The ensemble that trades particles

Science · Thermal & Statistical Physics · ages 22-23

Name ______________________ Date ____________

What does mu measure?
- Total energy of the system
- Free energy cost of one more particle
- The temperature of the reservoir
What does the grand ensemble hold fixed?
- Energy and particle number
- Temperature and chemical potential
- Pressure and volume only
The grand ensemble is the convenient choice when particle number wanders.
Circle one: True False
System P has higher mu than system Q, and they trade particles. Which way?
- From Q to P
- From P to Q
- Neither moves at all
The grand sum for a tiny system runs over what?
- Particle number only
- Energy only
- Both energy and particle number
How do you get the average particle number from the grand partition function?
- By differentiating with respect to mu
- By counting states by hand
- By fixing N and minimizing
Two chambers trade particles until nothing changes. What holds at the end?
- Equal volumes
- Equal temperatures only
- One shared chemical potential
A student builds a grand sum at strictly fixed N. What is wrong?
- Fixed N forbids the particle sum the ensemble needs
- Boltzmann factors never apply to particles
- Tiny systems have no states

Answer key

For grown-ups. Fold this page away before handing over the rest.

Free energy cost of one more particle · Mu prices each added particle.
Temperature and chemical potential · It trades both energy and particles with the world.
True · Wandering number is exactly what it handles.
From P to Q · Particles run down the mu slope.
Both energy and particle number · Both fluctuate, so both are summed.
By differentiating with respect to mu · One derivative with respect to mu yields it.
One shared chemical potential · Flow stops when mu matches everywhere.
Fixed N forbids the particle sum the ensemble needs · Grand means summing over N, not fixing it.