Phase space volume oscillator

This represents the sum of all the quantum states in the system in an energy range from 0 to E. Even though the phase space has been divided into quanta of action, this is still considered the classical sum of states because the classical phase space volume is first calculated and converted into quantum states only at the end. The distinction between the classical and quantum mechanical sum of states will become evident in the discussion of the harmonic oscillator which is best treated as a quantum mechanical system. 

The one-dimensional harmonic oscillator has two dimensions (p and q), so that its phase space volume for H < E is given by... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

The spectra from the 4-mode model coupled to 0, 5, 10, and 20 bath modes are plotted in Figs. 9(a)-(d). The addition of the bath clearly results in the structure of the spectrum being washed out. The experimental spectrum is, however, not obtained. The effect of the bath modes is made clear in Figs. 10(a)-(d), in which the absolute values of the autocorrelation function for the 4-mode system with 0, 5, 10, and 20 bath modes is plotted. Just the strongest 5 bath modes lead to a significant damping of the oscillations in the function. This is simply due to the extra volume of phase-space available... 

The microscopic state of the system defines coordinates, momenta, spins for every particle in the system. Each point in phase space corresponds to a microscopic state. There are, however, many microscopic states, in which the states of particular molecules or bonds are different, but values of the macroscopic observables are the same. For example, a very large number of molecular configurations and associated momenta in a fluid can correspond to the same number of molecules, volume, and energy. All points of the harmonic oscillator phase space that are on the same ellipse in Fig. 5 have the same total energy. 

If the system is larger or the diffusion coefficient is smaller, diffusion may be insufficient to maintain spatial homogeneity over all of space. The simplest manifestation of such lack of diffusive mixing is the desynchronization of the chaotic oscillations in different spatial regions local volumes (areas) of space containing many nodes oscillate in phase but the phase differs from local volume to volume. This is the analog of phase turbulence [28] for a periodic... 

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