Explanation:

1. simulate_phase_transition(N, BP_values):

   • This function simulates the dropping of N marbles for a range of binary probabilities (BP).

   • For each BP value, we use a binomial distribution to simulate how many marbles land in the left bin (N1). The remaining marbles fall into the right bin (N2=N−N1​).

   • The order parameter (M) is calculated as M=(N1−N2)/(N1+N2)​​, which measures the difference in the number of marbles on either side.

2. N = 1000:

   • We simulate 1000 marbles being dropped.

3. BP_values = np.linspace(0, 1, 100):

   • We generate 100 binary probability values, evenly spaced between 0 and 1, to model the rod angle in the Plinko experiment.

import numpy as np

import matplotlib.pyplot as plt

# Function to simulate marble drops and return N1 and N2

def simulate_N1_N2(N, BP_values):

    N1_list = []

    N2_list = []

    for BP in BP_values:

        # Simulate N marbles being dropped with probability BP to the left

        N1 = np.random.binomial(N, BP)  # Number of marbles to the left

        N2 = N - N1                     # Number of marbles to the right

        N1_list.append(N1)

        N2_list.append(N2)

    return N1_list, N2_list

# Parameters

N = 1000  # Number of marbles

BP_values = np.linspace(0, 1, 100)  # BP values from 0 to 1 in 100 steps

# Run the simulation to get N1 and N2

N1_list, N2_list = simulate_N1_N2(N, BP_values)

# Plot BP vs N1 and N2

plt.figure(figsize=(8, 6))

plt.plot(BP_values, N1_list, label='N1 (Marbles on Left)', color='r')

plt.plot(BP_values, N2_list, label='N2 (Marbles on Right)', color='g')

plt.title('BP vs N1 and N2 for N = 1000')

plt.xlabel('Binary Probability (BP)')

plt.ylabel('Number of Marbles (N1, N2)')

plt.grid(True)

plt.legend()

plt.show()

Output:

 The output is a graph showing the relationship between BP and the order parameter (M). When BP = 0.5, the system is symmetric (M = 0), and as BP approaches 0 or 1, the system becomes asymmetric, with M approaching +1 or -1.

 This simulation provides an intuitive way to visualize how the system transitions from symmetry to asymmetry, mimicking a phase transition.

import numpy as np

import matplotlib.pyplot as plt

# Function to simulate marble drops and calculate the order parameter (M)

def simulate_phase_transition(N, BP_values):

    order_params = []

    for BP in BP_values:

        # Simulate N marbles being dropped with probability BP to the left

        N1 = np.random.binomial(N, BP)  # Number of marbles to the left

        N2 = N - N1                     # Number of marbles to the right

        # Calculate the order parameter M

        M = (N1 - N2) / (N1 + N2)  # (N1 - N2) / (N1 + N2) normalizes the difference

        order_params.append(M)

    return order_params

# Parameters

N = 1000  # Number of marbles

BP_values = np.linspace(0, 1, 100)  # BP values from 0 to 1 in 100 steps

# Run the simulation

order_params = simulate_phase_transition(N, BP_values)

# Plot BP vs Order Parameter

plt.figure(figsize=(8, 6))

plt.plot(BP_values, order_params, label='Order Parameter (M)')

plt.title('BP vs Order Parameter for N = 1000')

plt.xlabel('Binary Probability (BP)')

plt.ylabel('Order Parameter (M)')

plt.grid(True)

plt.legend()

plt.show()