Recurrence times, Poincare

An example of such an initial state is provided by the situation in which the particles in one-half of V have much higher energy than those in the other half, at least if N is of macroscopic magnitude. Other such exceptional states will not necessarily appear so unusual. Nonetheless we speak of the system as non-equilibrium initially, as approaching equilibrium when A/ is large and decreasing, and as in equilibrium at subsequent times, when A/ is of order 1/N. While large fluctuations should reappear after the Poincare recurrence time, this time is enormous for fluid systems, at least if N is not too small. 

Interlude 3.2 Poincare Recurrence Times We have seen that Boltzmann s entropy theorem leads not only to an expression for the equilibrium distribution function, but also to a specific direction of change with time or irreversibility for a system of particles or molecules. The entropy theorem states that the entropy of a closed system can never decrease so, whatever entropy state the system is in, it will always change to a higher entropy state. At that time, Boltzmann s entropy theorem was viewed to be contradictory to a well-known theorem in dynamics due to Poincare. This theorem states that... 

Boltzmann attempted to resolve this paradox by considering the system particle dynamics from a probability sense. He estimated the Poincare recurrence time for a cubic centimeter of air containing about 10 molecules. His calculated result is that the Poincare recurrence time, i.e., the average time for the system to pass back through the initial state is on the order of the age of the universe ... 

In the case reported here, a chain with M = 10-15 modes should be able to accurately capture bath memory effects on the system dynamics, for times t relevant system dynamics lasts longer, a Markovian closure acting on the last member of the chain (with 7 = cl)r/2) provides a reasonably good approximation valid for all times. 

This second point is quite an interesting one, for there is a theorem known as the Poincare recurrence theorem which states that an isolated system (like our molecule left to itself) will in the course of time return to any of its previous states (e.g. the initial state), no matter how improbable that state may be. This recurrence can be observed with very small molecules but not with polyatomic molecules, because in the latter there are far too many levels of the final state the recurrence time is then far longer than any practicable observation time. 

The Poisson-stable trajectories may be sub-divided into two kinds depending on whether the sequence Tfc(e) of Poincare return times of a P-trajectory to its -neighborhood is bounded or not. Birkhoff named the trajectories of the first kind recurrent trajectories. Such a trajectory is remarkable because regardless of the choice of the initial point, given e > 0 the whole trajectory lies in an -neighborhood of the segment of the trajectory corresponding to a time interval L(e). Obviously, equilibrium states and periodic orbits are the closed recurrent trajectories. 

In the case of recurrent trajectories, there are certain statistics in Poincare return times which are weaker than that characterizing genuine Poisson-stable trajectories. Nevertheless, there is a particular sub-class of recurrent trajectories which is interesting in nonlinear dynamics. This is the class of the so-called almost-periodic motions. The remarkable feature which reveals the origin of these trajectories is that each component of an almost-periodic motion is an almost-periodic function (whose analytical properties are well studied, see for example [49, 66, 84]). 

Since the recurrent time of a nearby orbit to a cross-section is about uj p) (see the last section), it follows that the period of the orbits of the flow which corresponds to the fixed points of the Poincare map tends to infinity as p - -0 (typically, it is /y/pt ) Before the orbits return to the cross-section, each must make uj p) rotations in a small neighborhood of the just disappeared saddle-node L. Accordingly, the length of these periodic orbits is also increasing to infinity. Thus, Theorem 12.8 gives a positive answer to the following... 

The question stated above was formulated in two ways, each using an exact result from classical mechanics. One way, associated with the physicist Loschmidt, is fairly obvious. If classical mechanics provides a correct description of the gas, then associated with any physical motion of a gas, there is a time-reversed motion, which is also a solution of Newton s equations. Therefore if decreases in one of these motions, there ought to be a physical motion of the gas where H increases. This is contrary to the /f-theorem. The other objection is based on the recurrence theorem of Poincare [15], and is associated with the mathematician Zermelo. Poincare s theorem states that in a bounded mechanical system with finite energy, any initial state of the gas will eventually recur as a state of the gas, to within any preassigned accuracy. Thus, if H decreases during part of the motion, it must eventually increase so as to approach, arbitrarily closely, its initial value. 

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