Theorems of Liouville and Poincare & their relation to entropy

[Heirot]

If we have a system for which the Liouville's tm holds, can we automaticly say the Poincare's recurrence tm also holds? Presumably this is true in microcanonical ansable, but how about canonical, where the energy isn't constant?

 

[Andrew Mason]

If we have a system for which the Liouville's tm holds, can we automaticly say the Poincare's recurrence tm also holds? Presumably this is true in microcanonical ansable, but how about canonical, where the energy isn't constant?

Poincaré's recurrence theorem holds for any statistical system in which system's energy and the system's space do not change. It says, essentially, that such a system will return to a microstate that is to within an arbitrarily close approximation of its original microstate, if it is given enough time. So if energy is not conserved in the system, Poincaré's recurrence theorem does not hold.

AM

 

[astrokreng]

The theorems of Liouville and Poincare are fundamental principles in the field of dynamical systems, and they have important implications for the concept of entropy. Liouville's theorem states that the phase space volume of a closed system is conserved over time, while Poincare's recurrence theorem states that in a closed system, any state that has occurred will eventually recur infinitely many times.

In terms of entropy, these theorems have a direct relation to the second law of thermodynamics. Entropy is a measure of the disorder or randomness of a system, and according to the second law, the entropy of a closed system will tend to increase over time. Liouville's theorem implies that the phase space volume, which is related to the number of microstates of a system, remains constant over time. This means that in a closed system, the number of possible arrangements of particles or states does not change, but their distribution or organization may change, leading to an increase in entropy.

Poincare's recurrence theorem, on the other hand, suggests that in a closed system, there is a possibility for the system to return to a previous state. This has implications for the concept of entropy, as it suggests that a system can temporarily decrease in entropy and return to a more ordered state, before eventually increasing in entropy again.

In the microcanonical ensemble, where the energy is constant, both Liouville's theorem and Poincare's recurrence theorem hold. This is because the energy is conserved, and the system is closed, meaning that there is no exchange of energy or particles with the surroundings. In this case, we can say that Liouville's theorem implies Poincare's recurrence theorem.

However, in the canonical ensemble, where the energy is not constant and there is exchange of energy with the surroundings, Poincare's recurrence theorem may not hold. This is because the system is not closed, and there may be changes in the energy of the system over time. In this case, we cannot automatically say that Poincare's recurrence theorem holds, as it depends on the specific conditions of the system.

In summary, while Liouville's theorem and Poincare's recurrence theorem are related to each other and have implications for entropy, their applicability depends on the specific conditions of the system. In a closed system, both theorems hold and have implications for the increase of entropy over time, but in an open system, Poin