Non-Poisson statistics

After ruling out slow modulation as a possible approach to complexity, we are left with the search for a more satisfactory approach to complexity that accounts for the renewal BQD properties. Is it possible to propose a more exhaustive approach to complexity, which explains both non-Poisson statistics and renewal at the same time We attempt at realizing this ambitious task in Section XVII. In Section XVII.A we show that a non-Ohmic bath can regarded as a source of memory and cooperation. It can be used for a dynamic approach to Fractional Brownian Motion, which, is, however, a theory without critical events. In Section XVIII.B we show, however, that the recursion process is renewal and fits the requests emerging from the statistical analysis of real data afforded by the researchers in the BQD held. In Section XVII.C we explain why this model might afford an exhaustive approach to complexity. 

This equivalence between jumping trajectories and coherence relaxation is confined to the Poisson case. In this section we plan to study higher-order correlation functions, and we plan to prove the emergence from non-Poisson statistics of unexpected properties violating the condition for the trajectory-density equivalence. These properties, as we shall see in Section XV, weaken the conviction that the wave-function issue is settled. 

In recent times, the term superstatistics has been coined [153] to denote an approach to non-Poisson statistics, of any form, not only the Nutting (Tsallis) form, as in the original work of Beck [154]. We note that Cohen points out explicitly [153] that the time scale to change from a Poisson distribution to... 

Non-Poisson statistics due to pile-up rejection and loss-free counting... 

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

Equation (1) is the most general and rigorous analytical result. However, it does not take account of the aggregation of similar defects and hence it is applicable only at not very large irradiation doses (up to a concentration of defects (l/2)no, where no is concentration at saturation). In fact, here the existence of clusters of similar defects is allowed, but it is actually assumed that these clusters are statistical fluctuations of the Poisson distribution of similar defects, which does not reflect a real pattern of cluster formation with a substantially non-Poisson spectrum of fluctuations. It is assumed implicitly in equation (1) that, after each event of creating a new pair of defects, the entire system of defects is stirred to attain the Poisson distribution. In the case of the absence of the defect correlation in genetic pairs we arrive at equation (2). 

The kinetic complexity seen in oriented micelles persists in inverse micelles. The distribution of electron transfer quenchers within the water pool follows Poisson statistics and enables the kinetic data to describe migration rates to and from the aqueous subphase [65]. These orientation effects also make possible topological control of non-electron transfer photoreactions occurring within AOT micelles [66]. 

The crisis of the GME method is closely related to the crisis in the density matrix approach to wave-function collapse. We shall see that in the Poisson case the processes making the statistical density matrix become diagonal in the basis set of the measured variable and can be safely interpreted as generators of wave function collapse, thereby justifying the widely accepted conviction that quantum mechanics does not need either correction or generalization. In the non-Poisson case, this equivalence is lost, and, while the CTRW perspective yields correct results, no theoretical tool, based on density, exists yet to make the time evolution of a contracted Liouville equation, classical or quantum, reproduce them. 

At this stage, we are confident that a clear connection between Levy statistics and critical random events is established. We have also seen that non-Poisson renewal yields a class of GME with infinite memory, from within a perspective resting on trajectories with jumps that act as memory erasers. The non-Poisson and renewal character of these processes has two major effects. The former will be discussed in detail in Section XV, and the latter will be discussed in Section XVI. The first problem has to do with decoherence theory. As we shall see, decoherence theory denotes an approach avoiding the use of wave function collapses, with the supposedly equivalent adoption of quantum densities becoming diagonal in the pointer basis set. In Section XV we shall see that the decoherence theory is inadequate to derive non-Poisson renewal processes from quantum mechanics. In Section XVI we shall show that the non-Poisson renewal properties, revealed by the BQD experiments, rule out modulation as a possible approach to complexity. 

If we adopt the different walking rule of making the walker travel with constant velocity in between two unpredictable non-Poisson time events, subdiffusion is turned into superdiffusion. The CTRW method can be easily adapted to take care of this different walking rule [42,44]. However, in this case, as we shall see, there does not exist yet an exhaustive approach connecting the CTRW prescriptions to the GME structure discussed in Section III. This means that, not even in principle, it is yet known how to derive this kind of superdiffusion from the conventional Liouville prescriptions of nonequilibrium statistical physics. In this section we plan to make a preliminary illustration of this delicate issue. 

These results mean that, once the GME coinciding with the CTRW has been built up, we cannot look at it as a fundamental law of nature. If this GME were the expression of a law of nature, it would be possible to use it to study the response to external perturbations. The linear response theory is based on this fundamental assumption and its impressive success is an indirect confirmation that ordinary quantum and statistical mechanics are indeed a fair representation of the laws of nature. But, as proved by the authors of Ref. 104, this is no longer true in the non-Poisson case discussed in this review. 

The one-atom maser can be used to investigate the statistical properties of non-classical light [1298, 1299]. If the cavity resonator is cooled down to T < 0.5 K, the number of thermal photons becomes very small and can be neglected. The number of photons induced by the atomic fluorescence can be measured via the fluctuations in the number of atoms leaving the cavity in the lower level n — 1). It turns out that the statistical distribution does not follow Poisson statistics, as in the output of a laser with many photons per mode, but shows a sub-Poisson distribution with photon number fluctuations 70 % below the vacuum-state limit [1300]. In cavities with low losses, pure photon number states of the radiation field (Fock states) can be observed (Fig. 9.77) [1301], with photon lifetimes as high as 0.2 s At very low... 

It is not generally recognized that the uncertainty of counts in a spectrum may not be adequately described by Poisson statistics when there are non-random losses of counts, especially when counting rates are high. A good summary of the situation is given by Pomme et al. (2000). 

Counting process Equations for response statistical moments Jump processes Markov processes Non-Poisson processes Point process Probability density Random impulses Random vibrations Renewal processes... 

The discrepancy between high-elasticity classical theory and experimental curves o-e or o-X for elastoplastics indicated above is due to two factors firstly, by essentially non-Gaussian statistics of real polymer networks and, secondly, by the lack of coordination of two main postulates, lying in the basis of entropic high-elasticity classical theory - Gaussian statistics and elastoplastics incompressibility. The last condition is characterised by the criterion v = 0.5, where v is Poisson s ratio [46]. 

---