there is a photograph of john bell taken in 1982 in front of a blackboard in his office at cern .the famous clauser - horne - shimony - holt ( chsh ) bell inequality is written in the blackboard . at the right hand side of the maximum bound for local hidden variable theoriesit is written `` einstein . '' below that , it is written the maximum for quantum theory ( qt ) : `` . '' already in 1969 , chsh noticed that this was the maximum for two - qubit systems . in 1980 , tsirelson proved that it is also the maximum in qt , no matter the dimensionality of the state space .it took a lot of time until somebody asked the obvious question : why ?it took a surprising amount of time until somebody came with a compelling answer .however , it was soon proved that this answer can not explain the maximum quantum values for some tripartite bell inequalities .this leads us back to square one : why the quantum maxima of all bell inequalities ?what is the fundamental reason that limits quantum probabilities ?  in the summer of 1964 , before submitting the bell inequality paper , bell submitted other paper which , for several reasons , was not published until 1966 . there , bell discusses the implications for the hidden variables problem of gleason s theorem , which was directed to reducing the axiomatic basis of qt . the relevant corollary of gleason s theorem is that , if the dimensionality of the state space is grater than two , then there exists a set of elementary tests such that values (rue ) or (alse ) can not be assigned to them respecting that : ( i ) can not be assigned to two mutually exclusive tests , and ( ii ) must be assigned to exactly one of mutually exclusive tests .bell proved this corollary by constructing an explicit infinite set of elementary tests in for which such an assignment is impossible .a finite set was found by kochen and specker in 1962 , but not published until 1967 , making explicit a result anticipated by specker in 1960 .these sets prove the impossibility of reproducing qt with theories satisfying the assumption of non - contextuality of results , namely , the assumption that the `` measurement of an observable must yield the same value independently of what other measurements may be made simultaneously '' .bell considered that this assumption was not reasonable and finished his paper suggesting that it would be interesting to pursue some proof of impossibility of hidden variables replacing non - contextuality by some assumption of locality .one month later , bell submitted the bell inequality paper .however , the problem of hidden variables in qt can be mathematically formulated in a way which goes beyond whether or not non - contextuality is reasonable .the problem of hidden variables in qt is simply whether or not it is possible to recover the quantum probabilities from a joint probability distribution over a single probability space .what is proven by kochen - specker and bell s examples is that this is not possible in any scenario in which the dimensionality of the state space is three or higher , irrespective of whether or not locality can be invoked .what if bell would have derived a bell - like inequality violated by quantum systems of dimension three ?such an inequality was introduced in 2008 and is called the klyachko - can - biniciolu - shumovsky ( kcbs ) non - contextuality ( nc ) inequality .the kcbs inequality is the analog for quantum systems of dimension three of the chsh inequality .the kcbs inequality is the simplest nc inequality violated by quantum systems of dimension three ; the chsh inequality is the simplest nc inequality violated by quantum systems of dimension four .simplicity is here measured by the number of dichotomic observables used .the quantum violation of nc inequalities shows the impossibility to recover the quantum probabilities from a joint probability distribution over a single probability space .the kcbs inequality says that , for any non - contextual hidden variable ( nchv ) theory , with where are observables with two possible results and , and where , e.g. , denotes the joint probability of obtaining when measuring and when measuring .probabilities in eq .( [ mean ] ) are assumed to be well defined no matter in which order and are measured . however , for and this may not be the case .  similarly , the chsh inequality says that , for any local hidden variables ( lhv ) theory , with the difference between and is that , in , and can be measured on a subsystem while and are measured on a distant subsystem .therefore , in the choice of measurement on one subsystem and the result on the other subsystem can be space - like separated .this allows us to invoke locality to justify the assumption of non - contextuality .in contrast with ( [ kcbs ] ) and ( [ chsh ] ) , in qt , and the big question is not just why qt violates the inequalities for hidden variable theories , but rather why qt violates them _ exactly up to these limits_.consider non - demolition measurements that are repeatable ( i.e. , that give the same result when repeated ) and cause no disturbance on other measurements ( i.e. , when combined with these other measurements , all are repeatable ) .these measurements are called `` sharp '' and , in qt , are represented by projection operators .these are the measurements that von neumann called `` quantum observables '' .let us define an _event _ as the state of the system after some sharp measurements ( with certain results ) on some initial state .two events are equivalent when they correspond to indistinguishable states .two events are exclusive when there is a measurement that distinguishes between them .a theory satisfies the _ exclusivity ( e ) principle _ when any set of pairwise exclusive events is -wise exclusive . therefore ,if we assume the e principle , kolmogorov s axioms of probability lead us to the conclusion that the sum of the probabilities of any set of pairwise exclusive events can not be higher than  .however , the e principle can not be derived from kolmogorov s axioms . to see it ,consider the maximum value of the following sum of probabilities of events : where the notation is the same used above .the three events , and are pairwise exclusive .therefore , the only restrictions from kolmogorov s axioms are that the probabilities are non - negative and that    therefore , for theories satisfying kolmogorov s axioms the maximum is , since each of the three probabilities in ( [ spe ] ) can be .however , for theories satisfying the e principle , the maximum is , since the e principles forces that   the e principle can be derived from a variety of axioms .for example , from the axiom that pairwise co - measurability implies joint co - measurability , from the principle of fundamental sharpness of measurements , from axioms 1 and 2 in ref . , and from the principle of lack of irreducible third order interference .the e principle imposes limits to the sum of probabilities of pairwise exclusive events .therefore , in order to study the implications of the e principle for the limits of the kcbs and chsh inequalities , it is convenient to rewrite both inequalities in terms of sums of probabilities of events . for that , it is useful to notice that the condition of normalization of probabilities allows us to write    therefore , we can write    where    and is obtained from by changing the signs of all the results . then , we can write the kcbs and chsh inequalities and their quantum limits as follows :   first target is to explain why can not go beyond or , equivalently , why can not go beyond . for this purpose , consider two independent experiments both aiming the maximum of .suppose that one of the experiments is performed in vienna on a certain physical system , while the other experiment is performed in stockholm on a different physical system .let us denote by an event of the experiment in vienna , by the sum of the corresponding five probabilities , by an event of the experiment in stockholm and by the sum of the corresponding five probabilities .  since the experiments are independent , the probability of an event involving both experiments is the product of the probabilities of the corresponding ( single - city ) events .for example ,   exclusivity graphs of the five sets of five pairwise exclusive events used in the proof of the limit of the kcbs inequality .events are represented by nodes and exclusivity relations by edges .the black pentagons correspond to the exclusivity relations between the events .the coloured pentagrams correspond to the exclusivity relations between the events .any two graphs differ in a rotation of the pentagram . ]  having two copies , we can identify larger sets of pairwise exclusive events .for example , the set with the following events : , , , and .the e principle and assumption ( [ indep ] ) applied to this set imply that    similarly , by identifying sets of pairwise exclusive events , we can derive the following inequalities :   the geometry behind these sets is explained in fig .if we sum all five inequalities ( [ uno])([cinco ] ) , we obtain assuming that the maximum is the same in both experiments , i.e. , that , we can conclude that , for any theory satisfying the e principle , exactly as in qt .this is an arguably clearer presentation of a result introduced in ref .our second target is to explain why can not go beyond or , equivalently , why can not go beyond . for this purpose ,first notice that the state space on which and act is , at least , two - dimensional .second , notice that the conditions of normalization of probabilities allows us to write ,   now consider two independent experiments both testing .suppose that one of the experiments is performed in vienna and the other experiment in stockholm .as before , let us denote by one event in vienna and by one event in stockholm .  now notice that and are co - measurable and that the state space on which and act is , at least , four - dimensional .therefore , there must exist an observable co - measurable with and and such that the result for is if the results of and are equal , and if they are different . acts on a four - dimensional state space , but only distinguishes between two subspaces .therefore , there must be an observable co - measurable with that distinguishes between other two subspaces and such that since we have made no assumption about and ( other than each of them acts on a different at least two - dimensional subspace ) , we can relate to and , the same way we related to and .similarly , we can start with and and define and then define a co - measurable related to and .these observables allow us to identify larger sets of pairwise exclusive events .for example , the set with the following events : , , , , , , , and . since , by definition of , , the e principle and assumption ( [ indep ] ) applied to this setimply that   [ cols="^,^,^,^,^ " , ]   as explained in table [ table1 ] , there are 16 sets like this one . for each of them, there is a inequality like ( [ one ] ) .if we sum all of them we obtain , assuming that the maximum is the same in both experiments , we can conclude that , for any theory satisfying the e principle , exactly as in qt .this result is an improved version of an argument introduced in ref .a similar argument allows us to derive the quantum limits for -partite bell - like inequalities for non - local ( but not genuinely -partite non - local ) hidden variable theories .we have shown that some characteristic limits of qt have a simple explanation .indeed , we suspect that all the limits of quantum probabilities have the same explanation . if this would be the case , what should we learn about qt ?  