A system has three distinguishable molecules at rest, each with a quantized magnetic moment which can have \(z\)-components \(+1 / 2 \mu\) or \(-1 / 2 \mu\). Find an expression for the distribution function, \(f_{i}\) (i denotes the ith configuration), which maximizes entropy subject to the conditions \(\sum_{i} f_{i}=1\) and \(\sum_{i} M_{i, z} f_{i}=\gamma \mu\), where \(M_{i, z}\) is the magnetic moment of the system in the ith configuration. For the case \(y=1 / 2\), compute the entropy and compute \(f_{i}\).

Quantized magnetic moments in a system refer to the idea that magnetic moments can only take on discrete values. In this exercise, each molecule has a quantized magnetic moment that can be either \(+1/2 \mu\) or \(-1/2 \mu\). This means the magnetic moments are not continuous, but rather, they are restricted to specific levels. This quantization arises due to the intrinsic properties of the particles, commonly related to their quantum mechanical spin states. The system's magnetic configurations are all the possible ways these quantized moments can be arranged.

The distribution function, denoted as \(f_i\), is a mathematical expression that describes how probabilities are assigned among different possible states or configurations of a system. In this exercise, \(f_i\) represents the probability of the system being in the \i-th\ configuration. The two key conditions given are \(\textstyle{\sum_i \fi = 1}\), indicating that the total probability must sum to one, and \(\textstyle{\sum_i M_{i,z} \fi = \gamma \mu}\), which connects the expected value of the magnetic moment to a given system-wide constraint. These conditions ensure the probabilities are properly normalized and the system's overall magnetic moment aligns with specific criteria.

The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. In this problem, we use Lagrange multipliers to maximize the entropy of the system under the constraints \(\textstyle{\sum_i \fi = 1}\) and \(\textstyle{\sum_i M_{i,z} \fi = \gamma \mu}\). We introduce multiplier coefficients, \( \lambda_1\) and \( \lambda_2\), to create the Lagrangian function: \[ \mathcal{L} = -k_B \sum_i \fi \ln \fi + \lambda_1 \left( \sum_i \fi-1 \right) + \lambda_2 \left( \sum_i M_{i,z} \fi - \gamma \mu \right) \]
By differentiating this Lagrangian with respect to each \( \fi \) and setting these derivatives to zero, we derive the conditions under which the entropy is maximized.

Entropy maximization in statistical physics seeks the most probable distribution of system configurations, given a set of constraints. The principle of entropy maximization states that the configuration that maximizes entropy is the one that nature will most likely adopt, balancing all possible microstates. The entropy, \(S\), for our system, is given by \[ S = -k_B \sum_i \fi \ln \fi \]
To find the entropy-maximizing distribution, we use the constraints to form a Lagrangian as discussed previously and solve for the probabilities \( \fi \). Hence, by maximizing \(S\) subject to the conditions, we determine the distribution function that best represents the system's thermodynamic equilibrium.

Statistical configurations refer to the different possible arrangements of the individual states of the system's components. In our system of three distinguishable molecules with two possible magnetic moments each, the statistical configurations are \(2^3 = 8\) possible arrangements. Each configuration has a specific total magnetic moment, \(M_{i,z}\), which is the sum of magnetic moments for that configuration. List of possible configurations:

• \(+1/2 \mu, +1/2 \mu, +1/2 \mu\)
• \(+1/2 \mu, +1/2 \mu, -1/2 \mu\)
• \(+1/2 \mu, -1/2 \mu, +1/2 \mu\)
• \(-1/2 \mu, +1/2 \mu, +1/2 \mu\)
• \