Some questions and answers concerning the Grand canonical ensemble
What are the advantages of the grand canonical ensemble with respect to the canonical?
We will use the grand canoncial ensemble for calculating the properties of quantum gases (bosonic and fermionic quantum gases). As it turns out, this calculation is much more complicated in the case of the canonical ensemble. In other words, the easiest mathematical procedure to calculate the thermodynamic properties of quantum gases is via the grand canonical ensemble.
Do we need to have any kind of physical understanding of fugacity or shall we just see it as some independent variable of the partition function?
The central quantity is rather the chemical potential. The relation between fugacity and the chemical potential is similar to the relation between beta and the temperature.
Are we to understand that ultimately we can know the thermodynamic properties of systems with any number of particles inside of a volume of any size just with establishing the appropriate Hamiltonian and partition functions for some macroscopic but small region R?
You should understand the the thermodynamic properties of thermodynamic systems do not depend on the type of ensembles used. The ideal gas always is an ideal gas, it doesn't matter if we use the microcanonical, the canonical or the grand-canonical ensemble. The different ensembles rather mean that we study the same system in different situations.
Of course, there is one new thing in connection to the grand canonical ensemble, that is the chemical potential. This is because so fat we always assumed that the number of particles is fixed -- in a sense we refused to assign to N the status of a variable, we rather considered it as a fixed parameter. Nevertheless, if we consider any thermodynamic system, we are always able to calcualate the change in energy due to a change in the number of particles. That provides a way to calculate the chemical potential.
Can the boundaries of this region R be both physical and/or imaginary (for example, a spherical container with one small opening)?
Yes, I think that it doesn't matter whether we have particle exchange across a small hole or across an "imaginary" boundary.
Kerson Huang's book is mentioned in the notes as the book on which they are based on. Do you happen to have any other book recommendations to understand the subject better? Maybe something with detailed examples?
I am afraid the honest answer is "not really". I have a number of text books about Statistical Mechanics, but in any of these, the grand canonical ensemble is treated rather superficially. For instance it is almost always treated in connection to a gas with the thermodynamic variables, volume and pressure.
I would guess that the grand canonical ensemble may be appropriate in the case of interacting species of particles -- say in the case of chemical reactions and/or the coexistance of different phases. However, even there, I did not find cases of that type, where thegGrand canonical ensemble has been applied.
We will use the grand canoncial ensemble for calculating the properties of quantum gases (bosonic and fermionic quantum gases). As it turns out, this calculation is much more complicated in the case of the canonical ensemble. In other words, the easiest mathematical procedure to calculate the thermodynamic properties of quantum gases is via the grand canonical ensemble.
Do we need to have any kind of physical understanding of fugacity or shall we just see it as some independent variable of the partition function?
The central quantity is rather the chemical potential. The relation between fugacity and the chemical potential is similar to the relation between beta and the temperature.
Are we to understand that ultimately we can know the thermodynamic properties of systems with any number of particles inside of a volume of any size just with establishing the appropriate Hamiltonian and partition functions for some macroscopic but small region R?
You should understand the the thermodynamic properties of thermodynamic systems do not depend on the type of ensembles used. The ideal gas always is an ideal gas, it doesn't matter if we use the microcanonical, the canonical or the grand-canonical ensemble. The different ensembles rather mean that we study the same system in different situations.
Of course, there is one new thing in connection to the grand canonical ensemble, that is the chemical potential. This is because so fat we always assumed that the number of particles is fixed -- in a sense we refused to assign to N the status of a variable, we rather considered it as a fixed parameter. Nevertheless, if we consider any thermodynamic system, we are always able to calcualate the change in energy due to a change in the number of particles. That provides a way to calculate the chemical potential.
Can the boundaries of this region R be both physical and/or imaginary (for example, a spherical container with one small opening)?
Yes, I think that it doesn't matter whether we have particle exchange across a small hole or across an "imaginary" boundary.
Kerson Huang's book is mentioned in the notes as the book on which they are based on. Do you happen to have any other book recommendations to understand the subject better? Maybe something with detailed examples?
I am afraid the honest answer is "not really". I have a number of text books about Statistical Mechanics, but in any of these, the grand canonical ensemble is treated rather superficially. For instance it is almost always treated in connection to a gas with the thermodynamic variables, volume and pressure.
I would guess that the grand canonical ensemble may be appropriate in the case of interacting species of particles -- say in the case of chemical reactions and/or the coexistance of different phases. However, even there, I did not find cases of that type, where thegGrand canonical ensemble has been applied.
Thank you for your answer professor, I think I understand the subject better now.
ResponderBorrarI just have one last question:
You state that it doesn't matter whether we have particle exchange across a small hole or across an "imaginary" boundary, but in my question I was talking about comparing a complete imaginary boundary around a region (a spherical completely imaginary enclosure) and a complete physical enclosure with just one small opening.
I now understand we can analyze both cases with the grand canonical ensemble, but certainly both of those systems would have different rates of change of the variable N right? How can we take that into account when analyzing each system? Would it be through, for example, defining different hamiltonian functions for both?
It is true that they have different rates of exchange. However, the methods we are using, are based on the partition function and only deal with the equilibrium state. In that case the rates (neither that of particle exchange nor that of heat exchange) matter.
ResponderBorrarRemember the canonical ensemble, there we assumed that an interaction exists between the system and the environment, which allows the two to exchange energy. Depending on the strength of the interaction the rate of energy flow (in the case of a temperature difference) will vary with the interaction -- but for the equilibrium situation and for the equilibrium properties of the system the value of the rate does not matter.
Good morning. Excuse me, can we use the equipartition function on this ensemble too?
ResponderBorrar