added references to paper on arXiv (http://arxiv.org/abs/2405.18381)

PySTFP/symbolic.py

@@ -17,7 +17,7 @@
 
 [1] Short-time Fokker-Planck propagator beyond the Gaussian approximation,
     Julian Kappler
-	[update once the paper is on the arXiv]
+	arXiv: http://arxiv.org/abs/2405.18381
 
 ''';
 

README.md

@@ -79,4 +79,4 @@ PySTFP requires sympy and numpy, the example notebooks furthermore use matplotli
 
 ## References
 
-<a id="ref_1">[1] **Short-time Fokker-Planck propagator beyond the Gaussian approximation**. J. Kappler. arXiv: add link once paper has been uploaded to the arXiv.</a>
+<a id="ref_1">[1] **Short-time Fokker-Planck propagator beyond the Gaussian approximation**. J. Kappler. arXiv: [2405.18381](http://arxiv.org/abs/2405.18381).</a>

examples/derivations/entropy production rates.ipynb

@@ -6,7 +6,7 @@
    "source": [
     "## Introduction\n",
     "\n",
-    "We here calculate the various entropy production (rates) discussed in Appendix D of the paper *[add reference here once the paper is on the arXiv]*, which we in the following refer to as Ref. [1]."
+    "We here calculate the various entropy production (rates) discussed in Appendix D of the paper \"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), which we in the following refer to as Ref. [1]."
    ]
   },
   {

examples/derivations/perturbative propagators and moments.ipynb

@@ -26,7 +26,7 @@
    "source": [
     "## Introduction\n",
     "\n",
-    "This notebook contains the sympy code to derive the explicit analytical perturbative expressions discussed in [add reference once the paper is on arXiv], which in the following we refer to as Ref. [1].\n",
+    "This notebook contains the sympy code to derive the explicit analytical perturbative expressions discussed in the paper \"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), which in the following we refer to as Ref. [1].\n",
     "\n",
     "In this notebook, we symbolically calculate\n",
     "* the coefficients of the normalization-preserving propagator (NPP; Eq. (23) in Ref. [1])\n",

examples/entropy production in physical units.ipynb

@@ -9,7 +9,7 @@
     "In this notebook, we print the leading-order short-time behavior of the medium entropy production rate, the total entropy production rate, and the Gibbs entropy.\n",
     "\n",
     "The expressions we print are Eqs. (D15), (D16), (D17) in the paper\n",
-    "*[add arXiv reference once the paper has been uploaded]*, which in the following\n",
+    "\"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), which in the following\n",
     "we refer to as Ref. [1].\n",
     "\n",
     "For this, we first retrieve the dimensionless perturbative expressions from the module PySTFP, and then rewrite them in physical units.\n",

examples/numerical example/analytical solution.ipynb

@@ -41,7 +41,7 @@
    "source": [
     "## Introduction\n",
     "\n",
-    "In the paper [add reference once paper is on the arXiv], we derive an approximate short-time propagator for the Fokker-Planck equation. The validate our approximate propagator, we compare it to an exact analytical solution of the Fokker-Planck equation.\n",
+    "In the paper \"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), we derive an approximate short-time propagator for the Fokker-Planck equation. The validate our approximate propagator, we compare it to an exact analytical solution of the Fokker-Planck equation.\n",
     "\n",
     "In the present notebook, we evaluate both this exact analytical solution of the Fokker-Planck equation, and also use it to calculate some expectation values. \n",
     "\n",

examples/plot propagator.ipynb

@@ -10,7 +10,7 @@
     "drift and diffusivity profile.\n",
     "\n",
     "We use the normalization-preserving propagator Eq. (23) from \n",
-    "*[add arXiv reference once the paper has been uploaded]*, which in the following\n",
+    "\"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), which in the following\n",
     "we refer to as Ref. [1].\n",
     "\n",
     "The positivity-preserving propagator can be used by setting ''positivity_preserving'' \n",

examples/propagator in physical units.ipynb

@@ -12,7 +12,7 @@
     "propagator.\n",
     "\n",
     "The expressions we print reproduce the results from App. B of the paper\n",
-    "*[add arXiv reference once the paper has been uploaded]*, which in the following\n",
+    "\"Short-time Fokker-Planck propagator beyond the Gaussian approximation\" (arXiv: http://arxiv.org/abs/2405.18381), which in the following\n",
     "we refer to as Ref. [1].\n",
     "\n",
     "In the following, we first retrieve the dimensionless perturbative expressions \n",

examples/useful_definitions.py

@@ -1,6 +1,9 @@
 import sympy as sp
 from math import factorial 
 
+# References:
+# [1] "Short-time Fokker-Planck propagator beyond the Gaussian approximation"
+#     J. Kappler. arXiv: http://arxiv.org/abs/2405.18381
 
 epsilon = sp.symbols(r'\epsilon',real=True,positive=True)