Mixing statistical description

In the classical limit h - 0, the spectrum of the Landau-von Neumann superoperator tends to the spectrum of the classical Liouvillian operator. If the classical system is mixing, the classical Liouvillian spectrum is always continuous so that we may envisage an analytic continuation to define a discrete spectrum of classical resonances. It has been shown that such classical resonances are given by the zeros of the classical zeta function (2.44) and are called the Pollicott-Ruelle resonances sn(E) [63], These classical Liouvillian resonances characterize exponential decay and relaxation processes in the statistical description of classical systems. The leading Pollicott-Ruelle resonance defines the so-called escape rate of the system,... 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

The role of mixing has been studied in systems with more complex reaction schemes or considering more complex fluid-dynamical properties, and in the context of chemical engineering or microfluidic applications (for reviews on microfluidics see e.g. Squires (2005) or Ottino and Wiggins (2004)). Muzzio and Liu (1996) studied bi-molecular and so-called competitive-consecutive reactions with multiple timescales in chaotic flows. Reduced models that predict the global behavior of the competitive-consecutive reaction scheme were introduced by Cox (2004) and by Vikhansky and Cox (2006), and a method for statistical description of reactive flows based on a con-... 

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