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Particle in a Double well potential: A molecular dynamics toy example using openMM toolkit by dexterdev

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steemstem · @dexterdev · (edited)
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Particle in a Double well potential: A molecular dynamics toy example using openMM toolkit
<p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">In my previous articles, we have seen many examples which demonstrated classical molecular dynamics simulations of bigger systems like</p><ul style="margin: 0px 0px 1rem 2rem; padding: 0px; list-style-position: outside; line-height: 1.6; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/a-glimpse-from-my-cu" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">Protein complex in water box</a></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/let-us-cool-dmpc-bilayer-lipids-an-18-day-long-molecular-dynamics-experiment-on-hpc-facility" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">A lipid bilayer patch system in water box</a></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-4a-let-us-setup-a-simulation-and-run-it" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">Protein interacting with lipid bilayer patch in water box</a><br>etc</li></ul><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Today let us attempt something very simple. Very less math(or math that you can ignore and still you can understand most of the stuff!) and more intuitive explanation will be the focus of today's article. (Although this is what I strive always for... Don't know how successful I am in doing so. Do let me know via comments.) Let us try to get an intuition from this toy system.</p><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 26.88px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">The System</h2><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Think about a system where you have a hypothetical particle in a crowd of lot many smaller particles(ideally in the order of 10<span style="font-size: 14.4px; line-height: 0; position: relative; vertical-align: baseline; top: -0.5em;">23</span>, but lesser number will also work, which can be considered a large number. Why this number? Let that be a homework now.) The situation will be something like as below:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><img src="https://steemitimages.com/0x0/https://i.imgur.com/FizKy8z.gif" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br><em style="line-height: inherit;">Image Source:&nbsp;<a href="https://en.wikipedia.org/wiki/Brownian_motion#/media/File:Brownian_motion_large.gif" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">Wikimedia</a>, Author:&nbsp;<a href="https://commons.wikimedia.org/w/index.php?title=User:Lookang&amp;action=edit&amp;redlink=1" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">Lookang</a>, Author of computer model: Francisco Esquembre, Fu-Kwun and lookang, License:&nbsp;<a href="https://creativecommons.org/licenses/by-sa/3.0/" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">CC BY-SA 3.0</a></em></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">The small particles will have an inherent&nbsp;<em style="line-height: inherit;">random motion</em>&nbsp;in them where they collide each other and with the bigger particle too. Now this bigger particle will also exhibit a random motion due to this. This is typically called&nbsp;<em style="line-height: inherit;">Brownian motion</em>(The name from an 1827 biologist Robert Brown who observed via microscope that pollen grains jiggle in water) which emerges as a result of random walks. The classical molecular dynamics which we were doing in the past was very deterministic. Here we see randomness. Why so? Is there some inconsistency? There is no inconsistency. The thing is if each of these particles was allowed to move according to newtons laws(where we also know their initial positions and velocities) would exhibit some dynamics like this. The thing is when we have a lot of such trajectories we can try to model this uncertainty using randomness or noise terms. And gaussian noise is a good model to achieve this.</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">So if you don't want to track all of these nasty small particles you can resort to something called Langevin Dynamics. This is different from Classical Molecular dynamics. In classical molecular dynamics:<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/tVu6Vfj.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>is the equation of motion. mass times acceleration is the force which is the negative gradient of potential energy U. We have explained these things&nbsp;<a href="https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-2-the-force-field" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">before</a>.</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Now in the langevin formalism, the small particles become implicit. So the equation becomes:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><img src="https://steemitimages.com/640x0/https://i.imgur.com/FKPLfZc.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">where the second term in the right hand side of the equation is gamma(friction coefficient) times velocity and the third term is the noise term. R(t) is a zero mean, delta correlated stationary gaussian noise and scaling factor&nbsp;<img src="https://steemitimages.com/640x0/https://i.imgur.com/1Wtax3T.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;">&nbsp;is where the temperature T comes in the picture. Intuitively we can assume that when the temperature is high the diffusion coefficient(D) of the bigger particle will be high and if the viscosity is high the diffusion coefficient of the larger particle will be lesser. There is a relation which Einstein derived in 1905 for this:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><img src="https://steemitimages.com/640x0/https://i.imgur.com/tZBQvoW.