Homework exercise 4.3b)
Here is my attempt to the solution of the exercise 4.3b).
Pardon me for the hour I have had a lot of issues with the internet.
Work for an ideal gas in a 3D container (corrected)
Using the differential al work calculated on exercise 6.2: 1. How can we use the result to obtain DW for an ideal gas in a 3D container? What is the interpretation of the parameter "a" in that case. Solution. The differential of work calculated on exercise 6.2 is Here, d represents the dimension of the box. So The parameter a is the lenght, according to equation (67). Because we are working on 3 dimensions, a³=V . This might come from the definition of entropy*. 2. What do we get for DW for d=3 and n --> infinity? And how is this result related to the expected form DW = N k_B T dV/V? Solution. On the last expression we get DW for three dimensional case. Now, applying the limit on infinity, we expect that the differential of work would become on well potential one. So,
The expression for the entropy seems ok. May be you should have indicated where you applied the Stirling formula. Your last statement is an attempt to answer the question about the scaling of the trap-frequencies in the thermodynamic limit. Qualitatively it is ok. But quantitatively, one must require that "N* prod_k w_k" becomes constant in the thermodynamic limit. Which means that each frequency individually has to scale as N^{-1/d}
ResponderBorrar