Skip to content

Commit

Permalink
Merge pull request #169 from sandialabs/rescale-potential-stiffness
Browse files Browse the repository at this point in the history 
rescale potential stiffness
  • Loading branch information
@mrbuche
mrbuche authored Mar 21, 2023
2 parents 5b23566 + dff54f6 commit 6989d73
Show file tree
Hide file tree
Showing 19 changed files with 528 additions and 790 deletions.
45 changes: 13 additions & 32 deletions ...ource/physics/single_chain/fjc/thermodynamics/modified_canonical/example_asymptotic.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -27,17 +27,11 @@
},
{
"cell_type": "markdown",
"id": "8abeab96",
"metadata": {},
"source": [
"## Strong potential"
]
},
{
"cell_type": "markdown",
"id": "8c7dadae",
"id": "1592d36b",
"metadata": {},
"source": [
"## Strong potential\n",
"\n",
"For sufficiently strong potentials, the modified canonical ensemble can be accurately approximated using the reference system (the isometric ensemble) and an asymptotic correction. For example, the nondimensional force $\\eta$ as a function of the nondimensional potential distance $\\gamma$ is approximated as\n",
"\n",
"$$\n",
Expand Down Expand Up @@ -68,11 +62,10 @@
"source": [
"gamma = np.linspace(1e-3, 99e-2, 256)\n",
"for varpi in [1e0, 1e1, 1e2]:\n",
" w = varpi*fjc.number_of_links**2\n",
" eta = fjc.nondimensional_force(gamma, w)\n",
" eta = fjc.nondimensional_force(gamma, varpi)\n",
" line = plt.plot(gamma, eta, label=r'$\\varpi=$' + str(varpi))\n",
" eta_asymptotic = \\\n",
" fjc.asymptotic.strong_potential.nondimensional_force(gamma, w)\n",
" fjc.asymptotic.strong_potential.nondimensional_force(gamma, varpi)\n",
" plt.plot(gamma, eta_asymptotic, ':', color=line[0].get_color())\n",
"plt.legend()\n",
"plt.xlim([0, 1])\n",
Expand All @@ -84,17 +77,11 @@
},
{
"cell_type": "markdown",
"id": "1d21a64d",
"metadata": {},
"source": [
"## Weak potential"
]
},
{
"cell_type": "markdown",
"id": "dce6f3d2",
"id": "8a77ab06",
"metadata": {},
"source": [
"## Weak potential\n",
"\n",
"For sufficiently distant potentials, the modified canonical ensemble can be accurately approximated using the reference system (the isotensional ensemble) and an asymptotic correction. The potential is considered sufficiently distant when the length of center of the potential well to the end of the chain experiencing it is much larger than the expected end-to-end length of the chain. This disparity in length is only typically possible considering weak potentials. For example, if $\\eta/N_b\\varpi$ is the nondimensional potential distance, the nondimensional end-to-end length per link $\\gamma$ as a function of the effective nondimensional potential force $\\eta$ is approximated as\n",
"\n",
"$$\n",
Expand All @@ -105,14 +92,9 @@
"\n",
"$$\n",
" \\gamma(\\eta) = \\frac{1}{N_b}\\frac{\\partial}{\\partial\\eta}\\,\\ln\\left[\\iiint Q_0(\\boldsymbol{\\gamma}') \\, e^{N_b\\boldsymbol{\\eta}\\cdot\\boldsymbol{\\gamma}'} e^{-\\tfrac{\\varpi}{2} N_b^2\\left(\\boldsymbol{\\gamma}'\\right)^2} d^3\\boldsymbol{\\gamma}'\\right].\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"id": "8a77ab06",
"metadata": {},
"source": [
"$$\n",
"\n",
"\n",
"This exact relation is plotted below along with the asymptotic relation while varying $\\varpi$, the nondimensional potential stiffness. As $\\varpi$ decreases and/or the nondimensional force $\\eta$ increases (the nondimensional potential distance $\\eta/N_b\\varpi$ increases), the asymptotic approach appears to do increasingly well. Notably, the asymptoic approach appears to succeed for sufficiently distance potentials for any value of $\\varpi$. This is because the freely-jointed chain model has inextensible links, so even a stiff potential that is distant ($\\eta/N_b\\varpi\\gg 1$) will not stretch the chain past $\\gamma=1$. For chain models which have extensible links, the link stiffness will compete with the potential stiffness, such that the potential would need to be weak ($\\varpi\\ll 1$) in addition to distant ($\\eta/N_b\\varpi\\gg 1$) in order for the asymptotic approach to become accurate."
]
},
Expand All @@ -125,11 +107,10 @@
"source": [
"gamma_in = np.linspace(5e-1, 5, 256)\n",
"for varpi in [1e0, 1e-1, 1e-2]:\n",
" w = varpi*fjc.number_of_links**2\n",
" gamma = fjc.nondimensional_end_to_end_length_per_link(gamma_in, w)\n",
" gamma = fjc.nondimensional_end_to_end_length_per_link(gamma_in, varpi)\n",
" line = plt.plot(gamma, gamma_in, label=r'$\\varpi=$' + str(varpi))\n",
" gamma_asymptotic = \\\n",
" fjc.asymptotic.weak_potential.nondimensional_end_to_end_length_per_link(gamma_in, w)\n",
" fjc.asymptotic.weak_potential.nondimensional_end_to_end_length_per_link(gamma_in, varpi)\n",
" plt.plot(gamma_asymptotic, gamma_in, ':', color=line[0].get_color())\n",
"plt.legend()\n",
"plt.xlim([0, 1])\n",
Expand Down
15 changes: 7 additions & 8 deletions ...ics/single_chain/fjc/thermodynamics/modified_canonical/asymptotic/strong_potential/mod.rs
View file
Original file line number Diff line number Diff line change
Expand Up @@ -30,8 +30,7 @@ pub struct FJC
/// The expected force as a function of the applied potential distance, potential stiffness, and temperature, parameterized by the number of links and link length.
pub fn force(number_of_links: &u8, link_length: &f64, potential_distance: &f64, potential_stiffness: &f64, temperature: &f64) -> f64
{
let contour_length = (*number_of_links as f64)*link_length;
BOLTZMANN_CONSTANT*temperature/link_length*nondimensional_force(number_of_links, &(*potential_distance/contour_length), &(potential_stiffness*contour_length.powi(2)/BOLTZMANN_CONSTANT/temperature))
BOLTZMANN_CONSTANT*temperature/link_length*nondimensional_force(number_of_links, &(*potential_distance/((*number_of_links as f64)*link_length)), &(potential_stiffness*link_length.powi(2)/BOLTZMANN_CONSTANT/temperature))
}

