Question

For a (non-interacting) gas of bosons (any gas for that matter), the total particle number has to be a finite value and in the canonical description this is ensured by writing a constrained sum for the canonical partition function:

Z=∑{np}exp(−β∑pnpϵ(p))δN,∑pnp
Instead of computing this, one moves to the Grand Canonical Description where now there is no constraint on possible combinations of particles summing to some total N:
Zg=∑{np}exp(−β∑pnp(ϵ(p)−μ))
which can be re-written as:
Zg=∏p∑npexp(−βnp(ϵ(p)−μ))
Here, the np are the allowed occupations which can go to infinity for bosons.
What has happened I think is that somehow the condition of total particles being equal to N has been shoved in the chemical potential which takes care of it and therefore the np now become allowed occupations not constrained to total upto N. I only understand this vaguely and don't understand how the unconstrained sum vanishes in grand canonical picture due to the introduction of chemical potential.In short, the question then is:
How does the grand canonical description takes care of the fact that the total number of gas particles is some finite value N?

Answer 1

The grand canonical ensemble takes care of the fact that the total number of gas particles is finite by using the chemical potential to control the average number of particles, despite variability in individual occupation numbers.

Step-by-step explanation:

The question is asking how the grand canonical ensemble, which is a statistical approach in thermodynamics and statistical mechanics, ensures that the number of particles in a boson gas remains finite. The canonical ensemble uses a constrained sum where the total particle number N is fixed, which would ensure a finite number of particles. When switching to the grand canonical ensemble, the introduction of the chemical potential (μ) allows the partition function to be written in terms of the occupation numbers of particles, np, for different states p without explicitly constraining the total number of particles.

The key to understanding how the number of particles remains finite in the grand canonical ensemble is recognizing that the chemical potential controls the average number of particles in the system. Even though there is no explicit constraint, the value of the chemical potential is related to the average number of particles. When we calculate physical quantities such as the average number of particles, the effects of the chemical potential essentially act as a constraint, ensuring that the average number remains finite. Fluctuations around this average are possible, but the probability of finding an extremely large number of particles in the system decreases rapidly due to the exponential factors in the partition function.