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<!--
########################################
## @author Benjamin Thomas Schwertfeger (July 2021)
## copyright by Benjamin Thomas Schwertfeger (July 2021)
############
-->
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<title>Brownian Motion</title>
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="format.css">
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href="https://www.awi.de/typo3conf/ext/sms_boilerplate/Resources/Public/Images/AWI/awi_logo.svg"> -->
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<script src='https://cdn.jsdelivr.net/npm/chart.js@3.3.1/dist/chart.min.js'> </script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
</script>
<!-- PLOTLY FOR CONTOURPLOT -->
<script src="https://cdn.plot.ly/plotly-2.3.0.min.js"></script>
</head>
<body>
<!-- <img
src="https://upload.wikimedia.org/wikipedia/commons/f/f7/AWI_Logo_2017.svg"
style="position:absolute; right: 10px; top: 10px; width: 15%;" \> -->
<div class="main">
<div class="placeholder" style="width:100%;height:42px"></div>
<div class="headline" style="text-align:center;">
<h1>Brownian Motion</h1><br>
</div>
<!-- BROWNIAN MOTION -->
<div class="bm-wrapper">
<div class="description">
<h3>General</h3>
<ul>
<li>
<a href="https://www.tpi.uni-jena.de/~wipf/lectures/pfad/pfad7.pdf" target="_blank">Overview
article by A. Wipf,
University of Jena
</a>
</li>
<li>
<a href="https://people.bath.ac.uk/maspm/book.pdf" target="_blank"><i>Brownian
Motion</i></a> by
Peter Mörters and Yuval Peres
</li>
</ul>
<p>
<i>"The Brownian process describes the disordered motion
of
small particles suspended in a
liquid. It is believed that Brown studied pollen
particles floating in water under the microscope.
He observed minute particles executing a jittery
motion.
The theory of this motion has been
invented by Albert Einstein and Marian
Smoludchowski.
The mathematically rigorous construction of
the corresponding stochastic process has been
developed
by Norbert Wiener."</i> (Extraced from: A. Wipf
<i>"Path Integrals"</i> (2019), p. 60, Brownian Motion,
University of Jena)
</p>
<p>
#brownianMotion #multipleParticle
#forwardModeling
</p>
<ul>
<li>
<a href="BrownianMotion.ipynb" download>Download</a>
or <a href="BrownianMotion.html" target="_blank">show</a>
Jupyter Notebook
</li>
</ul>
</div>
<!-- BROWNIAN MOTION FIRST LINE AND HIST PLOT -->
<div class="bm-plot-content-wrapper">
<div id="bm_line_plot_container" class="bm-line-plot-container">
<canvas id="bm_line_plot" class="canvas"></canvas>
</div>
<div id="bm_hist_plot_container" class="bm-hist-plot-container">
<canvas id="bm_hist_plot" class="canvas"></canvas>
</div>
<div id="bm_input_section" class="bm-input-section">
<div class="bm-input-wrapper input-wrapper">
<div>\(C_a\)</div>
<div id="bm_Ca_value" name="bm_slide_value" class="bm-slide-value slide-value">10</div>
<input id="bm_Ca_slide" type="range" class="bm-slide slide" name="bm_slide" min="0" max="20"
step="0.5" value="10">
<!-- min="-0.001" max="-0.0001"
step="0.0001" value="-0.001"> -->
</div>
<div class="bm-input-wrapper input-wrapper">
<div>\(t\) (integration time)</div>
<div id="bm_T_value" name="bm_slide_value" class="bm-slide-value slide-value">1000</div>
<input id="bm_T_slide" type="range" class="bm-slide slide" name="bm_slide" min="5" max="6000"
step="5" value="1000">
</div>
<div class="bm-input-wrapper input-wrapper">
<div>\(N_{particles}\)</div>
<div id="bm_nlines_value" name="bm_slide_value" class="bm-slide-value slide-value">1</div>
<input id="bm_nlines_slide" type="range" class="bm-slide slide" name="bm_slide" min="1" max="50"
step="1" value="1">
</div>
<div class="bm-input-wrapper input-wrapper" style="min-width: 100px;">
<div>random start value</div>
<label class="switch">
<input id="bm_randomStartValueCheckbox" type="checkbox">
<span class="switch-slider"></span>
</label>
</div>
<button class="someBtn" id="bm_againBtn">Again</button>
<button class="someBtn" id="bm_resetBtn">Reset</button>
</div>
</div>
<div class="placeholder" style="width:100%;height:42px;"></div>
<div class="description">
<h3>Analysis of the densities</h3>
<p>
In the following you have the possibility to have a
closer look at the distributions in certain timesteps.
