Modeling Random Walks in Python for Fun

 Random walks are a fascinating and simple concept to model in Python, offering endless opportunities for exploration. Here’s how you can get started with a basic implementation and gradually enhance it for more fun.

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Basic 1D Random Walk

A one-dimensional random walk is the simplest version. Each step is either forward (+1) or backward (-1) along a line.

import random
import matplotlib.pyplot as plt

def random_walk_1d(steps):
    position = 0
    walk = [position]

    for _ in range(steps):
        step = random.choice([-1, 1])
        position += step
        walk.append(position)

    return walk

# Generate and plot a 1D random walk
steps = 100
walk = random_walk_1d(steps)

plt.figure(figsize=(10, 6))
plt.plot(walk)
plt.title("1D Random Walk")
plt.xlabel("Steps")
plt.ylabel("Position")
plt.show()

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2D Random Walk

In a 2D walk, the walker can move up, down, left, or right.

def random_walk_2d(steps):
    x, y = 0, 0
    walk = [(x, y)]

    for _ in range(steps):
        dx, dy = random.choice([(0, 1), (0, -1), (1, 0), (-1, 0)])
        x += dx
        y += dy
        walk.append((x, y))

    return walk

# Generate and plot a 2D random walk
steps = 200
walk = random_walk_2d(steps)

x_coords, y_coords = zip(*walk)
plt.figure(figsize=(8, 8))
plt.plot(x_coords, y_coords, marker="o", markersize=3)
plt.title("2D Random Walk")
plt.xlabel("X Position")
plt.ylabel("Y Position")
plt.axis("equal")
plt.show()

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3D Random Walk

In 3D, the walker can move along the x, y, or z axes.

from mpl_toolkits.mplot3d import Axes3D

def random_walk_3d(steps):
    x, y, z = 0, 0, 0
    walk = [(x, y, z)]

    for _ in range(steps):
        dx, dy, dz = random.choice([(1, 0, 0), (-1, 0, 0), 
                                    (0, 1, 0), (0, -1, 0), 
                                    (0, 0, 1), (0, 0, -1)])
        x += dx
        y += dy
        z += dz
        walk.append((x, y, z))

    return walk

# Generate and plot a 3D random walk
steps = 300
walk = random_walk_3d(steps)

x_coords, y_coords, z_coords = zip(*walk)
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection="3d")
ax.plot(x_coords, y_coords, z_coords)
ax.set_title("3D Random Walk")
plt.show()

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Ideas for Exploration

1. Boundaries: Simulate walks confined within a grid or sphere.
2. Biased Walks: Add probabilities to prefer certain directions.
3. Obstacles: Introduce barriers the walker must avoid.
4. Multiple Walkers: Model multiple entities walking and possibly interacting.
5. Physics-Based Walks: Incorporate inertia or forces for realistic modeling.
6. Fractal Patterns: Create self-similar patterns using random walks.

Would you like to dive into any of these ideas or explore enhancements?