for some people ,a fundamental message of qt is that the world is non - local , i.e. , that the results of quantum observables correspond to some reality and change non - locally .however , from this perspective , it is puzzling that the no - signaling principle allows for higher than quantum violations of the chsh inequality .why then qt is not more non - local ?the reason why qt is exactly as non - local as it is , apparently , the same reason why qt is exactly as contextual as it is .however , in the contextuality case , there is no alice and bob and no communication .why nobody paid attention to the e principle before ?arguably , because the e principle is trivial in classical deterministic theories and not well defined in non - local realistic theories .however , if one takes non - realism ( of the results of observables represented in qt by self - adjoint operators ) as a fundamental key of the world , then everything makes much more sense .the fundamental non - existence of results makes that not all conceivable combinations of observables allow for joint probability distributions ( i.e. , makes not all observables to be co - measurable ) .indeed , it makes all conceivable relations of co - measurability / non - co - measurability ( among sharp measurements ) realizable and , as a consequence , makes all conceivable relationships of exclusivity / non - exclusivity ( among events ) realizable .there is where the e principle makes a profound contribution : the possible sets of probabilities of a given scenario ( i.e. , a certain structure of co - measurability / non - co - measurability ) are restricted by the e principle applied to all conceivable embeddings of the scenario into a larger scenario . in this sense , the e principle acts in an holistic way . in particular , the limits of the probabilities of a given scenario follow from identifying the most ( or one of the most ) restrictive embedding(s ) ( details will be presented elsewhere ) .the resulting picture points out that non - realism is not `` a soft option '' , but rather a fundamental key of the world .qt is a probability theory about things that do not exist and are unpredictable at a fundamental level .one may argue that the view i have drawn before is just one of the possible options and that the predictions of qt are also compatible with contextual and non - local realistic views of the world .i disagree .common to all these other views is a certain degree of realism that ranges from hidden variables determining the results of all possible experiments to just taking the quantum state as real .it seems evident that any of these other views , when examined in detail , will lead to predictions that qt does not make .hopefully , we will soon identify these predictions and test them . time and experimentswill tell .this work was supported by the project fis2011 - 29400 ( mineco , spain ) with feder funds , the fqxi large grant project `` the nature of information in sequential quantum measurements '' and the program science without borders ( capes and cnpq , brazil ) .j. von neumann , _ mathematische grundlagen der quantenmechanik _ ( springer - verlag , berlin , 1932 ) .english version : _ mathematical foundations of quantum mechanics _( princeton university press , princeton , new jersey , 1955 ) .

Abstract: for years , the biggest unspeakable in quantum theory has been why quantum theory and what is quantum theory telling us about the world . recent efforts are unveiling a surprisingly simple answer . here we show that two characteristic limits of quantum theory , the maximum violations of clauser - horne - shimony - holt and klyachko - can - biniciolu - shumovsky inequalities , are enforced by a simple principle . the effectiveness of this principle suggests that non - realism is the key that explains why quantum theory .   = 1  john bell at cern . 1982 cern . ]


r. h. dicke in his pioneering work has shown that a system of number of inverted two - level atoms can spontaneously transit to the ground state during the time inversely proportional to the number of atoms .because the total energy emitted by the collection of atoms is proportional to , the radiation intensity reveals a square dependence on .this type of radiation was termed the `` collective spontaneous radiation '' , or the `` dicke superradiance '' .a remarkable progress has been achieved in the understanding of this phenomenon in  .the authors of  have developed a quantum theory of superradiance for a single - mode model .the simplicity of the model enables a detailed study of the kinetics and statistical properties of superradiance .later , it has been found that the radiation intensity undergoes appreciable statistical fluctuations and the rmsd of the superradiative instability time slowly decreases according to a logarithmic law as the number of particles is increased : .the elementary unit of the system we have just described is a two - level atom , the concept widely used in quantum electronics and optics to describe physical processes . in the excited state , the two - level atom interacting with the radiation fieldis also involved in relaxation processes caused by inter - particle collisions .if the radiation growth time is much greater than the relaxation time , then the pulse of superradiance is not formed , and the particles emit incoherently .this greatly restricts the selection of active medium for experimental investigation of superradiance and design of superradiance - based short - pulse electromagnetic sources .  in this connection ,a considerable attention has been paid recently to the generation of short superradiance pulses , also termed as `` self - amplified spontaneous emission '' , `` superradiance '' , or `` cooperative radiation '' , using short electron beams propagating in complex electrodynamical structures ( undulators , resonators , dielectric and corrugated waveguides , or photonic crystals ) . in such systems , the average relaxation time that depends on the collisions between charged particlesexceeds manifold the time needed for the formation of the cooperative - radiation pulse .cooperative radiation is generated via a nonlinear interaction of charged particles and the electromagnetic field , causing the electrons to bunch . as a result ,the whole of the electron beam becomes spatially modulated .spatial modulation is accompanied by a coherent emission of electromagnetic waves .  the possibility to produce short pulses of cooperative radiationwas first substantiated in for free electron lasers ( fels ) . however , the experimentally measured output characteristics of cooperative radiation ( power , radiation spectra , and energy in a pulse ) seemed really stochastic , which , according to theoretical and experimental studies , was closely related to the presence of the shot noise inherent in charged - particle beams .it is noteworthy that high frequency and time stability of the radiation pulse is crucial for many applications .for this reason , electron beams are premodulated at the radiation frequency to reduce the influence of shot noise on cooperative radiation .  in a microwave range ,single - pass crms and cherenkov generators are more commonly used .the generators of cooperative radiation operate in two regimes : traveling wave and backward wave .the distinguishing feature of the traveling - wave regime is that the group velocity of electromagnetic waves and the velocity of charged particles are co - directional .this regime was applied in first experiments with single - pass crms and cherenkov generators . theoretical consideration of radiation gain in short - pulse traveling - wave generators revealed an important peculiarity : the stability of the cooperative radiation parameters can be improved noticeably by injecting into the electrodynamical structure of a beam with a sharp front whose duration is comparable with the emission period . in this case , the fourier transform of the beam current contains quite a significant spectral component at the radiation frequency . as a result ,the generation of electromagnetic oscillations starts with coherent spontaneous emission of the whole beam instead of incoherent spontaneous emission from individual particles . as a result, the degree of fluctuations in cooperative radiation is decreased .though the early single - pass crms and cherenkov generators operated in the traveling - wave regime , the most impressive results were obtained in experiments with single - pass backward - wave tubes generating cooperative radiation whose peak power was appreciably greater than the beam power .for example , the peak radiated power of 1.2 gw was attained at ghz for beam power of 0.87 gw .the stability of output parameters of the cooperative - radiation pulse generated in the backward - wave regime , as in a single - pass traveling - wave generator , strongly depended on the beam front .the analysis of theoretical and experimental works on generation of cooperative radiation reveals that the main cause of statistical spread of the output characteristics is the shot noise inherent in the electron beam .the impact it produces can be diminished by premodulating the beam at the radiation frequency .naturally , one wonders what degree of beam premodulation is required for certain applications .this problem would be most prevalent for high - current generators with electron beams composed of ectons , individual portions of electrons that contain up to elementary charge carriers .taking into account the complex structure of the electron beam is a must for accurate estimations of the shot noise , which is essential in solving the problem of coherent summation of electromagnetic oscillations from several short - pulse sources of radiation .this paper considers the effect of shot noise on the generation of cooperative radiation from a premodulated short beam of particles .we shall consider this issue by the example of an ensemble of nonisochronous electrons - oscillators interacting with one another via the radiation field , used as one