second article uploaded

Author: steven-ja

content/posts/physics/percolation/images/lattice_illustration.png

@@ -8,43 +8,37 @@ menu:
     identifier: percolation
     parent: physics
     weight: 9
+hero: images/lattice_illustration.png
 tags: ["Science", ""]
 categories: ["Physics"]
 ---
-
 ## Introduction
 
-Percolation theory is a fundamental concept in statistical physics and mathematics that describes the behavior of connected clusters in a random graph. It is a model for understanding how a network behaves when nodes or links are added, leading to a phase transition from a state of disconnected clusters to a state where a large, connected cluster spans the system. This transition occurs at a critical threshold, known as the percolation threshold. The theory has applications in various fields, including material science, epidemiology, and network theory.
+[Percolation theory](https://en.wikipedia.org/wiki/Percolation_theory) is a fundamental concept in statistical physics and mathematics that describes the behavior of connected clusters in a random graph. It is a model for understanding how a network behaves when nodes or links are added, leading to a phase transition from a state of disconnected clusters to a state where a large, connected cluster spans the system. This transition occurs at a critical threshold, known as the percolation threshold. The theory has applications in various fields, including material science, epidemiology, and network theory.
 
 ## Why is Percolation Important? Useful Applications
 
-Percolation theory provides insights into the behavior of complex systems and phase transitions. Here are some key applications:
-
-### Material Science
-
-Percolation theory helps understand the properties of composite materials, such as electrical conductivity and strength. For example, the electrical conductivity of a composite material can change dramatically when the concentration of conductive filler reaches the percolation threshold.
-
-### Epidemiology 
-
-In studying disease spread, percolation models can predict the outbreak and spread of epidemics. The percolation threshold represents the critical point at which a disease becomes widespread in a population.
-
-### Network Theory
+Percolation theory is important because it provides insights into the behavior of complex systems and phase transitions. Here are some key applications:
 
-Percolation theory is used to study the robustness and connectivity of networks like the internet or social networks. It helps understand how networks can be disrupted and how to make them more resilient.
-
-### Geophysics
-
-In oil recovery, percolation theory models the flow of fluids through porous rocks, helping to optimize extraction processes.
-
-### Forest Fires
-
-Percolation models can simulate the spread of forest fires, aiding in the development of strategies for fire prevention and control.
+* **Material Science**: Percolation theory helps in understanding the properties of composite materials, such as conductivity and strength. For example, the electrical conductivity of a composite material can change dramatically when the concentration of conductive filler reaches the percolation threshold
+* **Epidemiology**: 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
+* **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
+* **Geophysics**: In oil recovery, percolation theory models the flow of fluids through porous rocks, helping to optimize extraction processes
+* **Forest Fires**: Percolation models can simulate the spread of forest fires, helping in the development of strategies for fire prevention and control
+  .
 
 ## Python Simulation Code
 
 Here is a simple example of a site percolation simulation on a square lattice in Python:
 
 ```python
+# -*- coding: utf-8 -*-
+"""
+Created on Wed Jun 19 07:15:50 2024
+
+@author: stefa
+"""
+
 import numpy as np
 import matplotlib.pyplot as plt
 import scipy.ndimage
@@ -55,34 +49,93 @@ def generate_lattice(n, p):
 
 def percolates(lattice):
     """Check if the lattice percolates."""
+    intersection = get_intersection(lattice)
+    return intersection.size > 1
+
+def get_intersection(lattice):
+    """Check if the bottom labels or top labels have more than 1 value equal    """
     labeled_lattice, num_features = scipy.ndimage.label(lattice)
     top_labels = np.unique(labeled_lattice[0, :])
     bottom_labels = np.unique(labeled_lattice[-1, :])
-    return np.intersect1d(top_labels, bottom_labels).size > 1
+    print(top_labels, bottom_labels, np.intersect1d(top_labels, bottom_labels).size)
+    return np.intersect1d(top_labels, bottom_labels)
 
 def plot_lattice(lattice):
     """Plot the lattice."""
     plt.imshow(lattice, cmap='binary')
     plt.show()
+  
+def lattice_to_image(lattice):
+    """Convert the lattice in an RGB image (even if will be black and white)"""
+    r_ch = np.where(lattice, 0, 255)
+    g_ch = np.where(lattice, 0, 255)
+    b_ch = np.where(lattice, 0, 255)
+  
+    image = np.concatenate((
+        np.expand_dims(r_ch, axis=2),
+        np.expand_dims(g_ch, axis=2),
+        np.expand_dims(b_ch, axis=2),
+        ), axis=2)
+  
+    return image
+
+def get_path(lattice):
+    intersection = get_intersection(lattice)
+    labeled_lattice, num_features = scipy.ndimage.label(lattice)
+    print(intersection)
+    return labeled_lattice == intersection[1]
 
-# Parameters
-n = 100  # Lattice size
-p = 0.6  # Site vacancy probability
+def add_path_img(image, path):
+    blank_image = np.zeros(image.shape)
+    # set the red channel to 255-> get a red percolation path
+    image[path, 0] = 255
+    image[path, 1] = 20
+    return image
+  
 
-# Generate and plot lattice
-lattice = generate_lattice(n, p)
-plot_lattice(lattice)
 
-# Check percolation
-if percolates(lattice):
-    print("The system percolates.")
-else:
-    print("The system does not percolate.")
+# Parameters
+n = 100  # Lattice size
+p_values = [0.2, 0.3, 0.4, 0.5, 0.58, 0.6]  # Site vacancy probabilities
+
+# Create a figure with subplots
+fig, axs = plt.subplots(2, 3, figsize=(15, 8))
+axs = axs.flatten()
+
+
+# Plot lattices for different p values
+for i, p in enumerate(p_values):
+    lattice = generate_lattice(n, p)
+    lattice_img = lattice_to_image(lattice)
+    if percolates(lattice):
+        axs[i].set_title(f"p = {p} (Percolates)")
+        path = get_path(lattice)
+        lattice_img = add_path_img(lattice_img, path)
+    else:
+        axs[i].set_title(f"p = {p} (Does not percolate)")
+    axs[i].imshow(lattice_img)
+    axs[i].axis('off')
+
+# Adjust spacing between subplots
+plt.subplots_adjust(wspace=0.1, hspace=0.1)
+
+# Show the plot
+plt.show()
 ```
 
