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@owlas
Last active May 29, 2022 04:09
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"# Thermal equilibrium of interacting dimer\n",
"\n",
"\n",
"In this notebook we simulate the thermal equilibrium (Boltzmann distrubion) of two interacting magnetic nanoparticles (dimer), coupled with dipolar interactions. The equilibrium distribution is computed with three different approaches\n",
" - Analytic solution is computed by hand (easy for two aligned particles)\n",
" - Markov chain Monte-Carlo (MCMC) sampling of the energy functions using PyMC3\n",
" - Simulating the stochastic Landau-Lifshitz-Gilbert equation until equilibrium using magpy\n",
" \n",
"\n",
"## Setup\n",
"\n",
"Two particles are aligned along their anisotropy axes at a distance of $R$. They have identical volume $V$, anisotropy constant $K$, saturation magnetisation $M_s$. The temperature of the environment is $T$. The angle of the magnetic moment to the anisotropy axis is $\\theta_1,\\theta_2$ for particle 1 and 2 respectively.\n",
"\n",
"![dimer](https://docs.google.com/drawings/d/1ljdYNWVgGOyGHJ5oRQp3_yOSHNxW7Kqkcp16TBl7xYw/pub?w=415&h=123)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"## Analytic solution\n",
"\n",
"The **solid angle** of the magnetisation angles follow a Boltzmann distribution, such that the angle $\\theta$ is distributed:\n",
"\n",
"\n",
"$$p\\left(\\theta_1,\\theta_2\\right) = \\frac{\\sin(\\theta_1)\\sin(\\theta_2)e^{-E\\left(\\theta_1,\\theta_2\\right)/\\left(K_BT\\right)}}{Z}$$\n",
"\n",
"where\n",
"\n",
"$$\\frac{E\\left(\\theta_1,\\theta_2\\right)}{K_BT}=\\sigma\\left(\\cos^2\\theta_1+\\cos^2\\theta_2\\right)\n",
"-\\nu\\left(3\\cos\\theta_1\\cos\\theta_2 - \\cos\\left(\\theta_1-\\theta_2\\right)\\right)$$\n",
"\n",
"$$\\sigma=\\frac{KV}{K_BT}$$\n",
"\n",
"$$\\nu=\\frac{\\mu_0V^2M_s^2}{2\\pi R^3K_BT}$$\n",
"\n",
"$\\sigma,\\nu$ are the normalised anisotropy and interaction strength respectively."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.axes_grid1 import ImageGrid\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"Individual energy terms"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"def dd(t1, t2, nu):\n",
" return -nu*(3*np.cos(t1)*np.cos(t2) - np.cos(t1-t2))\n",
"def anis(t1, t2, sigma):\n",
" return sigma*(np.sin(t1)**2 + np.sin(t2)**2)\n",
"def tot(t1, t2, nu, sigma):\n",
" return dd(t1, t2, nu) + anis(t1, t2, sigma)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"The unnormalised probability of state $\\theta_1\\theta_2$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"def p_unorm(t1,t2,nu,sigma):\n",
" return np.sin(t1)*np.sin(t2)*np.exp(-tot(t1,t2,nu,sigma))"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"2-dimensional Boltzmann distribution"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"from scipy.integrate import dblquad\n",
"def boltz_2d(ts, nu, sigma):\n",
" e = np.array([[p_unorm(t1,t2,nu,sigma) for t1 in ts] for t2 in ts])\n",
" Z = dblquad(lambda t1,t2: p_unorm(t1,t2,nu,sigma),\n",
" 0, ts[-1], lambda x: 0, lambda x: ts[-1])[0]\n",
"\n",
" return e/Z"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"## Analytical results\n",
"\n",
"We show resulting calculations for the following three cases:\n",
" - No interactions\n",
" - Strong interactions\n",
" - Strong interactions and weak anisotropy"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"nus = [0, 0.3, 0.3]\n",
"sigmas = [2.0, 2.0, 0.5]"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"### Energy landscape\n",
"\n",
"The plots below show the energy associated with each point of the 2-dimensional phase space (i.e. one angle per particle). Low energy (purple) states are energetically favourable wherea the high energy states (yellow) are not."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/plain": [
