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yay the analytic calculation works now

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Valentin Boettcher 2021-11-24 19:16:24 +01:00
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commit e9c75c3f27
22 changed files with 11396 additions and 1526 deletions
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178
python/energy_flow_proper/03_gaussian/analytic_calculation.org
View file

@ -33,14 +33,14 @@ First we need an exponential fit for our BCF.
#+begin_src jupyter-python
W, G_raw = utilities.fit_α(α, 3, 80, 10_000)
L, P = utilities.fit_α(α_0_dot, 5, 80, 10_000)
τ = np.linspace(0, t_max, 1000)
#+end_src
#+RESULTS:
Looks quite good.
#+begin_src jupyter-python
τ = np.linspace(0, t_max, 1000)
with utilities.hiro_style():
fig, ax = utilities.plot_complex(τ, α(τ), label="exact")
utilities.plot_complex(
τ, utilities.α_apprx(τ, G_raw, W), ax=ax, label="fit", linestyle="--"
@ -48,15 +48,20 @@ Looks quite good.
#+end_src
#+RESULTS:
:RESULTS:
| hline | <AxesSubplot:> |
[[file:./.ob-jupyter/327a32a75bc680a958a163026e6eee7921b2fa9d.svg]]
:END:
[[file:./.ob-jupyter/9c454eedaf97fa770850f6d4948856df8948b8df.svg]]
:RESULTS:
| hline | <AxesSubplot:> |
[[file:./.ob-jupyter/92e34b6c19b3044d4e45ef61b7b24e6a7dc9309f.svg]]
:END:
#+begin_src jupyter-python
τ = np.linspace(0, -t_max, 1000)
with utilities.hiro_style():
fig, ax = utilities.plot_complex(τ, α(τ), label="exact")
utilities.plot_complex(
τ, utilities.α_apprx(-τ, G_raw.conj(), W.conj()), ax=ax, label="fit", linestyle="--"
)
#+end_src
#+RESULTS:
[[file:./.ob-jupyter/37cc788143d55668dd7087df2e59b1e330223895.svg]]
#+begin_src jupyter-python
fig, ax = utilities.plot_complex(τ, α_0_dot(τ), label="exact")
@ -68,7 +73,7 @@ Looks quite good.
#+RESULTS:
:RESULTS:
| hline | <AxesSubplot:> |
[[file:./.ob-jupyter/de686898b2912464cac68188d8ea2b20260be6b9.svg]]
[[file:./.ob-jupyter/666257d29aba72e31b8e20527e5cf43a8c475808.svg]]
:END:
* Analytical Solution
@ -314,18 +319,16 @@ Now we can calculate \(G\).
#+RESULTS:
#+begin_src jupyter-python
with utilities.hiro_style():
plt.plot(τ, G_ex_new(τ).reshape(len(τ), 4))
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7fef38be9460> | <matplotlib.lines.Line2D | at | 0x7fef38be9490> | <matplotlib.lines.Line2D | at | 0x7fef38be95b0> | <matplotlib.lines.Line2D | at | 0x7fef38be96d0> |
[[file:./.ob-jupyter/a324792ec1f60f718e98393058c2a926e5d58396.svg]]
:END:
[[file:./.ob-jupyter/43540f787051db4b09fa82a1204956050edca117.svg]]
* Energy Flow
We adopt the terminology from my notes.
#+begin_src jupyter-python
#+begin_src jupyter-python :results none
A = G_11_ex
B = G_12_ex
Pk = P
@ -341,38 +344,16 @@ We adopt the terminology from my notes.
qp_0 = 1j
#+end_src
#+RESULTS:
First the coefficient matrices.
#+begin_src jupyter-python
# def Γ1(n, k):
# return (Bk[n] * Pk[k]) / (Lk[k] - Ck[n])]
First, we calculate the coefficient matrices.
#+begin_src jupyter-python :results none
Γ1 = (Bk[:, None] * Pk[None, :]) / (Lk[None, :] - Ck[:, None])
Γ2 = (Bk[:, None] * Gk[None, :]) / (Ck[:, None] - Wk[None, :])
Γ3 = (Bk[:, None] * Gk.conj()[None, :]) / (Ck[:, None] + Wk.conj()[None, :])
ΓA = (Ak[:, None] * Pk[None, :]) / (Lk[None, :] - Ck[:, None])
#+end_src
The convolution free part is easy.
#+begin_src jupyter-python
def q_corr_conv_free(t, s):
return (
q_s_0 * A(t) * A(s)
+ p_s_0 * B(t) * B(s)
+ qp_0 * A(t) * B(s)
+ qp_0.conjugate() * B(t) * A(s)
)
@np.vectorize
def flow_conv_free_old(t):
conv_free = quadpy.quad(
lambda s: (q_corr_conv_free(t, s) * α_0_dot(t - s)).imag, 0, t
)[0]
return -1 / 2 * conv_free
The convolution free part is easy to calculate.
#+begin_src jupyter-python :results none
def A_conv(t):
t = np.asarray(t)
result = np.zeros_like(t, dtype="complex128")
@ -403,54 +384,81 @@ The convolution free part is easy.
)
#+end_src
#+RESULTS:
#+RESULTS:
Then the auxiliary integrals.
#+begin_src jupyter-python
import numba
The part containing the convolution with the BCF is a bit mode
involved. The first version originates from my calculations and the
second one arose from sagemath. Thank God! They do aggree now.
#+begin_src jupyter-python :results none
def conv_part(t):
t = np.asarray(t)
result = np.zeros_like(t, dtype="float")
for (n, k, l, m) in itertools.product(
for (m, k, n, l) in itertools.product(
range(len(Bk)), range(len(Pk)), range(len(Bk)), range(len(Gk))
):
g_1_2 = (
Γ1[n, k]
,* Γ2[l, m]
Γ1[m, k]
,* Γ2[n, l]
,* (
(1 - np.exp(-(Ck[n] + Wk[m]) * t)) / (Ck[n] + Wk[m])
+ (np.exp(-(Ck[n] + Ck[l]) * t) - 1) / (Ck[n] + Ck[l])
+ (np.exp(-(Lk[k] + Wk[m]) * t) - 1) / (Lk[k] + Wk[m])
+ (1 - np.exp(-(Lk[k] + Ck[l]) * t)) / (Lk[k] + Ck[l])
(1 - np.exp(-(Ck[m] + Wk[l]) * t)) / (Ck[m] + Wk[l])
+ (np.exp(-(Ck[m] + Ck[n]) * t) - 1) / (Ck[m] + Ck[n])
+ (np.exp(-(Lk[k] + Wk[l]) * t) - 1) / (Lk[k] + Wk[l])
+ (1 - np.exp(-(Lk[k] + Ck[n]) * t)) / (Lk[k] + Ck[n])
)
)
g_1_3 = (
Γ1[n, k]
,* Γ3[l, m]
Γ1[m, k]
,* Γ3[n, l]
,* (
(1 / (Ck[n] + Ck[l]) - 1 / (Lk[k] + Ck[l]))
+ np.exp(-(Ck[n] + Ck[l]) * t)
,* (-1 / (Ck[n] + Ck[l]) + 1 / (Ck[n] + Wk[m].conj()))
+ np.exp(-(Ck[l] + Wk[m].conj()) * t)
,* (-1 / (Ck[n] + Wk[m].conj()) + 1 / (Lk[k] - Wk[m].conj()))
+ np.exp(-(Ck[l] + Lk[k]) * t)
,* (1 / (Lk[k] + Ck[l]) - 1 / (Lk[k] - Wk[m].conj()))
(1 - np.exp(-(Ck[m] + Ck[n]) * t)) / (Ck[m] + Ck[n])
- (1 - np.exp(-(Lk[k] + Ck[n]) * t)) / (Lk[k] + Ck[n])
- (np.exp(-(Ck[n] + Wk[l].conj() * t)) - np.exp(-(Ck[m] + Ck[n]) * t))
/ (Ck[m] - Wk[l].conj())
+ (np.exp(-(Ck[n] + Wk[l].conj() * t)) - np.exp(-(Lk[k] + Ck[n]) * t))
/ (Lk[k] - Wk[l].conj())
)
)
result += -1 / 2 * (g_1_2.imag + g_1_3.imag)
return result
import monster_formula
def conv_part_sage(t):
t = np.asarray(t)
result = np.zeros_like(t, dtype="float")
for (n, k, l, m) in itertools.product(
range(len(Bk)), range(len(Pk)), range(len(Gk)), range(len(Bk))
):
result += (
-1
/ 2
,* (
monster_formula.conv_part(
t,
Pk[k],
Lk[k],
Bk[n],
Ck[n],
Bk[m],
Ck[m],
Gk[l],
Wk[l],
Gk[l].conj(),
Wk[l].conj(),
)
).imag
)
return result
#+end_src
#+RESULTS:
Seems to be correct :P or at least plausible.
