Selforganization without conservation :
true or just apparent
scaleinvariance?
Abstract
The existence of true scaleinvariance in slowly driven models of selforganized criticality without a conservation law, as forestfires or earthquake automata, is scrutinized in this paper. By using three different levels of description  (i) a simple mean field, (ii) a more detailed meanfield description in terms of a (selforganized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation, it is shown on general grounds that nonconserving dynamics does not lead to bona fide criticality. Contrarily to conserving systems, a parameter, which we term “recharging” rate (e.g. the treegrowth rate in forestfire models), needs to be finetuned in nonconserving systems to obtain criticality. In the infinite size limit, such a finetuning of the loading rate is easy to achieve, as it emerges by imposing a second separation of timescales but, for any finite size, a precise tuning is required to achieve criticality and a coherent finitesize scaling picture. Using the approaches above, we shed light on the common mechanisms by which “apparent criticality” is observed in nonconserving systems, and explain in detail (both qualitatively and quantitatively) the difference with respect to true criticality obtained in conserving systems. We propose to call this selforganized quasicriticality (SOqC). Some of the reported results are already known and some of them are new. We hope the unified framework presented here helps to elucidate the confusing and contradictory literature in this field. In a second accompanying paper, we shall discuss the implications of the general results obtained here for models of neural avalanches in Neuroscience for which selforganized scaleinvariance in the absence of conservation has been claimed.
pacs:
05.50.+q,02.50.r,64.60.Ht,05.70.LnKeywords: Selforganized criticality. Generic scaleinvariance. Nonequilibrium statistical mechanics.
1 Introduction: Critical Selforganization with and without conservation
Powerlaw distributions are quite common in Nature: earthquakes and microfractures, solar flares, weather records, snow avalanches, crackling, and noise are just a few examples of systems displaying scaleinvariance [1]. Many of these, among many others, have been claimed to be critical, i.e. to lie at, or very close to, a critical point, which ensues scaleinvariance and the concomitant powerlaw distributions. It is worth stressing that powerlaws (or approximate powerlaws) emerging from the complex interactions of manycomponent systems placed at the vicinity of a critical point and, therefore, with diverging correlation lengths and the associated powerlaw decay of temporal and spatial correlations, should be clearly distinguished from other powerlaw distributions arising in many different contexts (word distributions, city populations, citations, etc). These latter can be generated by a wealth of noncritical mechanisms as, for instance, multiplicative noise or fragmentation processes (see [2, 3, 4] for recent reviews), without the need to invoke criticality.
For a system to be critical, it does not suffice to have powerlaw distributed observables but, more crucially, it has to obey finitesize scaling: measurements at various systemsizes (scales) can be related to each other by rescaling variables and quantities in some specific “scaleinvariant” way (i.e. scaling collapses can be performed).
Given that, in standard phase transitions (both in equilibrium and away from it), a precise parameter tuning is required to reach criticality and generate powerlaw distributed quantities, an alternative explanation for the emergence of generic critical scaleinvariance (i.e. criticality occurring without requiring finetuning) was historically much needed [5]. How does criticality emerge spontaneously?
In a seminal paper, Bak, Tang and Wiesenfeld (BTW) [6] introduced, back in , the concept of selforganized criticality (SOC) [7, 8, 9, 10, 11, 12] aimed at solving the previous conundrum. The research line opened by their breakthrough work continues to attract, more than twenty years after, a great deal of interest. From this perspective, the overwhelming activity generated in the last decade around scalefree networks [13] can be certainly considered as a prominent extension of preceding research on SOC.
A handful of mechanisms were proposed under the common name of SOC to justify the abundant presence of scaleinvariance in the natural world. The most successful among them is the one exemplified by the original BTW sandpile model. Many variations of the BTW sandpile were proposed: the Manna sandpile [14], the Oslo ricepile model [15], and the Zhang model [16], are just a few of them. For the sake of completeness and for future reference, a brief description of the bestknown prototypical SOC models is provided in Appendix A. Other possible mechanisms, as extremal dynamics [17], have been studied, but we shall not be concerned with them here.
The idea inspiring sandpiles, ricepiles, and related SOC models is that many systems in Nature, when pushed/driven slowly, respond very irregularly with rapid rearrangements (avalanches, bursts) of a broad variety of sizes. The distribution of such avalanches is scaleinvariant in many cases. Of course, sandpile SOC models are too simplistic to reproduce the detailed behavior of real sandpiles; they can be regarded as ”metaphors” capturing only some particular features of real systems in a stylized manner. Nevertheless, after many (partially) failed attempts [18] powerlaws for avalanche in real granular piles, obeying finitesize scaling, were experimentally measured [19]. Moreover, other physical situations, as vortex avalanches in typeII superconductors, can be mimicked as sandpiles, and their critical properties rationalized in terms of these [20].
The dynamics of a generic sandpile model can be synthesized as follows: some type of “energy” or “stress” (sandgrains) is progressively injected in discrete units (i.e. grains are dropped singly) at the sites of a spatially extended system (usually a twodimensional square lattice) at a slow timescale. Whenever a certain threshold of local energy is overcome, the corresponding site becomes unstable and its accumulated energy is redistributed at a much faster timescale among its neighbor sites. These, on their turn, can become unstable, and trigger a cascade of rearrangements, i.e. an avalanche or outburst of activity. Local redistribution rules are conserving in sandpiles: energy does not dissappear, but only diffuses around. Open boundaries are customarily considered to allow for energy release from the system. Once all activity ceases (i.e. the avalanche stops) new energy is injected (i.e. a grain is dropped) into the system, and so on, until a statistically stationary state is reached. In such a steady state avalanches are scale invariant, i.e their sizes/times are powerlaw distributed up to a maximum scale imposed by the system size: the system “selforganizes” to a critical point [7, 8, 9, 10, 11, 21, 22, 23]. Moreover, the associated power spectrum exhibit noise [24].
For the sake of generality, let us fix a nomenclature common to all models discussed in this paper: “energy” refers to the accumulated and transported magnitude and “activity” describes “energy above threshold”. In some cases, sites below threshold will be subclassified in two groups: critical, which can become active upon receiving one input of energy (for instance, one grain) and stable, which cannot [25].
An essential ingredient of SOC is slow driving (driving and dynamics operating at two infinitely separated timescales [7, 8, 9, 21, 26], i.e. avalanches are instantaneous relative to the timescale of driving). Such an infinite separation is usually achieved by driving the system only when all activity has stopped, but not during avalanches. For any finite separation of timescales, a finite characteristic (time/size) scale appears; hence, slow driving is a crucial requirement for generic scaleinvariance to emerge [21, 27, 28, 29].
It was soon emphasized that energy conservation is also a key player for criticality to emerge in sandpile models [21, 30, 31]. Note that by “conserving system” one can refer either to models with bulk conserving dynamics and boundary dissipation, or to cases with a bulk dissipation rate vanishing in the large system size limit [32]. Various relatively simple arguments were proposed to rationalize the existence of true criticality in the steady state of conserving selforganized systems (some of them are briefly discussed in the next section). However, these arguments cannot be easily extended to similar nonconserving systems. In particular, different works have shown that the level of dissipation acts as a relevant parameter in the renormalization group sense: any degree of bulk dissipation breaks criticality in sandpile models [21, 31, 33].
Still, given the large variety of natural phenomena exhibiting (exact or approximate) scaleinvariance in which some form of dissipation is inevitably present (i.e. systems without any obvious conserved quantity), alternative mechanisms for selforganization to criticality in the absence of conservation were needed to achieve a comprehensive picture of generic scaleinvariance [21].
Two acclaimed nonconserving selforganized models, or better, two families of models proposed to fill the gap between theoretical understanding and empirical facts are earthquake and forestfire models (see Appendix A for definitions). These are highly nontrivial, and interesting models with rich and complex phenomenology. Owing to the lack of solid theoretical arguments, analogous to the ones sketched above for conserving systems, and despite of numerical evidence showing powerlaws for some decades, the existence of true generic scaleinvariance in them has been long controversial.
It is beyond the scope of this paper to review exhaustively the large body of interesting literature devoted to nonconserving SOC models, some aspects of which remain unsettled. But, let us just underline that the stateoftheart is, as documented in the next section, that none of the considered nonconserving models is truly critical; they just show “apparent scaleinvariance” or “dirty criticality” for some decades.
However, this final conclusion has not been sufficiently stressed and it has certainly not permeated the literature. This is likely due to the absence of a general theory, which may suggest that the results discussed above are specific to each particular model. Indeed, works continue to be published assuming or claiming true criticality for SOC nonconserving systems. For instance, in [34] an interesting and solvable “nonconserving model of SOC” was proposed and studied analytically. In a more recent series of papers, Juanico and collaborators claim to have constructed different nonconserving selforganizing models with applications in various fields (as neuroscience, population dynamics, etc.) [35]. Also, in a recent work, Levina et al. propose a nonconserving SOC model for neural avalanches to capture the apparent scalefree behavior of avalanches of activity observed experimentally in networks of cortical neurons [36]. An exhaustive analysis of this last model, as well as a study of the possible relation between SOC and neural avalanches, is left for a separate publication.
The aim of the present paper is to put together some previously existing results, scattered in the literature, (although this is not intended to be an exhaustive review article) and, more importantly, to rationalize the conclusion that none of the above mentioned nonconserving models, nor variations of them, exhibits true criticality, within a unified framework. To this end, we rely on different types of analytical arguments complemented by computer simulations. In passing, we shall report on a number of new results and present a critical discussion on the existence of true scaleinvariance in Nature.
The rest of the paper is structured as follows: in Section 2, we briefly review conserving and nonconserving models of SOC, as well as some arguments to justify the existence of criticality in the first group. In the remaining Sections, we elucidate the existence or not of criticality in nonconserving systems using different approaches of increasing complexity. In particular, in Section 3, we discuss a simple meanfield approach based on an energy balance equation; it is useful to illustrate some key concepts as the “loading mechanism”. In Section 4, we study a selfconsistent meanfield approximation, namely the socalled selforganized branching process; it serves as an adequate benchmark to scrutinize the effects of dissipation and “loading” in critical selforganization. In Section 5, we present (and briefly review) the Langevin theory of conserving SOC systems. It constitutes a solid basis to implement dissipation and loading in a systematic way and to provide clear evidence on the lack of criticality in nonconserving models. Finally, the conclusions and a critical discussion of the implications of our main results are presented in Section 6.
2 Conserving versus nonconserving models of SOC
2.1 Conservation and criticality.
As said above, different type of arguments of different nature justify the existence of true criticality in conserving SOC models. Some of them are as follows:

The energy, introduced into a pile at generic sites, can reach the boundaries (and, thus, be dissipated) only by means of the diffusive transport of grains occurring during avalanches. Owing to this, and provided that a steady state exists, arbitrarily large avalanches (of all possible sizes) should exist for an arbitrarily large system size, ensuing a powerlaw sizedistribution. Contrarily, in the presence of nonvanishing bulkdissipation, energy disappears at some finite rate, and avalanches stop after some characteristic lifetime/size determined by the dissipation rate [21].
This type of argument, even if commonly used in the literature, is (at best) incomplete, and can be misleading. It does not consider the possibility of having a characteristic scale larger than the system size, which would allow for avalanches to reach the boundaries. Actually, this is what happens in many real sandpiles (with inertial effects): energy is dissipated quasiperiodically in large systemwide avalanches rather than in scaleinvariant avalanches (see the chapter on experimental setups of sandpiles in [8, 9, 37] or [21]).

From a more abstract viewpoint (not referring specifically to sandpiles or SOC), energy conservation follows from the existence of a continuous symmetry (in this case, temporal translational invariance) as a consequence of Noether’s theorem [38]. This also holds the other way around: time translational invariance implies energy conservation. When energy conservation is violated, the corresponding symmetry is broken and a characteristic (finite) timescale appears generically.

From a fieldtheoretical perspective, in order to have scaleinvariance, generic infrared divergences are required. But these are generically lost in the presence of a nonvanishing linear “mass” term (adopting the fieldtheory jargon) as, for instance, a dissipative term. In particular, if a term , is introduced into the simplest mesoscopic equation for a diffusive field
(1) where is a zeromean Gaussian white noise, a simple calculation reveals that the equal time twopoint correlation function can be written as
(2) for much larger than the correlation length . Accordingly, it is only for that diverges, the exponential cutoff disappears, and the correlation function decays algebraically; for any nonvanishing value of there is a sizeindependent exponential cutoff. Something similar occurs for other correlation functions.
Note that the noise in Eq.(1) is not fully conserving, but only conserving on average (i.e. conservation needs only to hold on average to preserve scale invariance [39]). The same conclusions can be also deduced for an equation analogous to Eq.(1) but with a strictly conserving noise (see [21, 27, 28]).

