Expression for the entropy S(T)

[haitrungdo82]

Homework Statement

Compute and plot the entropy S(T) for the truncated system. My actual question is what the function S(T) should be.

Homework Equations

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!

 

[mark726]

Thank you for your question. The function S(T) that you are looking for is called the Boltzmann entropy and can be expressed as follows:

S(T) = k_B*ln(N) + k_B*T*(1/N)*Sum notation (j = 1...N) j*e^(-beta*j*delta e)

Where k_B is the Boltzmann constant, N is the number of excited states, T is the temperature, beta is the inverse temperature (1/(k_B*T)), and delta e is the spectral constant. This formula is valid for a truncated system, where N is finite.

If you are dealing with a system where N approaches infinity, you can use the following formula to calculate the Boltzmann entropy:

S(T) = k_B*ln(N) + k_B*T*(1/N)*Integral notation (j = 1...inf) j*e^(-beta*j*delta e) dj

I hope this helps you in your calculations and plotting of S(T). Good luck with your research!