The contribution of the convolution part is pretty minor for zero
temperature.
#+begin_src jupyter-python
τ = np.linspace(0, t_max, 5000)
plt.plot(τ, conv_part(τ))
@ -458,32 +466,38 @@ Seems to be correct :P or at least plausible.
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7fef12c0b190> |
[[file:./.ob-jupyter/d090ca8d01f6283cee9f461f65dd3daf11f76cbf.svg]]
| <matplotlib.lines.Line2D | at | 0x7f85099ac190> |
[[file:./.ob-jupyter/ed8ce474b528ddf7104399b40e018db0180b3ee5.svg]]
:END:
Now we can define the flow:
#+begin_src jupyter-python
#+begin_src jupyter-python :results none
def flow(t):
return flow_conv_free(t) + conv_part(t)
#+end_src
#+RESULTS:
And plot it against the numerically obtained flow.
#+begin_src jupyter-python
plt.plot(τ, flow(τ))
with open('res.npy', 'rb') as f:
old_τ = np.load(f)
old_flow_τ = np.load(f)
with utilities.hiro_style():
plt.plot(old_τ, flow(old_τ))
plt.plot(old_τ, old_flow_τ)
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7fef12f25f70> |
[[file:./.ob-jupyter/73de08c911d754498fcd92847ff81b7e8183a205.svg]]
:END:
[[file:./.ob-jupyter/6c77bc1a9d844fa3e33a14c077310c32a7eec4a8.svg]]
They do aggree but I think the accuracy of the numerical integration
is getting worse with time.
#+begin_src jupyter-python
scipy.integrate.trapz(flow(τ), τ)
scipy.integrate.quad(flow, 0, 20)
#+end_src
#+RESULTS:
: 1.7594404753272874
| 1.6004236143630652 | 1.564508138231918e-08 |
Integresting... maybe the fact that it does not converge to zero
exactly has to do with this...

927
python/energy_flow_proper/03_gaussian/monster_formula.py Normal file
View file

@ -0,0 +1,927 @@
from numpy import exp
import numba
@numba.jit
def conv_part(t, P_k, L_k, B_n, C_n, B_m, C_m, G_l, W_l, Gc_l, Wc_l):
return -(
B_m * B_n * C_m ** 2 * C_n ** 2 * G_l * L_k ** 2 * P_k * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n ** 3 * Gc_l * L_k ** 2 * P_k * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m ** 2 * C_n * G_l * L_k ** 3 * P_k * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n ** 2 * Gc_l * L_k ** 3 * P_k * exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* L_k
* P_k
* W_l
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n ** 3 * Gc_l * L_k * P_k * W_l * exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m ** 2
* C_n
* G_l
* L_k ** 2
* P_k
* W_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n ** 3 * Gc_l * L_k ** 2 * P_k * W_l * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n * G_l * L_k ** 3 * P_k * W_l * exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * C_n * Gc_l * L_k ** 3 * P_k * W_l * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n ** 2 * Gc_l * L_k ** 3 * P_k * W_l * exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k
* P_k
* W_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n ** 2
* Gc_l
* L_k
* P_k
* W_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n ** 3 * Gc_l * L_k * P_k * W_l ** 2 * exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n
* G_l
* L_k ** 2
* P_k
* W_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n
* Gc_l
* L_k ** 2
* P_k
* W_l ** 2
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_n * Gc_l * L_k ** 3 * P_k * W_l ** 2 * exp(-C_m * t - C_n * t)
- B_m * B_n * C_n ** 2 * Gc_l * L_k * P_k * W_l ** 3 * exp(-C_m * t - C_n * t)
- B_m * B_n * C_n * Gc_l * L_k ** 2 * P_k * W_l ** 3 * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* L_k
* P_k
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * C_n ** 3 * Gc_l * L_k * P_k * Wc_l * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m ** 2 * G_l * L_k ** 3 * P_k * Wc_l * exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * C_n * G_l * L_k ** 3 * P_k * Wc_l * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n * Gc_l * L_k ** 3 * P_k * Wc_l * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * C_n ** 3 * Gc_l * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
- 2
* B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* Gc_l
* L_k
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_n ** 3 * Gc_l * L_k * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n
* Gc_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * G_l * L_k ** 3 * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
- B_m * B_n * C_n * G_l * L_k ** 3 * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * Gc_l * L_k ** 3 * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n * Gc_l * L_k ** 3 * P_k * W_l * Wc_l * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n ** 2
* G_l
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* Gc_l
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_n ** 3 * Gc_l * P_k * W_l ** 2 * Wc_l * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n ** 2
* G_l
* L_k
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_n ** 2
* Gc_l
* L_k
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* G_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n
* G_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* Gc_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_n
* Gc_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l
* exp(-C_m * t - C_n * t)
- B_m * B_n * Gc_l * L_k ** 3 * P_k * W_l ** 2 * Wc_l * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n ** 2 * Gc_l * P_k * W_l ** 3 * Wc_l * exp(-C_m * t - C_n * t)
- B_m * B_n * Gc_l * L_k ** 2 * P_k * W_l ** 3 * Wc_l * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* C_n
* G_l
* L_k
* P_k
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k
* P_k
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n ** 2
* Gc_l
* L_k
* P_k
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* G_l
* L_k ** 2
* P_k
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n
* Gc_l
* L_k ** 2
* P_k
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m * B_n * C_m * G_l * L_k ** 3 * P_k * Wc_l ** 2 * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* C_n
* G_l
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n ** 2
* Gc_l
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* G_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n
* G_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_n ** 2