Last but not least, a mesoscopic Langevin equation that captures the critical properties of stochastic sandpiles and related models with a conservation law has been proposed [22, 40, 41, 42]. It describes systems with many absorbing states (which correspond to the many stable microscopic configurations of a sandpile) and a conservation law. While a detailed description of this is left for a forthcoming section, we just stress here that it constitutes a sound field theoretical representation of conserving SOC, reproducing all critical exponents. The underlying mechanism of SOC, highlighted by this theory, cannot be straightforwardly generalized to nonconserving systems without including an additional fine tuning (as we shall show in Section 5).
In summary, there exists solid theoretical ground to underpin the existence of true criticality in conserving selforganized systems. The same type of arguments cannot be easily extended to nonconserving systems. In particular, as said already, nonconservative sandpiles have been explicitly shown to be noncritical.
2.2 Critical selforganization without conservation?
The two main prototypical nonconserving “selforganized” models (or families of models), studied profusely in the literature, are (see Appendix A):
Together with bulkdissipation, these two models have a common key ingredient, absent in conserving systems: there is an increase of the “background energy” at some (or at all) sites, occurring between avalanches; (i) the accumulated stress at each site grows continuously between quakes in earthquake models as the OFC, and (ii) new trees grow between two consecutive fires in forestfire models (see below).
The effect of these “loading mechanisms”, as we shall call them generically, is to counterbalance the loss of “energy” (grains, stress, trees) produced by dissipation and, in this manner, try to restore conservation on average and, thus, criticality [25, 34, 35]. However, let us caution that such a compensation needs to be exact and, therefore, unless a new mechanism giving rise to a perfect cancellation is devised, finetuning of the loading rate is the only obvious way in which conservation can be restored.
Let us now discuss in more detail these nonconserving archetypical models.
2.2.1 Earthquakes:
The OlamiFederChristensen cellular automaton [44] is a simplified version of a previously proposed fault dynamics models: the springblock of BurridgeKnopoff model [45] and related stickslip models [47, 48] designed to capture the essence of earthquakes (the GutenbergRichter law [52] for the distribution of magnitudes) as well as of similar systems with friction and jerky motion (see Appendix A).
At each time step, the “forces” (or energies, to stick to our generic terminology), defined at each site of a twodimensional lattice, are increased at a constant rate. Whenever at any site reaches the threshold value, , it is reset to zero and the forces at its nearest neighbors are increased by an amount . This might trigger cascades of rearrangements, i.e. avalanches. Observe that the bulk dynamics is conserving only for in twodimensions.
Early computer simulations and theoretical results [44, 53, 54] seemed to support the existence of criticality for values of as low as . It was also early reported that, imposing periodic boundary conditions, the OFC model enters a cycle of periodic configurations with no sign of criticality whatsoever [55, 28, 56]. This suggests that the bulk dynamics is profoundly influenced by boundary conditions. Actually, it was proposed that boundaryinduced heterogeneity is essential to obtain partial synchronization between clusters of different sizes and that such a partial synchronization or “phaselocking” mechanism is at the basis of the OFC complex behavior [55, 28, 56].
The role of different features (as, for instance, changes in the boundary conditions, introduction of quenched disorder in the local rules, lattice topology, etc.) on synchronization and their effects in the properties of the OFC model have been largely analyzed in the literature [53, 57, 58, 59, 60, 61, 62, 63, 64]. It has been also shown that results are affected by numerical precision [70]. The overall picture is that the “synchronization mechanism”, even if fascinating, is too fragile as to be a solid explanation for generic emergence of criticality. A nice and rather exhaustive review of the literature on the OFC model and variations of it can be found in [72].
On the analytical side, Bröker and Grassberger [65] and Chabanol and Hakim [66], in two independent papers, were able to calculate the energy distribution, the effective branching ratio, and the average avalanche size for a randomneighbor version of the OFC model, which turns out to be analytically solvable. Their main conclusion is that it is only in the conserving limit that the model becomes critical, while exponential cutoffs appear for any . Similarly, de Carvalho and Prado claimed, relying on an effective branching ratio analysis [67], that the OFC model is only critical in the conserving limit (see also [68]).
Remarkably, it is also shown in [65] and [66] that the average avalanche size is distributed as a powerlaw with a cutoff function, , which diverges in a very fast way when the conserving limit is approached. This provides an explanation for the relative large powerlaw regimes observed even in the nonconserving case. It would be certainly nice to have extensions of this result to other nonmeanfield like systems.
In a similar line of reasoning, Kinouchi and Prado introduced the concept of “robust criticality” or “almost criticality” [69]: for a fixed dissipation rate, systems with a loading mechanism are closer to criticality than systems without it. The reason for this is simple: moderate loading partially compensates energy dissipation.
Finally, the most recent and exhaustive analyses by Miller and Boulter [71], Grassberger [55], and Drossel and coauthors [72] conclude unambiguously, using a variety of arguments and largescale computer simulations, that the spatially extended version of the nonconserving OFC model is not critical.
As a consequence, the stateoftheart is that, despite of the apparent power law distributions spanning for a few decades, the OFC model is not truly scaleinvariant, except for its conservative limit. The question of whether real earthquakes are described or not by this type of SOC models or other type of mechanisms need to be invoked remains unsolved [73, 74, 75].
2.2.2 Forest fires:
The DrosselSchwabl model [50] (see also [76]) is an improved version of an older forestfire model [51] proposed to explain the apparent scaleinvariance of real forestfires [77] (see Appendix A).
Three type of states are defined: or empty, or occupied by a tree, and or burning tree. At each time step, new trees grow, at rate , at randomly chosen sites provided they were empty, and trees catch fire at a much smaller rate . Fire propagates deterministically to neighboring occupied sites and, after burning, trees become empty sites. The relevant parameter is [78], and the model has been claimed to be critical provided that the double limit , with ) is taken [50].
Observe that a double separation of time scales is imposed in the model definition: trees are born at a much faster rhythm than fires occur and fires propagate at a much faster pace than trees grow [50]. This is to be compared with the single time scale separation in sandpiles [50, 21]. We shall discuss later the consequences of such a double separation of timescales.
Analytical results and mappings into a branching process [79] first suggested some similarities with standard percolation models [50, 80, 81]. Given the limited analytical tractability of these models, the controversy about the existence of true criticality was mainly played on the ground of computer simulations [50, 82, 81, 83]. For sufficiently large systems, anomalies were reported to appear; among them: (i) the repulsive character of the fixed point (), (ii) the coexistence of largelysubcritical and supercritical clusters of trees, (iii) the existence of two length scales with different exponents into the system, (iv) the violation of standard scaling for the distribution of avalanche sizes , and (v) a pathological finitesize behavior [50, 81, 82, 83, 84].
Finally, when “massive” simulations of extremely large systems very close to the critical regime were accessible [85, 86], these anomalies turned into a lack of true critical behavior, beside of the apparent scaling observed for a few decades: the selforganized stationary state of the DrosselSchwabl model is not critical.
In the rest of the paper, we shall rationalize and generalize the above conclusion (i.e. absence of bonafide criticality of earthquake and forestfire SOC models) to generic nonconserving systems. to this end, we shall employ three different unified frameworks, as described in the three forthcoming sections.
3 A simple meanfield approach
As already mentioned, the randomneighbor version of the OFC model has been solved analytically, with the conclusion that, except for the conserving limit, it is not critical, but generically subcritical [65, 66]. On the other hand, in a subsequent work, Pruessner and Jensen [34] considered a modified version of such a model in which, by including a different (stronger) loading mechanism, they showed that criticality can be restored in the infinite size limit. In this section, we review the results in [34] and study the finitesize scaling of this and related meanfield systems.
The model in [34] is somewhere in between forestfire and earthquake models. It is defined as follows: consider a set of sites, each of them with an associated energy (with a continuous variable). As in the OFC model, three types of states exist: stable, with an energy ; susceptible, with ; and active sites, with . The main difference with respect to the OFC model is that, between avalanches, driving and loading operate as independent mechanisms [34]:

Triggering of an avalanche: A randomly chosen site, , is activated () provided it was susceptible.
The relaxation dynamics within avalanches is identical to that of the random neighbor OFC model with random neighbors: sites above threshold are emptied, , and the energy of its (randomly chosen) neighbors is increased by a fixed amount . Conservation holds for .
At a meanfield level, the condition for stationarity is given by the following energybalance equation [34]:
(3) 
where the different terms are as follows:
(i) l.h.s: For each relaxation event at site , the amount of dissipated energy is ; hence, the average dissipation during an avalanche is , where is the average avalanche size and the average energy of active sites.
(ii) r.h.s. first term: Triggering increases the energy of the selected susceptible site, , by an amount . The corresponding average increase is , where is the average energy of susceptible sites.
(iii) r.h.s., second term: Every time the background is loaded, the energy of a stable site, , is increased by an amount . This is attempted times and, the average number of triggering events before an avalanche is actually generated is (where and are the density of stable and susceptible sites, respectively). The average increase of background energy per avalanche is, finally, , where stands for the average energy of stable sites.
In this way, Eq.(3) establishes that, for a steady state to exist, the average dissipated energy should be compensated by the averaged energy increase of driving and loading. Now, imposing in Eq.(3) that diverges, one of the following two conditions must be obeyed for Eq.(3) to hold [34]:

