exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

on the entropy function in sociotechnical systems abstract the entropy function h pj log pj pj being the probability of a system being in state j and its continuum analogue h p x log p x dx are fundamental in shannon s theory of information transfer in communication systems it is here shown that the discrete form of h also appears naturally in single lane traffic flow theory in merchandising goods flow from a whole saler through a retailer to a customer certain features of the process may be deduced from price distribution functions derived from sears roebuck and company catalogues it is found that the dispersion in logarithm of catalogue prices of a given year has remained about constant independently of the year for over 75 years from this it may be inferred that the continuum entropy function for the variable logarithm of price had inadvertently through sears roebuck policies been maximized for that firm subject to the observed dispersion

dilatations and factorizable equations in this paper we show that the dilatation operator in one dimension can be used to enlarge the lie algebra generated by the raising and lowering operators for some classes of special functions as a result we are able to derive new recursion relations and addition formulae for these functions furthermore we derive generalized ladder operators for these functions under coordinate stretching and translations

thermodynamic treatment of nonphysical systems formalism and an example single lane traffic an effort is made to introduce thermodynamic and statistical thermodynamic methods into the treatment of nonphysical e g social economic etc systems emphasis is placed on the use of theentire thermodynamic framework not merely entropy entropy arises naturally related in a simple manner to other measurables but does not occupy a primary position in the theory however the maximum entropy formalism is a convenient procedure for deriving the thermodynamic analog framework in which undetermined multipliers are thermodynamic like variables which summarize the collective behavior of the system we discuss the analysis of levine and his coworkers showing that the maximum entropy formalism is the unique algorithm for achieving consistent inference of probabilities the thermodynamic like formalism for treating a single lane of vehicular traffic is developed and applied to traffic in which the interaction between cars is chosen to be a particular form of the follow the leader type the equation of state of the traffic the distributions of velocity and headway and the various thermodynamic like parameters e g temperature collective sensitivity pressure etc are determined for an experimental example holland tunnel nearest neighbor and pair correlation functions for the vehicles are also determined many interesting and suggestive results are obtained

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

world trade web topological properties dynamics and evolution this paper studies the statistical properties of the web of import export relationships among world countries using a weighted network approach we analyze how the distributions of the most important network statistics measuring connectivity assortativity clustering and centrality have coevolved over time we show that all node statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years and are likely to do so in the future conversely the distribution of positive link weights is slowly moving from a log normal density towards a power law we also characterize the autoregressive properties of network statistics dynamics we find that network statistics growth rates are well proxied by fat tailed densities like the laplace or the asymmetric exponential power finally we find that all our results are reasonably robust to a few alternative economically meaningful weighting schemes

exact large deviation functional of a stationary open driven diffusive system the asymmetric exclusion process we consider the asymmetric exclusion process asep in one dimension on sites i 1 n in contact at sites i 1 and i n with infinite particle reservoirs at densities a and b as a and b are varied the typical macroscopic steady state density profile x x a b obtained in the limit n l b a exhibits shocks and phase transitions here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile rho x text p n rho x sim exp l mathcal f a b rho x rho a rho b so that mathcal f is the large deviation functional a quantity similar to the free energy of equilibrium systems we find as in the symmetric purely diffusive case q 1 treated in an earlier work that mathcal f is in general a non local functional of x unlike the symmetric case however the asymmetric case exhibits ranges of the parameters for which mathcal f rho x is not convex and others for which mathcal f rho x has discontinuities in its second derivatives at x x in the latter ranges the fluctuations of order 1 sqrt n in the density profile near x are then non gaussian and cannot be calculated from the large deviation function

hierarchy in social organization abstract we find that area and population distributions of nations follow an inverse power law as is known for cities but with a different exponent to interpret this result we develop a growth model based on the geometrical properties of partitioning of the plane the substantial agreement between the model and the actual nation distributions motivates the hypothesis that the distribution of aggregates of organisms is related to land partitioning to test this hypothesis we follow the development of bacterial colonies of escherichia coli which compared to humans are on a completely different level of complexity we find that the distributions of e coli colonies follow an inverse power law with exponent similar to that of nations

table errata higher transcendental functions vol ii mcgraw hill new york 1953 mr 15 419 by a erdelyi w magnus f oberhettinger and f g tricomi

factorization of systems of differential equations it is shown that the classical infeld hull factorization method can be extended to coupled systems of second order equations a complete solution of the factorization equations in two dimensions is given and a partial enumeration of factorizable systems is made

theory of earthquakes interevent times applied to financial markets abstract we analyze the probability density function pdf of waiting times between financial loss exceedances the empirical pdfs are fitted with the self excited hawkes conditional poisson process with a long power law memory kernel the hawkes process is the simplest extension of the poisson process that takes into account how past events influence the occurrence of future events by analyzing the empirical data for 15 different financial assets we show that the formalism of the hawkes process used for earthquakes can successfully model the pdf of interevent times between successive market losses

factorisation of separable partial differential equations the author studies the factorisation of partial differential equations which admit a separation of variables as a particular application the author studies the algebraic structure of the resulting raising and lowering operators for the spherical harmonics in three and four dimensions