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br><em style="line-height: inherit;">This is one of the earliest so called fluctuation-dissipation relation.</em></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">The potential decides what force the particle experiences. Now think about a double well potential like this below:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><img src="https://steemitimages.com/640x0/https://i.imgur.com/nYzYT41.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">If you slice it in one dimension(at Y=0) you will see that it has 2 minimas:<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/ZHYyuyw.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>This is why we are calling it double well potential. The form of the potential energy function is as below:<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/wXMKKYa.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>where X and Y are particle position coordinates.</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Matlab code for the plotting potential:</p><pre style="font-size: 19.2px; margin-top: 0px; margin-bottom: 0px; padding: 0px; color: rgb(51, 51, 51); background-color: rgb(252, 252, 252);"><code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll; display: block;">clear; close all; clc
x=linspace(-2,2,25);
y=linspace(-2,2,25);
[X,Y] = meshgrid(x,y);
Z=4*(X-1).^2.*(X+1).^2 + Y.^2; % potential energy landscape
hs = surf(X,Y,Z); % Get surface object
</code></pre><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Now imagine that our particle which experiences friction and kicks(noise) from the other particles and is also under influence of this particular potential energy&nbsp;<img src="https://steemitimages.com/640x0/https://i.imgur.com/wXMKKYa.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;">. Depending on the temperature T, friction coefficient gamma and the barrier height of the potential(the peak between minimas) our particle will traverse in this energy landscape. Let us see that dynamics via a very simple simulation which we can setup in openMM.</p><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 26.88px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">The implementation of system using openMM toolkit</h2><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">OpenMM is a collection of python packages which help users to create python scripts to run MD simulations. It is a toolkit from&nbsp;<a href="https://pande.stanford.edu/" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">Vijay Pande group</a>&nbsp;which pioneered the&nbsp;<a href="https://foldingathome.org/" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">folding@home</a>&nbsp;project.</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Python Code(Which I wrote on Jupyter notebook initially):</p><pre style="font-size: 19.2px; margin-top: 0px; margin-bottom: 0px; padding: 0px; color: rgb(51, 51, 51); background-color: rgb(252, 252, 252);"><code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll; display: block;">%matplotlib inline
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
import simtk.openmm as mm#Imported numpy, plotting libraries and openmm
def run_simulation(n_steps=10000):
    "Simulate a single particle in the double well"
    system = mm.System()
    system.addParticle(1)# added particle with a unit mass
    force = mm.CustomExternalForce('2*(x-1)^2*(x+1)^2 + y^2')# defines the potential
    force.addParticle(0, [])
    system.addForce(force)
    integrator = mm.LangevinIntegrator(500, 1, 0.02)% Langevin integrator with 500K temperature, gamma=1, step size = 0.02
    context = mm.Context(system, integrator)
    context.setPositions([[0, 0, 0]])
    context.setVelocitiesToTemperature(500)
    x = np.zeros((n_steps, 3))
    for i in range(n_steps):
        x[i] = context.getState(getPositions=True).getPositions(asNumpy=True)._value
        integrator.step(1)
    return x
trajectory = run_simulation(25000)
ylabels = ['x', 'y']
for i in range(2):
    plt.subplot(2, 1, i+1)
    plt.plot(trajectory[:, i])
    plt.ylabel(ylabels[i])
plt.xlabel('Simulation time')
plt.show()
plt.hist2d(trajectory[:, 0], trajectory[:, 1], bins=(25, 25), cmap=plt.cm.jet)
plt.show()
</code></pre><h3 style="margin: 2rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 23.04px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">Explanation of the code</h3><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">What we are doing here is initiating a single particle which experiences the friction and kicks and evolves its positions via langevin integrator.&nbsp;<code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll;">system.addParticle(1)</code>&nbsp;means adding a particle with unit mass,&nbsp;<code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll;">force = mm.CustomExternalForce('2*(x-1)^2*(x+1)^2 + y^2')</code>&nbsp;is where you define the potential,&nbsp;<code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll;">integrator = mm.LangevinIntegrator(500, 1, 0.02)</code>&nbsp;is where you define the bath temperature, gamma(friction coefficient=1) and step size(0.02).&nbsp;<code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll;">trajectory = run_simulation(25000)</code>&nbsp;calls the simulation run for 25000 steps. and the rest is plotting step.</p><h3 style="margin: 2rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 23.04px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">Results</h3><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><img src="https://steemitimages.com/640x0/https://i.imgur.com/3zRlFHY.