/// The expected nondimensional force as a function of the applied nondimensional potential distance and nondimensional potential stiffness, parameterized by the number of links.
Expand All @@ -46,21 +45,19 @@ pub fn nondimensional_force(number_of_links: &u8, nondimensional_potential_dista
let sum_1: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p - 1)).sum();
let sum_2: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p - 2)).sum();
let sum_3: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p - 3)).sum();
(1.0/nondimensional_potential_distance + (0.5*number_of_links_f64 - 1.0)*sum_1/sum_0)/number_of_links_f64 + 0.5/nondimensional_potential_stiffness/number_of_links_f64*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0*((number_of_links_f64 - 2.0)*(sum_1/sum_0).powi(2) - (number_of_links_f64 - 3.0)*sum_2/sum_0) - (0.5*number_of_links_f64 - 1.5)*((0.5*number_of_links_f64 - 1.0)*sum_1*sum_2/sum_0.powi(2) - (0.5*number_of_links_f64 - 2.0)*sum_3/sum_0)) + 2.0*nondimensional_potential_distance.powi(-3) - 2.0*((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0 + nondimensional_potential_distance.powi(-1))*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*(sum_1/sum_0).powi(2) - (0.5*number_of_links_f64 - 1.5)*sum_2/sum_0) - nondimensional_potential_distance.powi(-2)))
(1.0/nondimensional_potential_distance + (0.5*number_of_links_f64 - 1.0)*sum_1/sum_0)/number_of_links_f64 + 0.5/nondimensional_potential_stiffness/number_of_links_f64.powi(3)*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0*((number_of_links_f64 - 2.0)*(sum_1/sum_0).powi(2) - (number_of_links_f64 - 3.0)*sum_2/sum_0) - (0.5*number_of_links_f64 - 1.5)*((0.5*number_of_links_f64 - 1.0)*sum_1*sum_2/sum_0.powi(2) - (0.5*number_of_links_f64 - 2.0)*sum_3/sum_0)) + 2.0*nondimensional_potential_distance.powi(-3) - 2.0*((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0 + nondimensional_potential_distance.powi(-1))*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*(sum_1/sum_0).powi(2) - (0.5*number_of_links_f64 - 1.5)*sum_2/sum_0) - nondimensional_potential_distance.powi(-2)))
}