</p>
<p>
Explanations:
<ul>
<li>
\(t\): integration time in time units (default:
1000)
</li>
<li>
\(n\): stepsize (default: 0.5)
</li>
<li>
\(N\): number of steps (default:
\(t/n = 2000\))
</li>
<li>
\(N_{particles}\): number of the particles
(default:
\(N_{particles} = 1000\))
</li>
<li>
\(h\): matrix with the shape \((N, 40)\)
where \(40\) is the number of equally spaced
categories in
which
the
values are grouped. Thus in a
histogram up to 40 bars would be possible.
</li>
<li>
\(C_a\): factor of random deviation
</li>
</ul>
</p>
<p>
Figure a) represents the distribution of the values
between the
steps \(N/2\) and \(N\).
</p>
<p>
Figure b) how often does which value occur?
</p>
<p>
Figure c) shows the distribution of the values in the
first step (R: h[1,]; Python: h[0]). Here it is
to be noted that with fixed starting value in the first
step always the same distribution occurs. However, if
the start is chosen randomly, variations can be seen
here as well.
</p>
<p>
Figure d), e) and f) show the distributions of
\(N_{particles}\)
in the second, fourth and eighth step, respectively.
</p>
<p>
Figure g) shows the distribution of \(N_{particles}\) at
step \(N/2\).
</p>
<p>
Figure h) shows the average of the distributions of the
last and penultimate step.
</p>
</div>
<!-- BROWNIAN MOTION DENSITY ANALYSIS PART 1 -->
<div class="bm-dens-plot-content-wrapper">
<div id="bm_dens_input_section" class="bm-dens-input-section">
<div class="bm-dens-input-wrapper input-wrapper">
<div>\(C_a\)</div>
<div id="bm_dens_Ca_value" name="bm_dens_slide_value" class="bm-dens-slide-value slide-value">10
</div>
<input id="bm_dens_Ca_slide" type="range" class="bm-dens-slide slide" name="bm_dens_slide"
min="-0" max="20" step="0.5" value="10">
</div>
<div class="bm-dens-input-wrapper input-wrapper">
<div>\(t\) (integration time)</div>
<div id="bm_dens_T_value" name="bm_dens_slide_value" class="bm-dens-slide-value slide-value">
1000</div>
<input id="bm_dens_T_slide" type="range" class="bm-dens_slide slide" name="bm_dens_slide"
min="40" max="2500" step="5" value="1000">
</div>
<div class="bm-dens-input-wrapper input-wrapper" style="min-width: 100px;">
<div>random start value</div>
<label class="switch">
<input id="bm_dens_randomStartValueCheckbox" type="checkbox">
<span class="switch-slider"></span>
</label>
</div>
<!-- <div class="bm-dens-input-wrapper input-wrapper"> -->
<button class="someBtn" id="bm_dens_againBtn">Again</button>
<button class="someBtn" id="bm_dens_resetBtn">Reset</button>
<!-- </div> -->
</div>
<div class="placeholder" style="width:100%;height:40px"></div>
<!-- FIRST TWO PLOTS - HISTOGRAM OF LAST 1000 STEPS AND TABLE PLOT -->
<div class="row-wrapper">
<div class="col-left">
<div id="bm_dens_line_plot_container" class="bm-dens-line-hist-plot-container
dens-plot-styling">
<canvas id="bm_dens_line_plot" class="canvas"></canvas>
</div>
</div>
<div class="col-right">
<div id="bm_dens_table_plot_container" class="bm-dens-table-plot-container
dens-plot-styling">
<canvas id="bm_dens_table_plot" class="canvas"></canvas>
</div>
</div>
</div>
<!-- THE 6 ADDITIONAL PLOTS -->
<div class="row-wrapper">
<div class="col-left">
<div id="bm_line_1_plot_container" class="bm-line-1-plot-container
dens-plot-styling">
<canvas id="bm_line_1_plot" class="canvas"></canvas>
</div>
</div>
<div class="col-right">
<div id="bm_line_2_plot_container" class="bm-line-2-plot-container
dens-plot-styling">
<canvas id="bm_line_2_plot" class="canvas"></canvas>
</div>
</div>
</div>
<div class="row-wrapper">
<div class="col-left">
<div id="bm_line_3_plot_container" class="bm-line-3-plot-container
