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yeracxam
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Hi there,
I am working through the preliminary chapters of "Concepts in Thermal Physics" by Blundell. I have not really studied statistics before, so have a lot to get to grips with, but I thought I was making headway until this question cast questions over my understanding again!
a) A system comprises N states which can have energy 0 or Δ. Show that the number of ways Ω(E) of arranging the total system to have energy E = rΔ (where r is an integer) is given by:
Ω(E) = N! / (r! * (N-r)! )
b) Now remove a small amount of energy sΔ from the system, where s << r. Show that:
[itex]\Omega(E - \epsilon) \approx \Omega(E) \frac{r^{s}}{(N-r)^{s}}[/itex]
c) And hence show that the system has temperature T given by
[itex]\frac{1}{k_{B}T} = \frac{1}{\Delta} ln(\frac{N-r}{r})[/itex]
d) Sketch kBT as a function of r from r=0 to r=N and explain the result.
[itex]\frac{1}{k_{B}T} = \frac{dln(\Omega)}{dE}[/itex]
a) Okay, no problems here.
b)
[itex]\Omega(E-\epsilon) = \frac{N!}{(r-s)!(N-r+s)!}[/itex]
from Stirling approx ln(N!) ≈ Nln(N) - N :
[itex]ln\Omega(E-\epsilon) \approx Nln(N) - (r-s)ln(r-s) - (N-r+s)ln(N-r+s)[/itex]
[itex]= Nln(N) - rln(r-s) - (N-r)ln(N-r+s) +s(ln(r-s) - ln(N-r+s))[/itex]
If I make the approximation ln(r-s) ≈ ln(r), ln(N-r+s) ≈ ln(N-r) because s<<r then I can rearrange to say:
[itex]ln\Omega(E-\epsilon) \approx ln\Omega(E) + ln(r^{s}) - ln((N-r)^{s})[/itex]
[itex]\Rightarrow \Omega(E-\epsilon) \approx \Omega(E)\frac{r^{s}}{(N-r)^{s}} [/itex]
Which gets me to the solution, but I am not sure why I am justified in saying ln(r-s) ≈ ln(r) but not (r-s) ≈ r?
c) I can get to this by simple differentiation of the stirling approximation, but this does not involve using my solution to b) to get there, so I am not sure if this is the correct approach?
d) Now comes the bulk of my confusion: Plotting the function seems to result in kBT → ∞ as r → N/2, and negative kBT for N/2 < r < N.
This seems to imply infinite and negative values for T, which I can't fathom!
My insights and questions:
- r = N/2 is clearly the macrostate which has the most microstates able to accommodate it
- When r changes, the energy in the system changes linearly... where does this energy come from / go?
- Shouldn't temperature have a linear relationship with the total energy in the system?
- Suppose this was a system connected to a heat bath / reservoir at temperature T. It will reach equilibrium with the reservoir and will therefore have temperature T and we can say something about the total energy the system is likely to contain. Does the result from this question suggest that it will, in fact, never contain more energy than NΔ/2, half the energy it is "capable" of containing?
- This does actually seem to fit with the shape of the Boltzmann distribution, with higher energies being exponentially less likely...
I think I am on the verge of understanding this, but at the moment my mind is going around in circles a bit, so I would appreciate some pointing in the right direction. Sorry for not posing a clearer question, but it's more help with understanding I am looking for if anybody would be able to oblige?
Many thanks!
Max
I am working through the preliminary chapters of "Concepts in Thermal Physics" by Blundell. I have not really studied statistics before, so have a lot to get to grips with, but I thought I was making headway until this question cast questions over my understanding again!
Homework Statement
a) A system comprises N states which can have energy 0 or Δ. Show that the number of ways Ω(E) of arranging the total system to have energy E = rΔ (where r is an integer) is given by:
Ω(E) = N! / (r! * (N-r)! )
b) Now remove a small amount of energy sΔ from the system, where s << r. Show that:
[itex]\Omega(E - \epsilon) \approx \Omega(E) \frac{r^{s}}{(N-r)^{s}}[/itex]
c) And hence show that the system has temperature T given by
[itex]\frac{1}{k_{B}T} = \frac{1}{\Delta} ln(\frac{N-r}{r})[/itex]
d) Sketch kBT as a function of r from r=0 to r=N and explain the result.
Homework Equations
[itex]\frac{1}{k_{B}T} = \frac{dln(\Omega)}{dE}[/itex]
The Attempt at a Solution
a) Okay, no problems here.
b)
[itex]\Omega(E-\epsilon) = \frac{N!}{(r-s)!(N-r+s)!}[/itex]
from Stirling approx ln(N!) ≈ Nln(N) - N :
[itex]ln\Omega(E-\epsilon) \approx Nln(N) - (r-s)ln(r-s) - (N-r+s)ln(N-r+s)[/itex]
[itex]= Nln(N) - rln(r-s) - (N-r)ln(N-r+s) +s(ln(r-s) - ln(N-r+s))[/itex]
If I make the approximation ln(r-s) ≈ ln(r), ln(N-r+s) ≈ ln(N-r) because s<<r then I can rearrange to say:
[itex]ln\Omega(E-\epsilon) \approx ln\Omega(E) + ln(r^{s}) - ln((N-r)^{s})[/itex]
[itex]\Rightarrow \Omega(E-\epsilon) \approx \Omega(E)\frac{r^{s}}{(N-r)^{s}} [/itex]
Which gets me to the solution, but I am not sure why I am justified in saying ln(r-s) ≈ ln(r) but not (r-s) ≈ r?
c) I can get to this by simple differentiation of the stirling approximation, but this does not involve using my solution to b) to get there, so I am not sure if this is the correct approach?
d) Now comes the bulk of my confusion: Plotting the function seems to result in kBT → ∞ as r → N/2, and negative kBT for N/2 < r < N.
This seems to imply infinite and negative values for T, which I can't fathom!
My insights and questions:
- r = N/2 is clearly the macrostate which has the most microstates able to accommodate it
- When r changes, the energy in the system changes linearly... where does this energy come from / go?
- Shouldn't temperature have a linear relationship with the total energy in the system?
- Suppose this was a system connected to a heat bath / reservoir at temperature T. It will reach equilibrium with the reservoir and will therefore have temperature T and we can say something about the total energy the system is likely to contain. Does the result from this question suggest that it will, in fact, never contain more energy than NΔ/2, half the energy it is "capable" of containing?
- This does actually seem to fit with the shape of the Boltzmann distribution, with higher energies being exponentially less likely...
I think I am on the verge of understanding this, but at the moment my mind is going around in circles a bit, so I would appreciate some pointing in the right direction. Sorry for not posing a clearer question, but it's more help with understanding I am looking for if anybody would be able to oblige?
Many thanks!
Max