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>The double well is very visible from the positions of x. The particle moves back and forth between 2 wells.<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/8ZkMp7s.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>This is a histogram of x versus y coordinates. Again you can see that there are 2 valleys here.</p><h4 style="margin: 1.5rem 0px 0.2rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 21.12px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">Different cases</h4><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Let us think about a case where the temperature is very low. Let us put T=10K. The particle should not leave one minima, right? Let us see:<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/XdpzYsG.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br><img src="https://steemitimages.com/640x0/https://i.imgur.com/XRdSeor.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>Indeed that is the case!</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">What should happen if we increase friction coefficient keeping T=500K? Let us put gamma=100.<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/dVBsMq0.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>Of course, the transitions became lesser in frequency! The effect of higher viscosity!</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Now, what happens if we raise the barrier of potential. This was the potential:&nbsp;<img src="https://steemitimages.com/640x0/https://i.imgur.com/wXMKKYa.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;">&nbsp;If I change 4 to 20, the barrier between wells will rise. Which means the particle will find it hard to jump the barrier. Let us repeat the experiment by only changing this parameter from the original experiment:<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/Wcwevb4.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br><img src="https://steemitimages.com/640x0/https://i.imgur.com/bgp8znR.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>Indeed that is the case! So if you try to reconstruct energy landscape from the above case, it will give a biased profile. So one solution can be running the simulation longer. Let us try?</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Instead of 25000 steps, let us run the simulation for 250000 steps.<br><img src="https://steemitimages.com/640x0/https://i.imgur.com/w4R2AaC.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br><img src="https://steemitimages.com/640x0/https://i.imgur.com/fHX6cK1.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>See the symmetric well is very visible now! If you reconstruct the energy surface from here it will be a good estimate.</p><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 26.88px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">Code Repository</h2><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">All code available here:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><a href="https://github.com/dexterdev/STEEMIT/tree/master/Particle_in_double_well_potential" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">https://github.com/dexterdev/STEEMIT/tree/master/Particle_in_double_well_potential</a></p><h3 style="margin: 2rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 23.04px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">Installing the openMM python library</h3><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">To install openMM, just do:<br><code style="font-family: Consolas, &quot;Liberation Mono&quot;, Courier, monospace; font-size: 16.32px; padding: 0.2rem; border: none; background-color: rgb(244, 244, 244); font-weight: inherit; border-radius: 3px; overflow: scroll;">conda install -c omnia -c conda-forge openmm</code><br>assuming you have installed anaconda python package in a 64 bit linux machine. For other cases and more details&nbsp;<a href="http://docs.openmm.org/latest/userguide/application.html#installing-openmm" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">see instructions here</a>.</p><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 26.88px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;">References, links, and further reading</h2><ul style="margin: 0px 0px 1rem 2rem; padding: 0px; list-style-position: outside; line-height: 1.6; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><li style="margin: 0px; padding: 0px; font-size: inherit;"><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility;">Some MD fundamentals:</p><ul style="margin: 0px 0px 0px 2rem; padding: 0px; list-style-position: outside; line-height: 1.6; list-style-type: disc;"><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-1-the-fundamentals" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-1-the-fundamentals</a></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-2-the-force-field" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-2-the-force-field</a></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><a href="https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-3-solving-the-molecular-dynamics-equation" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">https://steemit.com/steemstem/@dexterdev/classical-molecular-dynamics-series-part-3-solving-the-molecular-dynamics-equation</a></li></ul></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility;">A book reference: "Statistical Mechanics: Theory and Molecular Simulations" by Mark E. Tuckerman</p></li><li style="margin: 0px; padding: 0px; font-size: inherit;"><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility;"><a href="http://openmm.org/" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">http://openmm.org/</a></p></li></ul><hr style="box-sizing: content-box; overflow: visible; clear: both; max-width: 75rem; margin: 1.25rem auto; border-top: 0px; border-right: 