, i.e. there is strict conservation, or

diverges, which is a necessary condition for criticality.
In the second case, by studying the probability distribution function of and using a mapping into a branchingprocess, it has been shown that not only diverges, but also that the system is critical in the infinite size limit [34]. Moreover, as expected for a meanfield model, the sizeavalanche exponent is found to be [7, 8, 9, 25].
However, as already pointed out in [34], for any finite system size, , neither nor are infinite. In such a case, a finite value of the parameter must be finetuned to some precise value, , for Eq.(3) to hold. Such a value should diverge slower than , in order to achieve the right limit for , but there is no analytical prescription in [34] on how to fix it for each system size.
This is a wellknown problem, shared by forestfire models, where the number of trees grown between two fires () is a parameter which needs to be carefully tuned for any finite size: too small values lead to subcritical fires, while too large values generate supercritical fires spanning the whole system (and generating a bump for large values in the size distribution).
This is graphically illustrated in Fig. 1 (left), where the avalanche size distribution for the PruessnerJensen model is plotted for a system with sites, , and three different values of ; even if the values of are large, the sizedistributions are not pure powerlaws: they are either subcritical (with an exponential cutoff) or supercritical (with a bump for large avalanches). Nevertheless, partial scaling is observed in any case.
Given that the control parameter is an integer number, criticality cannot be tuned with arbitrary precision (specially for small system sizes) but, still, for each value of it is possible to find an almost critical value of , . In Fig. 1 (right) we show the avalanche size distribution at the finetuned critical point for different values of . Increasing the system size we have observed that such a value scales as suggesting
(4) 
A collapse of the critical sizedistribution curves for different values of (see the inset of Fig. 1) leads to , compatible with
(5) 
where the meanfield exponent is recovered.
In Section 5, we shall introduce a scaling Langevin theory for nonconserving SOC models, which explains in a straightforward way the two (formerly unknown) scaling laws, Eq.(4) and Eq.(5).
Let us comment on the peculiarity of the thermodynamic limit in this model (as well as in the DrosselSchwabl forestfire): the condition is automatically fulfilled by imposing the double separation of timescale discussed above, i.e. the second separation of timescales is tantamount to finetuning to its critical value . Note also that the infinite size limit is somehow pathological as the entire supercritical phase (as well as the critical point itself) collapses into a unique single point . Instead, for finite systems, a precise (not infinite) double separation of scales is required to set the system to the critical point, separating distinct subcritical and supercritical phases. This boils down to the need of finetuning for each value of to have a coherent finitesize scaling.
On the contrary, in the conserving limit, large avalanches spanning the whole system are observed for any size and criticality is reached without resorting to careful tuning.
Summing up, even if the randomneighbor model studied in [34] exhibits infinite avalanches and is critical in the infinite system size limit, it lacks of a welldefined finite size scaling and, therefore, it is not truly scaleinvariant: for any finite system, deviations from criticality are observed if the control parameter is not fine tuned to a precise dependent critical value. In conclusion, this model does not qualify as a bona fide selforganized critical system.
In this respect, the situation for the RNOFC model studied in [65, 66] is even worse: given that it lacks a parameter analogous to to be tuned, the degree of loading cannot be regulated and the model is generically subcritical for any nonvanishing dissipation rate even in the large system size limit [87].
4 Selforganized meanfield approach: SelfOrganized Branching Process
In this section, we complement the meanfield approach of the previous one by exploiting the selforganized branching process introduced by Zapperi, Lauritsen, and Stanley in [88]. First, in Subsection 4.1, we introduce this approach for a conserving sandpile model (the Manna cellular automaton). Then, following also Zapperi et al., in Subsect. 4.2 we move on to analyze dissipative models, showing that they are generically subcritical. Finally, as a last step, in Subsect. 4.3 we implement a loading mechanism in the selforganized branching process which captures the essence of earthquake and forestfire models, and explore under which circumstances the resulting model is critical.
4.1 Conserving case
In sufficiently high spatial dimensions (i.e. in the meanfield regime), avalanches in sandpiles rarely visit twice the same site; activity patterns are mostly treelike. An avalanche can be seen as a branching process [79] in which an individual (ancestor) creates a fixed number of descendants with probability . The average number of descendants per ancestor, , is called branching ratio. For avalanches propagate indefinitely (supercritical phase), for they stop after a typical number of generations (subcritical phase), while the process is critical in the marginal case, [79].
In this static branching process, fixing, without loss of generality, the number of descendants to , a given active site branches in with probability or has no offspring with probability (see Fig. 2), ensuing and a critical value .
Let us consider, as a simple example, the Manna sandpile with critical threshold (see Appendix A), and map its meanfield version into a selforganized branching process, in which itself is a dynamical variable [88]. In the Manna dynamics, each grain arriving at site can either generate activity (energy above threshold) if previously , or not if . An avalanche in high spatial dimensions can be, therefore, seen as a branching process with , i.e. the background energy density is nothing but the branching ratio [89]. A generation is defined as the set of sites probed for activation at each timestep; after generations there are involved sites (see Fig. 2).
For computer simulations, we fix a maximum number of generations and impose (to mimic boundary dissipation) that at the th generation all grains are lost. Note that, apart from such a dissipative boundary, the bulk dynamics is conserving.
Each avalanche modifies the background in which the next avalanche is to be started; i.e. it changes the value of . Thus, the branching probability becomes a fluctuating variable, as illustrated in Fig. 3. In the left part of the figure, the value of is plot as a function of the avalanche number for different system sizes, while in the right figure the statistically stationary distribution of values of is represented for various sizes; the width of the distributions decreases with increasing size and can be made as small as wanted.
To recover analytically these computational observations, let be the total number of grains into the system after avalanches; then, . If is the number of grains dissipated at the th (last) generation of the th avalanche, in order to have stationarity, the following balance equation:
(6) 
or
(7) 
must hold. The average number of grains dissipated at the boundary is ( sites at the boundary, each one occupied with probability [79]). For each avalanche, , where is a Gaussian white noise. Plugging this into Eq.(7) and taking the continuum limit for , one can formally write:
(8) 
whose deterministic part has a stable fixed point at .
Accordingly, in the thermodynamic limit (in which the effect of fluctuations can be neglected [88]), the dynamics attracts to its critical value (see Fig. 3 left) and the width of the fluctuations of around decreases with increasing system size (see Fig. 3 right). This simple (conserving) branching process selforganizes to its critical point.
4.2 Dissipative case
Borrowing still from Lauritsen, Zapperi and Stanley [90], let us introduce a nonvanishing bulkdissipation rate into the Manna model and, as a consequence, into its selforganized branching process representation. Each offspring (not necessarily in the last generation) is removed from the system with probability ; the effective branching probability becomes , and the criticality condition is , or
(9) 
and . Eq.(7) transforms into:
(10) 
where is the total amount of grains dissipated in the bulk. A simple calculation, synthesized in Appendix B, leads to the following equation for the evolution of
(11) 
where
(12) 
and, as above, the noise amplitude is Ndependent. After some simple algebra and omitting the noise term, Eq.(11) can be rewritten as
(13) 
with . It is straightforward to check that the only stable fixed point of Eq.(13) is [91]. Therefore, as , the selforganized dynamics leads to a subcritical point ; the fixedpoint branching ratio is less than unity, and the process propagates only for a finite number of generations for any nonvanishing value of [90]. It is only in the conserving limit, , that the selforganized value and the critical point coincide; otherwise there is selforganization to a subcritical point.
4.3 Dissipation and loading
In a recent series of papers [35], Juanico and collaborators introduced a background dynamics into the selforganized branching model with dissipation. These authors consider a dissipative version of the Manna sandpile rules in the following way: with probability , an active site transfers grains to different randomly chosen neighbors; with probability , only one grain is transferred to one neighbor while the other one is dissipated; finally, with probability , the two toppling grains are dissipated. For simplicity and to easy comparison with the calculations above, we fix ; the critical branching probability becomes .
A background dynamics is implemented in [35] by introducing a rate for a stable site to be turned into a critical one (), and a rate for the opposite transformation. From now on, and without loss of generality, we restrict ourselves to the “loading” process (which increases the energy) and fix . Neglecting the noise term, it is straightforward to arrive at the following evolution equation for ,
(14) 
with . It has a stable fixed point at
(15) 
Note that, at is equal to and thus, if is finetuned to
(16) 