/// The Helmholtz free energy as a function of the applied potential distance, potential stiffness, and temperature, parameterized by the number of links, link length, and hinge mass.
pub fn helmholtz_free_energy(number_of_links: &u8, link_length: &f64, hinge_mass: &f64, potential_distance: &f64, potential_stiffness: &f64, temperature: &f64) -> f64
{
let contour_length = (*number_of_links as f64)*link_length;
nondimensional_helmholtz_free_energy(number_of_links, link_length, hinge_mass, &(potential_distance/contour_length), &(potential_stiffness*contour_length.powi(2)/BOLTZMANN_CONSTANT/temperature), temperature)*BOLTZMANN_CONSTANT*temperature
nondimensional_helmholtz_free_energy(number_of_links, link_length, hinge_mass, &(potential_distance/((*number_of_links as f64)*link_length)), &(potential_stiffness*link_length.powi(2)/BOLTZMANN_CONSTANT/temperature), temperature)*BOLTZMANN_CONSTANT*temperature
}

/// The Helmholtz free energy per link as a function of the applied potential distance, potential stiffness, and temperature, parameterized by the number of links, link length, and hinge mass.
pub fn helmholtz_free_energy_per_link(number_of_links: &u8, link_length: &f64, hinge_mass: &f64, potential_distance: &f64, potential_stiffness: &f64, temperature: &f64) -> f64
{
let contour_length = (*number_of_links as f64)*link_length;
nondimensional_helmholtz_free_energy_per_link(number_of_links, link_length, hinge_mass, &(potential_distance/contour_length), &(potential_stiffness*contour_length.powi(2)/BOLTZMANN_CONSTANT/temperature), temperature)*BOLTZMANN_CONSTANT*temperature
nondimensional_helmholtz_free_energy_per_link(number_of_links, link_length, hinge_mass, &(potential_distance/((*number_of_links as f64)*link_length)), &(potential_stiffness*link_length.powi(2)/BOLTZMANN_CONSTANT/temperature), temperature)*BOLTZMANN_CONSTANT*temperature
}

/// The relative Helmholtz free energy as a function of the applied potential distance, potential stiffness, and temperature, parameterized by the number of links and link length.
Expand All @@ -74,10 +71,12 @@ pub fn relative_helmholtz_free_energy_per_link(number_of_links: &u8, link_length
{
helmholtz_free_energy_per_link(number_of_links, link_length, &1.0, potential_distance, potential_stiffness, temperature) - helmholtz_free_energy_per_link(number_of_links, link_length, &1.0, &(ZERO*(*number_of_links as f64)*link_length), potential_stiffness, temperature)
}

/// The nondimensional Helmholtz free energy as a function of the applied nondimensional potential distance, nondimensional potential stiffness, and temperature, parameterized by the number of links, link length, and hinge mass.
pub fn nondimensional_helmholtz_free_energy(number_of_links: &u8, link_length: &f64, hinge_mass: &f64, nondimensional_potential_distance: &f64, nondimensional_potential_stiffness: &f64, temperature: &f64) -> f64
{
let number_of_links_f64 = *number_of_links as f64;
let number_of_links_squared = number_of_links_f64.powi(2);
let contour_length = number_of_links_f64*link_length;
let n = *number_of_links as u128;
let p: i32 = (number_of_links - 2).into();
Expand All @@ -86,7 +85,7 @@ pub fn nondimensional_helmholtz_free_energy(number_of_links: &u8, link_length: &
let sum_0: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p)).sum();
let sum_1: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p - 1)).sum();
let sum_2: f64 = (0..=k-1).collect::<Vec::<u128>>().iter().map(|s| (-1.0_f64).powf(*s as f64)*(((1..=n).product::<u128>()/(1..=*s).product::<u128>()/(1..=n-s).product::<u128>()) as f64)*(m - (*s as f64)/number_of_links_f64).powi(p - 2)).sum();
-(0.125/PI/nondimensional_potential_distance*(n.pow(n as u32) as f64)/((1..=n-2).product::<u128>() as f64)*sum_0/contour_length.powi(3)).ln() - (number_of_links_f64 - 1.0)*(8.0*PI.powi(2)*hinge_mass*link_length.powi(2)*BOLTZMANN_CONSTANT*temperature/PLANCK_CONSTANT.powi(2)).ln() - 1.5*(2.0*PI/nondimensional_potential_stiffness).ln() - 3.0*(contour_length).ln() + 0.5/nondimensional_potential_stiffness*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*(sum_1/sum_0).powi(2) - (0.5*number_of_links_f64 - 1.5)*sum_2/sum_0) - nondimensional_potential_distance.powi(-2) - ((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0 + nondimensional_potential_distance.powi(-1)).powi(2))
-(0.125/PI/nondimensional_potential_distance*(n.pow(n as u32) as f64)/((1..=n-2).product::<u128>() as f64)*sum_0/contour_length.powi(3)).ln() - (number_of_links_f64 - 1.0)*(8.0*PI.powi(2)*hinge_mass*link_length.powi(2)*BOLTZMANN_CONSTANT*temperature/PLANCK_CONSTANT.powi(2)).ln() - 1.5*(2.0*PI/nondimensional_potential_stiffness/number_of_links_squared).ln() - 3.0*(contour_length).ln() + 0.5/nondimensional_potential_stiffness/number_of_links_squared*((0.5*number_of_links_f64 - 1.0)*((0.5*number_of_links_f64 - 1.0)*(sum_1/sum_0).powi(2) - (0.5*number_of_links_f64 - 1.5)*sum_2/sum_0) - nondimensional_potential_distance.powi(-2) - ((0.5*number_of_links_f64 - 1.0)*sum_1/sum_0 + nondimensional_potential_distance.powi(-1)).powi(2))
}

