EDITED: Consider a system of two Einstein solids, A and B, each containing 10 os

Question

EDITED: Consider a system of two Einstein solids, A and B, each containing 10 oscillators and sharing 20 units of energy. Assume the solids are weakly coupled and that the system is isolated. (a) How many different macrostates are available to the system? (b) How many different microstates are available to the system? (c) On time scales long enough for the system to thermalize, what is the probability of finding all of the energy in solid A? (d) What is the probability of finding the energy divided even between the two solids? (e) Let XA denote the macrostate with all of the energy in solid A and X1/2 denote the macrostate with the energy evenly divided. Briefly comment on the relative likelihood of the system starting in the X0 macrostate and some time later being found in the X1/2 macrostate versus the reverse happening. Specifically, what does this mean in terms of irreversibility? (Answer: P(X1/2) is aproxx equal to 850P(X0))

Explanation / Answer

a) Net rate heat is entering segment = (1/L)(dQ/dt) * (1 + x + x2)

b) Here, dQ/dt = C * dT/dt

=> By, Q' = (1/L)(dQ/dt) * (1 + x + x2)

By, double differentiating above equation

=> dT/dt = K * (d2T/dx2)

c)   As,   dT/dt = K * (d2T/dx2)

Integrating between limits 0 to x

=> T(x,t) = A + B/sqrt(t) * e-x2/4KT

d) This tell us that T(x) must be unevenly distributed throughout the body .

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