# Entropy Contradiction for a Single Harmonic Oscillator

• Snapu
In summary, the conversation discusses the calculation of entropy for a single quantum harmonic oscillator using both the partition function and multiplicity function approaches. While the partition function approach yields a non-zero entropy value, the multiplicity function approach results in an entropy of 0, which is contradictory. However, for a system of N harmonic oscillators, the two approaches are shown to be consistent. The conversation also mentions the temperature and energy of a classical harmonic oscillator in equilibrium and explains how the entropy is fixed at one Boltzmann constant.

#### Snapu

Making use of the partition function, it is straight forward to show that the entropy of a single quantum harmonic oscillator is:
$$\sigma_{1} = \frac{\hbar\omega/\tau}{\exp(\hbar\omega/\tau) - 1} - \log[1 - \exp(-\hbar\omega/\tau)]$$

However, if we look at the partition function for a single harmonic oscillator, then it is just g(1,n) = 1.
If we take the the logarithm of g(1,n), then we get the entropy is 0 which is in direct contradiction to our above result for σ1.

What is going on here?

This is all the more confusing because you can show that the partition function and multiplicity function approach are consistent for the N oscillator system. Namely, given that the partition function for the N-oscillator system is:
$$Z_{N} = Z_{1}^{N}$$
We can readily get the result:
$$\sigma_{N} = N\sigma_{1}$$

We can then check this result by starting with the multiplicity function for N harmonic oscillators:
$$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$
take the logarithm to get the entropy, and then use the definition for temperature
$$\frac{1}{\tau} = \frac{\partial\sigma}{\partial U}$$

Cranking through this, we can reproduce our above result for σN. Thus, for the N harmonic oscillator problem, calculating the entropy with a partition function approach is shown to be consistent with a multiplicity function approach even though it is not true for N=1. HOW??

EDIT: Above I wrote:
However, if we look at the partition function for a single harmonic oscillator, then it is just g(1,n) = 1
I meant to say that the multiplicity function is g(1,n) = 1. Not the partition function.

The energy of classical harmonic oscillator in a bath in equilibrium is kT where k is Boltzmann constant and T is the temperature of the bath. The entropy is S=Q/T=kT/T=k.
Therefore a classical harmonic oscillator has a fix entropy of one Boltzmann constant. Quantum oscillator has lower entropy that reduces exponentially with energy.

## 1. What is entropy contradiction for a single harmonic oscillator?

Entropy contradiction for a single harmonic oscillator refers to the concept in thermodynamics where the entropy of a system decreases as the energy increases. This contradicts the second law of thermodynamics, which states that the entropy of a closed system will never decrease.

## 2. What causes this contradiction?

This contradiction is caused by the fact that a single harmonic oscillator has a limited number of energy states, so as the energy increases, the system reaches a point where the number of available energy states decreases, leading to a decrease in entropy.

## 3. How does this relate to the second law of thermodynamics?

The second law of thermodynamics states that the entropy of a closed system will never decrease. However, the entropy contradiction for a single harmonic oscillator shows that in certain systems, the entropy can decrease as the energy increases, contradicting the second law.

## 4. Is this contradiction observed in other systems?

Yes, this contradiction has been observed in other systems besides a single harmonic oscillator. Examples include the ideal gas in a one-dimensional box and the Ising model in statistical mechanics.

## 5. What are the implications of this contradiction?

This contradiction challenges our understanding of the second law of thermodynamics and highlights the limitations of our current understanding of entropy. It also has implications for the behavior of complex systems and the possibility of exceptions to the second law in certain scenarios.