 This code generates a square lattice of size `n` with site vacancy probability `p`, checks if the lattice percolates (i.e., if there is a connected path from the top to the bottom), and plots the lattice.
 
-## GitHub Repositories
+In further version, also a connected path from left to right can be considered.
+
+### Results
+{{< img src="/posts/physics/percolation/images/percolation_plot.png" align="center" title="Results">}}
+
+## Conclusion
+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.
+
+In further articles we will explore some [python libraries](https://pypercolate.readthedocs.io/en/stable/) to develop a more advanced and practical example.
+
+<!-- ## GitHub Repositories
 
 For more advanced simulations and visualizations, you can refer to the following GitHub repositories:
 
@@ -92,7 +145,6 @@ For more advanced simulations and visualizations, you can refer to the following
 These resources provide a comprehensive starting point for understanding and simulating percolation processes using Python.
 
 Citations:
-[1] https://www.routledge.com/Introduction-To-Percolation-Theory-Second-Edition/Stauffer-Aharony/p/book/9780748402533
 [2] https://introcs.cs.princeton.edu/python/24percolation/
 [3] https://github.com/mrbrianevans/percolation
 [4] https://arxiv.org/pdf/1709.01141.pdf
@@ -101,4 +153,4 @@ Citations:
 [7] https://www.math.chalmers.se/~steif/perc.pdf
 [8] https://github.com/dh4gan/percolation-model
 [9] https://en.wikipedia.org/wiki/Percolation_theory
-[10] https://www.taylorfrancis.com/books/mono/10.1201/9781315274386/introduction-percolation-theory-ammon-aharony-dietrich-stauffer
+[10] https://www.taylorfrancis.com/books/mono/10.1201/9781315274386/introduction-percolation-theory-ammon-aharony-dietrich-stauffer -->