0px; border-left: 0px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; color: rgb(51, 51, 51); text-rendering: optimizeLegibility; font-size: 26.88px; background-color: rgb(252, 252, 252); line-height: 1.2 !important;"><a href="https://steemit.com/trending/steemstem" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">#steemSTEM</a></h2><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><a href="https://steemit.com/trending/steemstem" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">#steemSTEM</a>&nbsp;is a very vibrant community on top of STEEM blockchain for Science, Technology, Engineering and Mathematics (STEM). If you wish to support steemstem visit the links below:</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"></p><center style="color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><span style="font-weight: 600; line-height: inherit;">Have you voted for steemSTEM Witness?</span></center><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"></p><center style="color: rgb(51, 51, 51); 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text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"></p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Quick link for&nbsp;<a href="https://steemconnect.com/sign/account_witness_vote?approve=1&amp;witness=stem.witness" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">voting for the SteemSTEM Witness</a>(<a href="https://steemit.com/@stem.witness" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">@stem.witness</a>)</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; 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color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">100SP</a>&nbsp;|&nbsp;<a href="https://steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=steemstem&amp;vesting_shares=500%20SP" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">500SP</a>&nbsp;|&nbsp;<a href="https://steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=steemstem&amp;vesting_shares=1000%20SP" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">1000SP</a>&nbsp;|&nbsp;<a href="https://steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=steemstem&amp;vesting_shares=5000%20SP" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">5000SP</a>&nbsp;|&nbsp;<a href="https://steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=steemstem&amp;vesting_shares=10000%20SP" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">10000SP</a>).</p><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);">Also, visit and create your STEM posts using steemstem app here:&nbsp;<a href="https://www.steemstem.io/" rel="noopener" title="This link will take you away from steemit.com" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">https://www.steemstem.io</a><br>You can also set&nbsp;<a href="https://steemit.com/@steemstem" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">@steemstem</a>&nbsp;as beneficiary.</p><hr style="box-sizing: content-box; overflow: visible; clear: both; max-width: 75rem; margin: 1.25rem auto; border-top: 0px; border-right: 0px; border-left: 0px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"><span style="font-weight: 600; line-height: inherit;"></span></p><center style="color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; background-color: rgb(252, 252, 252);"><span style="font-weight: 600; line-height: inherit;">All images without image sources are my creations :)</span></center><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; color: rgb(51, 51, 51); font-family: &quot;Source Serif Pro&quot;, serif; background-color: rgb(252, 252, 252);"></p><h2 style="margin: 2.5rem 0px 0.3rem; padding: 0px; text-rendering: optimizeLegibility; background-color: rgb(252, 252, 252); line-height: 1.2 !important;"><center style="color: rgb(51, 51, 51); font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; font-size: 26.88px;">Follow me&nbsp;<a href="https://steemit.com/@dexterdev" style="line-height: inherit; color: rgb(6, 214, 169); cursor: pointer; transition: all 0.2s ease-in-out; outline-width: 0px;">@dexterdev</a></center><div style="color: rgb(51, 51, 51); font-family: &quot;Source Sans Pro&quot;, &quot;Helvetica Neue&quot;, Helvetica, Arial, sans-serif; font-size: 26.88px;">
<center style="font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; font-weight: 400;"><img src="https://steemitimages.com/DQmVM9FmLfAYojdB7Ex2ooGhkZMGHht3NzPLUSEvwSvpjTd/dexter.png" style="border-style: none; display: inline-block; vertical-align: middle; height: auto; width: auto; max-height: none;"><br>credit:&nbsp;<a href="https://steemit.com/@mathowl" style="line-height: inherit; color: rgb(31, 191, 143); cursor: pointer; transition: all 0.2s ease-in-out;">@mathowl</a></center><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; font-family: &quot;Source Serif Pro&quot;, serif; font-weight: 400; text-align: -webkit-center;"></p><hr style="box-sizing: content-box; overflow: visible; clear: both; max-width: 75rem; margin: 1.25rem auto; border-top: 0px; border-right: 0px; border-left: 0px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); font-family: &quot;Source Serif Pro&quot;, serif; font-size: 19.2px; font-weight: 400; text-align: -webkit-center;"><p style="margin-bottom: 1.5rem; padding: 0px; font-size: 19.2px; line-height: 28.8px; text-rendering: optimizeLegibility; font-family: &quot;Source Serif Pro&quot;, serif; font-weight: 400; text-align: -webkit-center;">Signing off now! Bye Bye friends! Keep on STEEMing!</p><br></div></h2>
👍  , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and 595 others
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vote details (660)
@chappertron · 
Wow, very nice. But I have to admit I have to read your other publications as well to get further into this subject. Where are these two minima from again? Don't get it!