* G_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n ** 2
* Gc_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- 2
* B_m
* B_n
* C_m
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* Gc_l
* L_k ** 2
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n
* Gc_l
* L_k ** 2
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m * B_n * G_l * L_k ** 3 * P_k * W_l * Wc_l ** 2 * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* C_n
* G_l
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_n ** 2
* G_l
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* C_n
* Gc_l
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_n ** 2
* Gc_l
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m
* G_l
* L_k
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* C_m
* Gc_l
* L_k
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
- B_m
* B_n
* G_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m
* B_n
* Gc_l
* L_k ** 2
* P_k
* W_l ** 2
* Wc_l ** 2
* exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n * Gc_l * P_k * W_l ** 3 * Wc_l ** 2 * exp(-C_m * t - C_n * t)
+ B_m * B_n * Gc_l * L_k * P_k * W_l ** 3 * Wc_l ** 2 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n * G_l * L_k * P_k * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * G_l * L_k ** 2 * P_k * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * C_n * G_l * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_m * G_l * L_k * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n * G_l * L_k * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * G_l * L_k ** 2 * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * C_n * G_l * P_k * W_l ** 2 * Wc_l ** 3 * exp(-C_m * t - C_n * t)
+ B_m * B_n * G_l * L_k * P_k * W_l ** 2 * Wc_l ** 3 * exp(-C_m * t - C_n * t)
- B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * C_n ** 3 * G_l * L_k ** 2 * P_k * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m ** 2 * C_n * G_l * L_k ** 3 * P_k * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * C_n ** 2 * G_l * L_k ** 3 * P_k * exp(-C_m * t - W_l * t)
- B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* L_k
* P_k
* W_l
* exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * C_n ** 3 * G_l * L_k * P_k * W_l * exp(-C_m * t - W_l * t)
- B_m
* B_n
* C_m ** 2
* C_n
* G_l
* L_k ** 2
* P_k
* W_l
* exp(-C_m * t - W_l * t)
- B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* L_k
* P_k
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m * B_n * C_m * C_n ** 3 * G_l * L_k * P_k * Wc_l * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m * B_n * C_n ** 3 * G_l * L_k ** 2 * P_k * Wc_l * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m ** 2 * G_l * L_k ** 3 * P_k * Wc_l * exp(-C_m * t - W_l * t)
+ B_m * B_n * C_n ** 2 * G_l * L_k ** 3 * P_k * Wc_l * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* C_n ** 2
* G_l
* P_k
* W_l
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m * B_n * C_m * C_n ** 3 * G_l * P_k * W_l * Wc_l * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m
* C_n ** 2
* G_l
* L_k
* P_k
* W_l
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m * B_n * C_n ** 3 * G_l * L_k * P_k * W_l * Wc_l * exp(-C_m * t - W_l * t)
- B_m
* B_n
* C_m ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_n ** 2
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* C_n
* G_l
* L_k
* P_k
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
- B_m * B_n * C_n ** 3 * G_l * L_k * P_k * Wc_l ** 2 * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* G_l
* L_k ** 2
* P_k
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m
* C_n
* G_l
* L_k ** 2
* P_k
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
+ B_m * B_n * C_m * G_l * L_k ** 3 * P_k * Wc_l ** 2 * exp(-C_m * t - W_l * t)
+ B_m * B_n * C_n * G_l * L_k ** 3 * P_k * Wc_l ** 2 * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* C_n
* G_l
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
- B_m * B_n * C_n ** 3 * G_l * P_k * W_l * Wc_l ** 2 * exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m ** 2
* G_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m
* C_n
* G_l
* L_k
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_m
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
+ B_m
* B_n
* C_n
* G_l
* L_k ** 2
* P_k
* W_l
* Wc_l ** 2
* exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * C_n * G_l * L_k * P_k * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_n ** 2 * G_l * L_k * P_k * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * G_l * L_k ** 2 * P_k * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_n * G_l * L_k ** 2 * P_k * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * C_n * G_l * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_n ** 2 * G_l * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_m * G_l * L_k * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - W_l * t)
- B_m * B_n * C_n * G_l * L_k * P_k * W_l * Wc_l ** 3 * exp(-C_m * t - W_l * t)
+ B_m * B_n * C_m ** 3 * C_n ** 2 * Gc_l * L_k * P_k * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m ** 2 * C_n ** 3 * Gc_l * L_k * P_k * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m ** 3 * C_n * Gc_l * L_k ** 2 * P_k * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m * C_n ** 3 * Gc_l * L_k ** 2 * P_k * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 2 * C_n * Gc_l * L_k ** 3 * P_k * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m * C_n ** 2 * Gc_l * L_k ** 3 * P_k * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m ** 3 * C_n ** 2 * Gc_l * P_k * W_l * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m ** 2 * C_n ** 3 * Gc_l * P_k * W_l * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 3 * Gc_l * L_k ** 2 * P_k * W_l * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_n ** 3 * Gc_l * L_k ** 2 * P_k * W_l * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m ** 2 * Gc_l * L_k ** 3 * P_k * W_l * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_n ** 2 * Gc_l * L_k ** 3 * P_k * W_l * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 3 * C_n * Gc_l * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m * C_n ** 3 * Gc_l * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 3 * Gc_l * L_k * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_n ** 3 * Gc_l * L_k * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m * Gc_l * L_k ** 3 * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_n * Gc_l * L_k ** 3 * P_k * W_l ** 2 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 2 * C_n * Gc_l * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m * C_n ** 2 * Gc_l * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 2 * Gc_l * L_k * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_n ** 2 * Gc_l * L_k * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_m * Gc_l * L_k ** 2 * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
