Method

Expanded Description: The Plinko board serves as an educational tool for understanding probability and statistics. As marbles are dropped from the top, they encounter a series of pegs arranged in a triangular pattern. Each time a marble hits a peg, it randomly deflects left or right, creating a distribution of marbles in bins at the bottom. This distribution, as more marbles are dropped, converges to a binomial distribution, providing a visual representation of statistical principles.

In this experiment, we extend the Plinko model beyond its basic statistical use. By introducing variables like the rod angle (or binary probability), we simulate a system akin to those seen in condensed matter physics. Specifically, we explore how the distribution of marbles changes as the system undergoes a phase transition. The marbles' movement through the grid mimics particle behavior in materials as external parameters (like temperature or magnetic field) are varied.

Our goal is to identify the critical phase transition point—the point where the system shifts from one phase to another. In doing so, we also calculate the order parameter, which quantifies how ordered or disordered the system becomes as it nears the transition. This approach provides insights into the symmetry breaking and critical phenomena commonly observed in physical systems undergoing phase transitions.

Detailed Experimental procedure incorporating the binary probability (BP) to define the rod angle, as shown below:

 Step-by-Step Procedure:

1. Initial Setup:

• Open the PhET Plinko Probability simulation.

• Set the total number of marbles (N) to 100.

• Set the Rows number as 11.

• Ensure the "Bounce around pegs" option is selected to simulate randomness in marble drops.

2. Define Binary Probability (BP) for Rod Angle:

• The rod angle is determined by the binary probability (BP), which corresponds to the following angles:

  • If BP = 0, the rod angle is 45°.

  • If BP = 0.5, the rod angle is 0°.

  • If BP = 1, the rod angle is -45°.

• By changing the BP value, you will adjust the rod angle and control the bias of marble distribution.

3. Drop Marbles and Record Data for Each BP Value:

• Start with BP = 0.5, corresponding to a rod angle of 0°. Drop all 100 marbles.

• Repeat this procedure 10 times at BP = 0.5 (angle = 0°), counting how many marbles fall in the left section N1​ and the right section N2​ for each trial.

• After 10 trials, calculate the mean and standard deviation for N1 and N2​:

  • The mean values of N1 and N2 are:

The standard deviations are:

• Record the mean and standard deviation for both N1​ and N2.

4. Repeat for Different BP Values (Rod Angles):

• Set BP = 0 (angle = 45°). Repeat the experiment:

  • Drop 100 marbles, perform 10 trials, and record the number of marbles falling into N1​ and N2​ for each trial.

  • Calculate the mean and standard deviation for N1​ and N2​.

• Set BP = 1 (angle = -45°), and repeat the experiment:

  • Drop 100 marbles, perform 10 trials, and calculate the mean and standard deviation for N1​ and N2​.

• You can also experiment with BP values between 0 and 1 (e.g., BP = 0.25, BP = 0.75) to explore intermediate rod angles and observe how the marble distribution changes.

5. Data Analysis:

• For each BP value, calculate the order parameter based on the mean values of N1​ and N2​: 

• Plot the order parameter MMM versus BP to visualize how the marble distribution changes with different BP values (and corresponding rod angles).

6. Estimate the Critical BP Value and Critical Exponent β:

• From the plot of MMM vs. BP, identify the critical BP value where the system transitions from a balanced distribution (N1 ≈ N2) to an unbalanced distribution (N1 ≠ N2).

• Fit the data near the critical BP value to a power-law equation

• Estimate the critical exponent β from this fit.

7. Statistical Analysis:

• Use the standard deviations from the 10 trials to assess the fluctuations in marble distribution at each BP value.

• Observe how fluctuations (standard deviations) increase near the critical BP value, indicating the system's sensitivity to phase transition.

• Summarize the results, including the critical BP value and critical exponent β.

• Discuss the statistical behavior of the system as BP (and the rod angle) changes, focusing on the phase transition and the associated fluctuations.

This procedure now incorporates the conversion from binary probability (BP) to the rod angle, ensuring a clear link between BP values and the phase transition in the system.

Data Analysis:

1. Calculate the Order Parameter:

• For each BP value, calculate the order parameter using the formula: 

Project 2: Critical Exponent and Order Parameter:

• As shown in Project 2, the system will conform to the following power law formula

• Here, α0 is a constant, BP is the binary probability, BPC is the critical binary probability (which will be determined), and β\betaβ is the critical exponent.

3. Determine the Critical Binary Probability BPC:

• Using the data for N1 and N2​ at different BP values, plot N1​ and N2​ versus BP.

• Identify the intersection point of the two curves for N1 and N2. This point gives you the critical binary probability (BPC), where the system undergoes a phase transition.

4. Logarithmic Fitting and Power Law:

• Use the table provided in Project 2 to calculate the logarithmic values

• Plot these values and fit them to a straight line to determine the critical exponent β\betaβ using the power-law relationship.

•  The method of the double-logarithmic plot can be found in the link: https://read01.com/zh-tw/DGRKn5L.html#.YLi8LfkzZhE

5. Report Results:

• From the fitting process, extract the values of the critical binary probability (BPC) and the critical exponent β. These values characterize the phase transition of the system.

By following these steps, you will calculate the order parameter and find the critical binary probability and describe the system’s behavior near the phase transition using the power law. This allows for a deeper understanding of how the binary probability controls the marble distribution and the critical phenomena in the system.