Counting States and The Einstein Solid – Thermodynamics Homework

1. The analysis of coin-flipping from the tutorial (attached below) may seem to be nothing more than an engaging game. However, it is also directly relevant to physics. Among the applications of this mathematical treatment is the system of an ideal paramagnet.

Consider a system of a set of N magnetic spins in the presence of an external magnetic field. Each spin can point either up or down, where ‘up’ refers to the direction of the external field.

Let the number of up spins be N_up and the number of down spins be N_down.

A.Write an equation relating N_up and N_down to N.

Rearrange your equation to express N_down in terms of N and N_up.

B.The magnetization of the system is related to the number of up and down spins. Each spin has a magnetic moment m. Write an expression for the total magnetization of the system M in terms of N, N_up, and m. (Hint: If there are equal numbers of up and down spins, what is M?)

Recall the mathematics of coin flipping. The number of microstates for a given macrostate can be determined using the combinatorial. A macrostate with M heads out of a set of N coins corresponds to a number of microstates given by .

C.Apply this method to the spin system described above. Write an expression for the number of microstates corresponding to a macrostate with N_up up spins and N total spins.

How many possible macrostates are there for this system?

D.Use a computer program (e.g., Excel, Mathematica) to produce a table with the number of microstates for each possible macrostate for a system of 20 spins. Attach the table to your homework.

E.Which of the macrostates is most probable for the system of 20 spins? What is the probability of this macrostate?

2. Below four situations are described in which fair coins are flipped. For each situation, state whether the description best represents a microstate or a macrostate, and explain briefly.

Case A: Seven distinguishable coins are flipped, with results THTHHTT

Case B: Seven distinguishable coins are flipped, resulting in four heads and three tails.

Case C: A coin is flipped seven times. The first three results are tails and the rest are heads.

Case D: A coin is flipped seven times, resulting in three tails and four heads.

Rank the probabilites of the outcomes in Cases A – D from largest to smallest, specifying if any are equal. (e.g., P_A = P_B > P_C > P_D)

3. Consider a system with three blue boxes and three red boxes. Eight balls may be placed in any of the boxes, and each box may hold any number of balls. Compare the probability of the following states: State A, with seven balls in blue boxes and one in red boxes, and State B, with three balls in red boxes and five in blue boxes.

Is State A more probable, equally probable, or less probable than State B? Explain briefly. Is this what you expect? Hopefully, you made this problem easier on yourself by noticing that it is analogous to what physical system?

4. Plot in Excel (or another graphing program of your choice) the probability distribution for the interacting system of 3 oscillators and 3 oscillators sharing 8 units of energy that you calculated in part IV of the tutorial on Einstein Solids (attached below). Also plot the probability distribution of 90 and 30 oscillators sharing 50 energy units from part V of the same tutorial. Remember to label axes.

Submit the graphs with your homework.

Compare them. What is the difference between the width of the peaks? What implication does this have for real macroscopic physical systems (like real gases and solids) that contain an Avagadro’s number’s worth of atoms and molecules?