Grand canonical and canonical ensemble

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Discussion Overview

The discussion revolves around the equivalence of the ground state energy ##E_0## calculated using the canonical ensemble and the grand canonical ensemble in statistical mechanics. Participants explore the implications of using each ensemble, particularly in the context of systems with fluctuating particle numbers versus fixed particle numbers.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the ground state energy ##E_0## would be the same when calculated using either the canonical or grand canonical ensemble.
  • Another participant clarifies that the grand canonical ensemble describes systems with fluctuating particle numbers, but in practice, these fluctuations tend to zero in the thermodynamic limit, allowing for equivalence with the canonical ensemble under certain conditions.
  • It is noted that the grand canonical ensemble can simplify calculations for many-particle systems compared to the canonical ensemble, where fixed particle numbers may complicate the partition function computation.
  • Participants discuss the conditions under which the two ensembles become equivalent, specifically in the thermodynamic limit where both volume and particle number approach infinity while maintaining a fixed density.
  • There is a reiteration of the importance of understanding the differences between the two ensembles in relation to the original question about their equivalence.

Areas of Agreement / Disagreement

Participants express differing views on the implications of using the grand canonical ensemble versus the canonical ensemble, particularly regarding the significance of particle number fluctuations. While some agree on the conditions for equivalence, others emphasize the importance of these differences in addressing the original question.

Contextual Notes

The discussion highlights the limitations of the grand canonical ensemble in describing systems with fixed particle numbers and the conditions under which the two ensembles can be considered equivalent. There is an acknowledgment of the need for careful consideration of particle number fluctuations in practical applications.

mimpim
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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
 
mimpim said:
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.
 
The two ensembles become usually equivalent in the thermodynamic limit, i.e. ##V \to \infty## and ## N \to \infty## with ##N/V## fixed.
 
king vitamin said:
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?
 
DrDu said:
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?").
 

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