# Thermodynamics equivalence between entropy in two ensembles

• nashed
In summary, in the thermodynamic limit, the entropy calculated using the canonical ensemble is equivalent to the entropy calculated using the micro-canonical ensemble.
nashed
So while practicing statistical mechanics problems I was faced with the following problem : calculate the entropy as function of energy for an ensemble of harmonic oscillators ( the Hamiltonian is ##\sum_{i=1}^N \frac {p_i^2} {2m} + \frac {m\cdot\omega\cdot q_i^2} 2##) ).

Now the official solution is given in the micro-canonical ensemble and turns out to be ## S \approx N\cdot K_b \ln {\frac {2\pi\cdot E} {Nh\omega}} ##

While using the canonical ensemble I get ## S=N\cdot K_b \ln {\frac {2\cdot\pi E} {Nh\omega}}+ 2NK_b - NKb\ln{N} ##

I'm not sure how to argue the equivalence under the thermodynamic limit, any suggestion on how to do that?

Last edited:
The thermodynamic limit is essentially the limit of large system size, where the macroscopic properties of the system remain unchanged. In this case, the entropy of the system should not depend on the size of the system, since the total energy is fixed. This means that the entropy in the canonical ensemble should be equal to the entropy in the micro-canonical ensemble in the thermodynamic limit. To show this, we can take the limit of large N, and compare the two expressions for the entropy. As N goes to infinity, the second term in the canonical expression will go to zero, and the two expressions will be equal.

## 1. What is the definition of entropy in thermodynamics?

Entropy is a measure of the disorder or randomness of a system. In thermodynamics, it is defined as the amount of energy in a system that is unavailable for doing work.

## 2. What are the two ensembles in thermodynamics?

The two ensembles in thermodynamics are the microcanonical ensemble and the canonical ensemble. The microcanonical ensemble describes a closed system with a fixed number of particles, energy, and volume. The canonical ensemble describes a system in contact with a heat reservoir at a constant temperature.

## 3. What is the relationship between entropy in the microcanonical and canonical ensembles?

The equivalence between entropy in the microcanonical and canonical ensembles states that in thermodynamic equilibrium, the entropy of a system is the same in both ensembles. This means that the two ensembles give equivalent descriptions of the system at equilibrium.

## 4. How does the thermodynamic equivalence between ensembles relate to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always increase over time. The thermodynamic equivalence between ensembles supports this law by showing that the total entropy of a system will remain constant at equilibrium, but will increase if the system is disturbed from equilibrium.

## 5. Can the thermodynamic equivalence between ensembles be applied to all systems?

The thermodynamic equivalence between ensembles is only applicable to systems at equilibrium. It does not apply to systems that are far from equilibrium, such as those undergoing rapid changes or non-equilibrium processes. In these cases, the two ensembles will give different descriptions of the system's entropy.

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