then the fixed point of Eq.(14) becomes , i.e. the pair , with given by Eq.(16) and , fulfills Eq.(15) and the selforganized branching process becomes critical. On the other hand, fixing (resp. ) the fixed point becomes supercritical (resp. subcritical) as illustrated in Fig. 4. In conclusion, by carefully tuning the loading parameter to exactly compensate the effect of dissipation, the system selforganizes to its critical point. This process, requiring an explicit parameter tuning, cannot be called bona fide selforganization.
Finally, note that both, the critical loading parameter and the driving rate , vanish in the large system size limit (), while the ratio “loading over driving” () diverges. These conditions are analogous to those usually imposed to forestfire models (). The calculation above illustrates that such conditions are necessary but not sufficient to achieve criticality in the absence of a conservation law in any finite system: a sizedependent fine tuning is also required.
Even if Juanico and coauthors claim to have designed critical selforganized models, all branching processes studied by them [35] are similar in spirit to the example above: close inspection of their rules reveals an underlying parameter fine tuning in all the different variations they study. For example, for the full model described above (i.e. with ), Juanico et al. explicitly make the “convenient” choice of parameters
(17) 
analogous to Eq.(16). Actually, in [35], Eq.(17) is obtained by finetuning the independent terms to the critical point, while the last term in the flow equation, proportional to , is neglected. Not surprisingly, the resulting model converges to its critical point in the thermodynamical limit; but, requiring careful tuning, it cannot be properly called critical selforganization.
Summing up, in this section we have illustrated that the conserving selforganized branching process shows asymptotically critical dynamics, while its dissipative counterpart selforganizes to a subcritical point. Introducing a loading mechanism, dissipation can be compensated and criticality restored if and only if the loading rate is finetuned to a precise sizedependent value. Otherwise, the system selforganizes generically either to a subcritical point or to supercritical one.
5 A full description: Langevin theory of SOC
Having already studied two different meanfield like approaches, in this section we discuss the complete Langevin theory of selforganizing systems. This theory explains the origin of the underlying critical point beyond meanfield, its universality in any dimension, as well as the key mechanism producing SOC.
First (Subsection 5.1), we review the existing absorbingphasetransition Langevin picture of conserving SOC models. Then (Subsect. 5.2), we introduce bulkdissipation and extend the theory to nonconserving systems. Finally (Subsect. 5.2), a loading mechanism is introduced to elucidate the behavior of nonconserving selforganized systems. We emphasize the substantial differences with respect to the conserving case.
5.1 Langevin theory with conservation
The main idea to construct a stochastic theory of conserving SOC is to “regularize” sandpiles (and related systems) by switching off both boundary dissipation and slow driving [22, 25, 40, 41, 92]. In this way, the total amount of sand or “energy”, , in the pile becomes a conserved quantity, and can be retained as a control parameter. Indeed, in the sodefined “fixedenergy ensemble” and for large values of , the system is in an active phase with neverending relaxation events. Instead, for small values of it gets trapped with certainty into some absorbing state [93] where all dynamics ceases (i.e. all sites are below threshold). Separating these two regimes there is a critical energy, , at which an absorbing phase transition takes place. In this way selforganized criticality is related to a standard phase transition [22, 94, 95].
It has been shown [22, 41, 96] that such a critical value, , coincides with the stationary energy density to which the original selforganizing sandpile converges. In other words, the energy around which the standard sandpile (i.e. including slowdriving and boundary dissipation) fluctuates is the critical point of the “fixedenergy sandpile”. Furthermore, the width of fluctuations becomes smaller and smaller with increasing system size (see left part of Fig. 5), guaranteeing that in the thermodynamic limit the original sandpile selforganizes to criticality.
This connection between “driven/dissipative” systems and their “fixedenergy” counterparts permits us to relate avalanche exponents to standard critical exponents (see [97] for scaling relations) and to rationalize the critical properties of SOC systems from the broader point of view of standard nonequilibrium (absorbingstate) phase transition [22, 41, 98].
Using this approach, it has been established that stochastic sandpiles do not belong to the robust directed percolation (DP) class, prominent among absorbing phase transitions, but to the socalled “conservingDP” (CDP hereafter) or Manna class. This class is characterized by the coupling of activity to a static conserved field representing the conservation of sandgrains [22, 41, 99, 100]. The field theory or set of mesoscopic Langevin equations proposed under phenomenological grounds to describe this class is:
(18) 
where is the activity field (characterizing the density of grains above threshold), is the locallyconserved energy field, , and are parameters and is a Gaussian white noise. Some dependences on have been omitted to unburden the notation.
Note that in the CDP class, two fields are required for a Langevin representation: the activity field representing grains/energy/force above threshold and the background or energy field describing the local amount of grains/energy/force. Nevertheless, it is possible to stick to a single field description by integrating out the energy equation. This generates two extra terms for the activity equation
(19) 
The second, history dependent (nonMarkovian) term describes the tendency of sites that have been less active than their neighbors in the past to be more susceptible for activation (e.g. in a sandpile, if the neighbors of a given site have toppled, the site is very likely to overcome the threshold). The single equation for the activity reads
(20) 
nonMarkovianity is the price to pay for removing the energy field [101].
The CDP class described either by Eq.(18) or by Eq.(20) has a critical dimension and embraces not only stochastic sandpiles, but also (among other examples) some conserving reactiondiffusion systems, for which the equations above can be explicitly derived from the microscopic dynamics [99, 100].
Eq.(18) can be studied either (i) in spatially extended systems, (ii) using random neighbors, or (iii) in a globally, alltoall, coupled version in which the Laplacian is replaced by . These last two are useful to construct meanfield approximations of the full (spatially extended) theory.
5.1.1 Relation with other universality classes
First, note that the wellknown theory for directed percolation (i.e. the Reggeon field theory [93]) is recovered upon fixing in Eq.(18). The additional conserved field turns out to be a relevant perturbation altering the critical behavior of systems in the DP class [22, 41, 100].
On the other hand, for the sake of completeness, let us just briefly mention that the CDP class is fully equivalent to the pinning/depinning transition of interfaces in random media, i.e. the Quenched Edwards Wilkinson [23, 102, 103]. The absorbing (resp. active) phase maps into the pinned (resp. depinned) one. Exploiting the mapping between these two descriptions (of a unique underlying physics), critical exponents for Eq.(18) can be deduced from existing renormalization group results for interfaces [104].
It is worth stressing that, in terms of interfacial models, conservation of energy is equivalent to interface translation invariance; for instance, a dissipative term like introduced in Eq.(18) would map into a term (where is the interface height) in the Quenched Edwards Wilkinson equation, which breaks such an invariance.
In this respect, recent experimental evidence of selforganized critical behavior (including finite size scaling), obtained for avalanches in type II superconductors [20], give critical exponents compatible with those of the CDP class. Barkhausen noise [105] and acoustic emission in fracture [106] are other related examples.
5.1.2 Conserved SOC as an absorbing state transition.
Within this framework, the way selforganized criticality works is as follows (see left part of Fig. 5) [22, 41]: if the sandpile is in its absorbing phase (, where stands for the spatially averaged value of in the steady state) then, owing to the driving mechanism, the energy is slowly increased until, eventually, the active phase () is reached. At this point avalanches are triggered and they restructure the sandpile energy configuration. Avalanches may dissipate energy at the open boundaries, until eventually the system falls back into an absorbing state, the avalanche stops, and slow driving acts again restarting the cycle. In this way, the sandpile is expected to fluctuate around its critical point, , with excursions to either the active or the absorbing phase as sketched in the upperleft part of Fig. 5.
To have a numerical confirmation of this, Eq.(18) can be interpreted as in SOC, i.e. one can implement slow driving and dissipation at infinitely separated timescales, and integrate the equation using the efficient algorithm introduced in [107]. In particular, one considers an absorbing configuration () and open boundaries, then add a small amount of activity/energy, , to a given site, :
(21) 
this generates an avalanche, which evolves according to Eq.(18). Iterating this process, one obtains the distribution of values sampled during avalanches, shown in the lowerleft panel of Fig. 5. Observe that, as dissipation and driving become arbitrarily small by increasing system size (actually, they are infinitesimally small in the thermodynamic limit), the degree of penetration into the active and absorbing phases is arbitrarily small, the distribution of becomes more and more peaked, and the system is arbitrarily close to its critical point. Moreover, the avalanche exponents measured by means of numerical integration of Eq.(18) at such a steady state coincide with (or can be related to) those obtained by performing standard fixedenergy simulations of Eq.(18) at its critical point [107, 103].