+ B_m * B_n * C_n * Gc_l * L_k ** 2 * P_k * W_l ** 3 * exp(-C_n * t - Wc_l * t)
- B_m * B_n * C_m ** 3 * C_n ** 2 * G_l * L_k * P_k
- B_m * B_n * C_m ** 2 * C_n ** 3 * G_l * L_k * P_k
+ B_m * B_n * C_m * C_n ** 3 * G_l * L_k ** 2 * P_k
- B_m * B_n * C_m ** 3 * C_n * Gc_l * L_k ** 2 * P_k
+ B_m * B_n * C_m * C_n ** 2 * G_l * L_k ** 3 * P_k
+ B_m * B_n * C_m ** 2 * C_n * Gc_l * L_k ** 3 * P_k
+ B_m * B_n * C_m ** 3 * C_n * G_l * L_k * P_k * W_l
- B_m * B_n * C_m ** 3 * C_n * Gc_l * L_k * P_k * W_l
+ B_m * B_n * C_m ** 3 * Gc_l * L_k ** 2 * P_k * W_l
- B_m * B_n * C_m * C_n * G_l * L_k ** 3 * P_k * W_l
- B_m * B_n * C_m ** 2 * Gc_l * L_k ** 3 * P_k * W_l
+ B_m * B_n * C_m * C_n * Gc_l * L_k ** 3 * P_k * W_l
+ B_m * B_n * C_m ** 2 * C_n * G_l * L_k * P_k * W_l ** 2
+ B_m * B_n * C_m ** 3 * Gc_l * L_k * P_k * W_l ** 2
- B_m * B_n * C_m ** 2 * C_n * Gc_l * L_k * P_k * W_l ** 2
- B_m * B_n * C_m * C_n * G_l * L_k ** 2 * P_k * W_l ** 2
+ B_m * B_n * C_m * C_n * Gc_l * L_k ** 2 * P_k * W_l ** 2
- B_m * B_n * C_m * Gc_l * L_k ** 3 * P_k * W_l ** 2
+ B_m * B_n * C_m ** 2 * Gc_l * L_k * P_k * W_l ** 3
- B_m * B_n * C_m * Gc_l * L_k ** 2 * P_k * W_l ** 3
+ B_m * B_n * C_m ** 3 * C_n ** 2 * G_l * P_k * Wc_l
+ B_m * B_n * C_m ** 2 * C_n ** 3 * G_l * P_k * Wc_l
- B_m * B_n * C_m ** 3 * C_n * G_l * L_k * P_k * Wc_l
+ B_m * B_n * C_m ** 3 * C_n * Gc_l * L_k * P_k * Wc_l
- B_m * B_n * C_n ** 3 * G_l * L_k ** 2 * P_k * Wc_l
+ B_m * B_n * C_m * C_n * G_l * L_k ** 3 * P_k * Wc_l
- B_m * B_n * C_n ** 2 * G_l * L_k ** 3 * P_k * Wc_l
- B_m * B_n * C_m * C_n * Gc_l * L_k ** 3 * P_k * Wc_l
- B_m * B_n * C_m ** 3 * C_n * G_l * P_k * W_l * Wc_l
+ B_m * B_n * C_m ** 3 * C_n * Gc_l * P_k * W_l * Wc_l
+ B_m * B_n * C_m ** 3 * G_l * L_k * P_k * W_l * Wc_l
- B_m * B_n * C_m ** 2 * C_n * G_l * L_k * P_k * W_l * Wc_l
- B_m * B_n * C_m ** 3 * Gc_l * L_k * P_k * W_l * Wc_l
+ B_m * B_n * C_m ** 2 * C_n * Gc_l * L_k * P_k * W_l * Wc_l
+ B_m * B_n * C_m * C_n * G_l * L_k ** 2 * P_k * W_l * Wc_l
- B_m * B_n * C_m * C_n * Gc_l * L_k ** 2 * P_k * W_l * Wc_l
- B_m * B_n * C_m * G_l * L_k ** 3 * P_k * W_l * Wc_l
+ B_m * B_n * C_n * G_l * L_k ** 3 * P_k * W_l * Wc_l
+ B_m * B_n * C_m * Gc_l * L_k ** 3 * P_k * W_l * Wc_l
- B_m * B_n * C_n * Gc_l * L_k ** 3 * P_k * W_l * Wc_l
- B_m * B_n * C_m ** 2 * C_n * G_l * P_k * W_l ** 2 * Wc_l
- B_m * B_n * C_m ** 3 * Gc_l * P_k * W_l ** 2 * Wc_l
+ B_m * B_n * C_m ** 2 * C_n * Gc_l * P_k * W_l ** 2 * Wc_l
+ B_m * B_n * C_m ** 2 * G_l * L_k * P_k * W_l ** 2 * Wc_l
- B_m * B_n * C_m ** 2 * Gc_l * L_k * P_k * W_l ** 2 * Wc_l
- B_m * B_n * C_m * G_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l
+ B_m * B_n * C_n * G_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l
+ B_m * B_n * C_m * Gc_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l
- B_m * B_n * C_n * Gc_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l
+ B_m * B_n * Gc_l * L_k ** 3 * P_k * W_l ** 2 * Wc_l
- B_m * B_n * C_m ** 2 * Gc_l * P_k * W_l ** 3 * Wc_l
+ B_m * B_n * Gc_l * L_k ** 2 * P_k * W_l ** 3 * Wc_l
+ B_m * B_n * C_m ** 3 * C_n * G_l * P_k * Wc_l ** 2
- B_m * B_n * C_m * C_n ** 3 * G_l * P_k * Wc_l ** 2
+ B_m * B_n * C_m ** 2 * C_n * G_l * L_k * P_k * Wc_l ** 2
+ B_m * B_n * C_n ** 3 * G_l * L_k * P_k * Wc_l ** 2
- B_m * B_n * C_m ** 2 * C_n * Gc_l * L_k * P_k * Wc_l ** 2
- B_m * B_n * C_m * C_n * G_l * L_k ** 2 * P_k * Wc_l ** 2
+ B_m * B_n * C_m * C_n * Gc_l * L_k ** 2 * P_k * Wc_l ** 2
- B_m * B_n * C_n * G_l * L_k ** 3 * P_k * Wc_l ** 2
- B_m * B_n * C_m ** 3 * G_l * P_k * W_l * Wc_l ** 2
+ B_m * B_n * C_m ** 2 * C_n * G_l * P_k * W_l * Wc_l ** 2
- B_m * B_n * C_m ** 2 * C_n * Gc_l * P_k * W_l * Wc_l ** 2
- B_m * B_n * C_m ** 2 * G_l * L_k * P_k * W_l * Wc_l ** 2
+ B_m * B_n * C_m ** 2 * Gc_l * L_k * P_k * W_l * Wc_l ** 2
+ B_m * B_n * C_m * G_l * L_k ** 2 * P_k * W_l * Wc_l ** 2
- B_m * B_n * C_n * G_l * L_k ** 2 * P_k * W_l * Wc_l ** 2
- B_m * B_n * C_m * Gc_l * L_k ** 2 * P_k * W_l * Wc_l ** 2
+ B_m * B_n * C_n * Gc_l * L_k ** 2 * P_k * W_l * Wc_l ** 2
+ B_m * B_n * G_l * L_k ** 3 * P_k * W_l * Wc_l ** 2
- B_m * B_n * C_m ** 2 * G_l * P_k * W_l ** 2 * Wc_l ** 2
+ B_m * B_n * C_m * C_n * G_l * P_k * W_l ** 2 * Wc_l ** 2
+ B_m * B_n * C_m ** 2 * Gc_l * P_k * W_l ** 2 * Wc_l ** 2
- B_m * B_n * C_m * C_n * Gc_l * P_k * W_l ** 2 * Wc_l ** 2
- B_m * B_n * C_n * G_l * L_k * P_k * W_l ** 2 * Wc_l ** 2
+ B_m * B_n * C_n * Gc_l * L_k * P_k * W_l ** 2 * Wc_l ** 2
+ B_m * B_n * G_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l ** 2
- B_m * B_n * Gc_l * L_k ** 2 * P_k * W_l ** 2 * Wc_l ** 2
+ B_m * B_n * C_m * Gc_l * P_k * W_l ** 3 * Wc_l ** 2
- B_m * B_n * Gc_l * L_k * P_k * W_l ** 3 * Wc_l ** 2
- B_m * B_n * C_m ** 2 * C_n * G_l * P_k * Wc_l ** 3
- B_m * B_n * C_m * C_n ** 2 * G_l * P_k * Wc_l ** 3
+ B_m * B_n * C_n ** 2 * G_l * L_k * P_k * Wc_l ** 3
+ B_m * B_n * C_n * G_l * L_k ** 2 * P_k * Wc_l ** 3
+ B_m * B_n * C_m ** 2 * G_l * P_k * W_l * Wc_l ** 3
- B_m * B_n * G_l * L_k ** 2 * P_k * W_l * Wc_l ** 3
+ B_m * B_n * C_m * G_l * P_k * W_l ** 2 * Wc_l ** 3
- B_m * B_n * G_l * L_k * P_k * W_l ** 2 * Wc_l ** 3
) / (
C_m ** 4 * C_n ** 3 * L_k ** 2
+ C_m ** 3 * C_n ** 4 * L_k ** 2
+ C_m ** 4 * C_n ** 2 * L_k ** 3
- C_m ** 2 * C_n ** 4 * L_k ** 3
- C_m ** 3 * C_n ** 2 * L_k ** 4
- C_m ** 2 * C_n ** 3 * L_k ** 4
+ C_m ** 4 * C_n ** 3 * L_k * W_l
+ C_m ** 3 * C_n ** 4 * L_k * W_l
