Chaos, molecular fluctuations, and the correspondence limit

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Abstract
Chaos is characterized by sensitive dependence on initial conditions. Trajectories determined by coupled, ordinary differential equations show sensitive dependence when their associated Liapunov exponent is positive. The Liapunov exponent is positive if the Jacobi matrix associated with the coupled differential equations has an eigenvalue with a positive real part, on the average, as the Jacobi matrix evolves along the trajectory. For macrovariable equations, there are also fluctuation equations which follow the macrovariable trajectories. The covariance matrix for these fluctuations evolves according to an equation in which the Jacobi matrix for the deterministic motion plays the dominant role. For a chaotic trajectory, the covariance matrix grows exponentially. This means that for macrovariable equations that imply chaos, the construction of the macrovariable equations out of an underlying master equation is no longer valid. The macrovariable equations cease to be physical, and the physical description must be done entirely at the master equation level of description where the fluctuations, which are very large scale, can be properly treated. In parallel with this analysis, the correspondence limit connecting the time evolution of the Wigner distribution with Liouville’s equation breaks down when the classical motion is strongly chaotic. This implies that strongly chaotic classical dynamics must be treated quantum mechanically in order to properly treat the quantum fluctuations which have grown macroscopically large. Experimental confirmation of these ideas is discussed.