of the basic models for the description of nonlinear generation of electromagnetic waves in crms .it has been shown in that in the absence of external action , the instability evolves in the ensemble of nonisochronous electrons - oscillators , accompanied at the initial stage by an exponential growth of the radiated power and autophasing of electrons - oscillators .this exponential growth is then suppressed due to nonlinearity , and the pulse of cooperative radiation is formed .the influence of nonisochronism on the peak power of cooperative radiation and the autophasing time of electrons - oscillators was studied in detail by vainstein and colleagues .however , the authors of assumed that the beam had been premodulated at the radiation frequency and left out the effects related to shot noise .in our analysis of statistical properties of cooperative radiation , the peak radiated power and autophasing time serve as random variables .the peak power is the major output characteristic of short - pulse electromagnetic sources , and the autophasing time is the parameter defining the minimum time of particle passage through the generator needed for cooperative instability to evolve .moreover , the rmsd in the autophasing time is of fundamental importance for coherent summation of oscillations from several cooperative - radiation sources .the paper s outline is as follows .first , we derive a system of equations describing the interaction of nonisochronous oscillators via the radiation field by the example of electron ensemble circulating in a uniform magnetic field .further comes a detailed consideration of statistical fluctuations of cooperative radiation in the presence of shot noise from the ensemble of nonisochronous electrons - oscillators with and without phase premodulation of charged particles .particular attention is given to finding the autophasing time distribution function .based on the described study , we shall then investigate in detail the statistical fluctuations in single - pass crms and discuss the limits imposed on the output characteristics thereof by the shot noise and the energy spread .furthermore , we shall discuss the possibility of coherent summation of oscillations coming from several single - pass sources of cooperative cyclotron radiation .let us consider the behavior of a nonrelativistic electron beam in a uniform magnetic field  directed along the axis in the presence of the radiative energy loss .( the behavior of the relativistic electron beam is discussed in appendix a. ) particle velocity components perpendicular and parallel to the magnetic - filed vector are denoted by  and , respectively .then the behavior of particles is described by the equations of motion in the form here , is the force acting on the particle from all particles and is its momentum related to the velocity  as   if the beam size is less than the radiation wavelength , and the coulomb repulsion force and induction fields can be neglected ( see appendix b ) , then the force acting on each particle is given by the sum ( see appendix c ) where is the radiation field induced by the particle .let us pay attention to an essential circumstance : the expression for is true if the radiative friction force is appreciably less than the lorentz force acting on each particle . otherwise , unphysical self - accelerated solutions may arise .the requirement that the radiative friction force should be much less than the lorentz force imposes limitation on the magnetic field strength ( in the opposite case the considered theory is invalid ) . for single particle : but in the case of a dense beam of coherently emitting particle with electrons , mass , and charge we can write by analogy with  with the present - day acceleration facilities , dense beams with electrons are available ; substitution of into gives  kgs .if this condition is violated , the equation set with the force is inapplicable .  thus , if the radiation reaction force is much less than the lorentz force and the beam size is less than the radiation wavelength , then the equations of motion describing the interaction between charged particles have the form :   in the absence of energy losses through emission , the particles are in circular motion with cyclic frequencies  depending on , which is responsible for nonisochronism of oscillations .  using the approximate relation where we shall write vector equations in a component - wise fashion where then we shall multiply the second equation by and add it to the first one :   assuming that all are equal , let us introduce the following notation : ( is the average tangential speed of particles ) . then takes the form  the equation set fully describes the behavior of the ensemble of electrons - oscillators moving in a uniform magnetic field in the presence of the radiative energy loss .the kinetic energy of transverse motion of the oscillators , the radiation power , and the time are related to the dimensionless quantities by  and where  and   let  kgs , , and , then ,  kw ,  ns , and  ghz . in this case , the nonisochronism parameter is   let us note that by adding the term of the form to the left - hand side of , we have that under the same initial conditions , all complex amplitudes are multiplied by the additional phase factor having no effect either on the radiation power or on the autophasing time .this enables analyzing different systems from a common standpoint , and so the equation set describing the behavior of the ensemble of weakly nonlinear electrons - oscillators oscillating in anharmonic potential has the form : here , the nonisochronism parameter is selected as .the normalized radiation power and electron energy are given by formulas  and respectively . in the absence of the energy spread ,the absolute values of the amplitudes equal unity at the initial time and the phases are uniformly distributed in the interval .shot noise in the ensemble of the electrons - oscillators is due to a random distribution of the initial phases .each set of corresponds to different initial conditions of the equation set .fluctuations in the initial conditions lead to fluctuations in the output characteristics of cooperative radiation which we study by a numerical experiment .as follows from , the behavior of the ensemble of nonisochronous oscillators in the absence of the energy spread is determined by two fixed parameters and and a random set of initial phases .because the distribution of initial phases is random , the numerical experiment with each pair of values of and must have many runs .this procedure will give information about statistical characteristics of cooperative radiation , the most important of which are peak power , autophasing time , and their relative rmsd  and .+  +   +  +  in numerical analysis of statistical fluctuations of cooperative radiation in the presence of shot noise , instead of the number of real electrons we took the number of simulated electrons with the charge and initial phases where are random variables uniformly distributed over the interval .it has been shown in that this procedure , boosting the performance of the program , correctly simulates the shot noise in the absence of energy spread .we selected the following values of controlling parameters : , .the numerical experiment with each pair was repeated one hundred times .1 and 2 show the results of our computation from which we can draw some very important conclusions .first , the relative rmsd of the dimensionless peak radiated power weakly dependent on the number of particles ( fig.1 ) decreases as .second , the autophasing time decreasing as the nonisochronism parameter is increased depends logarithmically on the number of particles .third , reduces according to the approximate formula , where slowly decreases as varies from 0 to 2 . extrapolating the obtained dependence of and to the region with large number of particles , , ( typical number of electrons in single - pass generators ) , we get the estimates and . from thiswe can deduce that shot noise leads to a fluctuation of the autophasing time at insignificant fluctuation of the peak radiated power .let us note here that phase premodulation of electrons - oscillators : having practically no effect on , , and , leads to a noticeable decrease in the fluctuation of autophasing time : no longer decreases logarithmically ( ) , but as ( fig .3 ) . in the expression, is the premodulation parameter .in the previous section we have shown that fluctuations in the cooperative radiation caused by shot noise depend on the degree to which the electrons - oscillators are premodulated .it has been found , in particular , that premodulation results in that the relative rmsd of the autophasing time becomes square - root dependent on ( ) instead of logarithmically dependent ( ) .let us show how this important result can be obtained analytically .  according to without premodulation , the contribution coming from each particle to the average oscillation amplitude is .the contribution coming from a premodulated particle ( the initial phases are uniformly distributed over the interval ) . by averaging over , we shall find the average value of as well as the rmsd thereof at and : where is the bessel function .   +   +  since , then in view of the central limiting theorem , and have a gaussian distribution : leading to that the radiation power distribution at the initial time has the form : where is the modified bessel function , .let us estimate the autophasing time by formula here , is the time period needed for the radiation power to increase by a factor of in the linear instability stage .equation implies , first , that the linear stage is much longer than the time during which nonlinear effects are essential , and , second , the radiated power in the saturation stage is .the latter condition means coherent summation of oscillations from all the particles .  using and, we can find the distribution function related to by thus we have from the distribution functions plotted in figure [ fig.4 ] follows that with growing premodulation , the autophasing time and its rmsd thereof decrease .using the distribution function , we shall compute the relative rmsd : here , is the euler constant and is the laguerre polynomial . if , then transforms to the form in the opposite case ( ) , the fig .[ fig.5 ] plots versus , respectively .it is obvious from [ fig.5 ] that when the number of particles exceeds a certain value depending on the degree of premodulation , the logarithmic dependence of the relative rmsd of the autophasing time ( ) goes to a square - root dependence ( ) .as follows from , the value of at and agree with the simulated ones ( fig .