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Mean-Field Approximation and Chaos

时间:2010-03-04 12:31:22  来源:  作者:

In the mean-field approximation the mathematical origin of dynamical chaos resides in the nonlinearity of the Hartree-Fock equations. These equations provide an approximate description, the best independent-particle description, which describes, for a certain time interval, the very complicated evolution of the true many-body system. Two questions then arise:
1) Does this chaotic behavior persist in time?
2) What is the best physical situation to observe this kind of nonlinearity?

To answer the first question, it should be stressed that quantum systems evolve according to a linear equation and this is an important feature which makes them different from classical systems. Since the Schrödinger equation is linear, so is any of its projections. Its time evolution follows the classical one, including chaotic behaviour, up to tH. After that, in contrast to the classical dynamics, we get localization (dynamical localization). The Liouville equation, on the other hand, is linear in classical and quantum mechanics. However, for bound systems, the quantum evolution operator has a purely discrete spectrum (therefore no long-term chaotic behaviour). By contrast, the classical evolution operator (Liouville operator) has a continuous spectrum (implying and allowing chaos). This means that persistent chaotic behaviour in the evolution of the states and observables is not possible. Loosely speaking, chaotic behaviour is possible in quantum mechanics only as a transient with lifetime tH [7,8].

The Heisenberg time, or break time, can be estimated from the Heisenberg indetermination principle and reads

 

tH $displaystyle simeq$ $displaystyle {hbar over Delta E}$  , (4)

 

where $ Delta$E is the mean energy level spacing and, according to the Thomas-Fermi rule, $ Delta$E $ propto$ $ hbar^{N}_{}$, where N is the number of degrees of freedom, i.e. the dimension of the configuration space. So, as $ hbar$ $ rightarrow$ 0, the Heisenberg time diverges as

 

tH $displaystyle sim$ $displaystyle hbar^{1-N}_{}$  , (5)

 

and it does so faster, the higher N is [9]. We observe that the limitation to persistent chaotic dynamics in quantum systems does not apply if the spectrum of the Hamiltonian operator $ hat{H}$ is continuous.

Concerning the second question, it is useful to remember that, in the thermodynamic limit, i.e. when the number N of particles tends to infinity at constant density, the spectrum is, in general, continuous and true chaotic phenomena are not excluded [10].

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