Friday, February 6, 2015
Tuesday, February 3, 2015
Helmholtz free energy *first draft
\( F=U-TS \) is that weird formula that we wanted to minimize. But why?
Note: This is my first attempt, I am not sure if its valid to imagine a system being built at constant temperature T.
To answer why, one must understand what U and TS are in this expression and the conditions our little crystal is in. It is in a huge temperature bath at T and our crystal has constant volume so no work can be done on or by it. Hence the only interaction with the outside system is thermal.
In general U in the Helmholtz free energy equation is just the internal energy of the system, I think that we made an approximation in class as we plugged in U at T=0 K.
If you want to be persuaded that in class the U we used is the energy of the system at T=0K read the following:
*Explanation: The above holds since we did not use any fermi-dirac statistics to calculate U instead we plugged in f(T)=1 in \( U= \int ED(E)f(T,E)dE \). Where in \(f(T)=\frac{1}{e^(E-\mu)/kT+1}\) by definition the fermi energy is greater than any occupied state making the exponent negative hence \(lim_{T\Rightarrow0}f(T)=1 \)
How can TS be interpreted?
First, \(Q=TdS\) by definition hence if a system is at constant temperature T and you raise its entropy from 0 to S you get \( \int_0^STds=TS \)
Therefore TS can be thought as the total heat added to a system at temperature T as it is raised from 0 to S entropy.
Now we combine these two quantities to get \(F=U-TS \)
Minimizing this quantity will allow us to have a minimum energy U internally while extracting as much heat from the system as we can!
I think that I need to think more and maybe redo all of it
Note: This is my first attempt, I am not sure if its valid to imagine a system being built at constant temperature T.
To answer why, one must understand what U and TS are in this expression and the conditions our little crystal is in. It is in a huge temperature bath at T and our crystal has constant volume so no work can be done on or by it. Hence the only interaction with the outside system is thermal.
In general U in the Helmholtz free energy equation is just the internal energy of the system, I think that we made an approximation in class as we plugged in U at T=0 K.
If you want to be persuaded that in class the U we used is the energy of the system at T=0K read the following:
*Explanation: The above holds since we did not use any fermi-dirac statistics to calculate U instead we plugged in f(T)=1 in \( U= \int ED(E)f(T,E)dE \). Where in \(f(T)=\frac{1}{e^(E-\mu)/kT+1}\) by definition the fermi energy is greater than any occupied state making the exponent negative hence \(lim_{T\Rightarrow0}f(T)=1 \)
How can TS be interpreted?
First, \(Q=TdS\) by definition hence if a system is at constant temperature T and you raise its entropy from 0 to S you get \( \int_0^STds=TS \)
Therefore TS can be thought as the total heat added to a system at temperature T as it is raised from 0 to S entropy.
Now we combine these two quantities to get \(F=U-TS \)
Minimizing this quantity will allow us to have a minimum energy U internally while extracting as much heat from the system as we can!
I think that I need to think more and maybe redo all of it
Special Projects.
Seems like people are interested in learning more about how Fermi statistics effect the electron-electron interaction and why we minimize the Helmholtz energy, F= E - TS, to find the equilibrium state of a system coupled to a heat bath (that is, at a well-defined temperature T).
For the latter issue, if someone(s) want to look into that and report back, I think people would be interested. I think that it is possible to connect that directly to the 2nd law or thermo, the one about entropy increasing. The Helmholtz energy plays a big role in a lot of physics so it is worth understanding.
For the e-e interaction issue, perhaps the following calculation might be helpful. Consider a square well from x=0 to 1 nm. Suppose there are two electrons in the wells and that the occupied states are sin(10 pi x) and sin(11 pi x), for example. I think that you can make a symmetric combination of those or an anti-symmetric combination. For those two cases, calculate the expectation value of \(|x_1-x_2|\). See of they are substantially different? Perhaps this will tell us if there are correlations built into these states and if electrons avoid each other more in one than the other? Does this make sense? Feel free to ask questions about it. Here is a normalization integral to get things started. (I could not get W-A to do the integral when the \(|x_1-x_2| = |x-y| \) term was added.)
Integrate [2 sin^2(9 pi x) 2 sin^2(8 pi y)] from x=0 to 1, y=0 to 1
For the latter issue, if someone(s) want to look into that and report back, I think people would be interested. I think that it is possible to connect that directly to the 2nd law or thermo, the one about entropy increasing. The Helmholtz energy plays a big role in a lot of physics so it is worth understanding.
For the e-e interaction issue, perhaps the following calculation might be helpful. Consider a square well from x=0 to 1 nm. Suppose there are two electrons in the wells and that the occupied states are sin(10 pi x) and sin(11 pi x), for example. I think that you can make a symmetric combination of those or an anti-symmetric combination. For those two cases, calculate the expectation value of \(|x_1-x_2|\). See of they are substantially different? Perhaps this will tell us if there are correlations built into these states and if electrons avoid each other more in one than the other? Does this make sense? Feel free to ask questions about it. Here is a normalization integral to get things started. (I could not get W-A to do the integral when the \(|x_1-x_2| = |x-y| \) term was added.)
