You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
Copy file name to clipboardExpand all lines: content/posts/physics/percolation/index.md
+34-2
Original file line number
Diff line number
Diff line change
@@ -9,7 +9,7 @@ menu:
9
9
parent: physics
10
10
weight: 9
11
11
hero: images/lattice_illustration.png
12
-
tags: ["Science", ""]
12
+
tags: ["Science", "Physics", "Statistics"]
13
13
categories: ["Physics"]
14
14
---
15
15
## Introduction
@@ -25,7 +25,37 @@ Percolation theory is important because it provides insights into the behavior o
25
25
***Network Theory**: Percolation theory is used to study the robustness and connectivity of networks, such as the internet or social networks. It helps in understanding how networks can be disrupted and how they can be made more resilient
26
26
***Geophysics**: In oil recovery, percolation theory models the flow of fluids through porous rocks, helping to optimize extraction processes
27
27
***Forest Fires**: Percolation models can simulate the spread of forest fires, helping in the development of strategies for fire prevention and control
28
-
.
28
+
29
+
30
+
## Mathematical and Physics Theory
31
+
32
+
Percolation theory can be studied using site percolation or bond percolation models. In site percolation, each site (or node) on a lattice is either occupied with probability $ p $ or empty with probability $ 1 - p $. In bond percolation, each bond (or edge) between sites is open with probability $ p $ or closed with probability $ 1 - p $.
33
+
34
+
### Step-by-Step Explanation:
35
+
36
+
1.**Define the Lattice**: Consider a 2D square lattice or a 3D cubic lattice. For simplicity, let's use a 2D square lattice.
37
+
38
+
2.**Assign Probabilities**: For each site (or bond), assign a probability $ p $ that it is occupied (or open).
39
+
40
+
3.**Cluster Formation**: Identify clusters of connected sites (or bonds). Two sites are in the same cluster if there is a path of occupied sites (or open bonds) connecting them.
41
+
42
+
4.**Critical Threshold $ p_c $**: Determine the critical probability $ p_c $ at which an infinite cluster first appears. For a 2D square lattice, it has been rigorously shown that $ p_c \approx 0.5927 $.
43
+
44
+
5.**Mathematical Formulation**: The percolation probability $ P(p) $ is the probability that a given site belongs to the infinite cluster. Near the critical threshold, this follows a power-law behavior:
45
+
$$
46
+
P(p) \sim (p - p_c)^\beta
47
+
$$
48
+
where $ \beta $ is a critical exponent and equal to $\frac{5}{36}$ for 2D squared lattice.
49
+
50
+
6.**Correlation Length $ \xi $**: The average size of finite clusters below $ p_c $ is characterized by the correlation length $ \xi $, which diverges as:
51
+
$$
52
+
\xi \sim |p - p_c|^{-\nu}
53
+
$$
54
+
where $ \nu $ is another critical exponent
55
+
56
+
7.**Conductivity and Other Properties**: In practical applications, properties like electrical conductivity in materials can be modeled by considering the effective medium theory or numerical simulations to calculate the likelihood of percolation and the size of clusters.
57
+
58
+
By analyzing these steps, percolation theory provides a comprehensive understanding of how macroscopic properties emerge from microscopic randomness, revealing universal behaviors that transcend specific systems.
29
59
30
60
## Python Simulation Code
31
61
@@ -119,6 +149,7 @@ for i, p in enumerate(p_values):
119
149
# Adjust spacing between subplots
120
150
plt.subplots_adjust(wspace=0.1, hspace=0.1)
121
151
152
+
122
153
# Show the plot
123
154
plt.show()
124
155
```
@@ -132,6 +163,7 @@ In further version, also a connected path from left to right can be considered.
132
163
133
164
## Conclusion
134
165
The previous plot shows that with _p>0.58_ a percolation path starts to be observed. However, this is so not alwasy happening for stochastical reasons. Hence that plot is the result of several iteration to find the most interesting plot. With _p>0.60_ percolation happens more than 90% of the time.
166
+
In general, this confirms the numerical value of $p_c$ that can be found in [literature](https://arxiv.org/abs/cond-mat/0005264) of 0.5927
135
167
136
168
In further articles we will explore some [python libraries](https://pypercolate.readthedocs.io/en/stable/) to develop a more advanced and practical example.