/// The nondimensional Helmholtz free energy per link as a function of the applied nondimensional potential distance, nondimensional potential stiffness, and temperature, parameterized by the number of links, link length, and hinge mass.
Expand Down
24 changes: 12 additions & 12 deletions ...cs/single_chain/fjc/thermodynamics/modified_canonical/asymptotic/strong_potential/test.jl
View file
Original file line number Diff line number Diff line change
Expand Up @@ -93,7 +93,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
force = model.force(potential_distance, potential_stiffness, temperature)
Expand Down Expand Up @@ -128,7 +128,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
helmholtz_free_energy = model.helmholtz_free_energy(
Expand Down Expand Up @@ -169,7 +169,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
helmholtz_free_energy_per_link = model.helmholtz_free_energy_per_link(
Expand Down Expand Up @@ -209,7 +209,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
relative_helmholtz_free_energy = model.relative_helmholtz_free_energy(
Expand Down Expand Up @@ -249,7 +249,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
relative_helmholtz_free_energy_per_link =
Expand Down Expand Up @@ -285,7 +285,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
helmholtz_free_energy = model.helmholtz_free_energy(
Expand Down Expand Up @@ -325,7 +325,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
relative_helmholtz_free_energy = model.relative_helmholtz_free_energy(
Expand Down Expand Up @@ -438,7 +438,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
helmholtz_free_energy = model.helmholtz_free_energy(
Expand Down Expand Up @@ -482,7 +482,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
helmholtz_free_energy_per_link = model.helmholtz_free_energy_per_link(
Expand Down Expand Up @@ -605,7 +605,7 @@ end
temperature =
parameters.temperature_reference + parameters.temperature_scale * (0.5 - rand())
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
relative_helmholtz_free_energy_0 = model.relative_helmholtz_free_energy(
Expand Down Expand Up @@ -633,7 +633,7 @@ end
temperature =
parameters.temperature_reference + parameters.temperature_scale * (0.5 - rand())
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
relative_helmholtz_free_energy_per_link_0 =
Expand Down Expand Up @@ -707,7 +707,7 @@ end
potential_distance =
nondimensional_potential_distance * number_of_links * link_length
potential_stiffness =
nondimensional_potential_stiffness / (number_of_links * link_length)^2 *
nondimensional_potential_stiffness / link_length^2 *
BOLTZMANN_CONSTANT *
temperature
force = model.force(potential_distance, potential_stiffness, temperature)
Expand Down
Loading

2 comments on commit 6989d73

@github-actions
Copy link

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

@JuliaRegistrator register

@JuliaRegistrator
Copy link

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Registration pull request created: JuliaRegistries/General/80059

After the above pull request is merged, it is recommended that a tag is created on this repository for the registered package version.

This will be done automatically if the Julia TagBot GitHub Action is installed, or can be done manually through the github interface, or via:

git tag -a v0.2.1 -m "<description of version>" 6989d73e1c49591f00132feae85bf3dff9d69792
git push origin v0.2.1

Please sign in to comment.