In condensed matter physics, phase transitions and critical phenomena, like the one you're modeling using PhET's Plinko Probability, are key concepts. Your experiment can be connected to several important condensed matter physics ideas, including symmetry breaking, critical exponents, and universality. Here’s a more detailed breakdown that ties the virtual experiment to condensed matter physics:

1. Phase Transitions and Symmetry Breaking:

• Phase Transition: In condensed matter physics, a phase transition is a fundamental change in the state of a system, typically driven by changes in an external parameter (like temperature or pressure). In your Plinko model, the external parameter is the rod angle, which influences the probability distribution of marbles falling to the left (N1) or right (N2).

• Symmetry Breaking: A key aspect of phase transitions in physics is the concept of symmetry breaking. For example, spins are disordered and symmetrical at high temperatures in a ferromagnet. The spins align below a critical temperature (the Curie point), breaking the symmetry. Similarly, in your experiment, the system is symmetrical when the rod angle is zero (balanced) (N1 ≈ N2). As you tune the angle, the symmetry is broken, and more marbles fall either to the left or right, analogous to an ordered state.

• Order Parameter: The difference between N1 and N2 is the order parameter for this transition, analogous to magnetization in ferromagnetic materials. The order parameter remains zero before the transition (symmetrical phase) and becomes non-zero after the transition (broken symmetry).

2. Critical Exponents and Universality:

• Critical Exponents: Near the phase transition point, various physical quantities follow power laws described by critical exponents. The most famous example is the magnetization in a ferromagnet near the Curie temperature, where the magnetization M behaves as:

where Tc​ is the critical temperature, and β is the critical exponent.

• • In your Plinko experiment, the critical exponent β\betaβ governs how the order parameter (the difference in marble counts) behaves as you tune the rod angle near the critical angle.

  • Near the critical angle θc​, the order parameter M would scale as:

Universality: The idea of universality states that different physical systems can exhibit the same critical behavior near their phase transitions despite differences in microscopic details. This is why a simplified model like Plinko, despite its simplicity, can mimic the critical phenomena observed in more complex systems such as magnets or liquid-gas transitions. The behavior of the system near criticality depends only on a few features like dimensionality and symmetry, not the specific interactions between particles.

3. Ising Model Connection:

• Ising Model Analogy: Your Plinko experiment is somewhat analogous to the Ising model in statistical mechanics, which describes phase transitions in systems of interacting spins.

  • In the Ising model, each spin can point up or down, much like your marbles can fall to the left (N1) or right (N2). The rod angle in Plinko plays the role of an external field in the Ising model, biasing the direction in which the spins (or marbles) fall.

  • The transition from a roughly equal number of marbles on both sides to a predominantly one-sided distribution mirrors the ferromagnetic phase transition, where spins align in one direction below the Curie temperature.

  • The order parameter in the Ising model is the net magnetization, analogous to the difference in marble counts in your experiment.

4. Fluctuations and Critical Point:

• Critical Point: At the critical point, fluctuations in the system are maximized. Large-scale fluctuations of the order parameter are observed in a physical system undergoing a second-order phase transition near the critical point. In your Plinko experiment, you should see larger fluctuations in the number of marbles falling to the left or right as you approach the critical rod angle. These fluctuations are analogous to spin fluctuations in the Ising model near the critical temperature.

• Correlation Length: In physical systems near a phase transition, the correlation length (the distance over which particles or spins are correlated) diverges. In Plinko, this might manifest as a growing sensitivity of the marble distribution to small changes in the rod angle near the critical point.

5. Mean-Field Theory and Critical Behavior:

• Mean-Field Theory: This approximation simplifies the interactions in a system by assuming each particle or spin interacts with an average (mean) field generated by all the others. In the context of your Plinko experiment, mean-field theory could be applied to describe how marbles are influenced by the overall probability landscape defined by the rod angle.

• Near the critical point, mean-field theory predicts certain values for the critical exponents. Still, these may differ from the true behavior in systems where fluctuations are important (like in two-dimensional systems, which is more applicable to the Plinko model).

