Derive Birkhoff Ergodic Theorem from Von Neumann

A
Thread starter wrobel
Start date Dec 20, 2018

In summary, the Birkhoff Ergodic Theorem is a mathematical theorem that states that the time average of a function of a system is equal to the space average of the function over the entire phase space of the system. It was proved by John von Neumann, a Hungarian-American mathematician, physicist, and computer scientist, in his work on the foundations of quantum mechanics. This theorem is also known as the Von Neumann Ergodic Theorem. Von Neumann derived the theorem using his own mathematical techniques and concepts, including the concept of ergodicity. The Birkhoff Ergodic Theorem has applications in physics, mathematics, economics, and computer science, particularly in the study of dynamical systems.

Dec 20, 2018

wrobel

Science Advisor

Insights Author

Is that possible to derive the Birkhoff Ergodic Theorem from or with the help of the Von Neumann ergodic theorem?

Mathematics news on Phys.org

Dec 31, 2018

WWGD

Science Advisor

Gold Member

It seems to come down to ( I just read about it now)to showing that a.e convergence implies conbegence in mean square. Is this correct?

1. What is the Birkhoff Ergodic Theorem?

The Birkhoff Ergodic Theorem is a mathematical theorem that states that in a dynamical system, the time average of a function of a system is equal to the space average of the function over the entire phase space of the system.

2. Who is Von Neumann?

John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to a wide range of fields, including mathematics, physics, computer science, and economics. He is particularly known for his work in quantum mechanics, game theory, and the development of the modern computer.

3. What is the relationship between Birkhoff Ergodic Theorem and Von Neumann?

Von Neumann proved the Birkhoff Ergodic Theorem in his work on the foundations of quantum mechanics, using his own mathematical techniques and concepts. This theorem is also often referred to as the Von Neumann Ergodic Theorem.

4. How does Von Neumann derive the Birkhoff Ergodic Theorem?

Von Neumann's proof of the Birkhoff Ergodic Theorem involves using the spectral theorem for compact operators and the properties of unitary operators. He also introduces the concept of ergodicity, which is essential to the theorem.

5. What are the applications of the Birkhoff Ergodic Theorem?

The Birkhoff Ergodic Theorem has applications in various fields, including physics, mathematics, economics, and computer science. It is used to study the behavior of dynamical systems and has been applied to areas such as statistical mechanics, chaos theory, and time series analysis.