[ fig.3 ] ) within 20% accuracy .let us note a very important circumstance .the duration of the cooperative radiation pulse from nonisochronous electrons - oscillators is  .the rmsd of the autophasing time in the absence of premodulation has the same order of magnitude : at .however , phase premodulation of particles leads to an appreciable decrease in the autophasing time spread : at .  in coherent summation of cooperative - radiation pulses ,the oscillation phases of all single - pass generators must differ by , thus posing the following limitation on the average statistical spread of autophasing time : giving , after the substitution of and  let and , then we have the following estimate of the required degree of premodulation : . at , the parameter of premodulation must be greater than .to take account of the electron energy spread , we assume the initial amplitudes to be gaussian random variables whose mean equals unity and the relative rmsd ( is the relative rmsd deviation of particle energy ) .it should be noted the penman - mcneil algorithm is nt applicable in the presence of energy spread . therefore instead of the number of simulated electrons we took the number of particles with the initial phases uniformly distributed in the interval .  analyzing the results of numerical experiments , we can see that the energy spread leads to a sharp drop in the radiated power ( fig .[ fig.6 ] ) .this is well - illustrated by the fig .[ fig.7 ] , where the growing influence of the energy spread with higher is seen clearly : the energy spread leads to a stronger suppression of radiation at large , and peak radiated power and radiated energy both decrease . at ( ) ,the maximum attainable radiated power reduces by more than a factor of two .in this paper we have studied the statistical properties of cooperative radiation from an ensemble of nonisochronous electrons - oscillators interacting with one another via the radiation field .it has been shown that for the number of electrons , typical of modern acceleration facilities , the relative rmsd of the autophasing time from its mean is .the fluctuations of the peak radiated power appear to be negligibly small .the autophasing time distribution function depending on the number of particles and the degree of premodulation has been obtained in the absence of energy spread .we used this function to show that when the number of particles exceeds a certain value depending on the degree of premodulation , the logarithmic dependence of the relative rmsd of the autophasing time on the number of electrons - oscillators ( ) goes to a square - root dependence ( ) .it has also been demonstrated that a slight energy spread ( ) results in a twofold drop of the maximum attainable power of cooperative radiation .the analysis made here indicates that shot noise , and electron velocity spread pose considerable constraints on the output characteristics of single - pass crms and limits the possibility of coherent summation of cooperative radiation pulses from several sources .let in laboratory reference there be a relativistic electron beam whose velocity is directed at a small angle to the magnetic - field vector . the particles in the beam move in a spiral under lorentz force .let us denote the particle velocity components perpendicular and parallel to vector by and , respectively .then the velocity component perpendicular to the magnetic field in the reference frame attached to the beam is given by : let us assume that , which leads to the limitations on the acceptable value of : in this section we shall focus on the distribution of the cooperative radiation from relativistic electrons - oscillators in the laboratory frame if the condition is fulfilled . in a moving frame of reference , the radiated power per unit solid angle is given by  where is the total power of electrons - oscillators .the question naturally arises of how the radiation intensity can be transformed into the lab reference frame .let us denote the radiated power per unit solid angle in the lab frame by . before seeking ,let us note that the quantity that equals the number of quanta emitted in solid angle during the time is the lorentz invariant , and is the lorentz invariant , too  the division of by gives another invariant by equating the invariants in both frames of reference we get because the absolute values of the wave vectors in moving ( ) and fixed ( ) frames of reference are related as  where is the angle between vectors and , whereas we can find   according to , the angular distribution of the power detected by the resting detector in the lab frame is related to as integration of with due account of finally gives ( ) us study the applicability boundaries of .the equation set was derived in the assumption that the beam size is much less than the radiation wavelength we also neglected the coulomb repulsion and induction fields . the latter assumption is often taken in solving specific problems , so we shall qualitatively outline its legitimacy .it has been shown in that the possibility of phasing electrons - oscillators without the external field arises because the particles oscillate nonisochronousally , i.e. , the particle oscillation frequency depends on energy ( see ) . as a result , a nonlinear term having the same order of magnitude as appears in .the coulomb and induction fields have no effect of autophasing if the acceleration induced to the electron by the coulomb forces ( induction fields are by a factor of less than the coulomb ones ) is much smaller than .let us write this condition in the form :   let  kgs , , and .then  cm ,  cm , and the particle orbit radius  cm is between and .that is why if in this specific case the beam size is the same as the particle orbit , we can satisfy both and requirements .let us consider a short beam of charged particles moving with nonrelativistic velocities .the beam particles produce the electromagnetic field that can be described using the scalar and vector retarded potentials : where is the distance from the volume element to the point of observation , where we seek the potential difference , and are the current and charge densities , respectively . the electric and magnetic fields relate to the potentials and as using these relation , we can readily find the force acting on each charge from all beam particles because the velocities of all charges are small compared to the speed of light , their distribution hardly can change a lot during the time ( is the size of the beam ) . that is why , to find forces acting on the charges , we can expand and into power series in . for the scalar potential , we find accurate up to the third order in terms of  : but is the constant total charge of the system ; then the second term in this expression equals zero , and so we can do the same with . but appearing in the expression for the vector potential is multiplied by when is substituted into the expression for the force . because we seek the potentials within the accuracy up to the third order in terms of  , in the expansion of suffice it to find the first two terms , i.e. ,   let us perform gauge transformation of the potentials : choosing the function such that the scalar potential becomes  then a new vector potential is passing from integration to summation over individual charges , we get  and where , .  the first term in determines the induction field of the system of charges and the scalar potential determines the electric field of the spatial charge .both fields are non - uniform and depend on the averaged charge distribution in the electron beam .the second term in is uniform and arises through radiation ; the related electric field is given by   199 r.h .dicke , phys .93 , 99 ( 1954 ) .r. bonifacio , p. schwendiman , and f. haake , phys .a 4 , 302 ( 1971 ) . v. de giorgio and f. ghielmetti , phys . rev .a 4 , 2415 ( 1971 ) . m. gross and s. haroche , phys .93 , 301 ( 1982 ) .andreev , v.i .emelyanov , and yu.a .ilinskii . _ cooperative phenomena in optics . _moscow : fizmatlit , 1988 [ in russian ] .n. skribanowitz , i.p .herman , j.c .macgillivray , and m.s .feld , phys .30 , 309 ( 1973 ) .macgillivray and m.s .feld , phys .a 14 , 1169 ( 1976 ) .r. bonifacio et .al . , rivista del nuovo cimento 13 , 1 ( 1990 ) .ginzburg et al . , ieee trans .plasma sci .41 , 646 ( 2013 ) .d.i . trubeckov and a.e .hramov , _ lectures on high - frequency electronics for physicists . v. 2 . _moscow , fizmatlit , 2004 .[ in russian ] s.v .anishchenko and v.g .baryshevsky , nucl .methods b 355 , 76 ( 2015 ) .rostov , a.a .elchaninov , i.v .romanchenko , and m.i .yalandin , appl .100 , 224102 ( 2012 ) .ginzburg et al . , phys . rev .115 , 114802 ( 2015 ) .a.m. kondratenko , e.l .saldin , part .10 , 207 ( 1980 ) .r. bonifacio , c. pellegrini , and l. narducci , opt .50 , 373 ( 1984 ) .r. bonifacio et al .73 , 70 ( 1994 ) .saldin , e.a .schneidmiller , and m.v .yurkov , opt .148 , 383 ( 1998 ) .m. hogan et al .79 , 289 ( 1998 ) .j. andruszkov et al . , phys .85 , 3825 ( 2000 ) .yurkov , nucl .methods a 483 , 51 ( 2002 ) .atvazyan et al .methods 507 , 368 ( 2003 ) .f. lehmkuhler et al ., scientific reports 4 , 05234 ( 2014 ) .yu , phys . rev .a 44 , 5178 ( 1991 ) .yu et al . , science 289 , 932 ( 2000 ) .e. allaria et al . , nature photon . 6 ,699 ( 2012 ) .j. amann et al . , nature photon . 6 ,693 ( 2012 ) . g. de ninno et al ., nature communications 6 , 8075 ( 2015 ) .ginzburg , i.v .zotova , a.s .sergeev , jetp lett .60 , 513 ( 1994 ) .ginzburg et al ., jetp lett .63 , 331 ( 1996 ) .ginzburg et al . ,78 , 2365 ( 1997 ) .ginzburg et al . , nucl .methods a. 393 , 352 ( 1997 ) . b.w.j .mcneil , g.r.m .robb , and d.a .jaroszynsky , optics comm .163 , 203 ( 1999 ) .wiggins et al . , phys .84 , 2393 ( 2000 ) .ginzburg et al . , phys .e 60 , 3297 ( 1999 ) n.s .ginzburg et al . , tech .72 , 83 ( 2002 ) .elchaninov , et al ., jetp lett .77 , 266 ( 2003 ) .elchaninov et al . ,laser and particle beams 21 , 187 ( 2003 ) .korovin et al . , phys .e 74 , 016501 ( 2006 ) .bugaev , e.a .litvinov , g.a .mesyats , and d.i .proskurovskii , sov .usp 18 , 51 ( 1975 ) .mesyats , plasma phys .fusion 47 , a109 ( 2005 ) .abubakirov , a.p .konjushkov , and a.s .sergeev , journal of communications technology and electronics 54 , 959 ( 2009 ) .gaponov , m.i .petelin , and v.k .yulpatov , radiophys .quantum electron .10 , 749 ( 1967 ) .zheleznyakov , v.v .kocharovskii , and vl.v .kocharovskii , radiophys .quantum electron .29 , 830 ( 1987 ) .yu.a . ilinskii and n.s .maslova , sov .jetp 67 , 96 ( 1988 ) .vainstein and a.i .kleev , dan sssr 311 , 862 ( 1990 ) .[ in russian ] yu.a .kobelev , l.a .ostrovskii , and i.a .soustova , sov .jetp 72 , 262 ( 1991 ) .s.v . anishchenko and v.g .baryshevsky , tech .61 , 934 ( 2016 ) .ginzburg et al . , tech .47 , 335 ( 2002 ) . c. penman and b.w.j .mcneil , opt .90 , 82 ( 1992 ) .landau and e.m .lifshiz , _ the classical theory of fields _ butterworth heineman , amsterdam , 1978 . v.l .ginzburg , _ theoretical physics and atrophysics _ , pergamon press , oxford , 1979 .

Abstract: shot noise , intrinsic to the ensemble of nonisochronous electrons - oscillators , is the cause of statistical fluctuations in cooperative radiation generated by single - pass cyclotron - resonance masers ( crms ) . autophasing time the time required for the cooperative radiation power to peak  is the critical parameter characterizing the dynamics of electrons - oscillators interacting via the radiation field . shot - noise related fluctuations of the autophasing time imposes appreciable limitations on the possibility of coherent summation of electromagnetic oscillations from several single - pass crms .   premodulation of charged particles leads to a considerable narrowing of the autophasing time distribution function . when the number of particles exceeds a certain value that depends on the degree to which the particles have been premodulated , the relative root - mean - square deviation ( rmsd ) of the autophasing time changes from a logarithmic dependence on ( ) to square - root ( ) . as a result , there is an increased probability of coherent summation of electromagnetic oscillations from several single - pass generators .   a slight energy spread ( % ) results in a twofold drop of the maximum attainable power of cooperative radiation .


the prisoner s dilemma ( pd ) is perhaps the most famous model in the field of game theory . roughly speaking , there are two sorts of pd : one - shot pd and iterated pd . nowadays a lot of studies on pd are focused on the latter case .for example , axelrod investigated the evolution of cooperative behavior in well - mixed populations of selfish agents by using pd as a paradigm .nowak and may induced spatial structure in pd , i.e. , agents were restricted to interact with his immediate neighbors .santos and pacheco found that when agents interacted following scale - free networks , cooperation would become a dominating trait throughout the entire range of parameters of pd .perc and szolnoki proposed that social diversity could induce cooperation as the dominating trait throughout the entire range of parameters of pd .  compared with the iterated pd, the one - shot pd is usually viewed simple . in the original version of one - shot pd ,two prisoners are arrested by a policeman .each prisoner must independently choose a strategy between `` confessing '' ( denoted as strategy `` _ _ defect _ _ '' ) and `` not confessing '' ( denoted as strategy `` _ _ cooperate _ _ '' ) .the payoff matrix of prisoners is shown in table 1 . as long as two agents are rational, the unique nash equilibrium shall be ( _ defect _ , _ defect _ ) , which results in a pareto - inefficient payoff .that is the dilemma ._ table 1 : the payoff matrix of pd , where _ , and ._ the first entry in the parenthesis denotes the payoff of agent 1 and the second entry stands for the payoff of agent 2_. +  [ cols="^,^,^",options="header " , ]   \1 ) : this strategy profile means two agents both participate the algorithmic model and submit parameters to the computer . according to ref . , for each agent , his dominant parameters are and , which result in a pareto - efficient payoff .+ 2 ) : this strategy profile means agent 1 participates the algorithmic model , but agent 2 takes back his channel and submits a message to the arbitrator directly . since agent 1 can observe agent 2 s action , in the end , both agents will take back their channels and submit messages to the arbitrator directly . obviously , the dominant message of each agent is , and the arbitrator will assign the pareto - inefficient payoff to agents .+ 3 ) : this strategy profile is similar to the above case .the arbitrator will assign to two agents .+ 4 ) : this strategy profile means two agents both take back their channels and send messages to the arbitrator directly .this case is similar to the case 2 .the arbitrator will assign to two agents .  from table 3, it can be seen that and are two nash equilibria , and the former is pareto - efficient .as specified by telser ( page 28 , line 2 , ) , `` _ _ a party to a self - enforcing agreement calculates whether his gain from violating the agreement is greater or less than the loss of future net benefits that he would incur as a result of detection of his violation and the consequent termination of the agreement by the other party . _ _ '' since two channels have been controlled by the computer in stage 1 , in the end is a self - enforcing nash equilibrium and the pareto - efficient payoff is the unique nash equilibrium outcome . in this sense ,the two agents escape the dilemma .  * type-5 pd * : + conditions 1 - 3 are the same as those in type-4 pd .+ 4 ) the last condition of type-4 pd does not hold .+ for this case , although the two agents can communicate before moving and agree that collaboration is good for each agent , they will definitely choose ( _ defect _ , _ defect _ ) as if they are separated .thus , the agents can not escape the dilemma .the algorithmic model revises common understanding on the one - shot pd . herewe will discuss some possible doubts about it .  _q1 _ : the type-4 pd seems to be a cooperative game because in condition 4 , the algorithmic model constructed by two agents acts as a correlation between agents .  _a1 _ : from the viewpoints of agents , the game is different from the original one - shot pd , since the payoff matrix confronted by the two agents has been changed from table 1 to table 3 . but from the viewpoints of the arbitrator , nothing is changed .thus , the so - called correlation between two agents is indeed _unobservable _ to the arbitrator .put differently , the arbitrator can not prevent agents from constructing the algorithmic model .+ on the other hand , since each agent can freely choose not to participate the algorithmic model and send a message to the arbitrator directly in stage 1 , the algorithmic model is self - enforcing and thus still a non - cooperative game .  _ q2 _ : after the algorithmic model is triggered , can it simply send , to the arbitrator instead of running steps 1 - 5 ?  _ a2 _ : the algorithmic model enlarges each agent s strategy space from the original strategy space _ \{cooperate , defect } _ to a two - dimensional strategy space \times[0,\pi/2]$ ] , and generates the pareto - efficient payoff in nash equilibrium .the enlarged strategy space includes the original strategy space of one - shot pd : the strategy ( _ cooperate , cooperate _ ) , ( _ cooperate , defect _ ) , ( _ defect , cooperate _ ) , ( _ defect , defect _ ) in the original pd correspond to the strategy , , , in the algorithmic model respectively , since , .+ however , the idea in this question restricts each agent s strategy space from the original strategy space _ \{cooperate , defect } _ to a single strategy _cooperate_. in this sense , two agents are required to sign a binding contract to do so .this is beyond the range of non - cooperative game ._ remark 2 _ : the algorithmic model is not suitable for type-1 and type-2 pd , because the computer can not perform actions on behalf of agents .the algorithmic model is not applicable for type-3 pd either because two agents are separated , thereby the algorithmic model can not be constructed . for the case of type-5 pd , the algorithmic model is not applicable because condition 4 in type-4 pd is vital and indispensable .in this paper , we categorize the well - known one - shot pd into five types and propose an algorithmic model to help two non - cooperative agents self - enforcingly escape a special type of pd , i.e. , the type-4 pd .the type-4 pd is justified when the arbitrator communicate with the agents indirectly through some channels , and each agent s strategy is not an action , but a message that can be sent to the arbitrator . with the rapid development of internet , more and more type-4 pd games will be seen .one point is important for the novel result : usually people may think the two payoff matrices confronted by agents and the arbitrator are the same ( i.e. , table 1 ) .however we argue that for the case of type-4 pd , the two payoff matrices can be different : the arbitrator still faces table 1 , but the agents can self - enforcingly change their payoff matrix to table 3 by virtue of the algorithmic model , which leads to a pareto - efficient payoff .the author is very grateful to ms .fang chen , hanyue wu ( _ apple _ ) , hanxing wu ( _ lily _ ) and hanchen wu ( _ cindy _ ) for their great support .

