self.Q_b_isometric(v, [self.lambda_TS, lambda_[1]]) /
self.Q_b_isometric(v, lambda_)
) / (
self.Q_b_isometric(1, [self.lambda_TS, lambda_[1]]) /
self.Q_b_isometric(1, lambda_)
)