sandialabs · mrbuche · Aug 9, 2023 · Aug 9, 2023
diff --git a/docs/source/main.bib b/docs/source/main.bib
@@ -2,93 +2,93 @@ @book{treloar1949physics
   title={The Physics of Rubber Elasticity},
   author={Treloar, Leslie Ronald George},
   year={1949},
-  location = {Cambridge, UK},
+  location={Cambridge, UK},
   publisher={Clarendon Press},
-  url = {https://global.oup.com/academic/product/the-physics-of-rubber-elasticity-9780198570271}
+  url={https://global.oup.com/academic/product/the-physics-of-rubber-elasticity-9780198570271}
 }
 
 @book{rubinstein2003polymer,
   title={Polymer Physics},
   author={Rubinstein, Michael and Colby, Ralph H},
   year={2003},
-  location = {New York},
+  location={New York},
   publisher={Oxford University Press},
-  url = {https://global.oup.com/academic/product/polymer-physics-9780198520597}
+  url={https://global.oup.com/academic/product/polymer-physics-9780198520597}
 }
 
 @book{mcq,
   title={Statistical Mechanics},
   author={McQuarrie, Donald A.},
   year={2000},
-  location = {Mill Valley, CA},
+  location={Mill Valley, CA},
   publisher={University Science Books},
-  url = {https://uscibooks.aip.org/books/statistical-mechanics/}
+  url={https://uscibooks.aip.org/books/statistical-mechanics/}
 }
 
 @book{zwanzig2001nonequilibrium,
   title={Nonequilibrium Statistical Mechanics},
   author={Zwanzig, Robert},
   year={2001},
-  location = {New York},
+  location={New York},
   publisher={Oxford University Press},
-  url = {https://global.oup.com/academic/product/nonequilibrium-statistical-mechanics-9780195140187}
+  url={https://global.oup.com/academic/product/nonequilibrium-statistical-mechanics-9780195140187}
 }
 
 @book{bender2013advanced,
   title={Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory},
   author={Bender, Carl M and Orszag, Steven A},
   year={2013},
-  location = {New York},
+  location={New York},
   publisher={Springer Science \& Business Media},
-  doi = {10.1007/978-1-4757-3069-2}
+  doi={10.1007/978-1-4757-3069-2}
 }
 
 @book{bleistein1975asymptotic,
   title={Asymptotic Expansions of Integrals},
   author={Bleistein, Norman and Handelsman, Richard A},
   year={1975},
-  location = {New York},
+  location={New York},
   publisher={Ardent Media},
   url={https://store.doverpublications.com/0486650820.html}
 }
 
 @article{buche2022freely,
-  title = {Freely jointed chain models with extensible links},
-  author = {Buche, Michael R. and Silberstein, Meredith N. and Grutzik, Scott J.},
-  journal = {Physical Review E},
-  shortjournal = {Phys. Rev. E},
-  volume = {106},
-  pages = {024502},
-  number = {024502},
-  numpages = {10},
-  year = {2022},
-  publisher = {American Physical Society},
-  doi = {10.1103/PhysRevE.106.024502},
+  title={Freely jointed chain models with extensible links},
+  author={Buche, Michael R. and Silberstein, Meredith N. and Grutzik, Scott J.},
+  journal={Physical Review E},
+  shortjournal={Phys. Rev. E},
+  volume={106},
+  pages={024502},
+  number={024502},
+  numpages={10},
+  year={2022},
+  publisher={American Physical Society},
+  doi={10.1103/PhysRevE.106.024502},
 }
 
 @article{buche2020statistical,
-  title = {Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble},
-  author = {Buche, Michael R. and Silberstein, Meredith N.},
-  journal = {Physical Review E},
-  shortjournal = {Phys. Rev. E},
-  volume = {102},
-  pages = {012501},
-  number = {012501},
-  numpages = {14},
-  year = {2020},
-  publisher = {American Physical Society},
-  doi = {10.1103/PhysRevE.102.012501},
+  title={Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble},
+  author={Buche, Michael R. and Silberstein, Meredith N.},
+  journal={Physical Review E},
+  shortjournal={Phys. Rev. E},
+  volume={102},
+  pages={012501},
+  number={012501},
+  numpages={14},
+  year={2020},
+  publisher={American Physical Society},
+  doi={10.1103/PhysRevE.102.012501},
 }
 
 @article{buche2021chain,
-  title = {Chain breaking in the statistical mechanical constitutive theory of polymer networks},
-  journal = {Journal of the Mechanics and Physics of Solids},
-  volume = {156},
-  pages = {104593},
-  year = {2021},
-  issn = {0022-5096},
-  doi = {10.1016/j.jmps.2021.104593},
-  author = {Michael R. Buche and Meredith N. Silberstein},
+  title={Chain breaking in the statistical mechanical constitutive theory of polymer networks},
+  journal={Journal of the Mechanics and Physics of Solids},
+  volume={156},
+  pages={104593},
+  year={2021},
+  issn={0022-5096},
+  doi={10.1016/j.jmps.2021.104593},
+  author={Michael R. Buche and Meredith N. Silberstein},
 }
 
 @phdthesis{buche2021fundamental,
@@ -115,27 +115,38 @@ @article{fiasconaro2019analytical
   title={Analytical results of the extensible freely jointed chain model},
   author={Fiasconaro, Alessandro and Falo, Fernando},
   journal={Physica A: Statistical Mechanics and its Applications},
-  shortjournal = {Physica A},
+  shortjournal={Physica A},
   volume={532},
   pages={121929},
   year={2019},
   publisher={Elsevier},
-  doi = {10.1016/j.physa.2019.121929}
+  doi={10.1016/j.physa.2019.121929}
 }
 
