Poincare Recurrence and the Infinite N Limit

However, this also means that the strictures of the recurrence theorem may not necessarily apply in this limit.
  • #1
NickJ
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(One version of) Poincare's Recurrence Theorem states that for any conservative system whose possible states S form a compact set in phase space, that system will "almost always" return arbitrarily close to its initial state, provided we wait long enough. ('Almost always' means 'all but a set of Lebesgue measure zero'.)

Carlo Cercignani, in a discussion of the Boltzmann equation, has this to say: the set of possible states for a conservative system is not compact in phase space for the limit where the number of particles in the system goes to infinity (N --> infty). (See The Mathematical Theory of Dilute Gases, Springer-Verlag: 1994, page 56 first paragraph.)

Why?

He also says that, in the N --> infty limit, the recurrence time for the system is expected to go to infinity (albeit at a much faster rate than N).

Why?

Can someone please explain? Does the non-compactness of S in the N --> infty limit have to do with S becoming unbounded? or not closed? Which one? And is there some intuitive reasoning to explain this? Or maybe a rigorous explanation too? I've been through lots of Cercignani's work; I can't find an explanation.

The answer is not this: the recurrence time is really really long, and that is why the Boltzmann equation is consistent with the recurrence theorem. Cercignani thinks that his claim *exempts* the Boltzmann equation from the strictures of the recurrence theorem, since he proves the validity of the Boltzmann equation for the N --> infty limit.

Thanks in advance!
 
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  • #2


I would like to offer some clarification on this topic. The non-compactness of the set of possible states in the N --> infty limit is indeed related to the system becoming unbounded. This can be seen from the fact that as the number of particles goes to infinity, the phase space also becomes infinitely large. This means that the system can explore a much larger range of possible states than in a system with a finite number of particles.

Additionally, the system may also become non-closed in the N --> infty limit. This means that the system may not have a well-defined boundary or boundary conditions, which can also contribute to the non-compactness of the set of possible states.

One intuitive way to understand this is to think about the behavior of individual particles in a gas. In a system with a finite number of particles, these particles will eventually collide and return to their initial positions and velocities, leading to a compact set of possible states. However, in a system with an infinite number of particles, there is always a chance that a particle may never collide and continue to explore an unbounded phase space.

In terms of recurrence time, as the number of particles goes to infinity, the time between recurrences also goes to infinity at a much faster rate. This is because with an infinite number of particles, the system has a much larger range of possible states to explore, making it less likely for the system to return to its initial state within a finite amount of time.

In summary, the non-compactness of the set of possible states in the N --> infty limit is due to the system becoming unbounded and possibly non-closed. This leads to a longer recurrence time, making the Boltzmann equation an effective tool for studying systems with a large number of particles.
 

1. What is Poincare Recurrence and the Infinite N Limit?

Poincare Recurrence and the Infinite N Limit is a mathematical concept that describes the behavior of a dynamical system, such as a physical or biological system, over an infinite amount of time. It is based on the idea that given enough time, any state of the system will eventually be repeated.

2. How does Poincare Recurrence and the Infinite N Limit relate to chaos theory?

In chaos theory, Poincare Recurrence and the Infinite N Limit is used to model the behavior of chaotic systems. It shows that even though these systems may seem random and unpredictable, there are underlying patterns and repetitions that occur over a long period of time.

3. What is the significance of Poincare Recurrence and the Infinite N Limit in physics?

In physics, Poincare Recurrence and the Infinite N Limit is important because it helps to explain the behavior of complex systems, such as the motion of planets in the solar system or the behavior of particles in a gas. It also has applications in thermodynamics, statistical mechanics, and quantum mechanics.

4. Can Poincare Recurrence and the Infinite N Limit be observed in real-world systems?

While it is a theoretical concept, there have been observations of Poincare Recurrence and the Infinite N Limit in real-world systems. For example, in the motion of particles in a gas, there will eventually be a recurrence of the exact same configuration of particles.

5. Is Poincare Recurrence and the Infinite N Limit a proven concept?

While there is strong evidence and mathematical proofs supporting the concept of Poincare Recurrence and the Infinite N Limit, it is still a theoretical concept and has not been definitively proven. It is still an area of ongoing research and study in mathematics and physics.

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