Abstract: the prisoner s dilemma is a simple model that captures the essential contradiction between individual rationality and global rationality . although the one - shot prisoner s dilemma is usually viewed simple , in this paper we will categorize it into five different types . for the type-4 prisoner s dilemma game , we will propose a self - enforcing algorithmic model to help non - cooperative agents obtain pareto - efficient payoffs . the algorithmic model is based on an algorithm using complex numbers and can work in macro applications .   prisoner s dilemma ; non - cooperative games .


several important real - world processes can be conceptualised as a series of events occurring in time and space , possibly spontaneously , where each event has the capacity to trigger subsequent events .for example , events in epidemiology correspond to the infection of an individual by a transmissible disease ; the infected individual may then go on to cause subsequent infections . in many settings it is possible to obtain large , detailed data that catalogue the occurrence of events but do not specify individual cause - effect relationships .for example , in epidemiology it is generally not possible to identify the cause of a specific infection .cascades of poisson processes ( copp ) have been successfully applied to facilitate inference and prediction in this setting .copp were originally developed by the geological community in the 1980s for statistical modelling of earthquake / aftershock data .recent and varied applications of copp models have included modelling the dynamics of retaliatory gang violence , terrorist activity , success of commercial book sales , military conflicts , disease at the cellular level , `` retweet cascades '' on the social network twitter and editing patterns on wikipedia .  in mathematical terms, copp may be viewed as branching processes with immigration .inference in general branching processes is challenging since complex , nonlinear , multidimensional models do not generally admit closed form solutions for estimators .moreover copp models can produce multimodal or very flat log - likelihood functions , precluding reliable numerical algorithms . to overcome these difficulties, viewed inference as an incomplete data problem and adapted the expectation - maximisation ( em ) algorithm to this setting . in brief ,the information about which events arose from the background , which events were triggered and what those triggers were ( collectively the `` branching structure '' ) is required to specify the likelihood , but is unobservable . by describing the branching structure probabilistically , the em algorithm seeks instead to maximise the expected likelihood , which can be achieved for certain parametric copp models ( see for examples ) .  in the absence of a mechanistic understanding of either the background or triggering processes ,nonparametric estimators play an important role . however showed that these estimators can exhibit large bias at small - to - moderate sample sizes ( ) ; it is therefore important to integrate as many samples as possible into nonparametric inference .unfortunately existing algorithms scale poorly , with computational and storage complexity of existing approaches being .for example , on earthquake data reported using a sample size of due to computational limitations , whereas seismological datasets now regularly exceed  .the `` big data '' now available in many application areas for copp models delivers a pressing need to develop scalable nonparametric estimation procedures .  in this paperwe develop accelerated nonparametrics for copp .our approach is based on truncation of kernel density estimators ( kdes ) to include only terms which are temporally and spatially local to an event of interest .the domains of the kdes are chosen adaptively based on nearest neighbour distances , rendering the approach essentially free from hyperparameters .the complete algorithm , described in section [ methods ] , enjoys computational complexity of just and requires storage .using seismological data , we demonstrate in section [ results ] that our estimators facilitate the analysis of larger datasets than was previously possible and provide an empirical assessment of performance in this setting .section [ discuss ] closes with a discussion of challenges and extensions to our methodology .we proceed as follows : sections [ notations ] and [ defs ] set up notation and formally define the copp model class .section [ algorithms ] surveys existing algorithms and nonparametric estimators for copp .section [ core ] contains the core of our methodology for scaling inference to large datasets .consider a metric space formed as the finite product where each coordinate is assumed to be a locally compact metric space . here is a product metric ; in this paper all product metrics are induced by the euclidean norm , so that for .consider a simple point process on adapted to a filtration .for clarity we equip the time domain with the euclidean metric , though an arbitrary locally compact metric may be used .write for the open ball of radius centred on .a measure on , known as the conditional intensity of the point process , is given by in this paper will be the natural filtration ; the history up to time , i.e. where is the time and are additional coordinates associated with event ( e.g. spatial coordinates ) .it can be shown that the finite - dimensional distributions of are characterised by the conditional intensity function ; in this paper inference and prediction for is carried out entirely through inference for .( we assume is such that the associated measure is finite on every compact subset of and contains no atoms , so that eqn .[ limit ] is well - defined .)   this paper concerns the class of point processes known variously as branching processes with immigration , copp models , `` self - exciting point processes '' , `` epidemic type aftershock sequences '' and `` hawkes processes '' .such models are characterised by a conditional intensity which is expressed as a superposition of point processes : here denotes a ( possibly time - varying ) background intensity and each represents a `` triggering '' intensity , giving rise to offspring from the trigger event . in this formulation ,all events are able , in principle , to trigger offspring at future times ; the generative model therefore can give rise to `` cascades '' of events , which explains the copp nomenclature ( fig . [ illustrate ] ) . estimation for copp modelsis equivalent to estimation for the background ( ) and triggering ( ) intensities . in an application to earthquake databelow , we model a time - independent background intensity , but for completeness we present the general case of time - varying background intensities .   in applications of copp models , data obtained over a time interval ] for thresholds .moreover , unlike , our methodology provides a mechanism to select adaptively , by implicitly solving for , thereby mitigating an important practical issue .the truncation parameter is not a hyperparameter _ per se _ and should be chosen sufficiently large that any further increase in leads to negligible variation in the estimated intensity functions , or indeed any derived quantities of interest .our preliminary empirical investigation demonstrated inferential and predictive performance that was competitive with parametric estimators , but also revealed systematic downward bias in estimation of reproductive ratios .this may be because the introduction of nonparametric uncertainty into the trigger function raises the evidence threshold to conclude that an event was triggered .further research will be required to address this methodological issue ; indeed , this contribution suggests a number of interesting extensions that are made possible by accelerated computation ; ( i ) reformulating the proposed estimators within the bayesian framework , allowing for ( a ) sequential updating of estimators , as new data arrive , and ( b ) regularising the reproductive ratio via a prior distribution , ( ii ) incorporating observation noise into kde , ( iii ) introducing latent variables to account for missing occurrence data , and ( iv ) migrating nearest neighbour computation to gpus .  whilst we focused on the popular class of processes with continuous state space , there exist a number of additional techniques to reduce computational complexity in discrete state spaces ( e.g. defined by networks ) ; see for details .  99 blundell , c. , beck , j. and heller , k.a .( 2012 ) modelling reciprocating relationships with hawkes processes ._ in proceedings of the 27th conference on advances in neural information processing systems ( nips ) _ , 2609 - 2617.          pan , j. , lauterbach , c. and manocha , d. ( 2010 ) efficient nearest - neighbor computation for gpu - based motion planning ._ in ieee / rsj international conference on intelligent robots and systems ( iros ) _ , 2243 - 2248 .sornette , d. and utkin , s. ( 2009 ) limits of declustering methods for disentangling exogeneous from endogeneous events in time series with foreshocks , main shocks and aftershocks ._ phys rev e _ * 79*:061110 .

Abstract: cascades of poisson processes are probabilistic models for spatio - temporal phenomena in which ( i ) previous events may trigger subsequent events , and ( ii ) both the background and triggering processes are conditionally poisson . such phenomena are typically `` data rich but knowledge poor '' , in the sense that large datasets are available yet a mechanistic understanding of the background and triggering processes which generate the data are unavailable . in these settings nonparametric estimation plays a central role . however existing nonparametric estimators have computational and storage complexity , precluding their application on large datasets . here , by assuming the triggering process acts only locally , we derive nonparametric estimators with computational complexity and storage complexity . our approach automatically learns the domain of the triggering process from data and is essentially free from hyperparameters . the methodology is applied to a large seismic dataset where estimation under existing algorithms would be infeasible .