dens-plot-styling">
<canvas id="bm_line_3_plot" class="canvas"></canvas>
</div>
</div>
<div class="col-right">
<div id="bm_line_4_plot_container" class="bm-line-4-plot-container
dens-plot-styling">
<canvas id="bm_line_4_plot" class="canvas"></canvas>
</div>
</div>
</div>
<div class="row-wrapper">
<div class="col-left">
<div id="bm_line_5_plot_container" class="bm-line-5-plot-container
dens-plot-styling">
<canvas id="bm_line_5_plot" class="canvas"></canvas>
</div>
</div>
<div class="col-right">
<div id="bm_line_6_plot_container" class="bm-line-6-plot-container
dens-plot-styling">
<canvas id="bm_line_6_plot" class="canvas"></canvas>
</div>
</div>
</div>
<div class="row-wrapper">
<div class="contour-plot-wrapper dens-plot-styling">
<div class="contour-plot-container">
<div id="contour_plot" style="width:100%; height:100%;">
</div>
</div>
</div>
</div>
<div class="placeholder" style="width:100%;height:20px;"></div>
</div>
<div class="placeholder" style="width:100%;height:42px;"></div>
<!-- DRIVNG FUNCTION Part 1 -->
<div class="description">
<h3>Driving function</h3>
<p>Following is the diving function. It is the backbone of
the
above results and can clarify its behavior.
</p>
<p>
<ul>
<li>
Driving function:
</li>
</ul>
</p>
<p style="text-align: center;">
\( \begin{align}
f_{(x)} &= dx^3+cx^2+bx-a \\ \\
F(x) = \int f(x) dx &=
\frac{d}{4}x^4+\frac{c}{3}x^3+\frac{b}{2}x^2-x
\end{align}
\)
</p>
</div>
<div class="df-plot-content-wrapper">
<div id="df_plot_container" class="df-plot-container">
<canvas id="df_plot" class="canvas"></canvas>
</div>
<div id="df_X-plot_container" class="df-X-plot-container">
<canvas id="df_X_plot" class="canvas"></canvas>
</div>
<div id="df_input_section" class="df-input-section">
<div class="df-input-wrapper input-wrapper">
<div>\(a\)</div>
<div id="df_a_value" name="df_slide_value" class="df-slide-value slide-value">1</div>
<input id="df_a_slide" type="range" class="df-slide slide" name="df_slide" min="-5" max="5"
step="0.01" value="1">
</div>
<div class="df-input-wrapper input-wrapper">
<div>\(b\)</div>
<div id="df_b_value" name="df_slide_value" class="df-slide-value slide-value">0.8</div>
<input id="df_b_slide" type="range" class="df-slide slide" name="df_slide" min="-5" max="5"
step="0.01" value="0.8">
</div>
<div class="df-input-wrapper input-wrapper">
<div>\(c\)</div>
<div id="df_c_value" name="df_slide_value" class="df-slide-value slide-value">0</div>
<input id="df_c_slide" type="range" class="df-slide slide" name="df_slide" min="-1" max="1"
step="0.001" value="0">
</div>
<div class="df-input-wrapper input-wrapper">
<div>\(d\)</div>
<div id="df_d_value" name="df_slide_value" class="df-slide-value slide-value">-0.001</div>
<input id="df_d_slide" type="range" class="df-slide slide" name="df_slide" min="-0.001"
max="-0.0001" step="0.0001" value="-0.001">
</div>
<button class="someBtn" id="df_resetBtn">Reset</button>
<div class="X-input-wrapper">
\(y_0\):
<input id="df_X_input_y0" type="text" class="df_X_input_field" value=1.25 />
</div>
</div>
</div>
</div>
<div id="footer" style="text-align:center; margin:0 auto;
padding-top: 6rem; width:
50%; font-size: 16px;">
Site developed by <a href="https://www.b-schwertfeger.de">Benjamin
T. Schwertfeger</a> (2021) <br>
<a href="https://www.awi.de" target="_blank">Alfred Wegener Institute, Helmholtz Centre for Polar and Marine
Research, Bremerhaven, Germany</a>
</div>
<div class="placeholder" style="width:100%;height:100px;"></div>
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<script src="layout_script.js" async defer></script>
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