Note the obvious analogies between this picture and the selforganized branching process described above: is the equivalent of , i.e. the selforganized control parameter; the critical point corresponds to the critical branching probability .
The advantage of Eq.(18) as a theory for SOC with respect to the selforganized branching process is that, while this last is a meanfield theory explaining qualitatively selforganization but failing to justify critical exponents in spatially extended systems, Eq.(18) is a full theory including fluctuations and spatialdimensionality. It provides accurate estimates for avalanche exponents in any dimension and opens the door to field theoretical analyses. Furthermore, Eq.(18), considered on a randomneighbor or an alltoall coupling, constitutes a sound meanfield description of conserving SOC, equivalent to those in the preceeding sections.
To end up, note, once again, the essential role played by conservation in this theory. The underlying phase diagram sketched in the left part of Fig. 5 relies on the averaged energy being a control parameter. If there was a nonvanishing bulk dissipation, the energy would change continuously during avalanche evolution. In the next subsection, we shall explore how this affects the absorbingphasetransition picture of SOC.
5.2 Langevin theory with bulkdissipation
To tackle the problem of nonconservation within the absorbingstate Langevin framework, we need to modify Eq.(18) to allow for bulk dissipation. Introducing in Eq.(18) the leading dissipative term, , and neglecting higher order corrections, the resulting set of equations becomes
(22) 
which is, obviously, nonconserving owing to the activitydependent energy leakage.
Integrating in time the equation for the energy field, the following extra terms for the activity equation are generated:
(23) 
The second term, dominant in Eq.(20), becomes a higher order correction here, i.e. it is irrelevant in the renormalization group sense as compared with the third, nonMarkovian, term. The last one is well known to be the leading nonlinearity in the dynamical percolation universality class [108]. From this perspective, it is no wonder that the critical behavior of some nonconserving SOC models (e.g. forestfires) has been related to (dynamical) percolation in the literature [109]. Such a class, whose full (onefield) Langevin equation is:
(24) 
describes the spreading properties of epidemics with immunization, etching of disordered solids [110], and some aspects of spreading in systems with many absorbing states without a conservation law (see [111] for more details). The term ensues that regions already visited by activity become less prompt to be active in the future. Owing to this term, Eq.(22) cannot sustain an active phase. However, even if it lacks a stable active phase, it exhibits a spreading phase transition separating a phase in which seeds of activity propagate indefinitely (in the form of rings of expanding activity; i.e. defining an “annular growth” phase) from an absorbing phase in which they do not [108] (see the diagram at the right part of Fig. 5).
This spreading transition, whose critical dimension is , is controlled by the initial state in which the seed of activity is placed; indeed, the initial energy, , is a mass term in Eq.(23). If the initial value of is large enough (i.e. favorable environment), then avalanches tend to propagate, while if it is small, they do not. Separating these two regimes there is a critical point for spreading propagation at some value .
In summary, the introduction of a nonvanishing dissipation rate affects in a relevant way the critical behavior of selforganizing systems; depending on the initial condition, the dissipative Eq.(22) can be in the propagating/supercritical or in the nonpropagating/subcritical phase of a dynamical percolation phase transition.
5.3 Full theory: dissipation and loading
Now, we are in a good position to understand in depth the role of the loading mechanism in dissipative models of SOC, within the Langevin framework. To do so, let us complement Eq.(22) with a specific prescription on how to change the background energy field between avalanches, i.e. let us consider a loading rule, as for instance:
(25) 
(where is a parameter and the average energy in the system) and a driving rule:
(26) 
Eq.(25) and Eq.(26) define one possible loading mechanism for Eq.(22); other choices are, of course, possible (for instance, the loading mechanism could also act “during” avalanches). The results presented in what follows are generic, essentially independent of such a choice.
As shown above, Eq.(22) can sustain avalanches propagating indefinitely (up to system size), provided that the initial energy is large enough. Therefore, considering large values of or of in Eq.(25)), the system becomes supercritical for avalanche propagation. Instead, small initial densities lead to subcritical propagation.
To illustrate this and the forthcoming discussion, we have performed computer simulations of Eq.(22) using the parameter values specified in the caption of Fig. 5, and complemented with the loading and driving rules Eq.(25) and Eq.(26). For the sake of simplicity, we have considered sites with an alltoall (meanfield) coupling. To check the robustness of our conclusions we have also studied a randomneighbor version, obtaining very similar results (not shown). As before, the equation has been integrated using the algorithm introduced in [107].
For a dissipation parameter and fixing , we find a critical point for spreading at some value of , , which generates, on average, an initial energy . At such a critical value, powerlaws for avalanche and spreading exponents are obtained. In complete analogy with Fig. 1, for smaller values of (and hence, smaller values of the initial average energy density) the avalanche size distribution has an exponential cutoff (subcritical), while for larger values the distribution develops a bump for large sizes (supercritical) (results not shown).
We have constructed histograms of the average energy by sampling during avalanches in computer simulations (see Fig. 6). Typically, for short times the system is in the right side of the distribution and, as the avalanche proceeds and dissipation acts, moves progressively leftward. This shifting generates a broad distribution of energy values for any system size. After the avalanche stops, the system is “loaded” again (Eq.(25)), a new avalanche is triggered (Eq.(26)), and so on. In this way, the system is kept hovering around the critical point, with a broad distribution of values, as illustrated in Fig. 5 and Fig. 6.
In Fig. 6 (left panel), histograms for and various values of are plot. As long as the loading is sufficiently strong () the distribution develops a doublepeak structure overlapping with both the propagating and the subcritical phases. Instead, for smaller values of (, in the figure), the loading is too small, and the histogram overlaps only with the subcritical phase. In Fig. 6, (right panel) histograms for and various values of are plot; in all cases, there is a doublepeak structure. Observe that, for large dissipation rates, the system gets deeper into the absorbing phase.
It is important to remark, that (as illustrated in Fig. 5) the dynamics is rather different from its conserving counterpart: while, in the conserving case, fluctuations around decrease in amplitude with systemsize (Fig. 5, leftpanels), in the nonconserving case the histograms remain broad, even in the thermodynamical limit (see Fig. 5, right panels); i.e. large variations around the critical spreading point persist for any system size.
We caution the reader that, within an avalanche, the process is not stationary (energy decreases) and, therefore, the histograms shown in Fig. 5 and Fig. 6 cannot be properly interpreted as probability distribution functions.
For this reason, we have also constructed stationary (steady state) histograms for (i) the distribution of the average initial energy for avalanches and (ii) the distribution of values after avalanches. Any of these can be used, as well, to illustrate the differences with the conserving case. For example, Fig.7 shows that the background in which avalanches are started is generically noncritical, but is broadly distributed around the critical point. Using this information, one can make the educated guess that the associated avalanche size distribution (or any other quantity measured for avalanches/spreading), with such a distribution of initial conditions, will be a convolution of different subcritical and supercritical curves weighted with the above distribution of initial energies.
Let us emphasize the lack of any mechanism tuning the system to criticality: the energy is initially set to some arbitrary value (controlled by the parameter ). If and only if the initial density is fine tuned to the critical value for spreading of activity, , the system is at the critical point for avalanche spreading. Otherwise, for larger values of the system is initially supercritical, while for smaller values it is subcritical.
In conclusion, finetuning of the loading mechanism is required to have critical spreading in nonconserving systems, and it is controlled by a dynamical percolation critical point. The important point to stress is that, even if the initial condition is not critical for generic values of and , the “hoveringaroundthecriticalpoint” mechanism, illustrated in Fig. 5 (right panels), keeps the dynamics effectively not far from criticality, but not at criticality, for a broad range of parameter values. Furthermore, contrarily to the conserving case (Fig. 5, left panels), the energy histograms do not tend to a deltapeak function for large system sizes. Large excursions into both the supercritical and the subcritical phases persist in the thermodynamic limit.
5.3.1 Revisiting the meanfield PruessnerJensen model.
Using this insight, we can now tackle the open question: how does scales at criticality in the model discussed in Section 3?
As is a background energy, it corresponds to a mass term in the Langevin equation for the activity, Eq.(23). Therefore, to preserve scaleinvariance, it needs to be scaled with as a distance to the critical point and, therefore:
(27) 
where is the correlation length and the correlation length exponent, whose meanfield value in the dynamical percolation class is [97].
At the upper critical dimension, (where hyperscaling meanfield relations are expected to hold), is limited by the system linear size, , where is the volume (i.e. total number of sites):
(28) 
Putting together these two last equations, we obtain the meanfield scaling result,
