Adding Chudley-Elliott Jump Diffusion Model

docs/docs/tutorials/diffusion_model.ipynb

@@ -6,7 +6,7 @@
    "metadata": {},
    "source": [
     "# Diffusion Model\n",
-    "We support several standard models of diffusion. Here we show an example of Brownian Translational Diffusion, where the scattering is a Lorentzian with width ($\\Gamma$) given by $\\Gamma = D Q^2$, where $D$ is the diffusion coefficient (in m$^2$/s) and $Q$ is the momentum transfer."
+    "We support several standard models of diffusion. Here we show examples of Brownian Translational Diffusion, Jump Translational Diffusion and Chudley-Elliott Jump Diffusion. In each model the scattering is a Lorentzian for every $Q$, and the models differ in how the width $\\Gamma(Q)$ depends on $Q$. For Brownian Translational Diffusion, the width is given by $\\Gamma = D Q^2$, where $D$ is the diffusion coefficient (in m$^2$/s) and $Q$ is the momentum transfer."
    ]
   },
   {
@@ -24,10 +24,26 @@
     "%matplotlib widget"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "2",
+   "metadata": {},
+   "source": [
+    "## Brownian Translational Diffusion"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3",
+   "metadata": {},
+   "source": [
+    "### Model for different Q values"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "2",
+   "id": "4",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -62,10 +78,18 @@
     "plt.title('Brownian Translational Diffusion Model')"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "5",
+   "metadata": {},
+   "source": [
+    "### Width as a function of Q"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "3",
+   "id": "6",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -81,6 +105,196 @@
     "plt.ylim(0, max(HWHM) * 1.1)\n",
     "plt.title('HWHM vs Q for Brownian Translational Diffusion')"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7",
+   "metadata": {},
+   "source": [
+    "## Jump Translational Diffusion\n",
+    "In Jump Translational Diffusion, the particle waits for a relaxation time $t$ between jumps. At low $Q$ the width follows the Brownian result, $\\Gamma \\approx \\hbar D Q^2$, but at high $Q$ it levels off:\n",
+    "\n",
+    "$$\\Gamma(Q) = \\frac{\\hbar D Q^2}{1 + D t Q^2},$$\n",
+    "\n",
+    "where $D$ is the diffusion coefficient (in m$^2$/s) and $t$ is the relaxation time (in ps)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8",
+   "metadata": {},
+   "source": [
+    "### Model for different Q values"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create Jump Translational Diffusion model\n",
+    "# and plot the model for different Q values.\n",
+    "# Q is in Angstrom^-1 and energy in meV.\n",
+    "\n",
+    "Q = np.linspace(0.5, 2, 7)\n",
+    "energy = np.linspace(-2, 2, 501)\n",
+    "scale = 1.0\n",
+    "diffusion_coefficient = 2.4e-9  # m^2/s\n",
+    "relaxation_time = 1.0  # ps\n",
+    "\n",
+    "jump_diffusion_model = edyn.JumpTranslationalDiffusion(\n",
+    "    display_name='JumpDiffusionModel',\n",
+    "    scale=scale,\n",
+    "    diffusion_coefficient=diffusion_coefficient,\n",
+    "    relaxation_time=relaxation_time,\n",
+    "    Q=Q,\n",
+    ")\n",
+    "\n",
+    "component_collections = jump_diffusion_model.get_component_collections()\n",
+    "\n",
+    "cmap = plt.cm.jet\n",
+    "nQ = len(component_collections)\n",
+    "plt.figure()\n",
+    "for Q_index in range(len(component_collections)):\n",
+    "    color = cmap(Q_index / (nQ - 1))\n",
+    "    y = component_collections[Q_index].evaluate(energy)\n",
+    "    plt.plot(energy, y, label=f'Q={Q[Q_index]} Å^-1', color=color)\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlabel('Energy (meV)')\n",
+    "plt.ylabel('Intensity (arb. units)')\n",
+    "plt.title('Jump Translational Diffusion Model')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10",
+   "metadata": {},
+   "source": [
+    "### Width as a function of Q"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "11",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Calculate and plot the half width at half maximum (HWHM) as function\n",