content/posts/physics/percolation/images/percolation_plot.png

@@ -0,0 +1,91 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Wed Jun 19 07:15:50 2024
+
+@author: stefano
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy.ndimage
+
+def generate_lattice(n, p):
+    """Generate an n x n lattice with site vacancy probability p."""
+    return np.random.rand(n, n) < p
+
+def percolates(lattice):
+    """Check if the lattice percolates."""
+    intersection = get_intersection(lattice)
+    return intersection.size > 1
+
+def get_intersection(lattice):
+    """Check if the bottom labels or top labels have more than 1 value equal    """
+    labeled_lattice, num_features = scipy.ndimage.label(lattice)
+    top_labels = np.unique(labeled_lattice[0, :])
+    bottom_labels = np.unique(labeled_lattice[-1, :])
+    print(top_labels, bottom_labels, np.intersect1d(top_labels, bottom_labels).size)
+    return np.intersect1d(top_labels, bottom_labels)
+
+def plot_lattice(lattice):
+    """Plot the lattice."""
+    plt.imshow(lattice, cmap='binary')
+    plt.show()
+    
+def lattice_to_image(lattice):
+    """Convert the lattice in an RGB image (even if will be black and white)"""
+    r_ch = np.where(lattice, 0, 255)
+    g_ch = np.where(lattice, 0, 255)
+    b_ch = np.where(lattice, 0, 255)
+    
+    image = np.concatenate((
+        np.expand_dims(r_ch, axis=2),
+        np.expand_dims(g_ch, axis=2),
+        np.expand_dims(b_ch, axis=2),
+        ), axis=2)
+    
+    return image
+
+def get_path(lattice):
+    intersection = get_intersection(lattice)
+    labeled_lattice, num_features = scipy.ndimage.label(lattice)
+    print(intersection)
+    return labeled_lattice == intersection[1]
+
+def add_path_img(image, path):
+    blank_image = np.zeros(image.shape)
+    # set the red channel to 255-> get a red percolation path
+    image[path, 0] = 255
+    image[path, 1] = 20
+    return image
+    
+
+
+# Parameters
+n = 100  # Lattice size
+p_values = [0.2, 0.3, 0.4, 0.5, 0.58, 0.6]  # Site vacancy probabilities
+
+# Create a figure with subplots
+fig, axs = plt.subplots(2, 3, figsize=(15, 8))
+axs = axs.flatten()
+
+
+# Plot lattices for different p values
+for i, p in enumerate(p_values):
+    lattice = generate_lattice(n, p)
+    lattice_img = lattice_to_image(lattice)
+    if percolates(lattice):
+        axs[i].set_title(f"p = {p} (Percolates)")
+        path = get_path(lattice)
+        lattice_img = add_path_img(lattice_img, path)
+    else:
+        axs[i].set_title(f"p = {p} (Does not percolate)")
+    axs[i].imshow(lattice_img)
+    axs[i].axis('off')
+
+# Adjust spacing between subplots
+plt.subplots_adjust(wspace=0.1, hspace=0.1)
+plt.tight_layout()
+
+# Show the plot
+plt.show()
+plt.savefig("images/percolation_plot.png", dpi=300)

content/posts/physics/percolation/index.md

@@ -0,0 +1,49 @@
+# section information
+section:
+  name: Skills
+  id: skills
+  enable: true
+  weight: 2
+  showOnNavbar: true
+  # Can optionally hide the title in sections
+  # hideTitle: true
+
+# Your Skills.
+# Give a summary of you each skill in the summary section.
+skills:
+- name: Kubernetes
+  logo: /images/sections/skills/kubernetes.png
+  summary: "Capable of deploying, managing application on Kubernetes. Experienced in writing Kubernetes controllers for CRDs."
+  url: "https://kubernetes.io/"
+
+- name: Go Development
+  logo: /images/sections/skills/go.png
+  summary: "Using as the main language for professional development. Capable of writing scalable, testable, and maintainable program."
+  url: "https://golang.org/"
+
+- name: Cloud Computing
+  logo: /images/sections/skills/cloud.png
+  summary: "Worked with most of the major clouds such as GCP, AWS, Azure etc."
+
+- name: Docker
+  logo: /images/sections/skills/docker.svg
+  summary: "Write most of the programs as dockerized container. Experienced with multi-stage, multi-arch build process."
+  url: "https://www.docker.com/"
+
+- name: Prometheus
+  logo: /images/sections/skills/prometheus.png
+  summary: "Capable of setup, configure Prometheus metrics. Experienced with PromQL, AlertManager. Also, experienced with writing metric exporters."
+  url: "https://prometheus.io/"
+
+- name: Linux
+  logo: /images/sections/skills/linux.png
+  summary: "Using as the main operating system. Capable of writing bash/shell scripts."
+
+- name: Git
+  logo: /images/sections/skills/git.png
+  summary: "Experienced with git-based development. Mostly, use Github. Also, have experience in working with GitLab."
+  url: "https://git-scm.com/"
+
+- name: C++
+  logo: /images/sections/skills/c++.png
+  summary: "Know basic C/C++ programming. Used for contest programming and problem solving."

content/posts/physics/percolation/percolation_example_0.py

@@ -156,6 +156,14 @@
 
 
 
+                <li class="nav-item">
+                  <a class="nav-link" href="/#skills">Skills</a>
+                </li>
+              
+            
+            
+              
+              
                 <li class="nav-item">
                   <a class="nav-link" href="/#experiences">Experiences</a>
                 </li>
@@ -371,6 +379,15 @@ <h5>Navigation</h5>
 
 
 
+                <li class="nav-item">
+                  <a class="smooth-scroll" href="http://localhost:1313/#skills">Skills</a>
+                </li>
+              
+              
+                
+                
+                  
+                
                 <li class="nav-item">
                   <a class="smooth-scroll" href="http://localhost:1313/#experiences">Experiences</a>
                 </li>