(29) 
This is in excellent agreement with the empirical result reported in Eq.(4). Analogously, the cutoff of the avalanche size distribution, , needs to scale as
(30) 
where we have used the fractal dimension, , for meanfield dynamical percolation [97]. Again, this prediction is in perfect agreement with the numerical results Eq.(5).
In conclusion, the loading parameter needs to be finetuned for each finite size to have true scaleinvariance; its scaling is inherited from a dynamical percolation critical point.
5.3.2 Spatially extended systems.
The numerical results reported in this section correspond, as already stressed, to an alltoall (as well as randomneighbors) coupling in Eq.(22). On the other hand, qualitatively similar results can be obtained for spatially extended systems: i.e. a broad distribution of the spatiallyaveraged energy, hovering around the critical point. The main difference is that, obviously, the background energy becomes heterogeneously distributed in space and, therefore, the situation becomes much more involved.
Eq.(22) generates spontaneously regions with higher and with lower values of , which have different propensities to activity propagation: there are locally supercritical and subcritical regions. Patches where avalanches have passed are typically less likely to propagate new activity owing to the nonMarkovian term in Eq.(22). The size distribution of such patches is, accordingly, inherited from the avalanche size distribution, creating a complex landscape for further avalanche propagation. This scenario is, of course, more complex than in the meanfield one discussed above, but the essence of the described phenomenology remains unaltered: the (local) control parameter hoversaround a dynamical percolation critical point, with fluctuations that do not vanish in the large systemsize limit.
Observe that, in order to tune the system to criticality (as done in the meanfield case) one would need in this case to define a more complicated loading mechanism which should get rid of the dynamically generated heterogeneities, leading to a homogeneous initialenergy state, tuned exactly to its critical value. Using the language of [22], one needs to “hire a babysitter” (or a “gardener” using the forestfire terminology [95]) to keep the spatially extended system sitting (everywhere) at criticality. Once such an efficient babysitter is at work, the initial condition is always at the (dynamical percolation) critical point in any dimension.
From this perspective, our theory provides additional support for the claim in [86] that the critical density of trees in a forest fire model should coincide with the percolation critical density, i.e. in order to observe critical propagation in the forestfire model one should tune the initial background (number of trees) to the corresponding dynamical percolation critical density.
As already pointed out by Grassberger some time ago [86], in the absence of an efficient gardener taking care of local finetuning, partial powerlaws are still observed in the forestfire model. This is due to the existence of patches with different densities of trees, which appear with a broad spectrum of sizes. Each of such patches lies at a different distance of the critical point. The convolution of avalanches propagating in such a variety of initial conditions originates a complex pseudoscaling picture which, obviously, does not correspond to strict criticality. A similar picture applies also to the (openboundaries) OFC earthquake model [72], and to selforganized models of neural activity (as will be illustrated in a forthcoming paper).
Summing up: In this section, we have reviewed the standard absorbing state phase transition picture of SOC, underlining the special role played by conservation. Then, we have introduced a bulk dissipation term and illustrated that the active phase disappears and the universality class of the involved spreading phase transition is changed from CDP to dynamical percolation. Dissipation needs to be compensated by a loading mechanism (which controls the initial conditions in which avalanches are started) to keep the energy balance. If and only if the loading mechanism is perfectly finetuned to generate a precise initial energy density, true criticality is observed. Otherwise, the system just hovers around a dynamical percolation critical point, with large excursions into the propagating and the absorbing phases. Contrarily to the conserving case, such fluctuations do not disappear in the thermodynamic limit. Strictly speaking, this mechanism of selforganization cannot be called critical, we propose to refer to it as selforganized quasicriticality (SOqC). Last but not least, our approach provides a way to rationalize the finite size scaling properties of nonconserving selforganized systems, as earthquake or forestfire models, and has allowed us to derive a number of previously unknown scaling relations.
6 Concluding Remarks
We have shown, by using different levels of description, that nonconserving models of selforganized criticality, as earthquake and fireforest models, are not truly critical.
First, we have studied a simple meanfield theory, based on an energy balance equation, for a nonconserving model introduced by Pruessner and Jensen. It permits us to illustrate that, even if one can construct nonconserving models that seem critical in the thermodynamic limit, there is no systematic way to have a coherent finitesize scaling description of them: a precise fine tuning is required for any finite size to observe criticality and to approach the thermodynamic limit in a scaleinvariant way.
Second, we have revisited the mapping of highdimensional (meanfield) avalanching systems into a selforganized branching process, introduced by Zapperi et al. some years ago. The underlying idea is that in high dimensions avalanches do not visit twice a given site and they can be described as a branching process, whose branching probability depends on the energy background. This allows us to write an evolution equation for the branching probability for models with slow driving and dissipation. While in the conserving case the branching probability converges to its critical value, in the presence of bulkdissipation the convergence is towards a subcritical point. Introducing a loading mechanism (which mimics the growth of new trees in forest fire automata or the continuous buildingup of stress in earthquake models), we have shown that the fixedpoint towards which the system selforganizes can be either critical, subcritical, or supercritical. Contrarily to previous claims, a finetuning of the loading mechanism is required to reach criticality within this approach.
Third, we have introduced a full stochastic description of SOC systems in terms of Langevin equations. We have reviewed how conserving systems selforganize to a critical point with well known critical exponents (in the conserveddirectedpercolation universality class). Instead, for dissipative systems with a loading term, the dynamics is found to hover around a critical point, with large excursions into the absorbing and the propagating phases, which do not disappear in the large system size limit. Therefore, such systems are not generically critical. Still, some traits of the underlying dynamical percolation critical point can be observed, depending on the loading parameter and systemsize, i.e depending on how large the excursions into the absorbing/propagating phase are.
All these three approaches provide overwhelming evidence that conservingdynamics is a necessary condition to observe selforganization to criticality. Instead, in nonconserving (dissipative) systems equipped with a loading mechanism, a fine tuning of a loading parameter is required to have the system sitting at a critical point. Otherwise, the system just hovers around a critical point, with broadly distributed fluctuations which do not dissappear in the thermodynamic limit: for a broad range of parameters, nonconserving systems can be fluctuating in the vicinity of a critical point, but not at the critical point. We propose to call this: selforganized quasicriticality..
This conclusion extends to some other nonconserving models of SOC as those
for synchronization of integrateandfire oscillators described in
[64, 112, 113], the model in [114], or the model of
neural avalanches in [36] (as will be explained in a separate
publication).
Is it sensible to refer to selforganized dissipative systems as “critical”?
The answer to this question is mostly a matter of taste, and depends on what one wants to define as criticality. Being strict and calling critical only to systems sitting at a critical point (allowing at most for fluctuations that vanish in the thermodynamic limit), then nonconserving systems are not truly critical models.
Being more permissive, one could accept, in principle, the term “critical” to refer to systems hovering around a critical point (with persistent excursions into the subcritical and supercritical regimes), which exhibit “dirty scaling”.
In order to avoid missunderstandings and missconceptions, we strongly favor the use of an alternative terminology like “almost criticality” [67], “pseudocriticality”, or, as we said, selforganized quasicriticality (SOqC) to refer to nonconserving selforganized systems, and suggest to restrict the term “critical” for truly scaleinvariant systems.
Actually, in many cases, strict criticality might not be required to explain empirical (truncated) powerlaws distributions observed in the real world and, therefore, “selforganized quasicriticality” remains a useful concept, despite of the somehow inappropriate and certainly confusing use (and abuse) of the word “critical” in the literature. Under this light, one could reconsider the empirical observations discussed in the introduction, as well as similar ones for which power law distributions have been reported. A critical inspection of them reveals in many cases that empirical data are better described by truncated powerlaws rather than by pure powerlaws [4].
Appendix A: Basic Models in a nutshell
For the sake of completeness, in this Appendix we present some of the basic toy models of selforganized criticality. All of them are defined in a dimensional lattice (generalizations to dimensions, to randomneighbor or alltoall couplings are straightforward).
Sandpilelike Models
Consider a (height or “energy”) variable , which takes integer (nonnegative) values at each site of a twodimensional lattice. The ingredients of sandpiles, ricepiles, and related models can be sketched as:

Slow Driving: A small input of energy is externally introduced into the system (grains are dropped), usually at a single site: .

Activation: A site receiving energy, stores it until a given threshold, , is exceeded and the site is declared active; otherwise nothing happens and the system is driven again.

Relaxation (or “toppling”): Each active site redistributes (all or a fraction of) its accumulated energy among its neighbors. A relaxation event can trigger a chain reaction or avalanche by activating its neighbors and so forth.

Boundary Dissipation: When redistribution events reach the (open) boundaries of the system, energy is dropped off.

Iteration: When activity has ceased, the avalanche stops, and a new external input is added. The driving/dissipation cycle is iterated until a statistically stationary state is reached.
For each specific model, the relaxation rules and some other details can change, giving rise to a zoo of models. We enumerate some of the more commonly studied ones:
The Deterministic BakTangWiesenfeld Sandpile Model: The threshold is fixed to ( in dimensions). Active sites relax according to:
(31) 
that is, an active site is emptied and its grains are deterministically redistributed amongst its nearest neighbors. The energy at sites out of the system is fixed to , enforcing boundary dissipation when boundary sites topple. Avalanches measured in such a steady state were originally claimed to be critical [6]. Later work showed that actually, owing to the deterministic nature of the model and, as a consequence, to the existence of many toppling invariants and breakdown of ergodicity [115] the system is not truly critical but exhibits anomalous multiscaling [116]. Other authors (see, for instance, [117]) suggested that avalanches do not obey any type of scaling whatsoever. To avoid the pathologies associated with deterministic rules, we focus all along this paper on stochastic models.
The Stochastic Manna Model: The dynamics of the Manna model is similar to the deterministic BTW, but the threshold is fixed to in any dimension and stochasticity is introduced in the redistribution rule [14]:
(32) 
This is the “inclusive” version of the model. In its “exclusive” version, the two grains are forced to go to different neighbors. Both of these versions selforganize the model to a critical point in the CDP (or “Manna”) class in any dimension [14, 99, 100].
The Stochastic Oslo Model: This (ricepile) model has annealed random thresholds at each site: every time a site topples, a new threshold is randomly chosen with equal probabilities [15]. Redistribution of energy units (grains) is done as in the Manna model, in a stochastic fashion. These rules lead to selforganization to a critical point in the CDP universality class with rather clean scaling [15, 99, 100].
The OlamiFederChristensen Earthquake Model
A continuous “force” or “energy” , randomly distributed between and a threshold value, , is initially assigned to each site of a twodimensional lattice. During the driving step, the site with maximum force, , is identified, and the force of all sites is increased by , creating (at least) one seed of activity (this is equivalent to advancing all sites at a fixed constant velocity until one of them, the maximum, reaches the threshold). Contrarily to sandpile models, this driving affects all sites. An active site relaxes according to:
(33) 
where is a bulkdissipation parameter. The model is conserving only for . The force at sites out of the system is fixed to , entailing boundary dissipation. The stationary state of such a system was claimed to be critical for a broad range of values of [44], but recent analyses disprove such a claim (see for instance [55, 72] and Section 1).
The DrosselSchwabl Forest Fire Model
Sites of a twodimensional lattice can be either empty (), occupied by a tree (), or by a tree on fire () [50]. The dynamics proceeds as follows:

At every empty site, a tree grows with probability , and the site becomes “occupied”:

Initial spark: A tree not surrounded by any fire becomes a “burning” tree (e.g. lightening) with probability :

A burning tree sets on fire all its nearest neighbors, and it becomes empty:

The “fire avalanche” proceeds burning all trees in contact with fires.

When fire ceases the dynamical processes is restarted.
The relevant parameter is which sometimes is called [83]. In the implementation of the model that we use, is the number of trees grown between two consecutive ignitions. Despite of the initial claims and various forms of reported “anomalous scaling”, the most recent studies revealed absence of generic scaleinvariance [85, 86].
Appendix B
To compute the mean number of grains lost in the bulk in the dissipative selforganized branching process [88], let us first consider the number of offsprings, , notoccupied owing either to absence of branching or to bulkdissipation:
(34) 
then, the average fraction corresponding to bulkdissipation is
(35) 
Noting that the avalanche size is and:
(36) 
then . Using the definition of :
(37) 
Plugging Eq.(37) into the equation for ,
(38) 
and, from this and Eq.(35):
(39) 
leading, once the continuum limit for has been taken and fluctuations included, to Eq.(11).
Acknowledgments: We acknowledge financial support from the Spanish Ministerio de Educación y Ciencia (FIS200500791) and Junta de Andalucía (FQM165). We also thank our friends and colleagues I. Dornic, F. de los Santos, P. Hurtado, P. Garrido, G. Pruessner, P. Grassberger, R. PastorSatorras, H. Chaté, M. Alava, S. Zapperi, A. Vespignani, R. Dickman for critical reading of the manuscript, for useful comments, and/or enjoyable collaboration in the past. This paper is dedicated, with admiration, to Geoff Grinstein, who introduced us to the art of Langevin equations, and suspected all this to be true long ago.
References
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