 @article{radiom2017influence,
-  title = {Influence of ligand-receptor interactions on force-extension behavior within the freely jointed chain model},
-  author = {Radiom, Milad and Borkovec, Michal},
-  journal = {Physical Review E},
-  shortjournal = {Phys. Rev. E},
-  volume = {96},
-  issue = {6},
-  pages = {062501},
-  numpages = {7},
-  year = {2017},
-  month = {Dec},
-  publisher = {American Physical Society},
-  doi = {10.1103/PhysRevE.96.062501}
+  title={Influence of ligand-receptor interactions on force-extension behavior within the freely jointed chain model},
+  author={Radiom, Milad and Borkovec, Michal},
+  journal={Physical Review E},
+  shortjournal={Phys. Rev. E},
+  volume={96},
+  issue={6},
+  pages={062501},
+  numpages={7},
+  year={2017},
+  month={Dec},
+  publisher={American Physical Society},
+  doi={10.1103/PhysRevE.96.062501}
+}
+
+@article{becker2010radial,
+  title={The radial distribution function of worm-like chains},
+  author={Becker, NB and Rosa, A and Everaers, R},
+  journal={The European Physical Journal E},
+  volume={32},
+  pages={53--69},
+  year={2010},
+  publisher={Springer},
+  doi={10.1140/epje/i2010-10596-0}
 }
 
 @article{morse1929diatomic,
@@ -148,7 +159,7 @@ @article{morse1929diatomic
   pages={57},
   year={1929},
   publisher={APS},
-  doi = {10.1103/PhysRev.34.57}
+  doi={10.1103/PhysRev.34.57}
 }
 
 @article{jones1924determinationii,

diff --git a/docs/source/physics/single_chain/wlc.rst b/docs/source/physics/single_chain/wlc.rst
@@ -6,7 +6,7 @@ WLC model
 
    Thermodynamics <wlc/thermodynamics>
 
-.. autoclass:: polymers.physics.single_chain.fjc::WLC(number_of_links, link_length, hinge_mass, persistance_length)
+.. autoclass:: polymers.physics.single_chain.wlc::WLC(number_of_links, link_length, hinge_mass, persistance_length)
 
    .. autoattribute:: number_of_links
    .. autoattribute:: link_length

diff --git a/docs/source/physics/single_chain/wlc/thermodynamics.rst b/docs/source/physics/single_chain/wlc/thermodynamics.rst
@@ -5,7 +5,7 @@ WLC model thermodynamics
    :maxdepth: 1
 
    Isometric <thermodynamics/isometric>
-   Isometric <thermodynamics/isotensional>
+   Isotensional <thermodynamics/isotensional>
 
 .. autoclass:: polymers.physics.single_chain.wlc.thermodynamics::WLC(number_of_links, link_length, hinge_mass, persistance_length)
 

diff --git a/docs/source/physics/single_chain/wlc/thermodynamics/isotensional.rst b/docs/source/physics/single_chain/wlc/thermodynamics/isotensional.rst
@@ -25,10 +25,3 @@ WLC model thermodynamics (isotensional)
    .. automethod:: nondimensional_gibbs_free_energy_per_link(nondimensional_force, temperature)
    .. automethod:: nondimensional_relative_gibbs_free_energy(nondimensional_force)
    .. automethod:: nondimensional_relative_gibbs_free_energy_per_link(nondimensional_force)
-
-.. raw::
- html
-
-   <hr>
-
-.. footbibliography::
diff --git a/src/physics/single_chain/wlc/thermodynamics/isometric/mod.jl b/src/physics/single_chain/wlc/thermodynamics/isometric/mod.jl
@@ -601,9 +601,11 @@ The equilibrium probability density of end-to-end lengths ``g_\\mathrm{eq}`` as
 parameterized by the number of links ``N_b``, link length ``\\ell_b``, and persistance length ``\\ell_p``,
 
 ```math
-g_\\mathrm{eq}(\\xi) = 4\\pi\\xi^2 P_\\mathrm{eq}(\\xi).
+g_\\mathrm{eq}(\\xi) = 4\\pi\\xi^2 P_\\mathrm{eq}(\\xi),
 ```
 
+which is calculated using the accurate analytic approximation provided by [Becker, Rosa, and Everaers](https://doi.org/10.1140/epje/i2010-10596-0).
+
 $(TYPEDSIGNATURES)
 """
 function equilibrium_radial_distribution(

diff --git a/src/physics/single_chain/wlc/thermodynamics/isometric/py.rs b/src/physics/single_chain/wlc/thermodynamics/isometric/py.rs
@@ -240,10 +240,12 @@ impl WLC
     {
         nondimensional_end_to_end_length_per_link.as_array().mapv(|nondimensional_end_to_end_length_per_link: f64| super::nondimensional_equilibrium_distribution(&self.nondimensional_persistance_length, &self.normalization_nondimensional_equilibrium_distribution, &nondimensional_end_to_end_length_per_link)).into_pyarray(py)
     }
-    /// The equilibrium probability density of end-to-end lengths as a function of the end-to-end length, given by :footcite:t:`treloar1949physics` as
+    /// The equilibrium probability density of end-to-end lengths as a function of the end-to-end length,
     ///
     /// .. math::
-    ///     g_\mathrm{eq}(\xi) = 4\pi\xi^2 P_\mathrm{eq}(\xi).
+    ///     g_\mathrm{eq}(\xi) = 4\pi\xi^2 P_\mathrm{eq}(\xi),
+    ///
+    /// which is calculated using the accurate analytic approximation provided by :footcite:t:`becker2010radial`.
     ///
     /// Args:
     ///     end_to_end_length (numpy.ndarray): The end-to-end length :math:`\xi`.