- C_m ** 4 * C_n * L_k ** 3 * W_l
- C_m * C_n ** 4 * L_k ** 3 * W_l
+ C_m ** 3 * C_n * L_k ** 4 * W_l
- C_m * C_n ** 3 * L_k ** 4 * W_l
- C_m ** 4 * C_n ** 2 * L_k * W_l ** 2
+ C_m ** 2 * C_n ** 4 * L_k * W_l ** 2
- C_m ** 4 * C_n * L_k ** 2 * W_l ** 2
- C_m * C_n ** 4 * L_k ** 2 * W_l ** 2
+ C_m ** 2 * C_n * L_k ** 4 * W_l ** 2
+ C_m * C_n ** 2 * L_k ** 4 * W_l ** 2
- C_m ** 3 * C_n ** 2 * L_k * W_l ** 3
- C_m ** 2 * C_n ** 3 * L_k * W_l ** 3
- C_m ** 3 * C_n * L_k ** 2 * W_l ** 3
+ C_m * C_n ** 3 * L_k ** 2 * W_l ** 3
+ C_m ** 2 * C_n * L_k ** 3 * W_l ** 3
+ C_m * C_n ** 2 * L_k ** 3 * W_l ** 3
- C_m ** 4 * C_n ** 3 * L_k * Wc_l
- C_m ** 3 * C_n ** 4 * L_k * Wc_l
+ C_m ** 4 * C_n * L_k ** 3 * Wc_l
+ C_m * C_n ** 4 * L_k ** 3 * Wc_l
- C_m ** 3 * C_n * L_k ** 4 * Wc_l
+ C_m * C_n ** 3 * L_k ** 4 * Wc_l
- C_m ** 4 * C_n ** 3 * W_l * Wc_l
- C_m ** 3 * C_n ** 4 * W_l * Wc_l
+ C_m ** 4 * C_n ** 2 * L_k * W_l * Wc_l
- C_m ** 2 * C_n ** 4 * L_k * W_l * Wc_l
+ C_m ** 4 * C_n * L_k ** 2 * W_l * Wc_l
+ C_m * C_n ** 4 * L_k ** 2 * W_l * Wc_l
- C_m ** 4 * L_k ** 3 * W_l * Wc_l
+ C_n ** 4 * L_k ** 3 * W_l * Wc_l
+ C_m ** 3 * L_k ** 4 * W_l * Wc_l
- C_m ** 2 * C_n * L_k ** 4 * W_l * Wc_l
- C_m * C_n ** 2 * L_k ** 4 * W_l * Wc_l
+ C_n ** 3 * L_k ** 4 * W_l * Wc_l
+ C_m ** 4 * C_n ** 2 * W_l ** 2 * Wc_l
- C_m ** 2 * C_n ** 4 * W_l ** 2 * Wc_l
+ C_m ** 3 * C_n ** 2 * L_k * W_l ** 2 * Wc_l
+ C_m ** 2 * C_n ** 3 * L_k * W_l ** 2 * Wc_l
- C_m ** 4 * L_k ** 2 * W_l ** 2 * Wc_l
+ C_m ** 3 * C_n * L_k ** 2 * W_l ** 2 * Wc_l
- C_m * C_n ** 3 * L_k ** 2 * W_l ** 2 * Wc_l
+ C_n ** 4 * L_k ** 2 * W_l ** 2 * Wc_l
- C_m ** 2 * C_n * L_k ** 3 * W_l ** 2 * Wc_l
- C_m * C_n ** 2 * L_k ** 3 * W_l ** 2 * Wc_l
+ C_m ** 2 * L_k ** 4 * W_l ** 2 * Wc_l
- C_n ** 2 * L_k ** 4 * W_l ** 2 * Wc_l
+ C_m ** 3 * C_n ** 2 * W_l ** 3 * Wc_l
+ C_m ** 2 * C_n ** 3 * W_l ** 3 * Wc_l
- C_m ** 3 * L_k ** 2 * W_l ** 3 * Wc_l
- C_n ** 3 * L_k ** 2 * W_l ** 3 * Wc_l
+ C_m ** 2 * L_k ** 3 * W_l ** 3 * Wc_l
- C_n ** 2 * L_k ** 3 * W_l ** 3 * Wc_l
- C_m ** 4 * C_n ** 2 * L_k * Wc_l ** 2
+ C_m ** 2 * C_n ** 4 * L_k * Wc_l ** 2
- C_m ** 4 * C_n * L_k ** 2 * Wc_l ** 2
- C_m * C_n ** 4 * L_k ** 2 * Wc_l ** 2
+ C_m ** 2 * C_n * L_k ** 4 * Wc_l ** 2
+ C_m * C_n ** 2 * L_k ** 4 * Wc_l ** 2
- C_m ** 4 * C_n ** 2 * W_l * Wc_l ** 2
+ C_m ** 2 * C_n ** 4 * W_l * Wc_l ** 2
- C_m ** 3 * C_n ** 2 * L_k * W_l * Wc_l ** 2
- C_m ** 2 * C_n ** 3 * L_k * W_l * Wc_l ** 2
+ C_m ** 4 * L_k ** 2 * W_l * Wc_l ** 2
- C_m ** 3 * C_n * L_k ** 2 * W_l * Wc_l ** 2
+ C_m * C_n ** 3 * L_k ** 2 * W_l * Wc_l ** 2
- C_n ** 4 * L_k ** 2 * W_l * Wc_l ** 2
+ C_m ** 2 * C_n * L_k ** 3 * W_l * Wc_l ** 2
+ C_m * C_n ** 2 * L_k ** 3 * W_l * Wc_l ** 2
- C_m ** 2 * L_k ** 4 * W_l * Wc_l ** 2
+ C_n ** 2 * L_k ** 4 * W_l * Wc_l ** 2
+ C_m ** 4 * C_n * W_l ** 2 * Wc_l ** 2
- C_m ** 3 * C_n ** 2 * W_l ** 2 * Wc_l ** 2
- C_m ** 2 * C_n ** 3 * W_l ** 2 * Wc_l ** 2
+ C_m * C_n ** 4 * W_l ** 2 * Wc_l ** 2
+ C_m ** 4 * L_k * W_l ** 2 * Wc_l ** 2
- C_n ** 4 * L_k * W_l ** 2 * Wc_l ** 2
+ C_m ** 3 * L_k ** 2 * W_l ** 2 * Wc_l ** 2
+ C_n ** 3 * L_k ** 2 * W_l ** 2 * Wc_l ** 2
- C_m ** 2 * L_k ** 3 * W_l ** 2 * Wc_l ** 2
+ C_n ** 2 * L_k ** 3 * W_l ** 2 * Wc_l ** 2
- C_m * L_k ** 4 * W_l ** 2 * Wc_l ** 2
- C_n * L_k ** 4 * W_l ** 2 * Wc_l ** 2
+ C_m ** 3 * C_n * W_l ** 3 * Wc_l ** 2
- C_m * C_n ** 3 * W_l ** 3 * Wc_l ** 2
+ C_m ** 3 * L_k * W_l ** 3 * Wc_l ** 2
+ C_n ** 3 * L_k * W_l ** 3 * Wc_l ** 2
- C_m * L_k ** 3 * W_l ** 3 * Wc_l ** 2
- C_n * L_k ** 3 * W_l ** 3 * Wc_l ** 2
+ C_m ** 3 * C_n ** 2 * L_k * Wc_l ** 3
+ C_m ** 2 * C_n ** 3 * L_k * Wc_l ** 3
+ C_m ** 3 * C_n * L_k ** 2 * Wc_l ** 3
- C_m * C_n ** 3 * L_k ** 2 * Wc_l ** 3
- C_m ** 2 * C_n * L_k ** 3 * Wc_l ** 3
- C_m * C_n ** 2 * L_k ** 3 * Wc_l ** 3
+ C_m ** 3 * C_n ** 2 * W_l * Wc_l ** 3
+ C_m ** 2 * C_n ** 3 * W_l * Wc_l ** 3
- C_m ** 3 * L_k ** 2 * W_l * Wc_l ** 3
- C_n ** 3 * L_k ** 2 * W_l * Wc_l ** 3
+ C_m ** 2 * L_k ** 3 * W_l * Wc_l ** 3
- C_n ** 2 * L_k ** 3 * W_l * Wc_l ** 3
- C_m ** 3 * C_n * W_l ** 2 * Wc_l ** 3
+ C_m * C_n ** 3 * W_l ** 2 * Wc_l ** 3
- C_m ** 3 * L_k * W_l ** 2 * Wc_l ** 3
- C_n ** 3 * L_k * W_l ** 2 * Wc_l ** 3
+ C_m * L_k ** 3 * W_l ** 2 * Wc_l ** 3
+ C_n * L_k ** 3 * W_l ** 2 * Wc_l ** 3
- C_m ** 2 * C_n * W_l ** 3 * Wc_l ** 3
- C_m * C_n ** 2 * W_l ** 3 * Wc_l ** 3
- C_m ** 2 * L_k * W_l ** 3 * Wc_l ** 3
+ C_n ** 2 * L_k * W_l ** 3 * Wc_l ** 3
+ C_m * L_k ** 2 * W_l ** 3 * Wc_l ** 3
+ C_n * L_k ** 2 * W_l ** 3 * Wc_l ** 3
) + (
B_m * B_n * C_n * Gc_l * L_k * P_k * exp(-C_n * t - L_k * t)
+ B_m * B_n * G_l * L_k ** 2 * P_k * exp(-C_n * t - L_k * t)
+ B_m * B_n * C_n * Gc_l * P_k * W_l * exp(-C_n * t - L_k * t)
+ B_m * B_n * G_l * L_k * P_k * W_l * exp(-C_n * t - L_k * t)
- B_m * B_n * Gc_l * L_k * P_k * W_l * exp(-C_n * t - L_k * t)
- B_m * B_n * Gc_l * P_k * W_l ** 2 * exp(-C_n * t - L_k * t)
- B_m * B_n * G_l * L_k * P_k * Wc_l * exp(-C_n * t - L_k * t)
- B_m * B_n * G_l * P_k * W_l * Wc_l * exp(-C_n * t - L_k * t)
- B_m * B_n * C_n * G_l * L_k * P_k * exp(-L_k * t - W_l * t)
- B_m * B_n * G_l * L_k ** 2 * P_k * exp(-L_k * t - W_l * t)
+ B_m * B_n * C_n * G_l * P_k * Wc_l * exp(-L_k * t - W_l * t)
+ B_m * B_n * G_l * L_k * P_k * Wc_l * exp(-L_k * t - W_l * t)
) / (
C_m * C_n ** 2 * L_k ** 2
+ C_m * C_n * L_k ** 3
- C_n ** 2 * L_k ** 3
- C_n * L_k ** 4
+ C_m * C_n ** 2 * L_k * W_l
- C_n ** 2 * L_k ** 2 * W_l
- C_m * L_k ** 3 * W_l
+ L_k ** 4 * W_l
- C_m * C_n * L_k * W_l ** 2
- C_m * L_k ** 2 * W_l ** 2
+ C_n * L_k ** 2 * W_l ** 2
+ L_k ** 3 * W_l ** 2
- C_m * C_n ** 2 * L_k * Wc_l
- C_m * C_n * L_k ** 2 * Wc_l
+ C_n ** 2 * L_k ** 2 * Wc_l
+ C_n * L_k ** 3 * Wc_l
- C_m * C_n ** 2 * W_l * Wc_l
+ C_n ** 2 * L_k * W_l * Wc_l
+ C_m * L_k ** 2 * W_l * Wc_l
- L_k ** 3 * W_l * Wc_l
+ C_m * C_n * W_l ** 2 * Wc_l
+ C_m * L_k * W_l ** 2 * Wc_l
- C_n * L_k * W_l ** 2 * Wc_l
- L_k ** 2 * W_l ** 2 * Wc_l
)

41
python/energy_flow_proper/03_gaussian/naive_integration.org
View file

@ -14,8 +14,6 @@
* System Parameters
#+begin_src jupyter-python :results none
Ω = 1.5
A = np.array([[0, Ω], [-Ω, 0]])
@ -528,7 +526,7 @@ Looks quite good.