Regards

Chapper
👍  
properties (23)
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vote details (1)
@dexterdev ·
$0.03
tw7wpl5wz
Hi @chappertron : Thank you for stopping by :) So the minimas are embedded in the potential function which I defined in the beginning. 
This is how the function is: ![](https://steemitimages.com/640x0/https://i.imgur.com/wXMKKYa.png)

You can see that there are 2 minimas at X=1 and X=-1 in this function.

See this plot(@ Y=0):
![](https://steemitimages.com/640x0/https://i.imgur.com/ZHYyuyw.png)

You saw the 2 minimas? Our particle toggles between these 2 states.
👍  
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vote details (1)
@chappertron · 
Oh of course! I have to look closer next time. Reminds me of my biophysics courses at the University used to. 

So this equation would explain why the yellow dot moves as it moves in a crowd of smaller particles?
👍  
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vote details (1)
@david-krug · 
Wow, and I thought I worked hard on my posts :D
👍  
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vote details (1)
@dexterdev · (edited)
:P I have to improve a lot though.
👍  
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vote details (1)
@david-krug · 
I don't know if you have to, but everyone can and progress is in big part what life is about I think. :)
👍  
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vote details (1)
@ferrate · 
Oooo my God...... Interesting things there in your blog and for so many years I never really met most of the things in said in your blog, which simply means, am really educated today. And I enjoyed the class even though I didn't pay an tuition..... Lol
Lovely work in there and every time spent on your blog was really worth it. Great pictures too. 
Great work and keep the science and technology spirit up

Posted using [Partiko Android](https://partiko.app/referral/ferrate)
👍  
properties (23)
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vote details (1)
@dexterdev ·
$0.02
😃
👍  ,
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vote details (2)
@ferrate · 
Cool

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@minnowsupport · 
<p>Congratulations!  This post has been upvoted from the communal account, @minnowsupport, by dexterdev from the Minnow Support Project. It's a witness project run by aggroed, ausbitbank, teamsteem, someguy123, neoxian, followbtcnews, and netuoso. The goal is to help Steemit grow by supporting Minnows.  Please find us at the <a href="https://discord.gg/HYj4yvw"> Peace, Abundance, and Liberty Network (PALnet) Discord Channel</a>.  It's a completely public and open space to all members of the Steemit community who voluntarily choose to be there.</p> <p>If you would like to delegate to the Minnow Support Project you can do so by clicking on the following links: <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=102530.639667%20VESTS">50SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=205303.639667%20VESTS">100SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=514303.639667%20VESTS">250SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=1025303.639667%20VESTS">500SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=2053030.639667%20VESTS">1000SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=10253030.639667%20VESTS">5000SP</a>. <br><strong>Be sure to leave at least 50SP undelegated on your account.</strong></p>
properties (22)
authorminnowsupport
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@steemitboard · 
Congratulations @dexterdev! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

