# A system is always in a well-defined state, so the entropy is constant?

1. Nov 9, 2011

### spocchio

This is my apparently-trivial problem, that probably means i don't understood what entropy is.

We all have faced the statement that entropy always increase.
With the power of statistical mechanics we can calculate the variation of entropy from two states, considering the difference of entropy of the final and the initial state.
To do it we suppose that the distribution of our system is

$\rho=exp(-\beta H(q,p))$

In the reality the distribution of a system is always a dirac delta in the phase space, and so there is no integral to do in the phase space to calculate entropy and so it is always constant.

$\rho=\Pi_i \delta (q_i-q_{i,t})\delta (p_i-p_{i,t})$

It seems to me that entropy depends on the knowledge that we have of the system. Observers with a deeper knowledge should see different variation of entropy, in particular, someone who know exactly the position and momentum of each particle should not see a variation of entropy.
Am I right?

Last edited: Nov 9, 2011
2. Nov 9, 2011

### Bill_K

Yes, you're exactly right. If the state of a system is specified completely in terms of the positions and momenta of each individual particle, it's called a "microstate". Such a state has zero entropy. If the state is specified in terms of macroscopic quantities such as density and pressure, it's called a "macrostate." A macrostate is a collection of microstates.
And even stronger, someone who follows the position and momentum of each particle will not see a variation of entropy. (Liouville's Theorem) Increase in entropy only happens when you fail to track the individual particles.

3. Nov 9, 2011

### Studiot

Should you not add the word 'closed' before system?

4. Nov 10, 2011

### spocchio

nice question, I dont know the answer, what should change?

can we say that is not the knowledge of H(q,p) that gives the desired entropy?. In conclusion (in statistical mechanics) only rho(q,p) that determinate uniquely the entropy, with the request that $\frac{d \rho}{dt}=0$.