"<mpl_toolkits.axes_grid1.axes_grid.Colorbar at 0x7f722ba3b128>"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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JsychpDDmjCZVXAxnpzUhNLP7zOz68vk24BZgzdBm5wJXWcF1wIGSDm2rjqPI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\n3GWwS6QqhdAdiWr6OFLODg4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"text/plain": [
"<matplotlib.figure.Figure at 0x7f722bcb0ef0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ts = np.linspace(0, np.pi, 100)\n",
"fg = plt.figure(figsize=(10,4))\n",
"axs = ImageGrid(\n",
" fg, 111, nrows_ncols=(1,3), axes_pad=0.15,\n",
" share_all=True,cbar_location=\"right\",\n",
" cbar_mode=\"single\",cbar_size=\"7%\",\n",
" cbar_pad=0.15,\n",
")\n",
"for nu, sigma, ax in zip(nus, sigmas, axs):\n",
" e = [[tot(t1, t2, nu, sigma) for t1 in ts] for t2 in ts]\n",
" cf=ax.contourf(ts, ts, e)\n",
" ax.set_xlabel('$\\\\theta_1$'); ax.set_ylabel('$\\\\theta_2$')\n",
" ax.set_aspect('equal')\n",
"axs[0].set_title('No interactions')\n",
"axs[1].set_title('Strong interactions')\n",
"axs[2].set_title('Weak anisotropy')\n",
"ax.cax.colorbar(cf) # fix color bar"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"### Probability state-space\n",
"\n",
"The probability of each point in the phase space is directily related to the energy through the Boltzmann distribution (above). The three different cases are described:\n",
" - With no interactions, the probability space is symetrical since both variables are identically distibuted and independent of one another. Both particle are much more likely to be found close to the anisotropy axis (either parallel or antiparallel).\n",
" - With interactions, the symmetry is broken and the particles prefer to be aligned with the anisotropy axis and one another. This increases the probability of the system being in aligned states around the axes rather.\n",
" - With weak anisotropy, the particles prefer to be aligned with one another but are less likely to be found near the anisotropy axis."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/plain": [
"<mpl_toolkits.axes_grid1.axes_grid.Colorbar at 0x7f722bc6a710>"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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8CpWPsfalGrp5eOoSuKDpda5znzSps5o2+cZanruii2igRwGdMTIrEVxGkwhT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\n/3vk91pU3WJ7AAAAAElFTkSuQmCC\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f722be1cb00>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ts = np.linspace(0, np.pi, 100)\n",
"fg = plt.figure(figsize=(10,4))\n",
"axs = ImageGrid(\n",
" fg, 111, nrows_ncols=(1,3), axes_pad=0.15,\n",
" share_all=True,cbar_location=\"right\",\n",
" cbar_mode=\"single\",cbar_size=\"7%\",\n",
" cbar_pad=0.15,\n",
")\n",
"for nu, sigma, ax in zip(nus, sigmas, axs):\n",
" b = boltz_2d(ts, nu, sigma)\n",
" cf=ax.contourf(ts, ts, b)\n",
" ax.set_xlabel('$\\\\theta_1$'); ax.set_ylabel('$\\\\theta_2$')\n",
" ax.set_aspect('equal')\n",
"axs[0].set_title('No interactions')\n",
"axs[1].set_title('Strong interactions')\n",
"axs[2].set_title('Weak anisotropy')\n",
"ax.cax.colorbar(cf) # fix color bar"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"## Markov-chain Monte-Carlo (MCMC)\n",
"\n",
"Computing the analytic solutions required the partition function $Z$ to be computed, which was achieved with numerical integration. For more than 2 particles this becomes extremely difficult. A more efficient approach is to use MCMC. In MCMC we randomly choose a system state and add it to a histogram depending on how energetically favourable the state is. The more states we try, the closer our histogram becomes to the true distribution.\n",
"\n",
"We use PyMC3 for MCMC sampling. The model is simple:\n",
"\n",
" - We sample uniformly on the unit sphere (i.e. draw solid angles uniformly), converting into the angle $\\theta_1$ and $\\theta_2$\n",
" - We compute the energy for both particles\n",
" - We pass this energy to PyMC3\n",
" \n",