how a disease spreads in a population is a question of much interest and relevance , and consequently has been extensively explored over the years .mathematically , epidemiological models have successfully captured the dynamics of infectious disease .one well known model for non - fatal communicable disease progression is the sirs cycle .this model appropriately describes the progression of diseases such as small pox , tetanus , influenza , typhoid fever , cholera and tuberculosis .the sirs cycle is described by the following stages . at the outsetan individual is _ susceptible _ to infection ( a stage denoted by s ) . on being infected by contact with other infected people in the neighbourhood , the individual moves on to the _ infectious _ stage ( denoted by i ) .this is followed by a_ refractory _ stage ( denoted by r ) . in the refractory stagethe individual becomes immune to disease and also does not infect others . butthis immunity is temporary as the individual becomes susceptible again after some time interval .specifically , in this work we consider a cellular automata model of the sirs cycle described above . in this model of disease progression, time evolves in discrete steps , with each individual , indexed by on a dimensional lattice , characterized by a counter    describing its _ phase _ in the cycle of the disease .here , where signifies the total length of the disease cycle . at any instant of time _, if phase (t ) = , then the individual at site is susceptible ; if , then it is infected ; if phase , it is in the refractory stage . for , phase the dynamics is given by the counter updating by every time step , and at the end of the refractory period the individual becomes susceptible again , which implies if then , .namely :   hence the disease progression is a _ cycle _ ( see fig.[schematic ] ) .we consider the typical condition where the refractory period is longer than the infective stage , i.e. .we now investigate the spread of epidemic in a group of spatially distributed individuals , where at the individual level the disease progresses in accordance with the sirs cycle . in particular , we consider a population of individuals on a -dimensional lattice where every node , representing the individual , has neighbors . unlike many studies with periodic boundary conditions , here the boundaries are fixed and there are no individuals outside the boundaries .so our model mimics a patch of population , and investigates the persistence of infection in such a patch ._ condition for infection : _ here we consider the condition that a susceptible individual ( s ) will become infected ( i ) _ if one or more of its nearest neighbours are infected_. that is , if , ( namely , the individual is susceptible ) , then , if any where .we first focus on the infection spreading patterns in the population .the principal question we ask is the following : when is infection persistent in a patch , and how this depends on the constitution of the initial population . in order to examine this, we study the spread of infection from a seed of infection ( namely one or two infected individuals ) across a patch of population composed of individuals at different stages in the disease cycle , and with varying degrees of heterogeneity in the population .  with no loss of generality we consider ; ; and a lattice of size . in our figureswe represent the state of an individual in the disease cycle ( namely s , i or r ) by a color , with white denoting a susceptible individual , black denoting a refractory individual and red denoting an infected individual .the fraction of susceptible individuals in the population at time is denoted by , the fraction of infected individuals by and the fraction of refractory individuals by . in the sections below wewill focus on the possibility of the prolonged existence of infection arising in different classes of initial populations , characterized by different , and .first we investigate the effect of an infected individual on a population patch where _ all individuals are entirely susceptible to infection_. namely , we consider the case where at the outset there is one infected individual and the rest of the population is in the susceptible state , with . fig .[ control ] displays the spreading patterns obtained in such a scenario .it is clearly evident that as time progresses the infection starts from the infected individual ( `` seed '' ) and spreads symmetrically .this symmetric spreading pattern is readily understood from the condition for infection , which turns susceptible individuals to infected if any one of their neighbors is infected .so the infected seed infects its four neighbors , and these newly infected individuals in turn infect their nearest four neighbours , and so on .this process leads to an isotropic wave of infection which stops at the boundaries .we confirmed the generality of these observations for different relative lengths of the infectious and refractory periods , namely varying and ( with ) .we further ascertained that the choice of the location of the infected individual did not affect these qualitative trends .now the key factor in infection spreading is the contact of susceptible individuals with infected ones .it is clear that such an interaction takes place only at the outer edge of the wave of infection , while the inner boundary of the infected zone is contiguous only to refractory individuals .so the infection only spreads outwards , and does not move back into the interior of the lattice again .importantly then , the infection is removed after a while from the patch of population , and all the individuals ( including the original infective seed ) comes to the end of the disease cycle and becomes susceptible again .so there is no infective site left in the population to perpetuate the infection and initiate another wave of disease spreading .thus a _ fully susceptible population does not allow the infection to persist . _, showing the spread of infection from one infected individual at , in a homogeneous initial population comprising entirely of susceptible individuals ( i.e. , , ) .the long bar shows the relative lengths of the susceptible ( s ) , infected ( i ) and refractory ( r ) stages in the disease cycle , where , and the total disease cycle is ( see text ) .the red box shows the fraction of s , i and r individuals in the population at that instant of time.,scaledwidth=100.0% ]   next we investigate the infection spread in the more realistic scenario where both refractory ( ) and susceptible individuals ( ) are present in the initial population , and are randomly distributed spatially .we first consider the case where the refractory individuals have phases , namely , they are at the start of the refractory stage of the disease cycle .we investigate the persistence of infection in heterogeneous populations , with the initial state having ( a ) a single seed of infection and ( b ) varying initial fractions of infected individuals . in both scenarios ,we analyze the effect of varying and on the persistence of infection .  to begin with , in fig .[ spread_s = r ] , we illustrate the effect of a single infected individual on an initial population with equal numbers of susceptible and refractory individuals , namely .it is evident from these representative results that in a well mixed population , consisting of a random collection of both susceptible and refractory individuals , introduction of a single infected individual can lead to _ persistent infection in the population ._  this can be rationalized as follows : the mixed presence of susceptible and refractory individuals , implies that the disease cycles of the individuals in the population are _ not synchronized_. so there are always some individuals in the infective stage of the disease cycle in the population , and these act as seeds for continued infection propagation , leading to persistent infection . _ counter - intuitively then , the presence of individuals who are ( temporarily ) immune to the disease amongst susceptible ones leads to sustained infection , while in a completely susceptible population the infection dies out . _  , and ) , with one infected individual at . herethe refractory individuals have phases ( namely , they are at the start of the refractory stage of the disease cycle ) .again , the long bar shows the relative lengths of the susceptible ( s ) , infected ( i ) and refractory ( r ) stages in the disease cycle , where , and the total disease cycle is ( see text ) .the red box shows the fraction of s , i and r individuals in the population at that instant of time .interestingly , the spatially random population evolves into a more regular pattern after a short transient time . , scaledwidth=100.0% ]  next we focus on the time evolution of an initial population consisting of a random mixture of , and states .in particular we investigate the nature of the persistent infection in the population under varying initial fractions of infected individuals .a typical random initial condition is shown in fig .[ time1 ] , with the initial fraction of infected sites being one - tenth and the initial fraction of susceptibile and refractory individuals being equal ( i.e. ) . heretoo we find that infection is sustained .   and . herethe refractory individuals have phases ( namely , they are at the start of the refractory stage of the disease cycle ) .again , the long bar shows the relative lengths of the susceptible ( s ) , infected ( i ) and refractory ( r ) stages in the disease cycle , where , and the total disease cycle is ( see text ) . the red box shows the fraction of s , i and r individuals in the population at that instant of time.,scaledwidth=100.0% ]  , , , in a heterogeneous population comprising initially of a random mixture of individuals , with and .,scaledwidth=60.0% ]  further , interestingly ,it is clear that there is an _ approximate recurrence of the complex patterns of infected individuals in the population_. fig .[ time2 ] shows the time evolution of the fraction of infected , refractory and susceptible individuals in the population , namely , and , in the case displayed in fig .[ time1 ] .it can be clearly seen that after transience , , and exhibit steady oscillatory dynamics , with period of oscillation close to the disease cycle length .this is consistent with the observed recurrence of the spatio - temporal patterns when persistent infection emerges .  a quantitative measure of the recurrence of patternscan also be obtained by calculating the difference of the state of the population from the initial state , as reflected by the hamming distance : where the sum is over all sites in the lattice .the time dependence of the hamming distance given above is shown in fig .[ hamming ] , and it clearly shows steady oscillations .this indicates that the fraction of the susceptible , infected and refractory individuals in the population , and more remarkably their _ locations _ , repeat almost periodically over time .it should be noted that the frequency of oscillation again approximately corresponds to the disease cycle length .   