Integrate [2 sin^2(9 pi x) 2 sin^2(8 pi y)] from x=0 to 1, y=0 to 1
Monday, February 2, 2015
Midterm Question
This looks great. Let's make this a 3-part problem.
part A: Calculate the spontaneous magnetization as a function of temperature for \(e^2/a = 1.5\; eV\) and \(E_{band}\) = 1.4 eV. (This part is for zero applied magnetic field). For this part create a graph of magnetization vs T. Define a \(T_c\) and look at the behavior in the vicinity of that. The part in the square root is a lot more important that the 1/T outside that. What is the critical exponent?
Part B: Calculate the magnetic susceptibility in the normal state, as Arjun has explained.
For both parts, the most important thing is your graph. That is what I will look at first. Do a really nice graph, hand drawn, not too large, with a nice title, labels and scales, and with an excellent caption that explains everything in a succinct and beautiful manner.
Part C. (added Feb 6) Calculate and plot the magnetic susceptibility as a function of T for \(e^2/a = 1.5\; eV\) and \(E_{band}\) = 1.7 eV.
===========================================
I decided to go back to my entropy calculation after something Zack mentioned, and I found a sign error in my arithmetic. \( C_0 \) remains the same, but the expression for the polynomial up to order \( O(x^m) \) is:
\( \sum_{k \in \text{evens} }^m 2^k \left[ \frac{1}{(k-1)N^{k-1} } - \frac{(1-N)}{kN^k} \right] \left( N_\uparrow - \frac{N}{2} \right)^k \)
Alternatively written as
\( \sum_{k \in \text{evens} }^m 2^k \left[ \frac{N - (k-1)}{k(k-1)N^k} \right] \left( N_\uparrow - \frac{N}{2} \right)^k \)
The first two terms (to order \( O(x^6) \) ) are:
\( \frac{N-1}{2N^2} 2^2(N_\uparrow -\frac{N}{2})^2 + \frac{N-3}{12N^4} 2^4(N_\uparrow -\frac{N}{2})^4 \)
And this is definitely 100% correct. I checked it against Mathematica and everything. This is a Taylor expansion from
\( 0.5 \ln (N^2 - N_\Delta ^2 ) + N_\Delta \tanh ^{-1} \left( \frac{N_\Delta }{N} \right) \quad \text{where} \quad N_\Delta = N_\uparrow - N_\downarrow \)
which is itself part of the Stirling Approximation for
\( \ln \left( \frac{N!}{N_\uparrow ! N_\downarrow ! } \right) \)
-Aaron
EDIT: Added the revised expression that Aaron derived for the entropy and changed the symbol for the band energy to avoid confusion. -Arjun
Sunday, February 1, 2015
Entropy
I think that if you set the bandwidth equal to zero, that will make the entropy calculation easier. I guess what we need to know is: for a given value of \(N_{\uparrow} - N_{\downarrow}\), how many states are there? That is, what is \(\Omega\)?
Then we can find the equilibrium state of the system by looking for the minima of F=E-TS,
where \( S= k ln(\Omega)\). Does that make sense?
Here is a suggestion. I think what we really want is just a graph of \( kT ln\Omega\) as a function of \(N_{\uparrow} - N_{\downarrow}\) and then to maybe add that to the e-e energy term and graph that to see which one controls the equilibrium. (or really, how they both influence it...) So instead of getting all caught up in formal math, and Stirling's formula etc. (which is fine, but might take too much time) maybe someone could just do that numerically for a particular case like N=1000 or so, and for a particular value of \(e^2/a\), like say 1 or 2 eV. Then graph the Free Energy as a function of \(N_{\uparrow} - N_{\downarrow}\) for a few values of T. Does that make any sense?
By whatever method, either analytically or numerically, what we need to keep moving forward is an approximate expression for S that includes terms up to 4th order in \(N_{\uparrow} - N_{\downarrow}\). I think S has a maximum at \(N_{\uparrow} - N_{\downarrow}=0\) and one can do a Taylor series expansion near there including the \((N_{\uparrow} - N_{\downarrow})^2\) and \((N_{\uparrow} - N_{\downarrow})^4\) terms. One can also fit a numerically generated S in that way.
Then we can find the equilibrium state of the system by looking for the minima of F=E-TS,
where \( S= k ln(\Omega)\). Does that make sense?
Here is a suggestion. I think what we really want is just a graph of \( kT ln\Omega\) as a function of \(N_{\uparrow} - N_{\downarrow}\) and then to maybe add that to the e-e energy term and graph that to see which one controls the equilibrium. (or really, how they both influence it...) So instead of getting all caught up in formal math, and Stirling's formula etc. (which is fine, but might take too much time) maybe someone could just do that numerically for a particular case like N=1000 or so, and for a particular value of \(e^2/a\), like say 1 or 2 eV. Then graph the Free Energy as a function of \(N_{\uparrow} - N_{\downarrow}\) for a few values of T. Does that make any sense?
By whatever method, either analytically or numerically, what we need to keep moving forward is an approximate expression for S that includes terms up to 4th order in \(N_{\uparrow} - N_{\downarrow}\). I think S has a maximum at \(N_{\uparrow} - N_{\downarrow}=0\) and one can do a Taylor series expansion near there including the \((N_{\uparrow} - N_{\downarrow})^2\) and \((N_{\uparrow} - N_{\downarrow})^4\) terms. One can also fit a numerically generated S in that way.
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