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# Some Problems in Statistical Mechanics

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Consider a lattice of N particles in an external homogeneous magnetic field,
where each particle has two possible energy states, depending on whether the spin of the particle points in the direction of the magnetic field ("down") or opposite to it ("up"). In the "down" state the particle has no energy e = 0 and in the "up" state it has the energy e =ε. The microstate of the system is specified by the energy states of all the particles, i.e. the list (e1, e2....eN), while the macrostate is specified by the total energy of the system E, i.e. the sum

(a) If N ¡ n particles are in the "down" state, what is the total energy of the system?

(b) Count the number of microstates that correspond to the macrostate in which the
total energy of the system is E =nε where is an integer.
(c) Count the number of all possible microstates for the system.