as a function of time , in a heterogeneous population comprising initially of a random mixture of individuals , with and .,scaledwidth=60.0% ]another pertinent observation here is the dependence of this dynamics on disease cycle. as the length of the infectious stage ( i.e. ) increases , keeping the total disease cycle length invariant , the fraction of infected individuals increases .the average is proportional to the fraction of the disease cycle occupied by the infectious stage , i.e the ratio .so the size of the infected population strongly depends on the nature of disease as reflected in the length of the infectious stage of the disease .we now attempt to gauge the statistically significant trends in , by averaging the fraction of infected individuals at asymptotic time , arising from a wide range of initial configurations at time .we denote this by . in terms of this quantity ,persistent infection is indicated by a non - zero value . however , after sufficient transient timesteps , if is zero , it indicates that the infection has died out .so can serve as an order parameter for the transition to sustained infection in a population .   for fixed and we have calculated , for different initial fractions of susceptible individuals .we explore the full possible range of  ] . for the remaining population, the initial fraction of susceptible and refractory individuals is set at different ratios .we consider an ensemble of initial conditions , with specific , and and find the time averaged , after long transience for each realization .the ensemble average of this quantity is displayed in fig .notably , we find that there is a definite _ window of persistence _ over the range of , where the infection never dies down and the fraction of infected individuals in the population is reasonably high on an average .( after transience ) with respect to the initial fraction of infected individuals in the population , and .the refractory sub - population consists of individuals with phase equal to . herethe disease cycle has ; , and is averaged over initial realizations .the specific case of a lattice is displayed .however note that different lattices sizes yield the same result.,scaledwidth=75.0% ]  in the state where infection is persistent , the individuals are unsynchronized and spread over the different stages of the disease cycle .so on an average the fraction of infected individuals is , namely the fraction of the total disease cycle occupied by the infected stage .for instance , in the example shown in fig .[ i0 ] with and , at the plateau of persistence , the infected fraction is approximately one - third of the population .the transition to persistent infection is sharp and occurs at .this implies that _ the infection can spread and persist even when there is only a single infected individual in the initial population_. this is consistent with the results we presented earlier ( cf .[ variation1 ] ) on infection spreading from a single infected individual .interestingly , the infection ceases to persist for higher values of , and the fall in persistence is rapid .that is , if the initial population has too many infected individuals , infection will not persist .this can be rationalized by noting that one needs a mix of susceptibles and refractory individuals in the population for persistent infection .for instance , considering the limiting case of all infected individuals in the initial state , it is clear that all individuals will go through the disease cycle in synchrony .so all individuals will become susceptible again together , but there will be no infective seed left in the population to perpetuate the infection .now we will explore the effect of non - uniformity within the refractory sub - population on the emergence of persistent infections .namely , we will consider the refractory individuals in the initial population to be in different stages of disease progression .we will consider two distinct ways of interpolating between the completely heterogeneous and completely uniform limiting cases , in order to gauge the effect of heterogeneity on sustaining infection .first we consider the initial refractory sub - population to be an admixture of subsets of individuals with uniform phase and with randomly distributed phases .specifically , we explore initial refractory sub - populations comprised of some fraction with phases randomly distributed over the range to , and the rest with fixed phase .we examine the spread and persistence of infection in such a scenario , under variation of the initial composition of the population .fig [ transition - p ] exhibits the persistence of infection , with respect to varying , arising in a population that had a single infected individual initially .different fractions of the initial refractory sub - population with randomized phases were explored , ranging from ( i.e. completely uniform ) , to ( i.e. completely heterogeneous ) .the trends clearly indicate a continuous cross - over from the condition where all refractory individuals are in the same phase , to the scenario where all are in random phases .( after transience ) with respect to initial fraction of susceptible individuals , for different fractions of the initial refractory sub - population having randomly distributed phases ( see key ) .here the disease cycle has ; , and is averaged over initial realizations and lattice size is .,scaledwidth=75.0% ]   ( after transience ) with respect to the initial fraction of infected individuals in the population , and .the initial refractory sub - population consists of different fractions with randomly distributed phases ( see key ) . herethe disease cycle has ; , and is averaged over initial realizations . while the specific case of a lattice is displayed , different lattices sizes yield the same result.,scaledwidth=75.0% ]  further , we explore the effect of varying the initial fraction of infected individuals , over the range  ] , ( where all refractory individuals have the same phase ) to [ , ( where heterogeneity is large as the phases of the refractory individuals are distributed over the entire refractory range ) .  fig .[ transition - r-1 ] exhibits representative results of as a function of the initial fraction of susceptibles , for the case where there is a single infected individual in the population at the outset .it can be clearly seen that a smooth cross - over takes place from the extremal case of all refractory individuals in the same phase , to the limit where the stages of the refractory individuals are spread randomly over the entire refractory period .the key observation here is that as the spread in phases increases , the range of persistent infection becomes larger .namely , when there is a large initial spread in the stages of disease among the individuals , at subsequent times there are always some individuals who can `` pick up the baton of infection '' , leading to persistent infection .( after transience ) with respect to initial fraction of susceptible individuals , for the refractory individuals having phases randomly distributed over different ranges in the refractory stage : [ 5,5 ] ; [ 5,7 ] ; [ 5,9 ] ; [ 5,11 ] ; [ 5,13 ] .here is averaged over realizations , lattice size is , and the disease cycle parameters , .,scaledwidth=75.0% ]   ( after transience ) with respect to initial fraction of infected individuals , for the refractory individuals having phases randomly distributed over different ranges in the refractory stage : [ 5,5 ] ; [ 5,7 ] ; [ 5,9 ] ; [ 5,11 ] ; [ 5,13 ] . here is averaged over realizations , lattice size is , and the disease cycle parameters , .,scaledwidth=65.0%,scaledwidth=45.0% ]  so we see that in the completely heterogeneous case , low susceptible and high refractory initial subpopulations favour persistent infection .but in a completely uniform population , a higher fraction of susceptibles leads to persistent infection .this has the following important implication : when refractory individuals are not synchronized at the same phase of disease progression , even if there are few susceptible individuals in the population initially , the infection grows substantially and the average size of the infected sub - population is large .so we have demonstrated that even when the entire population is susceptible to infection , the infection eventually dies out , while even a few susceptibles among an heterogeneous refractory population gives rise to a large persistent infected sub - population .  we can rationalize this counter - intuitive trend that persistent infection is more likely when the number of susceptible individuals in the initial population is low , as follows :when is low , there are many refractory individuals in the population surrounding the infected individual .these individuals are in various stages in the refractory period , and some become susceptible again while the seed is still infectious .if and the refractory individuals are uniformly distributed over the refractory range , the probability of the seed encountering a susceptible individual while still infectious is proportional to .since at least one neighbour in contact with the seed needs to be susceptible , this probability should be greater than for the infection to spread , on an average .so when the infective stage is sufficiently long ( as in our example of , in a disease cycle of length ) , extremely low initial can also lead to persistent infection .in summary , we have explored the emergence of persistent infection in a patch of population , where the disease progression of the individuals was given by the sirs model and an individual became infected on contact with another infected individual .we investigated the infection spreading qualitatively and quantitatively , under varying degrees of heterogeneity in the initial population .  specifically , we considered two scenarios extrapolating between the completely homogeneous and completely heterogeneous limit .one considers varying fractions of heterogeneous sub - populations and another examines varying ranges in the spread of the stages of disease progression . our central result is the following : we find that an infectious seed does not give rise to persistent infection in a homogeneous population consisting of individuals at the same stage of disease progression .rather , when the population is heterogeneous , and consists of randomly distributed individuals at various stages of the disease , infection becomes persistent in the population patch .the key to persistent infection is then the random admixture of refractory and susceptible individuals , leading to de - synchronization of the phases in the disease cycle of the individuals .so we have demonstrated that when the entire population is susceptible to infection , the infection eventually dies out , while even a few susceptibles among an heterogeneous refractory population gives rise to a large persistent infected sub - population .

Abstract: we explore the emergence of persistent infection in a patch of population , where the disease progression of the individuals is given by the sirs model and an individual becomes infected on contact with another infected individual . we investigate the persistence of contagion qualitatively and quantitatively , under varying degrees of heterogeneity in the initial population . we observe that when the initial population is uniform , consisting of individuals at the same stage of disease progression , infection arising from a contagious seed does not persist . however when the initial population consists of randomly distributed refractory and susceptible individuals , a single source of infection can lead to sustained infection in the population , as heterogeneity facilitates the de - synchronization of the phases in the disease cycle of the individuals . we also show how the average size of the window of persistence of infection depends on the degree of heterogeneity in the initial composition of the population . in particular , we show that the infection eventually dies out when the entire initial population is susceptible , while even a few susceptibles among an heterogeneous refractory population gives rise to a large persistent infected set .