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

the asymmetric simple exclusion process an integrable model for non equilibrium statistical mechanics the asymmetric simple exclusion process asep plays the role of a paradigm in non equilibrium statistical mechanics we review exact results for the asep obtained by the bethe ansatz and put emphasis on the algebraic properties of this model the bethe equations for the eigenvalues of the markov matrix of the asep are derived from the algebraic bethe ansatz using these equations we explain how to calculate the spectral gap of the model and how global spectral properties such as the existence of multiplets can be predicted an extension of the bethe ansatz leads to an analytic expression for the large deviation function of the current in the asep that satisfies the gallavotti cohen relation finally we describe some variants of the asep that are also solvable by the bethe ansatz

extremal projectors of q boson algebras we define the extremal projector of the q boson kashiwara algebra and study their basic properties applying their properties to the representation theory of the category whose objects are upper bounded modules we obtain its semi simplicity and the classification of simple modules

novel types of factorisable systems of differential equations this paper considers systems of factorisable equations for different forms of the factorisation kernel in one dimension some of these kernels lead only to trivial factorisable equations however in higher dimensions they yield some novel classes of factorisable equations a partial explicit classification of these new factorisable systems is given

family of commuting operators for the totally asymmetric exclusion process the algebraic structure underlying the totally asymmetric exclusion process is studied by using the bethe ansatz technique from the properties of the algebra generated by the local jump operators we explicitly construct the hierarchy of operators called generalized hamiltonians that commute with the markov operator the transfer matrix which is the generating function of these operators is shown to represent a discrete markov process with long range jumps we give a general combinatorial formula for the connected hamiltonians obtained by taking the logarithm of the transfer matrix this formula is proved using a symbolic calculation program for the first ten connected operators

current fluctuations in the exclusion process and bethe ansatz we use the bethe ansatz to derive analytical expressions for the current statistics in the asymmetric exclusion process with both forward and backward jumps the bethe equations are highly coupled and this fact has impeded their use in deriving exact results for finite systems we overcome this technical difficulty by reformulating bethe equations into a one variable polynomial problem akin to the functional bethe ansatz the perturbative solution of this equation leads to the cumulants of the current we calculate here the first two orders and derive exact formulae for the mean value of the current and its fluctuations

navier stokes like equations for traffic flow the macroscopic traffic flow equations derived from the reduced paveri fontana equation are closed starting with the maximization of the informational entropy the homogeneous steady state taken as a reference is obtained for a specific model of the desired velocity and a kind of chapman enskog method is developed to calculate the traffic pressure at the navier stokes level numerical solution of the macroscopic traffic equations is obtained and its characteristics are analyzed

dynamics of book sales endogenous versus exogenous shocks in complex networks we present an extensive study of the foreshock and aftershock signatures accompanying peaks of book sales the time series of book sales are derived from the ranking system of amazon com we present two independent ways of classifying peaks one based on the acceleration pattern of sales and the other based on the exponent of the relaxation they are found to be consistent and reveal the coexistence of two types of sales peaks exogenous peaks occur abruptly and are followed by a power law relaxation while endogenous sale peaks occur after a progressively accelerating power law growth followed by an approximately symmetrical power law relaxation which is slower than for exogenous peaks we develop a simple epidemic model of buyers connected within a network of acquaintances which propagates rumors and opinions on books the comparison between the predictions of the model and the empirical data confirms the validity of the model and suggests in addition that social networks have evolved to converge very close to criticality here in the sense of critical branching processes of opinion spreading we test in detail the evidence for a power law distribution of book sales and confirm a previous indirect study suggesting that the fraction of books density distribution p s of sales s is a power law p s ensuremath sim 1 s 1 ensuremath mu with ensuremath mu ensuremath approx 2

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

factorization of sturm liouville operators solvable potentials and underlying algebraic structure in this paper a general method of factorization of sturm liouville sl operators is provided a method to solve sl eigenvalue problems is presented new classes of exactly solvable potentials are obtained the supersymmetry and shape invariance approaches are generalized to the sl operators it is shown that the sl shape invariance potentials have an underlying algebraic structure this algebra is in general infinite dimensional the condition of finite algebra is obtained

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

spectral degeneracies in the totally asymmetric exclusion process we study the spectrum of the markov matrix of the totally asymmetric exclusion process tasep on a one dimensional periodic lattice at arbitrary filling although the system does not possess obvious symmetries except translation invariance the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules this behaviour is explained by a hidden symmetry property of the bethe ansatz assuming a one to one correspondence between the solutions of the bethe equations and the eigenmodes of the markov matrix we derive combinatorial formulae for the orders of degeneracy and the number of multiplets these results are confirmed by exact diagonalisations of small size systems this unexpected structure of the tasep spectrum suggests the existence of an underlying large invariance group

from gene families and genera to incomes and internet file sizes why power laws are so common in nature we present a simple explanation for the occurrence of power law tails in statistical distributions by showing that if stochastic processes with exponential growth in expectation are killed or observed randomly the distribution of the killed or observed state exhibits power law behavior in one or both tails this simple mechanism can explain power law tails in the distributions of the sizes of incomes cities internet files biological taxa and in gene family and protein family frequencies

