why tsallis statistics abstract it is a remarkable fact that the traditional teaching of thermodynamics as reflected in the textbooks and including the long developments about ensembles and thermodynamic functions is almost entirely about systems in equilibrium the time variable does not enter there is one exception however the single most important item the flagship of the thermodynamic navy the second law is about the irreversibility of the time evolution of systems out of equilibrium this is a bizarre situation to say the least a glaring case of the drunk man looking for his key under the lamp post when he knows that he lost it in the dark part of the street the moment has come for us to go looking in the dark part the behavior of systems as a function of time we have been given a powerful new flashlight chaos theory we should use it there on the formerly dark pavement we can find tsallis statistics

nonadditive conditional entropy and its significance for local realism based on the form invariance of the structures given by khinchin s axiomatic foundations of information theory and the pseudoadditivity of the tsallis entropy indexed by q the concept of conditional entropy is generalized to the case of nonadditive nonextensive composite systems the proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context indicating its utility for characterizing quantum entanglement a criterion deduced from it for separability of density matrices for validity of local realism is examined in detail by employing a bipartite spin 1 2 system it is found that the strongest criterion is obtained in the limit q

tutorial on seismic interferometry part 2 underlying theory and new advances in the 1990s the method of time reversed acoustics was developed this method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal when ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers they focus at the position of the original source even when the medium is very complex in seismic interferometry the time reversed responses are not physically sent into the earth but they are convolved with other measured responses the effect is essentially the same the time reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers a mathematical derivation based on reciprocity theory formalizes this principle the crosscorrelation of responses at two receivers integrated over different sources gives the green s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver this green s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving recent developments circumventing these assumptions include interferometric representations for attenuating and or moving media as well as unified representations for waves and diffusion phenomena bending waves quantum mechanical scattering potential fields elastodynamic electromagnetic poroelastic and electroseismic waves significant improvements in the quality of the retrieved green s functions have been obtained with interferometry by deconvolution a trace by trace deconvolution process compensates for complex source functions and the attenuation of the medium interferometry by multidimensional deconvolution also compensates for the effects of one sided and or irregular illumination

derivative pricing with non linear fokker planck dynamics abstract we examine how the black scholes derivative pricing formula is modified when the underlying security obeys non extensive statistics and fokker planck dynamics an unusual feature of such securities is that the volatility in the underlying ito langevin equation depends implicitly on the actual market rate of return this complicates most approaches to valuation here we show that progress is possible using variations of the cox ross valuation technique

on seismic interferometry the generalized optical theorem and the scattering matrix of a point scatterer we have analyzed the far field approximation of the green s function representation for seismic interferometry by writing each of the green s functions involved in the correlation process as a superposition of a direct wave and a scattered wave the green s function representation is rewritten as a superposition of four terms when the scattered waves are modeled with the born approximation it appears that a three term approximation of the green s function representation omitting the term containing the crosscorrelation of the scattered waves yields a nearly exact retrieval whereas the full four term expression leads to a significant nonphysical event this is because the born approximation does not conserve energy and therefore is an insufficient model to explain all aspects of seismic interferometry we use the full four term expression of the green s function representation to derive the generalized optical theorem unlike other recent derivations which use stationary phase analysis our derivation uses reciprocity theory from the generalized optical theorem we derive the nonlinear scattering matrix of a point scatterer this nonlinear model accounts for primary and multiple scattering at the point scatterer and conforms with well established scattering theory of classical waves the model is essential to explain fully the results of seismic interferometry even when it is applied to the response of a single point scatterer the nonlinear scattering matrix also has implications for modeling inversion and migration

hyperinflation in brazil israel and nicaragua revisited abstract the aim of the present work is to address the description of hyperinflation regimens in economy the spirals of hyperinflation developed in brazil israel and nicaragua are revisited this new analysis of data indicates that the episodes occurred in brazil and nicaragua can be understood within the frame of the model available in the literature which is based on a nonlinear feedback nlf characterized by an exponent 0 in the nlf model the accumulated consumer price index carries a finite time singularity of the type 1 t c t 1 determining a critical time t c at which the economy would crash it is shown that in the case of brazil the entire episode cannot be described with a unique set of parameters because the time series was strongly affected by a change of policy this fact gives support to the so called lucas critique who stated that model s parameters usually change once policy changes on the other hand such a model is not able to provide any t c in the case of the weaker hyperinflation occurred in israel it is shown that in this case the fit of data yields 0 this limit leads to the linear feedback formulation which does not predict any t c an extension for the nlf model is suggested

