Course: PH 307 Satistical Mechanics: Practice Problems: SET 3
Course: PH 307 Satistical Mechanics: Practice Problems: SET 3
Course: PH 307 Satistical Mechanics: Practice Problems: SET 3
Course: PH 307
Satistical Mechanics
2. Consider an isolated system consisting of a large number N of very weakly interacting localized
particles of spin 1/2. Each particle has a magnetic moment which can point either parallel or
anti-parallel to an applied field H. The energy E of the system is then E = (n1 n2 )H, where
n1 is the number of spins aligned parallel to H and n2 the number of spins aligned anti-parallel
to H.
(a) Consider the energy range between E and E + E where E is very small compared to E
but microscopically large so that E << H. What is the total number of states (E) lying in
this energy range?
(b) Write down an expression for ln (E) as a function of E. Simplify this expression by applying
Stirlings formula in its simplest form.
(c) Calculate temperature. Express energy in terms of temperature and magnetic field.
4. Consider the thermal interaction between two systems A, A0 . Find the probability P (E) of
finding the total system to be in a microstate such that the system A has energy between E and
E + E when the whole system is in equilibrium.
Take a small deviation << 1 of the equilibrium energy of A, i.e. E = E + . Expand
the probability P (E) around the equilibrium energy and retain upto 2 term. Show that the
probability distribution reduces to Gaussian one.
Show that the standard deviation is of the order Ef where f is the degrees of freedom of the
systems.
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5. Consider two spin systems A and A0 placed in an external field H. System A consists of N
weakly interacting localized particles of spin 1/2 and magnetic moment . Similarly, system A0
consists of N 0 weakly interacting localized particles of spin 1/2 and magnetic moment 0 . Two
systems are initially isolated with respective total energies E = bN H and E 0 = b0 N 0 0 H. They
are placed in thermal contact with each other (|b| << 1, |b0 | << 1).
(a) In the most probable situation corresponding to the final thermal equilibrium , how is the
energy E of system A related to the energy E 0 of system A0 ?
(b) What is the value of the energy E of system A?
(c) What is the heat Q absorbed by the system A in going from the initial situation to the final
situation when it is in equilibrium with A0 ?