Thermodynamics: Stats: Problems 3

Problem :

Write down the Gibbs sum. Be sure to get all of the indices correct.

Z_G(μ, τ) =

e^(Nμ-)/τ

Problem :

Give the expression for the absolute probability that a system will be found in the state with N₁ particles and energy .

P(N₁,

) =

Problem :

Give an expression for the average value of a property A for a system in diffusive and thermal contact with a "reservoir". A "reservoir" is a huge system next to our smaller system with large energy and number of particles.

< A > =

Problem :

Give an expression for the average number of particles in a system that is in thermal and diffusive contact with a reservoir.

We are looking for < N >, which we can calculate using the formula we just derived.

< N > =

Problem :

Suppose that we have a system that can be unoccupied or can have one particle in a state with energy . Write the Gibbs sum for this system.

One possible state has N = 0, for which we say that the energy is also zero. So the first term in the sum is 1. The second possible state has N = 1, and energy . We can write the total sum as: