Homework 6.2a,b)
6.2.b) Plot the potential for n = 2, 4, 8, 16. On the basis of this what kind of thermodynamic properties will you expect in the limit n → ∞?
We can observe from the plot that as n increases, the regions near x=0 flatten and the regions near x=a get steeper.
From that we conclude that in the limit, we will obtain an "wall potential" or "infinite well potential".
6.2.c) Calculate ln Ω(T, η, a) in the thermodynamic limit, for fixed n. What does the thermodynamic limit mean for η and a (are they intrinsic or extrinsic or none of the two)?
(I made a mistake in this pictures, the first integral must have only d^Np differetial, and d^Nq for the second one)
Then, we apply the logarithm to the expression, and we obtain
Based on the potential behaviour, the parameter "a" is acting as Volume, and as such should be affected by the thermodynamic limit as V→∞. So its extensive.
You forgot two parameters in the definition of the partition function, hbar^N and N!
ResponderBorrarhbar is needed in order to make the probability distribution a distribution over a countable number of microstate -- and it is only in that sense that the information-theoretic entropy is well defined. N! is needed in order to take into account that the particles are indistinguishable. In fact, the result you for \ln\Omega is not extensive because as one increases the system, N is increased but "a" also.
Your conclusion "Based on the potential behaviour, the parameter "a" ..." is correct, but it is impossible to conclude that on the basis of your result for \ln\Omega.