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Einstein solids: macrostates and microstates

Consider a system of two Einstein solids, A and B, each containing 10 oscillators, sharing a total of 20 units of energy. Assume that the solids are weakly coupled, and that the total energy is fixed.

(a) How many different macrostates are available to this system?

(b) How many different microstates are available to this system?

(c) Assuming that this system is in thermal equilibrium, what is the probability of
finding all the energy in solid A?

(d) What is the probability of finding exactly half of the energy in solid A?

(e) Under what circumstances would this system exhibit irreversible behavior

Solution Preview

Let's denote the quantum states of oscillator i of solid A as n_i and those of B by m_i.

The energy of the harmonic oscillators are then (n+1/2)h-bar omega. In this problem you are supposed to take your unit of energy to be h-bar omega. And the zero point energy must be disregarded. You can always shift your zero-point of the system by half a unit and decide to measure energy relative to that point. If we denote the total energy of solid A by N then we have:

n_1 + n_2 + n_3 + ...+ n_10 = N

and for solid B we denote the total energy by M:

m_1 + m_2 + m_3 + ...+ m_10 = M

N and M are then the macroscopic observable quantities. You can calculate the temperature of the system A and B in terms of N and M respectively. The number of macrostates is the number of different values that N and M can take. If the total energy is E (in your problem this is 20 but let's keep it general, setting E = 20 can lead to confusion with the total number of variables later on), then N + M has to be E so there are E+1 states ranging from N=0, M=E to N = E, M = 0. The number of microstates ...

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