Compute and plot the entropy S(T) for the truncated system. My actual question is what the function S(T) should be.
Consider a bound single-particle system with a discrete, equidistant (spectral constant = delta e) energy spectrum.
epsilon_j - epsilon_j-1 = delta e, (j = 1,2,...)
Partition function Z = Sum notation(j=0...N) e^(-beta*j*delta e), for N = 1...25
Partition function Z = 1/[1-e^(-1/t)] for N approaches infinitive (truncated system)
beta = 1/(k_B*T), k_B is a Boltzmann constant
t = (k_B*T)/delta e
N is the number of excited states
The Attempt at a Solution
I know 2 expressions for the entropy S:
a) S/k_B = lnZ + beta*E
With this expression, I know k_B, Z and beta (which can be expressed in terms of variable T). But I don't know E, which is the mean energy per configuration. Also, if I use the truncated formula above, I will face a problem with unknown t. Obviously, I can convert t into T by the above formula, but then I have to deal with the unknown delta e.
b) S = -k_B*[Sum notation (n = 1...omega) p_n*ln(p_n)], where omega is the number of available micro-states
Basically, I know that
p_j = (1/Z)*e^(-beta*j*delta e)
Then, again, I don't know delta e and I'm not sure what is the proper way to handle this kind of formula for the case of N approaches infinitive (truncated system).
Is there anyone who can show me an expression for S(T) with only variable T, either based on the above expressions or based on something else? I need to plot S(T), and you know that only T should be the independent variable.
Thank you in advance any input from you!!!