Grand canonical and canonical ensemble

[mimpim]

Suppose that we have a system of particles (I am assuming a general system), and I want to find the ground state energy $E_0$. We know that we can consider our system by canonical ensemble formalism OR by grand canonical ensemble so that $H_G=H-\mu N$ (in which $H$ is the Hamiltonian in canonical ensemble and $H_G$ is the grand canonical Hamiltonian).

My question is this: If I find the ground energy $E_0$ in either of them, then they would be the same?

Any explanation is welcomed
Thanks

 

[king vitamin]

My question is this: If I find the ground energy $E_0$ in either of them, then they would be the same?

To be pedantic, the grand canonical ensemble actually doesn't describe a system with a fixed number of particles. Rather, it describes a system where the number of particles is fluctuating (being exchanged with the environment). However, in practice, the number fluctuations usually tends to zero in the thermodynamic limit so you can use the GCE for a fixed number of particles anyways. The prescription is to compute the mean number of particles in your system, and then set your chemical potential to whatever value it needs to be such that the "mean number of particles" equals the actual number of particles. If your particle fluctuations are small (you should be able to check this), your resulting system will be totally equivalent to the canonical ensemble.

This actually turns out to be a godsend in many systems, since computing the grand partition function for a many-particle system is often much easier than computing the canonical partition function, where you need to sum over configurations with a fixed number of particles which may require some constraint due to quantum particle statistics.

 

[DrDu]

The two ensembles become usually equivalent in the thermodynamic limit, i.e. $V \to \infty$ and $N \to \infty$ with $N/V$ fixed.

 

[DrDu]

To be pedantic, the grand canonical ensemble actually doesn't describe a system with a fixed number of particles.

Yes, sure, but why should this be a problem in the context of the question?

 

[king vitamin]

Yes, sure, but why should this be a problem in the context of the question?

Since the CE does describe a system with a fixed number of particles, pointing out this difference is important when addressing the OP's question (roughly: "When are the two ensembles equivalent?").