model of macroeconomic evolution in stable regionally dependent economic fields abstract we develop a model for the evolution of economic entities within a geographical type of framework on a square symmetry lattice made of three economic regions firms described by a scalar fitness are allowed to move adapt merge or create spin offs under predetermined rules in a space and time dependent economic environment we only consider here one timely variation of the external economic field condition for the firm fitness evolution we take into account a constraint such that the disappearance of a firm modifies the fitness of nearest neighboring ones as in bak sneppen population fitness evolution model the concentration of firms the averaged fitness the regional distribution of firms and fitness for different time moments the number of collapsed merged and new firms as a function of time have been recorded and are discussed also the asymptotic values of the number of firms present in the three regions together with their average fitness as well as the number of respective births and collapses in the three regions are examined it appears that a sort of critical selection pressure exists a power law dependence signature of self critical organization is seen in the birth and collapse asymptotic values for a high selection pressure only a lack of self organization is also seen at region borders

universality in the chaotic dynamics associated with saddle centers critical points abstract we describe a new statistical pattern characteristic of saddle center critical points in the dynamics of non integrable hamiltonian systems associated with the partition of the hamiltonian into rotational motion energy and hyperbolic motion energy pieces in a linear neighborhood of the saddle center the distribution of hyperbolic energy of orbits visiting a neighborhood of the saddle center defines a statistical law that for a large range of values of the hyperbolic energy has the form p z z in a regime of high nonintegrability we present numerical evidence to support the conjecture that this pattern is universal for any hamiltonian system with one or several saddle centers the parameter appears as a universal characteristic associated with the chaotic dynamics of saddle centers

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

option pricing formulas based on a non gaussian stock price model options are financial instruments that depend on the underlying stock we explain their non gaussian fluctuations using the nonextensive thermodynamics parameter q a generalized form of the black scholes bs partial differential equation and some closed form solutions are obtained the standard bs equation q 1 which is used by economists to calculate option prices requires multiple values of the stock volatility known as the volatility smile using q 1 5 which well models the empirical distribution of returns we get a good description of option prices using a single volatility

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

contact forces in anisotropic frictional granular materials the probability density functions pdfs of contact forces in anisotropic cohesionless and frictional granular materials are studied numerically and theoretically using discrete element simulations of biaxial deformation of a large two dimensional assembly consisting of 200 000 disks it is observed that the pdfs for the normal and tangential components of the contact forces depend significantly on contact orientation the pdfs exhibit exponential decay and the pdf for the tangential component of the contact forces is not always symmetrical with respect to zero tangential force the shape of the pdf for the normal component of the contact forces changes with shear strain a qualitative explanation for this change is given that is related to the biaxial deformation mechanism in which the disrupted contacts are predominantly oriented in the direction of the minor principle stress a maximum entropy method is employed to study these pdfs theoretically using a prescribed stress tensor as constraint it is found that the theoretical results correspond qualitatively to many of the results obtained from the discrete element simulations discrepancies between theory and simulations are attributed to the fact that the kinematics have not been taken into account in the theory

granular contact force density of states and entropy in a modified edwards ensemble a method has been found to analyze edwards granular contact force probability functional for a special case as a result the granular contact force probability density functions are obtained from first principles for this case the results are in excellent agreement with the experimental and simulation data the derivation assumes edwards flat measure a density of states dos that is uniform within the metastable regions of phase space the enabling assumption supported by physical arguments and empirical evidence is that correlating information is not significantly recursive through loops in the packing maximizing a state counting entropy results in a transport equation that can be solved numerically for the present this has been done using the mean structure approximation projecting the dos across all angular coordinates to more clearly identify its predominant nonuniformities these features are 1 the grain factor psi related to grain stability and strong correlation between the contact forces on the same grain and 2 the structure factor upsilon related to newton s third law and strong correlation between neighboring grains

characterization of gaussian self similar stochastic processes using wavelet based informational tools efficient tools to characterize stochastic processes are discussed quantifiers originally proposed within the framework of information theory like entropy and statistical complexity are translated into wavelet language which renders the above quantifiers into tools that exhibit the important localization advantages provided by wavelet theory two important and popular stochastic processes fractional brownian motion and fractional gaussian noise are studied using these wavelet based informational tools exact analytical expressions are obtained for the wavelet probability distribution finally numerical simulations are used to validate our analytical results

generalized optical theorem for reflection transmission and extinction of power for scalar fields we present a derivation of the optical theorem that makes it possible to obtain expressions for the extinguished power in a very general class of problems not previously treated the results are applied to the analysis of the extinction of power by a scatterer in the presence of a lossless half space applications to microscopy and tomography are discussed