+    "# of Q\n",
+    "Q = np.linspace(0.1, 2, 101)\n",
+    "HWHM = jump_diffusion_model.calculate_width(Q)\n",
+    "plt.figure()\n",
+    "plt.plot(Q, HWHM)\n",
+    "plt.xlabel('Q (Å$^{-1}$)')\n",
+    "plt.ylabel('HWHM (meV)')\n",
+    "plt.xlim(0, 2.5)\n",
+    "plt.ylim(0, max(HWHM) * 1.1)\n",
+    "plt.title('HWHM vs Q for Jump Translational Diffusion')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12",
+   "metadata": {},
+   "source": [
+    "## Chudley-Elliott Jump Diffusion\n",
+    "The Chudley-Elliott model of\n",
+    "\n",
+    "> C. T. Chudley and R. J. Elliott, *Neutron Scattering from a Liquid on a Jump Diffusion Model*,\n",
+    "> Proc. Phys. Soc. **77**, 353 (1961),\n",
+    "> [doi:10.1088/0370-1328/77/2/319](https://doi.org/10.1088/0370-1328/77/2/319).\n",
+    "\n",
+    "describes a particle that sits on a site for a residence time $\\tau$ and then jumps a fixed distance $l$. The width is\n",
+    "\n",
+    "$$\\Gamma(Q) = \\frac{\\hbar}{\\tau}\\left(1 - \\frac{\\sin(Ql)}{Ql}\\right),$$\n",
+    "\n",
+    "where $\\tau$ is the residence time (in ps) and $l$ is the jump length (in Å). The width oscillates with $Q$ and approaches $\\hbar/\\tau$ at high $Q$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "13",
+   "metadata": {},
+   "source": [
+    "### Model for different Q values"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "14",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create Chudley-Elliott Jump Diffusion model\n",
+    "# and plot the model for different Q values.\n",
+    "# Q is in Angstrom^-1 and energy in meV.\n",
+    "\n",
+    "Q = np.linspace(0.5, 2, 7)\n",
+    "energy = np.linspace(-2, 2, 501)\n",
+    "scale = 1.0\n",
+    "residence_time = 1.0  # ps\n",
+    "jump_length = 1.5  # Å\n",
+    "\n",
+    "chudley_elliott_model = edyn.ChudleyElliottJumpDiffusion(\n",
+    "    display_name='ChudleyElliottModel',\n",
+    "    scale=scale,\n",
+    "    residence_time=residence_time,\n",
+    "    jump_length=jump_length,\n",
+    "    Q=Q,\n",
+    ")\n",
+    "\n",
+    "component_collections = chudley_elliott_model.get_component_collections()\n",
+    "\n",
+    "cmap = plt.cm.jet\n",
+    "nQ = len(component_collections)\n",
+    "plt.figure()\n",
+    "for Q_index in range(len(component_collections)):\n",
+    "    color = cmap(Q_index / (nQ - 1))\n",
+    "    y = component_collections[Q_index].evaluate(energy)\n",
+    "    plt.plot(energy, y, label=f'Q={Q[Q_index]} Å^-1', color=color)\n",
+    "\n",
+    "plt.legend()\n",
+    "plt.xlabel('Energy (meV)')\n",
+    "plt.ylabel('Intensity (arb. units)')\n",
+    "plt.title('Chudley-Elliott Jump Diffusion Model')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "15",
+   "metadata": {},
+   "source": [
+    "### Width as a function of Q"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "16",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Calculate and plot the half width at half maximum (HWHM) as function\n",
+    "# of Q\n",
+    "Q = np.linspace(0.1, 4, 201)\n",
+    "HWHM = chudley_elliott_model.calculate_width(Q)\n",
+    "plt.figure()\n",
+    "plt.plot(Q, HWHM)\n",
+    "plt.xlabel('Q (Å$^{-1}$)')\n",
+    "plt.ylabel('HWHM (meV)')\n",
+    "plt.xlim(0, 4.2)\n",
+    "plt.ylim(0, max(HWHM) * 1.1)\n",
+    "plt.title('HWHM vs Q for Chudley-Elliott Jump Diffusion')\n",
+    "plt.show()"
+   ]
   }
  ],
  "metadata": {

docs/docs/tutorials/index.md

@@ -49,7 +49,8 @@ tutorials.
 - [Detailed balance](detailed_balance.ipynb) – Learn about detailed
   balancing.
 - [Diffusion model](diffusion_model.ipynb) – Learn how to create and use
-  a model of diffusion.
+  models of diffusion: Brownian translational, jump translational and
+  Chudley-Elliott jump diffusion.
 - [DeltaLorentz](delta_lorentz.ipynb) – Learn how to create and use a
   model with a Delta function and a Lorentzian that have a shared
   Debye-Waller-like Q-dependence.