How can I calculate entropy using different methods?

[weiss_tal]

Hello all,

I have a question regarding entropy which I'm sure you guys will have no problem with. I can't get it though... :)

In the literature there are several ways for calculating the entropy, and that really confuses me. for example in the canonical ensemble. One way is to define the the multiplicity as a function of the populated states. i.e g(N,n1,n2,n3...)=N!/(n1!n2!n3!...) and then taking log(g). The population n1,n2... are of course constrained so that the total energy is fixed. And also N=n1+n2+n3...(Stanford Uni. lectures)

Another way is defining the multiplicity g as a function of E_i (possible energy of the system) and again taking the log of it. (Kittel).

Both ways are defining the entropy and from both one can derive the temperature with ds/dE.

My problem with it that they are not the same function. The first one for a given n1,n2... is bound to have some energy E_i but there are more permutations of n1,n2... which gives the same energy E_i. Therefore g(E_i) which is given in the second method is bigger then g(n1,n2..) of the first method.

Which is true? what should I do? or maybe the differences, taking the log of it, are just negligible?

Thank you all very very very much, and forgive me for my awful english,

Tal.

 

[eljose79]

Dear Tal,

Thank you for your question regarding entropy and the different ways of calculating it. Entropy is a concept that can be defined in various ways, depending on the context and the system being studied. Both methods you mentioned are valid ways of calculating the entropy, but they may apply to different systems or situations.

In the first method, the multiplicity is defined as a function of the populated states, which is useful for systems with a fixed number of particles and a fixed total energy. This method is commonly used in statistical mechanics and is known as the Boltzmann entropy. It takes into account the different ways that the particles can be arranged in the system while still maintaining the same total energy. This method is often used in the canonical ensemble, as you mentioned.

In the second method, the multiplicity is defined as a function of the possible energies of the system. This method is known as the Gibbs entropy and is commonly used in thermodynamics and statistical mechanics. It takes into account the different ways that the particles can be distributed among the different possible energy levels of the system. This method is useful for systems with a variable number of particles and a variable total energy, and it is often used in the grand canonical ensemble.

In both cases, taking the logarithm of the multiplicity is necessary in order to define the entropy. This is because the multiplicity itself can become very large, and taking the logarithm helps to make the values more manageable.

In summary, both methods are valid ways of calculating entropy, but they apply to different systems and situations. It is important to use the appropriate method for the system you are studying. I hope this helps to clarify the differences between the two methods. If you have any further questions, please don't hesitate to ask.