<table><tr><td>https://steemitimages.com/60x70/http://steemitboard.com/@dexterdev/payout.png?201904110056</td><td>You received more than 1000 as payout for your posts. Your next target is to reach a total payout of 2000</td></tr>
</table>

<sub>_You can view [your badges on your Steem Board](https://steemitboard.com/@dexterdev) and compare to others on the [Steem Ranking](http://steemitboard.com/ranking/index.php?name=dexterdev)_</sub>
<sub>_If you no longer want to receive notifications, reply to this comment with the word_ `STOP`</sub>



**Do not miss the last post from @steemitboard:**
<table><tr><td><a href="https://steemit.com/steemitboard/@steemitboard/the-steem-blockchain-survived-its-first-virus-plague"><img src="https://steemitimages.com/64x128/https://steemitimages.com/0x0/![](https://cdn.steemitimages.com/DQmYKnawrtqFCShyVvVs4QS7548eretcMUKQJc3D2f69pb9/image.png)"></a></td><td><a href="https://steemit.com/steemitboard/@steemitboard/the-steem-blockchain-survived-its-first-virus-plague">The Steem blockchain survived its first virus plague!</a></td></tr></table>

###### [Vote for @Steemitboard as a witness](https://v2.steemconnect.com/sign/account-witness-vote?witness=steemitboard&approve=1) to get one more award and increased upvotes!
properties (22)
authorsteemitboard
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@steemstem ·
re-dexterdev-particle-in-a-double-1554311912-20190403t191012068z
<div class='text-justify'> <div class='pull-left'> <center> <br /> <img width='200' src='https://res.cloudinary.com/drrz8xekm/image/upload/v1553698283/weenlqbrqvvczjy6dayw.jpg'> </center>  <br/> <br /> </div>

This post has been voted on by the **SteemSTEM** curation team and voting trail. It is elligible for support from <b><a href='https://www.steemstem.io/#!/@curie'>@curie</a></b> and <b><a href='https://www.steemstem.io/#!/@utopian-io'>@utopian-io</a></b>.<br /> 

If you appreciate the work we are doing, then consider supporting our witness [**stem.witness**](https://steemconnect.com/sign/account_witness_vote?approve=1&witness=stem.witness). Additional witness support to the [**curie witness**](https://steemconnect.com/sign/account_witness_vote?approve=1&witness=curie) and [**utopian-io witness**](https://steemconnect.com/sign/account_witness_vote?approve=1&witness=utopian-io) would be appreciated as well.<br /> 

For additional information please join us on the [**SteemSTEM discord**]( https://discord.gg/BPARaqn) and to get to know the rest of the community!<br />

Thanks for having added <b><a href='https://www.steemstem.io/#!/@steemstem'>@steemstem</a></b> as a beneficiary to your post. This granted you a stronger support from SteemSTEM.<br />

Thanks for having used the <b><a href='https://www.steemstem.io'>steemstem.io</a></b> app. You got a stronger support!</div>
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@steevc · 
This reminds me of a very simple atomic model I produced at school. That was many years ago and the computers were not up to what can be done now.
👍  
properties (23)
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vote details (1)
@tking77798 ·
$0.01
jyo50ag2g
The animated title pic really does a fantastic job of drawing the viewer in.  I'll have to try that at some point.  Also, very cool introduction to Brownian motion and how coding can help analyze this phenomena.
👍  
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vote details (1)
@tking77798 ·
ue3hdx7tc
The animated title pic really does a fantastic job of drawing the viewer in.  I'll have to try that at some point.  Also, very cool introduction to Brownian motion and how coding can help analyze this phenomena.
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@utopian-io · 
#### Hi @dexterdev!

Your post was upvoted by Utopian.io in cooperation with @steemstem - supporting knowledge, innovation and technological advancement on the Steem Blockchain.

#### Contribute to Open Source with utopian.io
Learn how to contribute on <a href='https://join.utopian.io'>our website</a> and join the new open source economy.

**Want to chat? Join the Utopian Community on Discord https://discord.gg/h52nFrV**
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