Partition Function for Phonons

[Master J]

In looking at phonons, and their energy, I came across the Partition function. THis was needed to calculate the internal energy of the solid.

But howcome the Partition function is used, and not the GRAND Partition function? The number of phonons is not conserved, I know that, but isn't N, the number, fixed in the Partition function? Surely one should used the GPF to take into acount the varying number??


I guess I have misunderstood something here. Any help appreciated!

 

[kanato]

Well, within the thermodynamic limit, both partition functions give equivalent results, so which one you use is a matter of convenience. Varying particle numbers can be taken into account in the canonical ensemble, it just is not as convenient. Since particle number for phonons is never conserved, the chemical potential is always zero, which makes the grand partition function for phonons trivially related to the partition function for a collection of harmonic oscillators.

 

[charng]

The Partition function is a fundamental concept in statistical mechanics and is used to calculate the thermodynamic properties of a system. In the case of phonons, the Partition function takes into account the different energy levels and the number of phonons occupying each level. This allows us to calculate the internal energy of the solid, which is a measure of the average energy of all the phonons in the system.

The Grand Partition function, on the other hand, is used in systems where the number of particles is not fixed, such as in a gas or plasma. In these systems, the number of particles can change due to interactions with the environment. In the case of phonons, the number of phonons is not conserved as they can be created or destroyed through interactions with other phonons or with the lattice. Therefore, the Grand Partition function is not necessary for calculating the internal energy of a solid.

It is important to note that the Partition function and the Grand Partition function are both valid approaches to calculating thermodynamic properties, but they are used in different contexts depending on the system being studied. In the case of phonons, the Partition function is the appropriate approach as it takes into account the fixed number of phonons in the solid. I hope this clarifies your understanding of the use of the Partition function in relation to phonons.