Thermodynamic entropy of system of any size.

by IttyBittyBit
Tags: entropy, size, thermodynamic
IttyBittyBit is offline
Aug18-11, 03:24 AM
P: 159
After a bit of calculation, I came up with the following quantity for the bit-entropy of a thermodynamic system.

We have the following assumptions:

1. System at thermal equilibrium.
2. Ideal gas.
3. Monatomic gas (i.e. no internal degrees of freedom for particles).
4. All particles have equal mass.
5. Units are such that k_B (boltzmann constant) normalized to 1.

Using just information-theoretic arguments (no assumptions from thermodynamics!) I calculated the raw entropy of such a system to be:

S = (Q/T)[(log(T/T0) - 1) + (log(V/V0) - 1) + log(Q/T)]

(T=temperature, Q=thermal energy, V=volume of system, T0,V0=unknown normalizing constants).

This can be simplified to:

S = (Q/T)[log(Q/T0) + log(V/V0) - 2]

Further, I suspect it works for any system size, even systems that wouldn't be called 'ensembles' in the thermodynamic sense (like just a single particle, in which case Q=0. In general, we take Q = total energy - kinetic energy of center of mass of system).

In addition, we find that dS is proportional to dQ/T (i.e. Clausius law of entropy), in the limit where Q >> T (which is always true in thermodynamic ensembles) and volume is held constant.

Yet another interesting thing about this is that the entropy is not zero at the limit of T=0 (because then Q=0 too). Thus it appears the third law of thermodynamics need not apply from a purely information-theoretic standpoint.

Is my formula correct?
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DrDu is offline
Aug18-11, 04:17 AM
Sci Advisor
P: 3,371
IttyBittyBit is offline
Aug18-11, 04:51 AM
P: 159
Thanks for that link, I didn't know about that. I was just trying to satisfy my own curiosity.

The Sackur-Tetrode equation appears only to work under the assumption of the uncertainty principle, whereas I derived my formula without any QM assumptions. Therefore I don't think they can be directly compared. However, this part:

He showed it to consist of stacking the entropy (missing information) due to four terms: positional uncertainty, momenta uncertainty, quantum mechanical uncertainty principle and the indistinguishability of the particles
It very similar to my own method (except for the QM part obviously).

DrDu is offline
Aug18-11, 06:41 AM
Sci Advisor
P: 3,371

Thermodynamic entropy of system of any size.

If I am not wrong, Sakur and Tetrode derived their formula in a classical mechanics context (in 1912 QM was not yet discovered), i.e. without involving the uncertainty principle.
IttyBittyBit is offline
Aug18-11, 07:05 PM
P: 159
No, the Sackur-Tetrode equation requires Planck's constant, which was of course discovered in 1899.

From :

It remained for quantum mechanics to introduce a reasonable value for the value of Φ which yields the Sackur-Tetrode equation for the entropy of an ideal gas. It too suffers from a divergent entropy at absolute zero, but is a good approximation to an ideal gas over a large range of densities.
Φ is really just a convoluted way of introducing HUP. It's kind of interesting in itself that the HUP was already 'realized' 15 years before Heisenberg published his paper.

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