origin of pareto like spatial distributions in ecosystems recent studies of cluster distribution in various ecosystems revealed pareto statistics for the size of spatial colonies these results were supported by cellular automata simulations that yield robust criticality for endogenous pattern formation based on positive feedback we show that this patch statistics is a manifestation of the law of proportionate effect mapping the stochastic model to a markov birth death process the transition rates are shown to scale linearly with cluster size this mapping provides a connection between patch statistics and the dynamics of the ecosystem the first passage time for different colonies emerges as a powerful tool that discriminates between endogenous and exogenous clustering mechanisms imminent catastrophic shifts such as desertification manifest themselves in a drastic change of the stability properties of spatial colonies

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

exact eigenspectrum of the symmetric simple exclusion process on the complete complete bipartite and related graphs we show that the infinitesimal generator of the symmetric simple exclusion process recast as a quantum spin ferromagnetic heisenberg model can be solved by elementary techniques on the complete complete bipartite and related multipartite graphs some of the resulting infinitesimal generators are formally identical to homogeneous as well as mixed higher spins models the degeneracies of the eigenspectra are described in detail and the clebsch gordan machinery needed to deal with the arbitrary spin s representations of the su 2 is briefly developed we mention in passing how our results fit within the related questions of a ferromagnetic ordering of energy levels and a conjecture according to which the spectral gaps of the random walk and the interchange process on finite simple graphs must be equal

o n symmetries sum rules for generalized hermite polynomials and squeezed states quantum optics has been dealing with coherent states squeezed states and many other non classical states the associated mathematical framework makes use of special functions as hermite polynomials laguerre polynomials and others in this connection we here present some formal results that follow directly from the group o n of complex transformations motivated by the squeezed states structure we introduce the generalized hermite polynomials which include as particular cases the hermite polynomials as well as the heat polynomials using generalized raising operators we derive new sum rules for the which are covariant under o n transformations the and the associated sum rules become useful for evaluating wigner functions in a straightforward manner as a byproduct we use one of these sum rules on the operator level to obtain raising and lowering operators for the laguerre polynomials and show that they generate an sl 2 r su 1 1 algebra

memory and burstiness in dynamic networks a discrete time random process is described which can generate bursty sequences of events a bernoulli process where the probability of an event occurring at time t is given by a fixed probability x is modified to include a memory effect where the event probability is increased proportionally to the number of events that occurred within a given amount of time preceding t for small values of x the interevent time distribution follows a power law with exponent 2 x we consider a dynamic network where each node forms and breaks connections according to this process the value of x for each node depends on the fitness distribution x from which it is drawn we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions and for both cases where the memory effect either is or is not present this work can potentially lead to methods to uncover hidden fitness distributions from fast changing temporal network data such as online social communications and fmri scans

percolation model for growth rates of aggregates and its application for business firm growth motivated by recent empirical studies of business firm growth we develop a dynamic percolation model which captures some of the features of the economical system i e merging and splitting of business firms represented as aggregates on a d dimensional lattice we find the steady state distribution of the aggregate size and explore how this distribution depends on the model parameters we find that at the critical threshold the standard deviation of the aggregate growth rates increases with aggregate size s as s where can be explained in terms of the connectedness length exponent and the fractal dimension df with 1 2df 0 20 for d 2 and 0 125 for d the distributions of aggregate growth rates have a sharp peak at the center and pronounced wings extending over many standard deviations giving the distribution a tent shape form the laplace distribution the distributions for different aggregate sizes scaled by their standard deviations collapse onto the same curve

collective purchase behavior toward retail price changes a b s t r a c t by analyzing a huge amount of point of sale data collected from japanese supermarkets we find power law relationships between price and sales numbers the estimated values of the exponents of these power laws depend on the category of products however they are independent of the stores thereby implying the existence of universal human purchase behavior the rate of sales numbers around these power laws are generally approximated by log normal distributions implying that there are hidden random parameters which might proportionally affect the purchase activity

universal features in the growth dynamics of complex organizations we analyze the fluctuations in the gross domestic product gdp of 152 countries for the period 1950 1992 we find that i the distribution of annual growth rates for countries of a given gdp decays with fatter tails than for a gaussian and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with results on firm growth these results are consistent with the hypothesis that the evolution of organizations with complex structure is governed by similar growth mechanisms s0031 9007 98 07339 6 pacs numbers 02 50 ey 01 75 m 05 40 j 05 45 b in the study of physical systems the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior in contrast most studies on the time evolution of economic time series have concentrated on average growth rates 1 18 here we investigate the possibility that the study of fluctuations in economics may also lead to a better understanding of the mechanisms responsible for the observed dynamics 19 23 we therefore analyze the fluctuations in the growth rate of the gross domestic product gdp of 152 countries during the period 1950 1992 24 we will show that i the distribution of annual growth rates for countries of a given gdp is consistent for a certain range with an exponential decay and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with