I think Z_n = (Z_1)^n/n! should be a general result deduced from the integral. There is no other constraints in the deduction. If we consider interactions, this is not a noninteracting system any more. A general result should also be applicable to discrete cases. But if we exclude the points...
Thank you, mfb. Even they are not distinguishable, the n-particle partition function (canonical) can still be expressed in single-particle partition function: Z_n = (Z_1)^n/n!
I can figure it out mathematically that the single points (which correspond to multiple particles at same locations) do...
The P170 of this book deduce the partition function of n noninteracting particles in a canonical ensemble. The result is
Z_n = (Z_1)^n/n! ----nondistinguishable
Z_n = (Z_1)^n -- distinguishable
Z_1 is the one-particle partition function. There is no other assumption. So this is why I get...
Dear Orodruin, I know this cannot be an issue because it is quite fundamental. It is where I get confused when I read the textbook. I know my understanding most probably went wrong, but I don't know where. I understand what you mean, but this does not seem to have made it clear. I am reading the...
Yes, if the particles are distinguishable, I missed the states of exchanging the particles. This is not the topic here. The integral should be a generalized form which can be reduced to any special case like what I just used.
Just transform the integral into summation considering only two particles and three discrete allowed spatial coordinates. "1-3" are three coordinates. "x" means one particle occupies one coordinate. "xx" means two particle occupy the same coordinate. "0" is null.
Thank you. I think I may take the calculation of partition function as an example. Just simplify the case. Three spatial coordinates (three energy levels), two particle, no inter-particle interaction. There are 6 combinations
1 2 3
x x 0 E1
x 0 x E2
0 x x E3
xx 0 0 E4
0 xx 0...
Thank you!
Why should we separate the N particles into single particles and do integrals in different copies of generalized coordinates. I mean, considering the N particle in a same container, I cannot figure it out.
Your second reply is very interesting. is there any mathematical evidence or...
In statistical physics, the partition function should be calculated in the whole phase space.This is finally an integral over the phase space, like ∫d3Nqd3Np...
The problem is that the integral covers some cases that different particles have the same generalized coordinates and momenta. So, is...
Thank you for reply Dr. Du. But in the program (e.g. phon), the number of k point is variable. That means we can choose any k wave vector we want in the BZ.
The tutorial of this program says the limitation of this method is we need a large supercell to make Hellmann-Feynman force negligible...
Dear All,
I have trouble to understand the calculation of phonon using small displacement method. I found people said the limitation of this method was that it requires the wave vector k orthogonal to the supercell size(?). Does it mean the wave vector is not any you want? why? I got really...
The definition I provided is the general definition in multi-component system. It of course serves the single component system. That is the increment of Gibbs energy brought about by increasing unit quantity of, e.g. CO. This definition of chemical potential can be found in any textbooks of...
Thank you. Do you mean we need more than one order parameter? But, in the Landau expansion, because of the symmetry, the sign (positive or negative) doesn't bring about any difference.
Hello everyone,
As is known to all, the order parameter for the ferromagnetic case in the Landau expression is chosen as the magnetization. This is easy to understand. But for the antiferromagnetic case, what is the order parameter? People told me it was magnetic moment on the sublattice. But...
The chemical potential is defined usually by the partial derivative of Gibbs energy with respect to the qantity of the component, under the condition that the temperature, pressure and the quantities of other components are constant.