Question

Using matlab, evaluate the following system:

Consider two Einstein solids \(A\) and \(B\) that can exchange energy (but not oscillators/particles) with one another but the combined composite system is isolated from the surroundings. Suppose systems \(A\) and \(B\) have \(N_{A}\) and \(N_{B}\) oscillators, and \(q_{A}\) and \(q_{B}\) units of energy respectively. The total number of microstates for this macrostate for the macrostate \(N_{A}, N_{B}, q, q_{A}\) is given by

$$ \Omega\left(N_{A}, N_{B}, q, q_{A}\right)=\Omega\left(N_{A}, q_{A}\right) \Omega\left(N_{B}, q_{B}\right) $$

where

$$ \Omega\left(N_{i}, q_{i}\right)=\frac{\left(q_{i}+N_{i}-1\right) !}{q_{i} !\left(N_{i}-1\right) !} $$

(a) If \(N_{A}=2\) and \(N_{B}=4\), and initially \(q_{A}=6\) and \(q_{B}=0\) :

(i) What is the initial number of microstates for the composite system?

(ii) Once energy can be exchanged between the two subsystems, plot the multiplicity of the various possible macrostates of the combined system as a function of \(q_{A}\).

(iii) What is the macrostate which has the highest number of microstates, and what is the number of microstates.

(iv) What is the probability that the system will return to its initial state?

(b) Repeat the above for: \(N_{A}=50\) and \(N_{B}=100\), and initially \(q_{A}=150\) and \(q_{B}=0 .\) Also, what quantity is common to both systems in the macrostate with the highest multiplicity? What do you notice about the width of the peak?

(c) Consider the special case where \(N_{A}=1\). Suppose that \(N_{B}=20, q_{A}=0\), and \(q_{B}=30\). If we assume that the subsystem \(A\) can exchange energy with the much larger system \(B\) :

(i) Calculate, and plot the probability that system \(A\) has energy \(q_{A}\) for all possible macrostates of the system?

(ii) Postulate the functional form of the probability distribution (Hint: Try various log plots).

(iii) The probability in this case is called the Boltzmann probability distribution. Why is the form of this probability different than the probability that you found in the other problems?

Answer 1

To solve this problem the details calculations are given below