general representations for wavefield modeling and inversion in geophysics acoustic electromagnetic elastodynamic poroelastic and electroseismic waves are all governed by a unified matrix vector wave equation the matrices in this equation obey the same symmetry properties for each of these wave phenomena this implies that the wave vectors for each of these phenomena obey the same reciprocity theorems by substituting green s matrices in these reciprocity theorems unified wavefield representations are obtained analogous to the well known acoustic wavefield representations these unified representations find applications in geophysical modeling migration inversion multiple elimination and interferometry

chaos suppression in the large size limit for long range systems we consider the class of long range hamiltonian systems first introduced by anteneodo and tsallis and called the xy model this involves n classical rotators on a d dimensional periodic lattice interacting all to all with an attractive coupling whose strength decays as r r being the distance between sites using a recent geometrical approach we estimate for any d dimensional lattice the scaling of the largest lyapunov exponent lle with n as a function of in the large energy regime where rotators behave almost freely we find that the lle vanishes as n with 1 3 for 0 d 1 2 and 2 3 1 d for 1 2 d 1 these analytical results present a nice agreement with numerical results obtained by campa et al including deviations at small n

the mechanism of double exponential growth in hyper inflation abstract analyzing historical data of price indices we find an extraordinary growth phenomenon in several examples of hyper inflation in which price changes are approximated nicely by double exponential functions of time in order to explain such behavior we introduce the general coarse graining technique in physics the monte carlo renormalization group method to the price dynamics starting from a microscopic stochastic equation describing dealers actions in open markets we obtain a macroscopic noiseless equation of price consistent with the observation the effect of auto catalytic shortening of characteristic time caused by mob psychology is shown to be responsible for the double exponential behavior

simple models for granular force networks abstract a remarkable feature of static granular matter is the distribution of force along intricate networks even regular inter particle contact networks produce wildly inhomogeneous force networks where certain chains of particles carry forces far larger than the mean although the full information of each inter particle contact requires analysis of the stress tensor potentially including friction and elasticity there is a rich tradition of approximating these interactions with simple scalar models that focus on the geometry of the system in this paper we briefly review past theoretical approaches to understanding the geometry of force networks we then investigate the structure of experimentally derived granular force networks using a simple algorithm to obtain corresponding graphs we compare our observations with the results of geometric models including random bond percolation which show similar spatial distributions without enforcing vector force balance our findings suggest that some aspects of the mean geometry of granular force networks may be captured by these simple descriptions

empirical regularities of opening call auction in chinese stock market abstract we study the statistical regularities of an opening call auction using the ultra high frequency data of 22 liquid stocks traded on the shenzhen stock exchange in 2003 the distribution of the relative price defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price the detrended fluctuation analysis dfa method is adopted to investigate the long term memory of relative order prices we further study the statistical regularities of order sizes in the opening call auction and observe a phenomenon of number preference known as order size clustering the probability density function pdf of order sizes could be well fitted by a q gamma function and the long term memory also exists in order sizes in addition both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit order book lob established immediately after the opening call auction and a price clustering phenomenon is observed

forces on bins the effect of random friction the q model of coppersmith et al phys rev e 53 4673 1996 has renewed interest in understanding the forces generated along the walls and at the bottom of a silo filled with a granular material fluctuations in the mean stress have been characterized for the q model and related to experimental work on stress chains the classical engineering approach to bin loads follows from janssen s analysis z ver dtsch ing 39 1045 1895 predicting a saturation of stress as a function of depth in a tall silo in this paper we reexamine the janssen theory introducing randomness into the important parameters in the theory the janssen analysis relies on assumptions not met in practice for this reason we numerically solve the partial differential equations governing the equilibrium of forces in a bin again including randomness in parameters we show that the most important of these parameters is a coefficient of friction at the wall of the bin this random friction model combines some features of fluctuations as seen in experiments with a classical continuum mechanics approach to describing granular materials

probability density functions of contact forces for cohesionless frictional granular materials a theory is developed for the probability density functions of contact forces for cohesionless frictional granular materials in quasi static equilibrium this theory is based on a maximum information entropy principle with an expression for information entropy that is appropriate for granular materials entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by coulomb friction the theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient the theoretical predictions are compared with results from discrete element simulations on isotropic two dimensional assemblies under hydrostatic stress good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions while the quantitative agreement is fairly good discrepancies between theory and simulations such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces are attributed to the fact that the method does not take into account any kinematics which are essential in relation to elastic effects

