|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Second Law of Thermodynamics, Macroscopic Observables within Boltzmann's Principle but without Thermodynamic Limit | | D.H.E.Gross
; | | Date: |
18 Dec 2000 | | Subject: | Statistical Mechanics; Chaotic Dynamics | cond-mat.stat-mech nlin.CD nucl-th | | Abstract: | Boltzmann’s principle S=k ln W allows to extend equilibrium thermo-statistics to ``Small’’ systems without invoking the thermodynamic limit. The clue is to base statistical probability on ensemble averaging and not on time averaging. It is argued that due to the incomplete information obtained by macroscopic measurements thermodynamics handles ensembles or finite-sized sub-manifolds in phase space and not single time-dependent trajectories. Therefore, ensemble averages are the natural objects of statistical probabilities. This is the physical origin of coarse-graining which is not anymore a mathematical ad hoc assumption. From this concept all equilibrium thermodynamics can be deduced quite naturally including the most sophisticated phenomena of phase transitions for ``Small’’ systems. Boltzmann’s principle is generalized to non-equilibrium Hamiltonian systems with possibly fractal distributions ${cal{M}}$ in 6N-dim. phase space by replacing the conventional Riemann integral for the volume in phase space by its corresponding box-counting volume. This is equal to the volume of the closure $ar{cal{M}}$. With this extension the Second Law is derived without invoking the thermodynamic limit. The irreversibility in this approach is due to the replacement of the phase-space volume by the volume of its closure $ar{cal{M}}$. The physical reason for this replacement is that macroscopic measurements cannot distinguish ${cal{M}}$ from $ar{cal{M}}$. Whereas the former is not changing in time due to Liouville’s theorem, the volume of the closure can be larger. In contrast to conventional coarse graining the box-counting volume is defined in the limit of infinite resolution. I.e. there is no artificial loss of information. | | Source: | arXiv, cond-mat/0101281 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER