Search

When at equilibrium, large-scale systems obey conventional thermodynamics because they belong to microscopic configurations (or states) that are typical. Crucially, the typical states usually represent only a small fraction of the total number of possible states, and yet the characterization of the set of typical states -- the typical set -- alone is sufficient to describe the macroscopic behavior of a given system. Consequently, the concept of typicality, and the associated Asymptotic Equipartition Property allow for a drastic reduction of the degrees of freedom needed for system's statistical description. The mathematical rationale for such a simplification in the description is due to the phenomenon of concentration of measure. The later emerges for equilibrium configurations thanks to very strict constraints on the underlying dynamics, such as weekly interacting and (almost) independent system constituents. The question naturally arises as to whether the concentration of measure and related typicality considerations can be extended and applied to more general complex systems, and if so, what mathematical structure can be expected in the ensuing generalized thermodynamics. In this paper we illustrate the relevance of the concept of typicality in the toy model context of the "thermalized" coin and show how this leads naturally to Shannon entropy. We also show an intriguing connection: The characterization of typical sets in terms of Renyi and Tsallis entropies naturally leads to the free energy and partition function, respectively, and makes their relationship explicit. Finally, we propose potential ways to generalize the concept of typicality to systems where the standard microscopic assumptions do not hold.

The Ultimatum Game is conventionally formulated in the context of two players. Nonetheless, real-life scenarios often entail community interactions among numerous individuals. To address this, we introduce an extended version of the Ultimatum Game, called the Multi-Proposer-Multi-Responder Ultimatum Game. In this model, multiple responders and proposers simultaneously interact in a one-shot game, introducing competition both within proposers and within responders. We derive subgame-perfect Nash equilibria for all scenarios and explore how these non-trivial values might provide insight into proposal and rejection behavior experimentally observed in the context of one vs. one Ultimatum Game scenarios. Additionally, by considering the asymptotic numbers of players, we propose two potential estimates for a "fair" threshold: either 31.8% or 36.8% of the pie (share) for the responder., Comment: 11 pages main text, 10 pages supplementary information, 3 + 1 figures

Homophily, the tendency of humans to attract each other when sharing similar features, traits, or opinions has been identified as one of the main driving forces behind the formation of structured societies. Here we ask to what extent homophily can explain the formation of social groups, particularly their size distribution. We propose a spin-glass-inspired framework of self-assembly, where opinions are represented as multidimensional spins that dynamically self-assemble into groups; individuals within a group tend to share similar opinions (intra-group homophily), and opinions between individuals belonging to different groups tend to be different (inter-group heterophily). We compute the associated non-trivial phase diagram by solving a self-consistency equation for 'magnetization' (combined average opinion). Below a critical temperature, there exist two stable phases: one ordered with non-zero magnetization and large clusters, the other disordered with zero magnetization and no clusters. The system exhibits a first-order transition to the disordered phase. We analytically derive the group-size distribution that successfully matches empirical group-size distributions from online communities., Comment: 6 pages, 5 pages of SI, to appear in Phys. Rev. Lett

Social and biological collectives need to exchange information to persist and to function. This happens across internal networks, whose structure represents static channels through which information flows. Less studied is the quantity and variety of information transmitted. We characterize a part of the information flow, the information going into organizations, primarily business firms. We measure what firms read using a data set of hundreds of millions of records of news articles accessed by employees across millions of firms. We measure and relate quantitatively three essential aspects: reading volume, reading variety, and firm size. First we compare volume with firm size, showing that firms grow sublinearly with the volume of their reading. The scaling means that inequality in information volume exaggerates the classic Zipf's law inequality in firm size, pointing to an economy of scale in information consumption. Then, by connecting variety and volume, we show that the firms vary in their reading habits to a limited degree. Firms above a certain size become repetitive readers, consistent with the sudden onset of a coordination cost between teams, not individual employees. Finally, we relate information variety to size to show that large firms tend to increase investments in existing areas of interest instead of divesting from them to move to new areas. We argue that this reflects structural constraints in growth. The results indicate how information consumption reflects the role of internal structure, beyond individual employees, analogous to information processing in other social and biological systems.

Complex systems that are characterized by strong correlations and fat-tailed distribution functions have been argued to be incompatible within the framework of Boltzmann-Gibbs entropy. As an alternative, so-called generalized entropies were proposed and intensively studied. Here we show that this incompatibility is a misconception. For a broad class of processes, Boltzmann entropy the log multiplicity remains the valid entropy concept, however, for non-i.i.d., non-multinomial, and non-ergodic processes, Boltzmann entropy is not of Shannon form. The correct form of Boltzmann entropy can be shown to be identical with generalized entropies. We derive this result for all processes that can be mapped reversibly to adjoint representations where processes are i.i.d.. In these representations the information production is given by the Shannon entropy. We proof that over the original sampling space this yields functionals that are identical to generalized entropies. The problem of constructing adequate context-sensitive entropy functionals therefore can be translated into the much simpler problem of finding adjoint representations. The method provides a comprehensive framework for a statistical physics of strongly correlated systems and complex processes., Comment: 14 pages paper + SI, 1 figure

The existence of the {\em typical set} is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given the enormous consequences for the understanding of the system's dynamics, and its role underlying the presence of stable, almost deterministic statistical patterns, a question arises whether typical sets exist in much more general scenarios. We demonstrate here that the typical set can be defined and characterized from general forms of entropy for a much wider class of stochastic processes than it was previously thought. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces; suggesting that typicality is a generic property of stochastic processes, regardless of their complexity. Our results impact directly in the understanding of the stability of complex systems, open the door to new data compression strategies and points to the existence of statistical mechanics-like approaches to systems arbitrarily away from equilibrium with dynamic phase spaces. We argue that the potential emergence of robust properties in complex stochastic systems provided by the existence of typical sets has special relevance to biological systems., Comment: 15 pages, 4 figures

Recent attempts to understand the origin of social fragmentation are based on spin models which include terms accounting for two social phenomena: homophily -- the tendency for people with similar opinions to establish positive relations -- and social balance -- the tendency for people to establish balanced triadic relations. Spins represent attribute vectors that encode multiple (binary) opinions of individuals and social interactions between individuals can be positive or negative. Recent work suggests that large systems of N >> 1 individuals never reach a balanced state (where unbalanced triads with one or three hostile links remain), provided the number of attributes for each agent is less than O(N^2) [Phys. Rev. Lett. 125, 078302]. Here we show that this statement is overly restrictive. Within a Hamiltonian framework that minimizes individuals' social stress, we demonstrate that stationary, balanced, but fragmented states can be reached for any number of attributes, if, in addition to homophily, individuals take into account a significant fraction, q, of their triadic relations. Above a critical value q_c, balanced states result. This result also holds for sparse realistic social networks. Finally, in the limit of small q, our result agrees with that of [Phys. Rev. Lett. 125, 078302]., Comment: 8 pages, 7 figures

Many countries have passed their first COVID-19 epidemic peak. Traditional epidemiological models describe this as a result of non-pharmaceutical interventions that pushed the growth rate below the recovery rate. In this new phase of the pandemic many countries show an almost linear growth of confirmed cases for extended time-periods. This new containment regime is hard to explain by traditional models where infection numbers either grow explosively until herd immunity is reached, or the epidemic is completely suppressed (zero new cases). Here we offer an explanation of this puzzling observation based on the structure of contact networks. We show that for any given transmission rate there exists a critical number of social contacts, $D_c$, below which linear growth and low infection prevalence must occur. Above $D_c$ traditional epidemiological dynamics takes place, as e.g. in SIR-type models. When calibrating our corresponding model to empirical estimates of the transmission rate and the number of days being contagious, we find $D_c\sim 7.2$. Assuming realistic contact networks with a degree of about 5, and assuming that lockdown measures would reduce that to household-size (about 2.5), we reproduce actual infection curves with a remarkable precision, without fitting or fine-tuning of parameters. In particular we compare the US and Austria, as examples for one country that initially did not impose measures and one that responded with a severe lockdown early on. Our findings question the applicability of standard compartmental models to describe the COVID-19 containment phase. The probability to observe linear growth in these is practically zero., Comment: 10 pages, 7 figures

With the availability of cell phones, internet, social media etc. the interconnectedness of people within most societies has increased drastically over the past three decades. Across the same timespan, we are observing the phenomenon of increasing levels of fragmentation in society into relatively small and isolated groups that have been termed filter bubbles, or echo chambers. These pose a number of threats to open societies, in particular, a radicalisation in political, social or cultural issues, and a limited access to facts. In this paper we show that these two phenomena might be tightly related. We study a simple stochastic co-evolutionary model of a society of interacting people. People are not only able to update their opinions within their social context, but can also update their social links from collaborative to hostile, and vice versa. The latter is implemented such that social balance is realised. We find that there exists a critical level of interconnectedness, above which society fragments into small sub-communities that are positively linked within and hostile towards other groups. We argue that the existence of a critical communication density is a universal phenomenon in all societies that exhibit social balance. The necessity arises from the underlying mathematical structure of a phase transition phenomenon that is known from the theory of a kind of disordered magnets called spin glasses. We discuss the consequences of this phase transition for social fragmentation in society., Comment: 15 pages, 12 figures

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations, the approach is equivalent to the grand-canonical ensemble; for small systems, we find significant deviations. We derive the detailed fluctuation theorem and Crooks' work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition., Comment: 9 pages, 3 figures, 10 pages of supplementary material, accepted to Nature Communications

In the current COVID19 crisis many national healthcare systems are confronted with an acute shortage of tests for confirming SARS-CoV-2 infections. For low overall infection levels in the population, pooling of samples can drastically amplify the testing efficiency. Here we present a formula to estimate the optimal pooling size, the efficiency gain (tested persons per test), and the expected upper bound of missed infections in the pooled testing, all as a function of the populationwide infection levels and the false negative/positive rates of the currently used PCR tests. Assuming an infection level of 0.1 % and a false negative rate of 2 %, the optimal pool size is about 32, the efficiency gain is about 15 tested persons per test. For an infection level of 1 % the optimal pool size is 11, the efficiency gain is 5.1 tested persons per test. For an infection level of 10 % the optimal pool size reduces to about 4, the efficiency gain is about 1.7 tested persons per test. For infection levels of 30 % and higher there is no more benefit from pooling. To see to what extent replicates of the pooled tests improve the estimate of the maximal number of missed infections, we present all results for 1, 3, and 5 replicates., Comment: 2 figures; figure one with 2, figure 2 with 3 panes

Even though irreversibility is one of the major hallmarks of any real life process, an actual understanding of irreversible processes remains still mostly semiempirical. In this paper we formulate a thermodynamic uncertainty principle for irreversible heat engines operating with an ideal gas as a working medium. In particular, we show that the time needed to run through such an irreversible cycle multiplied by the irreversible work lost in the cycle, is bounded from below by an irreducible and process-dependent constant that has the dimension of an action. The constant in question depends on a typical scale of the process and becomes comparable to Planck's constant at the length scale of the order Bohr-radius, i.e., the scale that corresponds to the smallest distance on which the ideal gas paradigm realistically applies., Comment: 15 pages, 4 figures, supplemental material 4 pages, accepted to Philos. Trans. R. Soc. A

Many stochastic complex systems are characterized by the fact that their configuration space doesn't grow exponentially as a function of the degrees of freedom. The use of scaling expansions is a natural way to measure the asymptotic growth of the configuration space volume in terms of the scaling exponents of the system. These scaling exponents can, in turn, be used to define universality classes that uniquely determine the statistics of a system. Every system belongs to one of these classes. Here we derive the information geometry of scaling expansions of sample spaces. In particular, we present the deformed logarithms and the metric in a systematic and coherent way. We observe a phase transition for the curvature. The phase transition can be well measured by the characteristic length r, corresponding to a ball with radius 2r having the same curvature as the statistical manifold. Increasing characteristic length with respect to the size of the system is associated with sub-exponential sample space growth is associated with strongly constrained and correlated complex systems. Decreasing of the characteristic length corresponds to super-exponential sample space growth that occurs for example in systems that develop structure as they evolve. Constant curvature means exponential sample space growth that is associated with multinomial statistics, and traditional Boltzmann-Gibbs, or Shannon statistics applies. This allows us to characterize transitions between statistical manifolds corresponding to different families of probability distributions., Comment: 20 pages, 3 figures, 3 appendices

Sentence formation is a highly structured, history-dependent, and sample-space reducing (SSR) process. While the first word in a sentence can be chosen from the entire vocabulary, typically, the freedom of choosing subsequent words gets more and more constrained by grammar and context, as the sentence progresses. This sample-space reducing property offers a natural explanation of Zipf's law in word frequencies, however, it fails to capture the structure of the word-to-word transition probability matrices of English text. Here we adopt the view that grammatical constraints (such as subject--predicate--object) locally re-order the word order in sentences that are sampled with a SSR word generation process. We demonstrate that superimposing grammatical structure -- as a local word re-ordering (permutation) process -- on a sample-space reducing process is sufficient to explain both, word frequencies and word-to-word transition probabilities. We compare the quality of the grammatically ordered SSR model in reproducing several test statistics of real texts with other text generation models, such as the Bernoulli model, the Simon model, and the Monkey typewriting model., Comment: 8 pages, 3 figures

In the world of generalized entropies---which, for example, play a role in physical systems with sub- and super-exponential phasespace growth per degree of freedom---there are two ways for implementing constraints in the maximum entropy principle: linear- and escort constraints. Both appear naturally in different contexts. Linear constraints appear e.g. in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of $\phi$-deformed exponential families that correspond to linear constraints (as studied by J. Naudts), with those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of $(c,d)$-entropy that covers all situations that are compatible with the first three Shannon-Khinchin axioms, and that include Shannon, Tsallis, Anteneodo-Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality, and mention that the escort distributions arising in the two dualities are generally different and only coincide for the case of the Tsallis deformation., Comment: 9 pages, 2 figures

Depending on context, the term entropy is used for a thermodynamic quantity, a~measure of available choice, a quantity to measure information, or, in the context of statistical inference, a maximum configuration predictor. For systems in equilibrium or processes without memory, the mathematical expression for these different concepts of entropy appears to be the so-called Boltzmann--Gibbs--Shannon entropy, H.For processes with memory, such as driven- or self-reinforcing-processes, this is no longer true: the different entropy concepts lead to distinct functionals that generally differ from H. Here we focus on the maximum configuration entropy (that predicts empirical distribution functions) in the context of driven dissipative systems. We develop the corresponding framework and derive the entropy functional that describes the distribution of observable states as a function of the details of the driving process. We do this for sample space reducing (SSR) processes, which provide an analytically tractable model for driven dissipative systems with controllable driving. The fact that a consistent framework for a maximum configuration entropy exists for arbitrarily driven non-equilibrium systems opens the possibility of deriving a full statistical theory of driven dissipative systems of this kind. This provides us with the technical means needed~to derive a thermodynamic theory of driven processes based on a statistical theory. We discuss the Legendre structure for driven systems., Comment: 11 pages, 2 figures

The nature of statistics, statistical mechanics and consequently the thermodynamics of stochastic systems is largely determined by how the number of states $W(N)$ depends on the size $N$ of the system. Here we propose a scaling expansion of the phasespace volume $W(N)$ of a stochastic system. The corresponding expansion coefficients (exponents) define the universality class the system belongs to. Systems within the same universality class share the same statistics and thermodynamics. For sub-exponentially growing systems such expansions have been shown to exist. By using the scaling expansion this classification can be extended to all stochastic systems, including correlated, constraint and super-exponential systems. The extensive entropy of these systems can be easily expressed in terms of thee scaling exponents. Systems with super-exponential phasespace growth contain important systems, such as magnetic coins that combine combinatorial and structural statistics. We discuss other applications in the statistics of networks, aging, and cascading random walks., Comment: 16 pages including supplementary material, revised version

Sample space reducing (SSR) processes offer a simple analytical way to understand of the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to cascading pro- cesses. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function gives the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process., Comment: 7 pages, 2 figures

There are three ways to conceptualize entropy: entropy as an extensive thermodynamic quantity of physical systems (Clausius, Boltzmann, Gibbs), entropy as a measure for information production of ergodic sources (Shannon), and entropy as a means for statistical inference on multinomial Bernoulli processes (Jaynes maximum entropy principle). Even though these notions are fundamentally different concepts, the functional form of the entropy for thermodynamic systems in equilibrium, for ergodic sources in information theory, and for independent sampling processes in statistical systems, is degenerate, $H(p)=-\sum_i p_i\log p_i$. For many complex systems, which are typically history-dependent, non-ergodic and non-multinomial, this is no longer the case. Here we show that for such processes the three entropy concepts lead to different functional forms of entropy. We explicitly compute these entropy functionals for three concrete examples. For Polya urn processes, which are simple self-reinforcing processes, the source information rate is $S_{\rm IT}=\frac{1}{1-c}\frac1N \log N$, the thermodynamical (extensive) entropy is $(c,d)$-entropy, $S_{\rm EXT}=S_{(c,0)}$, and the entropy in the maxent principle (MEP) is $S_{\rm MEP}(p)=-\sum_i \log p_i$. For sample space reducing (SSR) processes, which are simple path-dependent processes that are associated with power law statistics, the information rate is $S_{\rm IT}=1+ \frac12 \log W$, the extensive entropy is $S_{\rm EXT}=H(p)$, and the maxent result is $S_{\rm MEP}(p)=H(p/p_1)+H(1-p/p_1)$. Finally, for multinomial mixture processes, the information rate is given by the conditional entropy $\langle H\rangle_f$, with respect to the mixing kernel $f$, the extensive entropy is given by $H$, and the MEP functional corresponds one-to-one to the logarithm of the mixing kernel., Comment: 12 pages, 2 figures

Sample Space Reducing (SSR) processes are simple stochastic processes that offer a new route to understand scaling in path-dependent processes. Here we define a cascading process that generalises the recently defined SSR processes and is able to produce power laws with arbitrary exponents. We demonstrate analytically that the frequency distributions of states are power laws with exponents that coincide with the multiplication parameter of the cascading process. In addition, we show that imposing energy conservation in SSR cascades allows us to recover Fermi's classic result on the energy spectrum of cosmic rays, with the universal exponent -2, which is independent of the multiplication parameter of the cascade. Applications of the proposed process include fragmentation processes or directed cascading diffusion on networks, such as rumour or epidemic spreading., Comment: 10 pages, 6 figures

It has been repeatedly stated that maximum likelihood (ML) estimates of exponents of power-law distributions can only be reliably obtained for exponents smaller than minus one. The main argument that power laws are otherwise not normalizable, depends on the underlying sample space the data is drawn from, and is true only for sample spaces that are unbounded from above. Here we show that power-laws obtained from bounded sample spaces (as is the case for practically all data related problems) are always free of such limitations and maximum likelihood estimates can be obtained for arbitrary powers without restrictions. Here we first derive the appropriate ML estimator for arbitrary exponents of power-law distributions on bounded discrete sample spaces. We then show that an almost identical estimator also works perfectly for continuous data. We implemented this ML estimator and discuss its performance with previous attempts. We present a general recipe of how to use these estimators and present the associated computer codes., Comment: 11 pages, 3 figures

Species or population that proliferate faster than others become dominant in numbers. Catalysis allows catalytic sets within a molecular reaction network to dominate the non catalytic parts of the network by processing most of the available substrate. As a consequence one may consider a 'catalytic fitness' of sets of molecular species. The fittest sets emerge as the expressed chemical backbone or sub-network of larger chemical reaction networks employed by organisms. However, catalytic fitness depends on the systemic context and the stability of systemic dynamics. Unstable reaction networks would easily be reshaped or destroyed by fluctuations of the chemical environment. In this paper we therefore focus on recognizing systemic stability as an evolutionary selection criterion. In fact, instabilities of regulatory systems dynamics become predictive for associated evolutionary forces driving the emergence large reaction networks that avoid or control inherent instabilities. Systemic instabilities can be identified and analyzed using relatively simple mathematical random networks models of complex regulatory systems. Using a statistical ensemble approach one can identify fundamental causes of instable dynamics, infer evolutionary preferred network properties, and predict evolutionary emergent control mechanisms and their entanglement with cell differentiation processes. Surprisingly, what systemic stability tells us here is that cells (or other non-linear regulatory systems) never had to learn how to differentiate, but rather how to avoid and control differentiation. For example, in this framework we can predict that regulatory systems will evolutionary favor networks where the number of catalytic enhancers is not larger than the number of suppressors., Comment: 8 pages 4 figures

Despite its simplicity, it seems to my best of knowledge that the possibly simplest approach towards deriving equations governing irreversible thermodynamics from gas-kinetic considerations within the framework of classical mechanics has never been pursued. In this paper we address this omission and derive the equations describing the irreversible thermodynamics of a gas in a piston and associated thermodynamic cycles performed in finite time. What we find is a thermodynamic action principle: The irreversible work we require for performing a thermodynamic cycle in finite time times the time we require to run through the cycle, a isothermal compression/decompression cycle for instance, will always be larger or equal to a lower bound given by a system specific constant with the dimension of an action. This process specific action constants can take values of the order of Plank's constant for microscopic processes, such as displacing a Hydrogen atom by one atom diameter. For macroscopic processes (e.g. a bicycle pump) also the action constant takes macroscopically observable values. We discuss the finite time Carnot cycle, and estimate the lower bound for irreversible work required when rewriting physically embodied bits of information and compare the resulting values with Landauer's bond, the irreversible amount of energy that needs to be dissipated for re-writing a bit., Comment: 7 pages, 3 figures

It has been shown recently that a specific class of path-dependent stochastic processes, which reduce their sample space as they unfold, lead to exact scaling laws in frequency and rank distributions. Such Sample Space Reducing processes (SSRP) offer an alternative new mechanism to understand the emergence of scaling in countless processes. The corresponding power law exponents were shown to be related to noise levels in the process. Here we show that the emergence of scaling is not limited to the simplest SSRPs, but holds for a huge domain of stochastic processes that are characterized by non-uniform prior distributions. We demonstrate mathematically that in the absence of noise the scaling exponents converge to $-1$ (Zipf's law) for almost all prior distributions. As a consequence it becomes possible to fully understand targeted diffusion on weighted directed networks and its associated scaling laws law in node visit distributions. The presence of cycles can be properly interpreted as playing the same role as noise in SSRPs and, accordingly, determine the scaling exponents. The result that Zipf's law emerges as a generic feature of diffusion on networks, regardless of its details, and that the exponent of visiting times is related to the amount of cycles in a network could be relevant for a series of applications in traffic-, transport- and supply chain management., Comment: 11 pages, 5 figures

Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, non-interacting systems, and to the ensemble picture; It loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for self-reinforcing P\'olya urn processes, which explicitly generalise multinomial statistics. We demonstrate the adequacy of this constructive approach towards non-multinomial pendants of entropy by computing frequency and rank distributions of P\'olya urn processes. We show how microscopic update rules of a path-dependent process allow us to explicitly construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function., Comment: 13 pages, 3 figures

The formation of sentences is a highly structured and history-dependent process. The probability of using a specific word in a sentence strongly depends on the 'history' of word-usage earlier in that sentence. We study a simple history-dependent model of text generation assuming that the sample-space of word usage reduces along sentence formation, on average. We first show that the model explains the approximate Zipf law found in word frequencies as a direct consequence of sample-space reduction. We then empirically quantify the amount of sample-space reduction in the sentences of ten famous English books, by analysis of corresponding word-transition tables that capture which words can follow any given word in a text. We find a highly nested structure in these transition tables and show that this `nestedness' is tightly related to the power law exponents of the observed word frequency distributions. With the proposed model it is possible to understand that the nestedness of a text can be the origin of the actual scaling exponent, and that deviations from the exact Zipf law can be understood by variations of the degree of nestedness on a book-by-book basis. On a theoretical level we are able to show that in case of weak nesting, Zipf's law breaks down in a fast transition. Unlike previous attempts to understand Zipf's law in language the sample-space reducing model is not based on assumptions of multiplicative, preferential, or self-organised critical mechanisms behind language formation, but simply used the empirically quantifiable parameter 'nestedness' to understand the statistics of word frequencies., Comment: 7 pages, 4 figures. Accepted for publication in the Journal of the Royal Society Interface

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample-space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power-laws, $p(x)\sim x^{-\lambda}$, where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to how much a given process reduces its sample-space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from $\alpha = 2$ to $\infty$. We discuss several applications showing how SSR processes can be used to understand Zipf's law in word frequencies, and how they are related to diffusion processes in directed networks, or ageing processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organised critical processes., Comment: 7 pages, 5 figures in Proceedings of the National Academy of Sciences USA (published ahead of print April 13, 2015)

The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there exists an ongoing controversy whether the notion of the maximum entropy principle can be extended in a meaningful way to non-extensive, non-ergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the (c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept for non-ergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process., Comment: 6 pages, 1 figure. To appear in PNAS

Complex systems are often inherently non-ergodic and non-Markovian for which Shannon entropy loses its applicability. In particular accelerating, path-dependent, and aging random walks offer an intuitive picture for these non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived from three of the Shannon-Khinchin axioms, and by violating the fourth -- the so-called composition axiom. The corresponding entropy is of the form $S_{c,d} \sim \sum_i \Gamma(1+d,1-c\ln p_i)$ and depends on two system-specific scaling exponents, $c$ and $d$. This entropy contains many recently proposed entropy functionals as special cases, including Shannon and Tsallis entropy. It was shown that this entropy is relevant for a special class of non-Markovian random walks. In this work we generalize these walks to a much wider class of stochastic systems that can be characterized as `aging' systems. These are systems whose transition rates between states are path- and time-dependent. We show that for particular aging walks $S_{c,d}$ is again the correct extensive entropy. Before the central part of the paper we review the concept of $(c,d)$-entropy in a self-contained way., Comment: 8 pages, 5 eps figures. arXiv admin note: substantial text overlap with arXiv:1104.2070

Motivation Gene transcription requires the orchestrated binding of various proteins to the promoter of a gene. The binding times and binding order of proteins allow to draw conclusions about the proteins' exact function in the recruitment process. Time-resolved ChIP experiments are being used to analyze the order of protein binding for these processes. However, these ChIP signals do not represent the exact protein binding patterns. Results We show that for promoter complexes that follow sequential recruitment dynamics the ChIP signal can be understood as a convoluted signal and propose the application of deconvolution methods to recover the protein binding patterns from experimental ChIP time-series. We analyze the suitability of four deconvolution methods: two non-blind deconvolution methods, Wiener deconvolution and Lucy-Richardson deconvolution, and two blind deconvolution methods, blind Lucy-Richardson deconvolution and binary blind deconvolution. We apply these methods to infer the protein binding pattern from ChIP time-series for the pS2 gene.

The q-Gaussian function emerges naturally in various applications of statistical mechanics of non-ergodic and complex systems. In particular it was shown that in the theory of binary processes with correlations, the q-Gaussian can appear as a limiting distribution. Further, there exist several problems and situations where, depending on procedural or algorithmic details of data-processing, q-Gaussian distributions may yield distinct values of q, where one value is larger, the other smaller than one. To relate such pairs of q-Gaussians it would be convenient to map such distributions onto one another, ideally in a way, that any value of q can be mapped uniquely to any other value q'. So far a (duality) map from q -> q'=(7-5q)/(5-3q) was found, mapping q from the interval q\in [-\infty, 1] -> q'\in [1, 5/3]. Here we complete the theory of transformations of q-Gaussians by deriving a general map \gamma_{qq'}, that transforms normalizable q-Gaussian distributions onto one another for which q and q' are in the range of [1,3). By combining this with the previous result, a mapping from any value of q \in [-\infty,3) is possible to any other value q'\in [-\infty,3). We show that the action of \gamma_{qq'} on the set of q-Gaussian distributions is a transformation groupoid., Comment: 11 pages, 3 figures

For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or non-ergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort probabilities, the other not. The two ways are connected through a duality. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized logarithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy., Comment: 4 pages, 1 page supporting information

Systemic properties of living cells are the result of molecular dynamics governed by so-called genetic regulatory networks (GRN). These networks capture all possible features of cells and are responsible for the immense levels of adaptation characteristic to living systems. At any point in time only small subsets of these networks are active. Any active subset of the GRN leads to the expression of particular sets of molecules (expression modes). The subsets of active networks change over time, leading to the observed complex dynamics of expression patterns. Understanding of this dynamics becomes increasingly important in systems biology and medicine. While the importance of transcription rates and catalytic interactions has been widely recognized in modeling genetic regulatory systems, the understanding of the role of degradation of biochemical agents (mRNA, protein) in regulatory dynamics remains limited. Recent experimental data suggests that there exists a functional relation between mRNA and protein decay rates and expression modes. In this paper we propose a model for the dynamics of successions of sequences of active subnetworks of the GRN. The model is able to reproduce key characteristics of molecular dynamics, including homeostasis, multi-stability, periodic dynamics, alternating activity, differentiability, and self-organized critical dynamics. Moreover the model allows to naturally understand the mechanism behind the relation between decay rates and expression modes. The model explains recent experimental observations that decay-rates (or turnovers) vary between differentiated tissue-classes at a general systemic level and highlights the role of intracellular decay rate control mechanisms in cell differentiation., Comment: 16 pages, 5 figures

Democratic societies are built around the principle of free and fair elections, that each citizen's vote should count equal. National elections can be regarded as large-scale social experiments, where people are grouped into usually large numbers of electoral districts and vote according to their preferences. The large number of samples implies certain statistical consequences for the polling results which can be used to identify election irregularities. Using a suitable data collapse, we find that vote distributions of elections with alleged fraud show a kurtosis of hundred times more than normal elections on certain levels of data aggregation. As an example we show that reported irregularities in recent Russian elections are indeed well explained by systematic ballot stuffing and develop a parametric model quantifying to which extent fraudulent mechanisms are present. We show that if specific statistical properties are present in an election, the results do not represent the will of the people. We formulate a parametric test detecting these statistical properties in election results. Remarkably, this technique produces similar outcomes irrespective of the data resolution and thus allows for cross-country comparisons., Comment: For data see also http://www.complex-systems.meduniwien.ac.at/elections/election.html

We show how the dependence of phase space volume $\Omega(N)$ of a classical system on its size $N$ uniquely determines its extensive entropy. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a {\em generalized} (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and is practically statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for binomial processes and argue that generalized entropies could be relevant for self organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion., Comment: 5 pages, 2 figures

Shannon and Khinchin showed that assuming four information theoretic axioms the entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here we note that in physical systems one of these axioms may be violated. For non-ergodic systems the so called separation axiom (Shannon-Khinchin axiom 4) will in general not be valid. We show that when this axiom is violated the entropy takes a more general form, $S_{c,d}\propto \sum_i ^W \Gamma(d+1, 1- c \log p_i)$, where $c$ and $d$ are scaling exponents and $\Gamma(a,b)$ is the incomplete gamma function. The exponents $(c,d)$ define equivalence classes for all interacting and non interacting systems and unambiguously characterize any statistical system in its thermodynamic limit. The proof is possible because of two newly discovered scaling laws which any entropic form has to fulfill, if the first three Shannon-Khinchin axioms hold. $(c,d)$ can be used to define equivalence classes of statistical systems. A series of known entropies can be classified in terms of these equivalence classes. We show that the corresponding distribution functions are special forms of Lambert-${\cal W}$ exponentials containing -- as special cases -- Boltzmann, stretched exponential and Tsallis distributions (power-laws). In the derivation we assume trace form entropies, $S=\sum_i g(p_i)$, with $g$ some function, however more general entropic forms can be classified along the same scaling analysis., Comment: 8 pages, 2 figures, 4 theorems

Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures $\beta$, so that the probability distribution is $p(\epsilon_i) \propto \int_{0}^{\infty} f(\beta) e^{-\beta \epsilon_i}d\beta$, where the `kernel' $f(\beta)$ is nonnegative and normalized ($\int f(\beta)d \beta =1$). We discuss the relation between this distribution and the generalized entropic form $S=\sum_i s(p_i)$. The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension., Comment: 5 pages, no figure

One of the virtues of peer review is that it provides a self-regulating selection mechanism for scientific work, papers and projects. Peer review as a selection mechanism is hard to evaluate in terms of its efficiency. Serious efforts to understand its strengths and weaknesses have not yet lead to clear answers. In theory peer review works if the involved parties (editors and referees) conform to a set of requirements, such as love for high quality science, objectiveness, and absence of biases, nepotism, friend and clique networks, selfishness, etc. If these requirements are violated, what is the effect on the selection of high quality work? We study this question with a simple agent based model. In particular we are interested in the effects of rational referees, who might not have any incentive to see high quality work other than their own published or promoted. We find that a small fraction of incorrect (selfish or rational) referees can drastically reduce the quality of the published (accepted) scientific standard. We quantify the fraction for which peer review will no longer select better than pure chance. Decline of quality of accepted scientific work is shown as a function of the fraction of rational and unqualified referees. We show how a simple quality-increasing policy of e.g. a journal can lead to a loss in overall scientific quality, and how mutual support-networks of authors and referees deteriorate the system., Comment: 5 pages 4 figures

Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the average connectivity., Comment: 11 pages, 5 figures

To characterize strongly interacting statistical systems within a thermodynamical framework - complex systems in particular - it might be necessary to introduce generalized entropies, $S_g$. A series of such entropies have been proposed in the past, mainly to accommodate important empirical distribution functions to a maximum ignorance principle. Until now the understanding of the fundamental origin of these entropies and its deeper relations to complex systems is limited. Here we explore this questions from first principles. We start by observing that the 4th Khinchin axiom (separability axiom) is violated by strongly interacting systems in general and ask about the consequences of violating the 4th axiom while assuming the first three Khinchin axioms (K1-K3) to hold and $S_g=\sum_ig(p_i)$. We prove by simple scaling arguments that under these requirements {\em each} statistical system is uniquely characterized by a distinct pair of scaling exponents $(c,d)$ in the large size limit. The exponents define equivalence classes for all interacting and non interacting systems. This allows to derive a unique entropy, $S_{c,d}\propto \sum_i \Gamma(d+1, 1- c \ln p_i)$, which covers all entropies which respect K1-K3 and can be written as $S_g=\sum_ig(p_i)$. Known entropies can now be classified within these equivalence classes. The corresponding distribution functions are special forms of Lambert-$W$ exponentials containing as special cases Boltzmann, stretched exponential and Tsallis distributions (power-laws) -- all widely abundant in nature. This is, to our knowledge, the first {\em ab initio} justification for the existence of generalized entropies. Even though here we assume $S_g=\sum_ig(p_i)$, we show that more general entropic forms can be classified along the same lines., Comment: 6 pages, 3 pages Supplementary Information, 2 Figs

We demonstrate with a thought experiment that fitness-based population dynamical approaches to evolution are not able to make quantitative, falsifiable predictions about the long-term behavior of evolutionary systems. A key characteristic of evolutionary systems is the ongoing endogenous production of new species. These novel entities change the conditions for already existing species. Even {\em Darwin's Demon}, a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time, could not pre-state the impact of these novelties on established populations. We argue that fitness is always {\it a posteriori} knowledge -- it measures but does not explain why a species has reproductive success or not. To overcome these conceptual limitations, a variational principle is proposed in a spin-model-like setup of evolutionary systems. We derive a functional which is minimized under the most general evolutionary formulation of a dynamical system, i.e. evolutionary trajectories causally emerge as a minimization of a functional. This functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean-field approximation. We test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves. The model is further able to reproduce stylized facts of timeseries from several man-made and natural evolutionary systems. Light will be thrown on how species and their fitness landscapes dynamically co-evolve., Comment: 13 pages, 3 figures, 2 tables

We propose a simple quantitative model of Schumpeterian economic dynamics. New goods and services are endogenously produced through combinations of existing goods. As soon as new goods enter the market they may compete against already existing goods, in other words new products can have destructive effects on existing goods. As a result of this competition mechanism existing goods may be driven out from the market - often causing cascades of secondary defects (Schumpeterian gales of destruction). The model leads to a generic dynamics characterized by phases of relative economic stability followed by phases of massive restructuring of markets - which could be interpreted as Schumpeterian business `cycles'. Model timeseries of product diversity and productivity reproduce several stylized facts of economics timeseries on long timescales such as GDP or business failures, including non-Gaussian fat tailed distributions, volatility clustering etc. The model is phrased in an open, non-equilibrium setup which can be understood as a self organized critical system. Its diversity dynamics can be understood by the time-varying topology of the active production networks., Comment: 21 pages, 11 figures

Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and width constraints yields the Gaussian distribution. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with additive noise), as well as the Central Limit Theorem attractors, are Gaussians. The simplest stochastic model with such features is N to infinity independent binary random variables, as first proved by de Moivre and Laplace. What happens for strongly correlated random variables? Such correlations are often present in physical situations as e.g. systems with long range interactions or memory. Frequently q-Gaussians become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic processes, allow a systematical analysis of the relation between correlations and non-Gaussian distributions. In particular, a generalized stochastic model yielding q-Gaussians for all q (including q>1) was missing. This is achieved here by using the Laplace-de Finetti representation theorem, which embodies strict scale-invariance of interchangeable random variables. We demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be BG., Comment: 6 pages, 1 fig, to appear in EPJ B

We study the robustness of functionals of probability distributions such as the R\'enyi and nonadditive S_q entropies, as well as the q-expectation values under small variations of the distributions. We focus on three important types of distribution functions, namely (i) continuous bounded (ii) discrete with finite number of states, and (iii) discrete with infinite number of states. The physical concept of robustness is contrasted with the mathematically stronger condition of stability and Lesche-stability for functionals. We explicitly demonstrate that, in the case of continuous distributions, once unbounded distributions and those leading to negative entropy are excluded, both Renyi and nonadditive S_q entropies as well as the q-expectation values are robust. For the discrete finite case, the Renyi and nonadditive S_q entropies and the q-expectation values are robust. For the infinite discrete case, where both Renyi entropy and q-expectations are known to violate Lesche-stability and stability respectively, we show that one can nevertheless state conditions which guarantee physical robustness., Comment: 6 pages, to appear in Euro Phys Lett

Traditionally evolution is seen as a process where from a pool of possible variations of a population (e.g. biological species or industrial goods) a few variations get selected which survive and proliferate, whereas the others vanish. Survival probability is typically associated with the 'fitness' of a particular variation. In this paper we argue that the notion of fitness is an a posteriori concept, in the sense that one can assign higher fitness to species that survive but one can generally not derive or even measure fitness - or fitness landscapes - per se. For this reason we think that in a 'physical' theory of evolution such notions should be avoided. In this spirit, here we propose a random matrix model of evolution where selection mechanisms are encoded in the interaction matrices of species. We are able to recover some key facts of evolution dynamics, such as punctuated equilibrium, i.e. the existence of intrinsic large extinctions events, and, at the same time, periods of dramatic diversification, as known e.g. from fossil records. Further, we comment on two fundamental technical problems of a 'physics of evolution', the non-closedness of its phase space and the problem of co-evolving boundary conditions, apparent in all systems subject to evolution., Comment: 7 pages, 6 figures

In statistical physics lately a specific kind of average, called the q-expectation value, has been extensively used in the context of q-generalized statistics dealing with distributions following power-laws. In this context q-expectation values appear naturally. After it has been recently shown that this non-linear functional is instable, under a very strong notion of stability, it is therefore of high interest to know sufficient conditions for when the results of q-expectations are robust under small variations of the underlying distribution function. We show that reasonable restrictions on the domain of admissible probability distributions restore uniform continuity for the q-expectation. Bounds on the size of admissible variations can be given. The practical usefulness of the theorems for estimating the robustness of the q-expectation value with respect to small variations is discussed., Comment: 17 pages, no figures

We formulate three famous, descriptive essays of C.N. Parkinson on bureaucratic inefficiency in a quantifiable and dynamical socio-physical framework. In the first model we show how the use of recent opinion formation models for small groups can be used to understand Parkinson's observation that decision making bodies such as cabinets or boards become highly inefficient once their size exceeds a critical 'Coefficient of Inefficiency', typically around 20. A second observation of Parkinson - which is sometimes referred to as Parkinson's Law - is that the growth of bureaucratic or administrative bodies usually goes hand in hand with a drastic decrease of its overall efficiency. In our second model we view a bureaucratic body as a system of a flow of workers, which enter, become promoted to various internal levels within the system over time, and leave the system after having served for a certain time. Promotion usually is associated with an increase of subordinates. Within the proposed model it becomes possible to work out the phase diagram under which conditions bureaucratic growth can be confined. In our last model we assign individual efficiency curves to workers throughout their life in administration, and compute the optimum time to send them to old age pension, in order to ensure a maximum of efficiency within the body - in Parkinson's words we compute the 'Pension Point'., Comment: 15 pages, 5 figures

Reverse engineering of gene regulatory networks presents one of the big challenges in systems biology. Gene regulatory networks are usually inferred from a set of single-gene over-expressions and/or knockout experiments. Functional relationships between genes are retrieved either from the steady state gene expressions or from respective time series. We present a novel algorithm for gene network reconstruction on the basis of steady-state gene-chip data from over-expression experiments. The algorithm is based on a straight forward solution of a linear gene-dynamics equation, where experimental data is fed in as a first predictor for the solution. We compare the algorithm's performance with the NIR algorithm, both on the well known E.Coli experimental data and on in-silico experiments. We show superiority of the proposed algorithm in the number of correctly reconstructed links and discuss computational time and robustness. The proposed algorithm is not limited by combinatorial explosion problems and can be used in principle for large networks of thousands of genes., Comment: 10 pages, 3 figures

We propose a model for evolution aiming to reproduce statistical features of fossil data, in particular the distributions of extinction events, the distribution of species per genus and the distribution of lifetimes, all of which are known to be of power law type. The model incorporates both species-species interactions and ancestral relationships. The main novelty of this work is to show the feasibility of k-core percolation as a selection mechanism. We give theoretical predictions for the observable distributions, confirm their validity by computer simulation and report good agreement with the fossil data. A key feature of the proposed model is a co-evolving fitness landscape determined by the topology of the underlying species interactions, ecological niches emerge naturally. The predicted distributions are independent of the rate of speciation, i.e. whether one adopts an gradualist or punctuated view of evolution., Comment: 6 pages, 3 eps figures

Hilhorst and Schehr recently presented a straight forward computation of limit distributions of sufficiently correlated random numbers \cite{hilhorst}. Here we present the analytical form of entropy which --under the maximum entropy principle (with ordinary constraints)-- provides these limit distributions. These distributions are not $q$-Gaussians and can not be obtained with Tsallis entropy., Comment: 4 pages 1 fig