Thanks for pointing that out. I haven't run the corrected computation yet but this might as well resolve my issues with the physical intepretation of what happens for ## k<<1##.
edit
So, the result of the previous integral which war wrong is
$$ \frac{\pi L^{2k+1}}{(1-k)} (R^{2(1-k)} -...
Hi everyone,
I want to post this exercise and my attempt to a solution since there are a couple of points I am not entirely sure of and I might need your help. I'll address them while posting my solution
1) ##Z_N = (Z_1)^N##. I can evaluate ##Z_1## integrating with respect of both parts of the...
This is hopefully wrapping it up :)
I computed the values both for ##z_{CM}## and for ##F(z_2)##.
As for the CM I have done the following (thanks to TSny's adivce):
$$p(z) = Ke^{-\beta m g z} \text{ and } \int_{z_1}^{z_2}p(z)dz =1.$$
substituting ##p## in the integral I obtain...
I was thinking about my solution to the problem for ##N >1## sites this morning and realized it was all wrong. Thanks for the hint! Grand Free Energy scales in an extensive fashion with the number of sites, if there is no interaction. As such if
$$Z_{G1} = 1 + e^{ \beta (2 \varepsilon +2 \mu)}...
A rather important notation remark before I discuss my solution,
$$\langle n \rangle = - \frac{\partial \Phi_G}{\partial \mu} .$$
I seem to have skipped both the sign and the differentiation in the statement.
1) If I have ##N_{11}## sites hosting two atoms and ##N_{10}## absorbing one then the...
You're right,
but I have to admit that the text here war really not clear enough (at least to my opinion). BTW I foung a similar problem in the Statistical and Thermal Physics by Gould and Tobochnik (here). The problem here is an extension of example 4.5* in chapter 4. I will discuss my...
in this case ## A=Z_N ^{-1}e^{-\beta \frac{\langle p \rangle ^2}{2m}}## (##Z_N ## is the partition function for the canonical ensemble, the momentum is on average ). I can now find the maximum through:
$$ \frac{dp}{dz} = 0 \text{ and solving for z.}$$
Hi everyone,
even before addressing the following points I have a serious issue in understandig the text of the Exercise.My idea was to model this system with a lattice gas. Given that each site can host 2 atoms I have 3 possibilities for each site: I'll call'em ##S_{11} S_{00}## and ##...
First of all thanks again for the helpful insights in the subject matter. As you mentioned the typos in the writing are due to silly mistakes (e.g. the dimensional problem with energy was simply cause by the fact that on my homework I had everything written in terms of ##k_B T## whereas this...
your're right I should probably address the problem with my results as well so that you get a better understanding of what I did.
Let's start from the beginning:
1) ##\mathcal{H} = \sum_{i=1}^{N} \frac{p_i ^2}{2m} + mg q_i## hence I get for the partition function (for a single particle):
$$ Z=...
Hi everyone,
this is my first message after presentation so please be merciful if the notation is somewhat messy. Here's my attempt at a solution:
As for points 1) and 2) I used the definition of partition function
$$Z = \frac{1}{h^{3N}} \int e^{-\beta \mathcal{H}} d^3p d^3q$$
and the fact that...
Hi everyone
My name is Giacomo, I am Italian but I live in Germany since 2013. I started to study physics as a second degree after the one I already have in Philosophy am I am glad to have found this forum since I will probably have a load of questions :P