@@ -385,9 +390,43 @@ <h2 id="why-is-percolation-important-useful-applications">Why is Percolation Imp
385
390
<li><strong>Epidemiology</strong>: In the study of disease spread, percolation models can predict the outbreak and spread of epidemics. The percolation threshold can represent the critical point at which a disease becomes widespread in a population</li>
386
391
<li><strong>Network Theory</strong>: Percolation theory is used to study the robustness and connectivity of networks, such as the internet or social networks. It helps in understanding how networks can be disrupted and how they can be made more resilient</li>
387
392
<li><strong>Geophysics</strong>: In oil recovery, percolation theory models the flow of fluids through porous rocks, helping to optimize extraction processes</li>
388
-
<li><strong>Forest Fires</strong>: Percolation models can simulate the spread of forest fires, helping in the development of strategies for fire prevention and control
389
-
.</li>
393
+
<li><strong>Forest Fires</strong>: Percolation models can simulate the spread of forest fires, helping in the development of strategies for fire prevention and control</li>
390
394
</ul>
395
+
<h2id="mathematical-and-physics-theory">Mathematical and Physics Theory</h2>
396
+
<p>Percolation theory can be studied using site percolation or bond percolation models. In site percolation, each site (or node) on a lattice is either occupied with probability $ p $ or empty with probability $ 1 - p $. In bond percolation, each bond (or edge) between sites is open with probability $ p $ or closed with probability $ 1 - p $.</p>
<p><strong>Define the Lattice</strong>: Consider a 2D square lattice or a 3D cubic lattice. For simplicity, let’s use a 2D square lattice.</p>
401
+
</li>
402
+
<li>
403
+
<p><strong>Assign Probabilities</strong>: For each site (or bond), assign a probability $ p $ that it is occupied (or open).</p>
404
+
</li>
405
+
<li>
406
+
<p><strong>Cluster Formation</strong>: Identify clusters of connected sites (or bonds). Two sites are in the same cluster if there is a path of occupied sites (or open bonds) connecting them.</p>
407
+
</li>
408
+
<li>
409
+
<p><strong>Critical Threshold $ p_c $</strong>: Determine the critical probability $ p_c $ at which an infinite cluster first appears. For a 2D square lattice, it has been rigorously shown that $ p_c \approx 0.5927 $.</p>
410
+
</li>
411
+
<li>
412
+
<p><strong>Mathematical Formulation</strong>: The percolation probability $ P(p) $ is the probability that a given site belongs to the infinite cluster. Near the critical threshold, this follows a power-law behavior:
413
+
$$
414
+
P(p) \sim (p - p_c)^\beta
415
+
$$
416
+
where $ \beta $ is a critical exponent and equal to $\frac{5}{36}$ for 2D squared lattice.</p>
417
+
</li>
418
+
<li>
419
+
<p><strong>Correlation Length $ \xi $</strong>: The average size of finite clusters below $ p_c $ is characterized by the correlation length $ \xi $, which diverges as:
420
+
$$
421
+
\xi \sim |p - p_c|^{-\nu}
422
+
$$
423
+
where $ \nu $ is another critical exponent</p>
424
+
</li>
425
+
<li>
426
+
<p><strong>Conductivity and Other Properties</strong>: In practical applications, properties like electrical conductivity in materials can be modeled by considering the effective medium theory or numerical simulations to calculate the likelihood of percolation and the size of clusters.</p>
427
+
</li>
428
+
</ol>
429
+
<p>By analyzing these steps, percolation theory provides a comprehensive understanding of how macroscopic properties emerge from microscopic randomness, revealing universal behaviors that transcend specific systems.</p>
</span></span></code></pre></div><p>This code generates a square lattice of size <code>n</code> with site vacancy probability <code>p</code>, checks if the lattice percolates (i.e., if there is a connected path from the top to the bottom), and plots the lattice.</p>
@@ -495,7 +535,8 @@ <h3 id="results">Results</h3>
495
535
>
496
536
497
537
<h2id="conclusion">Conclusion</h2>
498
-
<p>The previous plot shows that with <em>p>0.58</em> a percolation path starts to be observed. However, this is so not alwasy happening for stochastical reasons. Hence that plot is the result of several iteration to find the most interesting plot. With <em>p>0.60</em> percolation happens more than 90% of the time.</p>
538
+
<p>The previous plot shows that with <em>p>0.58</em> a percolation path starts to be observed. However, this is so not alwasy happening for stochastical reasons. Hence that plot is the result of several iteration to find the most interesting plot. With <em>p>0.60</em> percolation happens more than 90% of the time.
539
+
In general, this confirms the numerical value of $p_c$ that can be found in <ahref="https://arxiv.org/abs/cond-mat/0005264" target="_blank" rel="noopener">literature</a> of 0.5927</p>
499
540
<p>In further articles we will explore some <ahref="https://pypercolate.readthedocs.io/en/stable/" target="_blank" rel="noopener">python libraries</a> to develop a more advanced and practical example.</p>
500
541
<!-- ## GitHub Repositories
501
542
@@ -648,6 +689,11 @@ <h5 class="text-center ps-3">Table of Contents</h5>
648
689
<ul>
649
690
<li><ahref="#introduction">Introduction</a></li>
650
691
<li><ahref="#why-is-percolation-important-useful-applications">Why is Percolation Important? Useful Applications</a></li>
692
+
<li><ahref="#mathematical-and-physics-theory">Mathematical and Physics Theory</a>
0 commit comments