6. Connection to Percolation Theory:

• Percolation Theory: Another model from statistical mechanics, percolation theory, describes how connected clusters form in a random system. In your Plinko experiment, as the rod angle is tuned, you can think of the marbles as part of a percolating cluster that favors one side. When the system reaches the critical point, this “cluster” of marbles dominates one side, forming a giant cluster in percolation theory.

• The transition in percolation theory from a small cluster-dominated system to one where a giant cluster dominates is similar to the transition in Plinko, where one side accumulates more marbles beyond the critical angle.

The Virtual PhET Plinko Probability experiment simulates key aspects of phase transitions in condensed matter physics, particularly symmetry breaking and critical phenomena. The critical exponent β\betaβ characterizes the behavior of the order parameter near the phase transition. The tuning of the rod angle mimics an external field driving the system toward a phase transition, much like temperature or magnetic fields in physical systems. Concepts like the Ising model, mean-field theory, and percolation theory can be applied to understand how the distribution of marbles behaves near the critical angle, connecting this simple model to deeper physical principles.

To give a more detailed theoretical framework for the experiment with PhET's Plinko Probability and to connect it to condensed matter physics, more formalized theories and equations were included, as shown below: 

1. Phase Transition Theory:

In condensed matter physics, phase transitions are often described by the behavior of an order parameter near a critical point. In your Plinko experiment, the order parameter can be defined as:

where:

• N1​ is the number of marbles on the left side.

• N2​ is the number of marbles on the right side.

The order parameter M is analogous to magnetization in a ferromagnet, which becomes non-zero below the Curie temperature, representing spontaneous symmetry breaking.

 Symmetry Breaking:

 When the rod angle θ is zero, the system is symmetrical, with N1≈N2​, meaning M=0. As θ deviates from zero, the symmetry is broken, and MMM begins to take on a non-zero value, signaling a phase transition.

2. Mean-Field Theory and Critical Exponent:

In mean-field theory, the behavior of the order parameter near the critical point can be modeled by a power-law relation. Near the critical rod angle θc\theta_cθc​, the order parameter MMM follows:

where:

• M is the order parameter.

• θ is the rod angle.

• θc is the critical angle at which the phase transition occurs.

• β is the critical exponent associated with the order parameter.

In physical systems, the critical exponent β\betaβ characterizes how the order parameter vanishes as the system approaches the critical point. In many systems, this value is around β=1/2​, but it can vary depending on the dimensionality and nature of interactions.

3. Fluctuations and Susceptibility:

 Near the critical point, fluctuations in the order parameter become large, and the susceptibility χ, which measures the e to changes in external parameters (in this case, the rod angle), diverges. The susceptibility is defined as:

where γ is another critical exponent associated with the divergence of the susceptibility. This reflects how sensitive the system becomes to small changes in the rod angle near θc​, just like a ferromagnet becomes highly responsive to external magnetic fields near its critical temperature.

4. Correlation Length and Critical Behavior:

The correlation length ξ\xiξ describes how far correlations between different parts of the system extend. As the system approaches the critical point θc\theta_cθc​, the correlation length diverges, following:

where ν is the critical exponent for the correlation length.

 In the virtual Plinko experiment, you may not directly measure ξ, but the divergence of ξ implies that the system becomes highly correlated near the critical angle, with large-scale fluctuations in the number of marbles falling to the left or right.

6. Connection to the Ising Model:

The Plinko experiment can be related to the 1D or 2D Ising model, which is used to describe phase transitions in systems with discrete variables (like spins or, in your case, marbles). In the Ising model, spins can point either up or down, analogous to marbles falling left or right. The total magnetization M (or difference between left and right marbles) is the order parameter.

In the Ising model, at the critical temperature Tc​ (or critical angle θc​ in your experiment), the system transitions from a disordered phase (equal spins/marbles on both sides) to an ordered phase (predominantly spins/marbles on one side). Near the critical point, the magnetization (or marble imbalance) scales as:

or equivalently, in your experiment:

7. Scaling Laws and Universality:

The scaling laws for critical exponents, such as β, γ, and ν, often follow certain relations known as scaling laws. One important scaling law is:

where η\etaη is another critical exponent related to the decay of correlations at the critical point. These scaling laws are part of the theory of universality, which states that systems with similar symmetry and dimensionality exhibit the same critical behavior, regardless of microscopic details.