Non-Poisson

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 ... 

For example, a) in (radioactivity) counting experiments a non-Poisson random error component, equal in magnitude (variance) to the Poisson component, will not be detected until there are 46 degrees of freedom ( ), and b) it was necessary for a minor component in a mixed Y-ray spectrum to exceed its detection limit by -50 , before its absence was detected by lack-of-fit (x, model error) (7). 

Figures 1.20 and 1.21 show that at long time t 105 aggregates of similar particles are well pronounced. Their formation is associated with emergence of the non-Poisson fluctuation spectrum in a system. Similar particle aggregation leads to the effective dissimilar particle separation (as compared to their random distribution) and thus - to a reduced reaction rate which is essentially less than expected by the linear approximation.

The temporal evolution of spatial correlations of both similar and dissimilar particles for d = 1 is shown in Fig. 6.15 (a) and (b) for both the symmetric, Da = Dft, and asymmetric, Da = 0 cases. What is striking, first of all, is rapid growth of the non-Poisson density fluctuations of similar particles e.g., for Dt/r = 104 the probability density to find a pair of close (r ro) A (or B) particles, XA(ro,t), by a factor of 7 exceeds that for a random distribution. This property could be used as a good aggregation criterion in the study of reactions between actual defects in solids, e.g., in ionic crystals, where concentrations of monomer, dimer and tetramer F centres (1 to 3 electrons trapped by anion vacancies which are 1 to 3nn, respectively) could be easily measured by means of the optical absorption [22], Namely in this manner non-Poissonian clustering of F centres was observed in KC1 crystals X-irradiated for a very long time at 4 K [23],... 

One more formalism used for the solving of this problem is scaling [52, 53]. The conclusion was drawn that for d = 1 and 2 aggregation occurs but it is not true for d = 3 which contradicts all findings above. The possible explanation is that scaling approach is not sensitive to those non-Poisson fluctuations where a system remains macroscopically homogeneous (i.e., the correlation length is finite). 

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). 

IX. Non-Poisson Dichotomous Noise and Higher-Order Correlation Functions... 

XV. Non-Poisson and Renewal Processes A Problem for Decoherence Theory... 

C. Further Problems Caused by Non-Poisson Physics Transition from the Quantum to the Classical Domain... 

XVI. Non-Poisson Renewal Processes A Property Conflicting with Modulation Theories... 

We show that the GME, with aging, even if it is properly derived so as to yield results equivalent to the corresponding CTRW picture, might fail the purpose of predicting correctly the system s response to an external perturbation. We show that to derive a correct result we have to adopt the single trajectory perspective and to study the field action on the single trajectory rather than on a set of trajectories. This means that the non-Poisson renewal condition renders the GME a theoretical tool of limited validity. 

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. 

Note that the crisis of the GME method, based on the density approach, is generated by non-Poisson renewal processes. Therefore, we find it important to illustrate, by means of Section V, the model for intermittence, on which our theoretical arguments are based. 

Section IX is devoted to the important task of revealing the physical reason of the breakdown of the equivalence between density and trajectory perspective in the non-Poisson case. First of all, in Section IX.A, we show that several theoretical approaches to the generalized fluctuation-dissipation process rest implicitly on the assumption that the higher-order correlation functions of a dichotomous noise are factorized. In Section IX.B we show that the non-Poisson condition violates this factorization property, thereby explaining the departure of the density from the trajectory approach in the non-Poisson case. 

In Section XIII we show that a GME of a given age, difficult if not impossible to build up using the density perspective, can be satisfactorily derived from the CTRW perspective. Yet, there are reasons to believe that this kind of GME is of limited utility, given the fact that the important problem of absorption and emission in the non-Poisson case requires the use of a genuinely CTRW approach. This important fact is discussed in detail in Section XIV. 

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. 

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. 

In this review we show that there are two main sources of memory. One of them correspond to the memory responsible for Anderson localization, and it might become incompatible with a representation in terms of trajectories. The fluctuation-dissipation process used here to illustrate Anderson localization in the case of extremely large Anderson randomness is an idealized condition that might not work in the case of correlated Anderson noise. On the other hand, the non-Poisson renewal processes generate memory properties that may not be reproduced by the stationary correlation functions involved by the projection approach to the GME. Before ending this subsection, let us limit ourselves to anticipating the fundamental conclusion of this review The CTRW is a correct theoretical tool to address the study of the non-Markov processes, if these correspond to trajectories undergoing unpredictable jumps. 

In the special case where there are only two sites, the CTRW procedure, supplemented by the Poisson assumption of Eq. (69), yields the Pauli master equation of Eq. (2). This means that the Pauli master equation is compatible with a random picture, where a particle with erratic motion jumps back and forth from the one to the other state, with a condition of persistence expressed by the exponential waiting time distribution of Eq. (69). Recent fast technological advances are allowing us to observe in mesoscopic systems analogous intermittent properties, with distinct nonexponential distribution of waiting times. This is the reason why in this section we focus our attention on how to derive a v(/(t) with a non-Poisson character. 

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. 

The second term is responsible for the fluctuations of and the symbol R denotes the set of variables necessary to assign to the variable S, the proper intermittent properties. This is a crucial assumption. The model might rest, for instance, on a double-well potential, within which the variable E, moves, virtually attaining only the values corresponding to the bottoms of the two wells. The crucial issue is to make the distributions of time of sojourn at the bottom of these two wells distinctly non-Poisson and renewal at the same time. Here we limit ourselves to assuming that the theoretical waiting time distribution i(t) has the form of Eq. (92) and that /exp(f) is related to it via Eq. (73). In the specific case that we are here describing, a convenient form for the projection operator P is... 

We shall be referring to these prescriptions for dichotomous noise as the Dichotomous Factorization (DF) assumption. We notice that the factorization assumption is widely used regardless of whether the correlation function is an exponential or not. See, for instance, the work of Fulinski [62] for the use of this assumption in the non-Poisson case. We shall come back to these important issues in later sections. For the time being, let us limit ourselves to noticing that in the Poisson case,... 

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

IX. NON-POISSON DICHOTOMOUS NOISE AND HIGHER-ORDER CORRELATION FUNCTIONS... 

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. 

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