# (181ac) Analysis of Pressure Fluctuations In Fluidized Beds. II Reconstruction of the Data by the Fokker-Planck and Langevin Equations

Authors:
University of Southern California
University of Minnesota

We develop a continuum representation of the pressure fluctuation time series

p(t) for a fluidized bed (FB) by using stochastic methods based on the

Markov processes. It is shown that the data may be represented by Markov

series with a Markov time scale t_M that is estimated based on the data.

Using the relation between the Markov processes and the Kramers-Moyal (KM)

expansion that is a continuum equation that involves, in principle, an

infinite number of coefficients, we represent the pressure fluctuation time

series by a KM expansion. However, since the third and higher-order

coefficients of the expansion are very small, the KM expansion reduces to a

Fokker-Planck (FP) equation that represents p(t) in terms of a drift and a

diffusion coefficients that are computed based on the data. The FP equation is,

in turn, equivalent to a Langevin equation, which is used to {\it reconstruct}

the data, i.e. generate the time series that mimic, in a statistical sense,

the original data. Thus, the Langevin equation may also be used to make

probabilistic predictions for the future values of the pressure over time

scales that are on the order of the Markov time scale t_M. The accuracy of the

reconstructed series and, hence, their continuum representation, is

demonstrated. We also compute the frequency of level-crossing at a given level

alpha, i.e. the relative frequency (probability) of occurence of the event

defined, for two times t_{i-1} and t_i, by, nu_alpha=P[p(t_i)>\alpha,

p(t_{i-1})<\alpha], where P(x) is the probability of the event. nu_\alpha

yields the frequency that a given pressure fluctuation level may be expected.

The average time that one should wait in order to observe the pressure at a

given level again is also computed. The two quantities may be particularly

important for modeling of the FBs. A relation is presented between the drift

and diffusion coefficients of the FP equation and the Hurst exponent that has

previously been used to describe the pressure fluctuation time series in terms

of self-affine stochastic distributions.