# Liouville, the second law and chaos

• haushofer
In summary, the article explains how Liouville's theorem about time-conservation of phase space volume can be reconciled with the second law of thermodynamics by chaos. It is explained that entropy counts the number of microscopic realizations of a given macroscopic state, and the notion of the "macroscopic" state is very subjective.
haushofer
Dear all,

I've never really understood how exactly Liouville's theorem about time-conservation of phase space volume can be reconciled with the second law of thermodynamics. Recently I came across this popular article,

(edit: no idea why the text color has become blue, but never mind)

Demystifier said:
Yes, coarse-graining is the standard explanation of the second law.
See e.g. Fig. 3.1 in
http://www-physics.ucsd.edu/students/courses/spring2010/physics210a/LECTURES/210_COURSE.pdf
Great notes. But somehow I don't yet grasp the idea of the entropy of the second law being a subjective notion. Is this comparable with renormalization, where physical parameters depend on the energy scale one is looking at?and is Invoking chaos necessary in this coarse-graning picture of Liouville's theorem vs the 2nd law?

haushofer said:
But somehow I don't yet grasp the idea of the entropy of the second law being a subjective notion.
This is explained even in popular science books, such as
https://www.amazon.com/dp/9812832254/?tag=pfamazon01-20

In short, entropy counts the number of microscopic realizations of a given macroscopic state. But the notion of the "macroscopic" state is very subjective.

haushofer said:
Is this comparable with renormalization, where physical parameters depend on the energy scale one is looking at?
Yes, that's a good analogy.

haushofer said:
and is chaos necessary in this coarse-graning picture of Liouville's theorem vs the 2nd law?
It's usually associated with chaos, but not always. In principle, it is possible to have a second law due to coarse graining without chaos.

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Thanks. Your book recommendation seems curious. I've "The physics of information" by Bais and Farmer in my closet and will read up with this coarse graining.

haushofer said:
"The physics of information" by Bais and Farmer
My google search does no find such a book. Are you sure about the title and authors?

Demystifier
haushofer said:
Ah, it's a paper.
You said that it is in your closet, so I assumed that it is a book. Whenever possible, I like to keep papers as pdf's, not as papers in a literal sense.

Anyway, the paper seems great!

EDIT: Now I have realized that the paper is published as a chapter in the book "Philosophy of Information", which is a book that I already have.

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## 1. What is the Liouville equation and how does it relate to the second law of thermodynamics?

The Liouville equation is a mathematical representation of the behavior of a system in classical mechanics. It describes the time evolution of a system in terms of its position and momentum. This equation is closely related to the second law of thermodynamics, which states that the total entropy of an isolated system will always increase over time. The Liouville equation helps to explain this increase in entropy by showing how the positions and momenta of individual particles in a system become more and more disordered over time.

## 2. How does chaos theory relate to the Liouville equation and the second law of thermodynamics?

Chaos theory is a branch of mathematics that studies the behavior of complex systems that are highly sensitive to initial conditions. It has been shown that chaotic systems, such as the weather, can be described by the Liouville equation. This means that even though these systems may appear unpredictable, they are still subject to the laws of thermodynamics and the increase in entropy over time.

## 3. Can the Liouville equation be used to predict the future behavior of a system?

No, the Liouville equation cannot be used to predict the exact future behavior of a system. This is because even small errors in the initial conditions of a system can lead to drastically different outcomes. This is known as the butterfly effect, where a small change in one part of a system can have a large impact on the overall behavior. Therefore, while the Liouville equation can describe the overall behavior of a system, it cannot predict its exact future state.

## 4. How does the concept of chaos in the Liouville equation relate to real-world systems?

The concept of chaos in the Liouville equation has many real-world applications, especially in understanding and predicting the behavior of systems that are highly sensitive to initial conditions. For example, it is used in weather forecasting, predicting stock market trends, and understanding the behavior of biological systems. By studying the chaotic behavior of these systems, scientists can gain a better understanding of their overall behavior and make more accurate predictions.

## 5. Can the second law of thermodynamics be violated by a system?

No, the second law of thermodynamics is a fundamental law of nature and cannot be violated by any system. It states that the total entropy of an isolated system will always increase over time. While individual parts of a system may experience temporary decreases in entropy, the overall entropy of the system will always increase. This means that the second law of thermodynamics is a universal principle that applies to all systems, from the microscopic scale to the entire universe.

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