#+RESULTS:
:RESULTS:
| hline | <AxesSubplot:> |
[[file:./.ob-jupyter/e9507967be2dea9da1f4042b39d2ec65865b45c1.svg]]
[[file:./.ob-jupyter/afa5b41a1ab07e555bddc54be560b06c11bf3906.svg]]
:END:
** Calculate the Magic Numbers
@ -780,8 +778,8 @@ Now we can calculate \(G\).
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7fcc1281a6a0> | <matplotlib.lines.Line2D | at | 0x7fcc1281a6d0> | <matplotlib.lines.Line2D | at | 0x7fcc1281a7f0> | <matplotlib.lines.Line2D | at | 0x7fcc1281a910> |
[[file:./.ob-jupyter/e7a19f95ebdd3a7fb2b1e78861af91ab06b7db79.svg]]
| <matplotlib.lines.Line2D | at | 0x7fcc1162c8b0> | <matplotlib.lines.Line2D | at | 0x7fcc1162cdc0> | <matplotlib.lines.Line2D | at | 0x7fcc10d36760> | <matplotlib.lines.Line2D | at | 0x7fcc10d36940> |
[[file:./.ob-jupyter/caa8f4b42f6de38f8733ac5169a063de1d5c5d53.svg]]
:END:
#+begin_src jupyter-python
@ -790,8 +788,8 @@ Now we can calculate \(G\).
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7fcc127214f0> |
[[file:./.ob-jupyter/ec188f7ddf18303562f421a80ae70ffa801eeb82.svg]]
| <matplotlib.lines.Line2D | at | 0x7fcc10516610> |
[[file:./.ob-jupyter/5cf02217f725f3e631bb65db0df160a7584ebf65.svg]]
:END:
* Correlation Function
@ -893,7 +891,7 @@ We have to integrate over a wedge.
conv_integrand, 0, t, lambda s: 0, lambda s: s, lambda _, __: 0, lambda _, __: t
)[0]
return -1 / 2 * (conv_part + conv_free)
return -1 / 2 * (conv_part + conv_free), -1/2 * conv_part, -1/2 * conv_free
# conv_part = scheme.integrate(
@ -979,10 +977,6 @@ We have to integrate over a wedge.
#+end_src
#+RESULTS:
:RESULTS:
| <matplotlib.lines.Line2D | at | 0x7f0e56b7d310> |
[[file:./.ob-jupyter/20eec8889b78c5fefdbed39d830b177d3ffcf49e.svg]]
:END:
@ -991,7 +985,6 @@ We have to integrate over a wedge.
#+end_src
#+RESULTS:
: 1.4995572317677925
#+begin_src jupyter-python
with open("res.npy", 'wb') as f:
@ -999,21 +992,23 @@ We have to integrate over a wedge.
np.save(f, flow_τ)
#+end_src
#+begin_src jupyter-python
from numpy.random import default_rng
def generate_wedge_samples(t, n):
rng = default_rng()
points = rng.uniform(0, t, size=n * 3)
r, l, s = points[::3], points[1::3], points[2::3]
mask = (r < s)
r, l, s = r[mask], l[mask], s[mask]
return r, l, s
with open("res.npy", 'rb') as f:
τ = np.load(f)
flow_τ = np.load(f)
#+end_src
#+RESULTS:
#+begin_src jupyter-python
with utilities.hiro_style():
plt.plot(τ, flow_τ)
#+end_src
#+RESULTS:
[[file:./.ob-jupyter/f8925d0fb5e2dcf292b99f7784b208b718bc92cd.svg]]
* Scratch
#+begin_src jupyter-python

BIN
python/energy_flow_proper/03_gaussian/res.npy Normal file
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Binary file not shown.

168
tex/junk/test.tex Normal file
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@ -0,0 +1,168 @@
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left(B_{n} C_{n}
G_{l} + B_{n} G_{l} \overline{W_{l}}\right)} e^{\left(W_{l} r + C_{n} t
+ r \overline{W_{l}}\right)} - {\left(B_{n} C_{n} \overline{G_{l}} -
B_{n} W_{l} \overline{G_{l}} + {\left(B_{n} C_{n} G_{l} - B_{n} C_{n}
\overline{G_{l}} + B_{n} W_{l} \overline{G_{l}} + B_{n} G_{l}
\overline{W_{l}}\right)} e^{\left(C_{n} r + r
\overline{W_{l}}\right)}\right)} e^{\left(W_{l} t\right)}\right)}
e^{\left(-C_{n} t - W_{l} t - r \overline{W_{l}}\right)}}{C_{n}^{2} -
C_{n} W_{l} + {\left(C_{n} - W_{l}\right)} \overline{W_{l}}}\]
\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{B_{m} B_{n} C_{m}^{2}
C_{n}^{2} G_{l} L_{k}^{2} P_{k} e^{\left(-C_{m} t - C_{n} t\right)} +
B_{m} B_{n} C_{m} C_{n}^{3} G_{l} L_{k}^{2} P_{k} e^{\left(-C_{m} t -
C_{n} t\right)} + B_{m} B_{n} C_{m}^{2} C_{n} G_{l} L_{k}^{3} P_{k}
e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n} C_{m} C_{n}^{2} G_{l}
L_{k}^{3} P_{k} e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} L_{k}^{2} P_{k} W_{l} e^{\left(-C_{m} t - C_{n}
t\right)} + B_{m} B_{n} C_{n}^{3} G_{l} L_{k}^{2} P_{k} W_{l}
e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n} C_{m}^{2} G_{l}
L_{k}^{3} P_{k} W_{l} e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n}
C_{n}^{2} G_{l} L_{k}^{3} P_{k} W_{l} e^{\left(-C_{m} t - C_{n}
t\right)} - B_{m} B_{n} C_{m}^{2} C_{n}^{2} G_{l} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n} C_{m} C_{n}^{3} G_{l}
P_{k} W_{l}^{2} e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} L_{k} P_{k} W_{l}^{2} e^{\left(-C_{m} t - C_{n}
t\right)} - B_{m} B_{n} C_{m} C_{n}^{2} G_{l} L_{k} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n} C_{m} C_{n} G_{l}
L_{k}^{2} P_{k} W_{l}^{2} e^{\left(-C_{m} t - C_{n} t\right)} - B_{m}
B_{n} C_{n}^{2} G_{l} L_{k}^{2} P_{k} W_{l}^{2} e^{\left(-C_{m} t -
C_{n} t\right)} - B_{m} B_{n} C_{m} G_{l} L_{k}^{3} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n} C_{n} G_{l} L_{k}^{3}
P_{k} W_{l}^{2} e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} P_{k} W_{l}^{3} e^{\left(-C_{m} t - C_{n}
t\right)} - B_{m} B_{n} C_{n}^{3} G_{l} P_{k} W_{l}^{3} e^{\left(-C_{m}
t - C_{n} t\right)} - B_{m} B_{n} C_{m}^{2} G_{l} L_{k} P_{k} W_{l}^{3}
e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n} C_{n}^{2} G_{l} L_{k}
P_{k} W_{l}^{3} e^{\left(-C_{m} t - C_{n} t\right)} - 2 \, B_{m} B_{n}
C_{n} G_{l} L_{k}^{2} P_{k} W_{l}^{3} e^{\left(-C_{m} t - C_{n}
t\right)} - 2 \, B_{m} B_{n} G_{l} L_{k}^{3} P_{k} W_{l}^{3}
e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n} C_{m} C_{n} G_{l}
P_{k} W_{l}^{4} e^{\left(-C_{m} t - C_{n} t\right)} + B_{m} B_{n}
C_{n}^{2} G_{l} P_{k} W_{l}^{4} e^{\left(-C_{m} t - C_{n} t\right)} +
B_{m} B_{n} C_{m} G_{l} L_{k} P_{k} W_{l}^{4} e^{\left(-C_{m} t - C_{n}
t\right)} + B_{m} B_{n} C_{n} G_{l} L_{k} P_{k} W_{l}^{4}
e^{\left(-C_{m} t - C_{n} t\right)} + 2 \, B_{m} B_{n} C_{n} G_{l} P_{k}
W_{l}^{5} e^{\left(-C_{m} t - C_{n} t\right)} + 2 \, B_{m} B_{n} G_{l}
L_{k} P_{k} W_{l}^{5} e^{\left(-C_{m} t - C_{n} t\right)} - B_{m} B_{n}
C_{m}^{2} C_{n}^{2} G_{l} L_{k}^{2} P_{k} e^{\left(-C_{m} t - W_{l}
t\right)} - B_{m} B_{n} C_{m} C_{n}^{3} G_{l} L_{k}^{2} P_{k}
e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{m}^{2} C_{n} G_{l}
L_{k}^{3} P_{k} e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{m}
C_{n}^{2} G_{l} L_{k}^{3} P_{k} e^{\left(-C_{m} t - W_{l} t\right)} -
B_{m} B_{n} C_{m}^{2} C_{n} G_{l} L_{k}^{2} P_{k} W_{l} e^{\left(-C_{m}
t - W_{l} t\right)} + B_{m} B_{n} C_{n}^{3} G_{l} L_{k}^{2} P_{k} W_{l}
e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{m}^{2} G_{l}
L_{k}^{3} P_{k} W_{l} e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n}
C_{n}^{2} G_{l} L_{k}^{3} P_{k} W_{l} e^{\left(-C_{m} t - W_{l}
t\right)} + B_{m} B_{n} C_{m}^{2} C_{n}^{2} G_{l} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n} C_{m} C_{n}^{3} G_{l}
P_{k} W_{l}^{2} e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} L_{k} P_{k} W_{l}^{2} e^{\left(-C_{m} t - W_{l}
t\right)} + B_{m} B_{n} C_{m} C_{n}^{2} G_{l} L_{k} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n} C_{m} C_{n} G_{l}
L_{k}^{2} P_{k} W_{l}^{2} e^{\left(-C_{m} t - W_{l} t\right)} + B_{m}
B_{n} C_{n}^{2} G_{l} L_{k}^{2} P_{k} W_{l}^{2} e^{\left(-C_{m} t -
W_{l} t\right)} + B_{m} B_{n} C_{m} G_{l} L_{k}^{3} P_{k} W_{l}^{2}
e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n} C_{n} G_{l} L_{k}^{3}
P_{k} W_{l}^{2} e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} P_{k} W_{l}^{3} e^{\left(-C_{m} t - W_{l}
t\right)} - B_{m} B_{n} C_{n}^{3} G_{l} P_{k} W_{l}^{3} e^{\left(-C_{m}
t - W_{l} t\right)} + B_{m} B_{n} C_{m}^{2} G_{l} L_{k} P_{k} W_{l}^{3}
e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{n}^{2} G_{l} L_{k}
P_{k} W_{l}^{3} e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{m}
C_{n} G_{l} P_{k} W_{l}^{4} e^{\left(-C_{m} t - W_{l} t\right)} - B_{m}
B_{n} C_{n}^{2} G_{l} P_{k} W_{l}^{4} e^{\left(-C_{m} t - W_{l}
t\right)} - B_{m} B_{n} C_{m} G_{l} L_{k} P_{k} W_{l}^{4}
e^{\left(-C_{m} t - W_{l} t\right)} - B_{m} B_{n} C_{n} G_{l} L_{k}
P_{k} W_{l}^{4} e^{\left(-C_{m} t - W_{l} t\right)} + B_{m} B_{n}
C_{m}^{3} C_{n}^{2} G_{l} L_{k} P_{k} e^{\left(-C_{n} t - W_{l}
t\right)} + B_{m} B_{n} C_{m}^{2} C_{n}^{3} G_{l} L_{k} P_{k}
e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{m}^{3} C_{n} G_{l}
L_{k}^{2} P_{k} e^{\left(-C_{n} t - W_{l} t\right)} - B_{m} B_{n} C_{m}
C_{n}^{3} G_{l} L_{k}^{2} P_{k} e^{\left(-C_{n} t - W_{l} t\right)} -
B_{m} B_{n} C_{m}^{2} C_{n} G_{l} L_{k}^{3} P_{k} e^{\left(-C_{n} t -
W_{l} t\right)} - B_{m} B_{n} C_{m} C_{n}^{2} G_{l} L_{k}^{3} P_{k}
e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{m}^{3} C_{n}^{2}
G_{l} P_{k} W_{l} e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n}
C_{m}^{2} C_{n}^{3} G_{l} P_{k} W_{l} e^{\left(-C_{n} t - W_{l}
t\right)} - B_{m} B_{n} C_{m}^{3} G_{l} L_{k}^{2} P_{k} W_{l}
e^{\left(-C_{n} t - W_{l} t\right)} - B_{m} B_{n} C_{n}^{3} G_{l}
L_{k}^{2} P_{k} W_{l} e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n}
C_{m}^{2} G_{l} L_{k}^{3} P_{k} W_{l} e^{\left(-C_{n} t - W_{l}
t\right)} - B_{m} B_{n} C_{n}^{2} G_{l} L_{k}^{3} P_{k} W_{l}
e^{\left(-C_{n} t - W_{l} t\right)} - B_{m} B_{n} C_{m}^{3} C_{n} G_{l}
P_{k} W_{l}^{2} e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{m}
C_{n}^{3} G_{l} P_{k} W_{l}^{2} e^{\left(-C_{n} t - W_{l} t\right)} -