"PyMC3 uses the NUTS algorithm to efficicently sample trial states with a high acceptance rate."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"ERROR (theano.sandbox.cuda): Failed to compile cuda_ndarray.cu: libcublas.so.8.0: cannot open shared object file: No such file or directory\n"
]
}
],
"source": [
"import pymc3 as pm"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"Setting up the PyMC3 model is simple! Specify the priors and the energy function using `pm.Potential`"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"nu = 0.3\n",
"sigma = 1.5\n",
"\n",
"with pm.Model() as model:\n",
" z1 = pm.Uniform('z1', -1, 1)\n",
" theta1 = pm.Deterministic('theta1', np.arccos(z1))\n",
" z2 = pm.Uniform('z2', -1, 1)\n",
" theta2 = pm.Deterministic('theta2', np.arccos(z2))\n",
" \n",
" energy = tot(theta1, theta2, nu, sigma)\n",
" \n",
" like = pm.Potential('energy', -energy)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"Chose the NUTS algorithm as our MCMC step and request a large number of random samples. These are returned in the `trace` variable."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"100%|██████████| 100500/100500 [00:53<00:00, 1863.73it/s]\n"
]
}
],
"source": [
"with model:\n",
" step = pm.NUTS()\n",
" trace = pm.sample(100000, step=step)"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"### Compare results\n",
"\n",
"The `trace` contains a large number of samples of the system states. The distribution (histogram) over this large sample of states should be close to the true distribution. We can compare the 2D histogram of the system angles to the analytic solution we computed previously. "
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"image/png": 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h4i+dRWoa03dvp7Cte1mVC+xcjtLhIxz+0iGyR0ZwFK5WmCYi4KeMjVu2kA23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epc5AM6+4JRepw66+pv/Gk4PlXl8ZdHWdUZ4d03dy28Qwuz2NER4/ffto73ru1gYKzEv/x4mL9/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\nksdGMQt5XE2juG4Fu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"text/plain": [
"<matplotlib.figure.Figure at 0x7f721ffbe240>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"b = boltz_2d(ts, nu, sigma)\n",
"plt.hist2d(trace['theta1'], trace['theta2'], bins=70, normed=True)\n",
"plt.contour(ts, ts, b, cmap='Greys')\n",
"plt.gca().set_aspect('equal')\n",
"plt.xlabel('$\\\\theta_1$'); plt.ylabel('$\\\\theta_2$');"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
},
"source": [
"### Marginalisation over one variable\n",
"\n",
"The results look good! We can also check that the marginal distributions of the two angles are correct. The marginal distributions are also easier to check by eye.\n",
"\n",
"$$p(\\theta_1) = \\int p(\\theta_1, \\theta_2) \\mathrm{d}\\theta_2$$\n",
"$$p(\\theta_2) = \\int p(\\theta_1, \\theta_2) \\mathrm{d}\\theta_1$$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": true,
"deletable": true,
"editable": true
},
"outputs": [],
"source": [
"from scipy.integrate import trapz\n",
"b_marginal = -b.sum(axis=0) / trapz(ts, b.sum(axis=0))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
},
"outputs": [
{
"data": {
"text/plain": [
"<matplotlib.text.Text at 0x7f722be1ca58>"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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IkphUlU9nM1lJmXZgeoALEyVrivbk6dhRhEX5e/HThMj7XY5bbJ1CetkU+Poh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\nAAAASUVORK5CYII=\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7211ad8400>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fg, axs = plt.subplots(ncols=2, figsize=(9,3))\n",
"for i in range(2):\n",
" axs[i].hist(trace['theta{}'.format(i+1)], bins=100, normed=True)\n",
" axs[i].plot(ts, b_marginal, lw=2)\n",
" axs[i].set_xlabel('$\\\\theta_{}$'.format(i+1))\n",
" axs[i].set_ylabel('$p(\\\\theta_{})$'.format(i+1))\n",
"plt.suptitle('Marginal distributions', fontsize=18)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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