B_{m} B_{n} C_{m}^{3} G_{l} L_{k} P_{k} W_{l}^{2} e^{\left(-C_{n} t -
W_{l} t\right)} - B_{m} B_{n} C_{n}^{3} G_{l} L_{k} P_{k} W_{l}^{2}
e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{m} G_{l} L_{k}^{3}
P_{k} W_{l}^{2} e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{n}
G_{l} L_{k}^{3} P_{k} W_{l}^{2} e^{\left(-C_{n} t - W_{l} t\right)} -
B_{m} B_{n} C_{m}^{2} C_{n} G_{l} P_{k} W_{l}^{3} e^{\left(-C_{n} t -
W_{l} t\right)} - B_{m} B_{n} C_{m} C_{n}^{2} G_{l} P_{k} W_{l}^{3}
e^{\left(-C_{n} t - W_{l} t\right)} - B_{m} B_{n} C_{m}^{2} G_{l} L_{k}
P_{k} W_{l}^{3} e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n}
C_{n}^{2} G_{l} L_{k} P_{k} W_{l}^{3} e^{\left(-C_{n} t - W_{l}
t\right)} + B_{m} B_{n} C_{m} G_{l} L_{k}^{2} P_{k} W_{l}^{3}
e^{\left(-C_{n} t - W_{l} t\right)} + B_{m} B_{n} C_{n} G_{l} L_{k}^{2}
P_{k} W_{l}^{3} e^{\left(-C_{n} t - W_{l} t\right)} - B_{m} B_{n}
C_{m}^{3} C_{n}^{2} G_{l} L_{k} P_{k} - B_{m} B_{n} C_{m}^{2} C_{n}^{3}
G_{l} L_{k} P_{k} - B_{m} B_{n} C_{m}^{3} C_{n} G_{l} L_{k}^{2} P_{k} +
B_{m} B_{n} C_{m} C_{n}^{3} G_{l} L_{k}^{2} P_{k} + B_{m} B_{n}
C_{m}^{2} C_{n} G_{l} L_{k}^{3} P_{k} + B_{m} B_{n} C_{m} C_{n}^{2}
G_{l} L_{k}^{3} P_{k} + B_{m} B_{n} C_{m}^{3} C_{n}^{2} G_{l} P_{k}
W_{l} + B_{m} B_{n} C_{m}^{2} C_{n}^{3} G_{l} P_{k} W_{l} + B_{m} B_{n}
C_{m}^{3} G_{l} L_{k}^{2} P_{k} W_{l} - B_{m} B_{n} C_{n}^{3} G_{l}
L_{k}^{2} P_{k} W_{l} - B_{m} B_{n} C_{m}^{2} G_{l} L_{k}^{3} P_{k}
W_{l} - B_{m} B_{n} C_{n}^{2} G_{l} L_{k}^{3} P_{k} W_{l} + B_{m} B_{n}
C_{m}^{3} C_{n} G_{l} P_{k} W_{l}^{2} - B_{m} B_{n} C_{m} C_{n}^{3}
G_{l} P_{k} W_{l}^{2} + B_{m} B_{n} C_{m}^{3} G_{l} L_{k} P_{k}
W_{l}^{2} + B_{m} B_{n} C_{n}^{3} G_{l} L_{k} P_{k} W_{l}^{2} - B_{m}
B_{n} C_{m} G_{l} L_{k}^{3} P_{k} W_{l}^{2} - B_{m} B_{n} C_{n} G_{l}
L_{k}^{3} P_{k} W_{l}^{2} - 2 \, B_{m} B_{n} C_{m}^{3} G_{l} P_{k}
W_{l}^{3} - B_{m} B_{n} C_{m}^{2} C_{n} G_{l} P_{k} W_{l}^{3} - B_{m}
B_{n} C_{m} C_{n}^{2} G_{l} P_{k} W_{l}^{3} + B_{m} B_{n} C_{m}^{2}
G_{l} L_{k} P_{k} W_{l}^{3} + B_{m} B_{n} C_{n}^{2} G_{l} L_{k} P_{k}
W_{l}^{3} - B_{m} B_{n} C_{m} G_{l} L_{k}^{2} P_{k} W_{l}^{3} + B_{m}
B_{n} C_{n} G_{l} L_{k}^{2} P_{k} W_{l}^{3} + 2 \, B_{m} B_{n} G_{l}
L_{k}^{3} P_{k} W_{l}^{3} + 2 \, B_{m} B_{n} C_{m} G_{l} P_{k}
W_{l}^{5} - 2 \, B_{m} B_{n} G_{l} L_{k} P_{k} W_{l}^{5}}{C_{m}^{4}
C_{n}^{3} L_{k}^{2} + C_{m}^{3} C_{n}^{4} L_{k}^{2} + C_{m}^{4}
C_{n}^{2} L_{k}^{3} - C_{m}^{2} C_{n}^{4} L_{k}^{3} - C_{m}^{3}
C_{n}^{2} L_{k}^{4} - C_{m}^{2} C_{n}^{3} L_{k}^{4} - C_{m}^{4}
C_{n}^{3} W_{l}^{2} - C_{m}^{3} C_{n}^{4} W_{l}^{2} - C_{m}^{4}
C_{n}^{2} L_{k} W_{l}^{2} + C_{m}^{2} C_{n}^{4} L_{k} W_{l}^{2} -
C_{m}^{4} C_{n} L_{k}^{2} W_{l}^{2} - C_{m} C_{n}^{4} L_{k}^{2}
W_{l}^{2} - C_{m}^{4} L_{k}^{3} W_{l}^{2} + C_{n}^{4} L_{k}^{3}
W_{l}^{2} + C_{m}^{3} L_{k}^{4} W_{l}^{2} + C_{m}^{2} C_{n} L_{k}^{4}
W_{l}^{2} + C_{m} C_{n}^{2} L_{k}^{4} W_{l}^{2} + C_{n}^{3} L_{k}^{4}
W_{l}^{2} + C_{m}^{4} C_{n} W_{l}^{4} + C_{m}^{3} C_{n}^{2} W_{l}^{4} +
C_{m}^{2} C_{n}^{3} W_{l}^{4} + C_{m} C_{n}^{4} W_{l}^{4} + C_{m}^{4}
L_{k} W_{l}^{4} - C_{n}^{4} L_{k} W_{l}^{4} - C_{m}^{3} L_{k}^{2}
W_{l}^{4} - C_{n}^{3} L_{k}^{2} W_{l}^{4} + C_{m}^{2} L_{k}^{3}
W_{l}^{4} - C_{n}^{2} L_{k}^{3} W_{l}^{4} - C_{m} L_{k}^{4} W_{l}^{4} -
C_{n} L_{k}^{4} W_{l}^{4} - C_{m}^{2} C_{n} W_{l}^{6} - C_{m} C_{n}^{2}
W_{l}^{6} - C_{m}^{2} L_{k} W_{l}^{6} + C_{n}^{2} L_{k} W_{l}^{6} +
C_{m} L_{k}^{2} W_{l}^{6} + C_{n} L_{k}^{2} W_{l}^{6}} + \frac{B_{m}
B_{n} C_{n} G_{l} L_{k} P_{k} e^{\left(-C_{n} t - L_{k} t\right)} +
B_{m} B_{n} G_{l} L_{k}^{2} P_{k} e^{\left(-C_{n} t - L_{k} t\right)} +
B_{m} B_{n} C_{n} G_{l} P_{k} W_{l} e^{\left(-C_{n} t - L_{k}
t\right)} - B_{m} B_{n} G_{l} L_{k} P_{k} W_{l} e^{\left(-C_{n} t -
L_{k} t\right)} - 2 \, B_{m} B_{n} G_{l} P_{k} W_{l}^{2} e^{\left(-C_{n}
t - L_{k} t\right)} - B_{m} B_{n} C_{n} G_{l} L_{k} P_{k}
e^{\left(-L_{k} t - W_{l} t\right)} - B_{m} B_{n} G_{l} L_{k}^{2} P_{k}
e^{\left(-L_{k} t - W_{l} t\right)} + B_{m} B_{n} C_{n} G_{l} P_{k}
W_{l} e^{\left(-L_{k} t - W_{l} t\right)} + B_{m} B_{n} G_{l} L_{k}
P_{k} W_{l} e^{\left(-L_{k} t - W_{l} t\right)}}{C_{m} C_{n}^{2}
L_{k}^{2} + C_{m} C_{n} L_{k}^{3} - C_{n}^{2} L_{k}^{3} - C_{n}
L_{k}^{4} - C_{m} C_{n} L_{k}^{2} W_{l} - C_{m} L_{k}^{3} W_{l} + C_{n}
L_{k}^{3} W_{l} + L_{k}^{4} W_{l} - C_{m} C_{n}^{2} W_{l}^{2} - C_{m}
C_{n} L_{k} W_{l}^{2} + C_{n}^{2} L_{k} W_{l}^{2} + C_{n} L_{k}^{2}
W_{l}^{2} + C_{m} C_{n} W_{l}^{3} + C_{m} L_{k} W_{l}^{3} - C_{n} L_{k}
W_{l}^{3} - L_{k}^{2} W_{l}^{3}}\]
\end{document}