what should a statistical mechanics satisfy to reflect nature there is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual boltzmann gibbs prescriptions more specifically ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it in any geometrically simple subset of the a priori allowed phase space in any even extended trivial sense a vast class of such systems appears nevertheless to be tractable within thermostatistical methods completely analogous to the usual ones the question posed in the title arises then naturally some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

tsallis deformation parameter q quantifies the classical quantum transition abstract we investigate the classical limit of a type of semiclassical evolution the pertinent system representing the interaction between matter and a given field on using tsallis q entropy as a quantifier of the ensuing dynamics we find that it not only appropriately describes the quantum classical transition but that the associated deformation parameter q itself characterizes the different regimes involved in the process detecting the most salient fine details of the changeover

phase transition of dynamical herd behaviors for yen dollar exchange rates abstract we study the herd behavior and the phase transition for the yen dollar exchange rate in the japanese financial market it is obtained that the probability distribution of returns satisfies the power law behavior p r r with scaling exponents 3 11 2 81 and 2 29 at time intervals 1 min 30 min and 1 h the crash region in which the probability density increases with the increasing return appears when the herding parameter h satisfies h 2 33 for the case of 30 min we especially obtain that no crash occurs 30 min and that the probability distribution of price returns occurs in the phase transition at 30 min

a simple and exact acoustic wavefield modeling code for data processing imaging and interferometry applications improvements in industrial seismic seismological acoustic or interferometric theory and applications often result in quite subtle changes in sound quality seismic images or information which are nevertheless crucial for improved interpretation or experience when evaluating new theories and algorithms using synthetic data an important aspect of related researchisthereforethat numerical errors due towavefield modeling are reduced to a minimum we present a new matlab code based on the foldy method that models theoretically exact direct and scattered parts of a wavefield its main advantage lies in the fact that while all multiple scattering interactions aretakenintoaccount unlikefinite difference or finite element methods numerical dispersion errors are avoided the method is therefore ideal for testing new theory in industrial seismics seismology acoustics and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions we present the theory behind the foldy acoustic modeling method and provide examples of its implementation we also benchmark the code against a good finite difference code because our foldy code was written and optimized to test new theory in seismic interferometry examples of its application to seismicinterferometryarealsopresented showing its validity and importance when exact modeling results are needed

application of the beck model to stock markets value at risk and portfolio risk assessment abstract we apply the beck model developed for turbulent systems that exhibit scaling properties to stock markets our study reveals that the beck model elucidates the properties of stock market returns and is applicable to practical use such as the value at risk estimation and the portfolio analysis we perform empirical analysis with daily intraday data of the s p500 index return and find that the volatility fluctuation of real markets is well consistent with the assumptions of the beck model the volatility fluctuates at a much larger time scale than the return itself and the inverse of variance or inverse temperature obeys distribution as predicted by the beck model the distribution of returns is well fitted by q gaussian distribution of tsallis statistics the evaluation method of value at risk var one of the most significant indicators in risk management is studied for q gaussian distribution our proposed method enables the var evaluation in consideration of tail risk which is underestimated by the variance covariance method a framework of portfolio risk assessment under the existence of tail risk is considered we propose a multi asset model with a single volatility fluctuation shared by all assets named the single model and empirically examine the agreement between the model and an imaginary portfolio with dow jones indices it turns out that the single model gives good approximation to portfolios composed of the assets with non gaussian and correlated returns

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

representation theorems and green s function retrieval for scattering in acoustic media reciprocity theorems for perturbed acoustic media are provided in the form of convolution and correlation type theorems these reciprocity relations are particularly useful in the general treatment of both forward and inverse scattering problems using green s functions to describe perturbed and unperturbed waves in two distinct wave states representation theorems for scattered waves are derived from the reciprocity relations while the convolution type theorems can be manipulated to obtain scattering integrals that are analogous to the lippmann schwinger equation the correlation type theorems can be used to retrieve the scattering response of the medium by cross correlations unlike previous formulations of green s function retrieval the extraction of scattered wave responses by cross correlations does not require energy equipartitioning allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and or attenuative scatterers these concepts are illustrated with a number of examples including analytic solutions to a one dimensional scattering problem and a numerical example in the context of seismic waves recorded on the ocean bottom

on distribution of contact forces in random granular packings abstract in an earlier paper ngan phys rev e 68 2003 011301 proposed a statistical mechanics based theory to predict the mathematical form of the probability force distribution of contact forces in stressed granular packings however the wrong choice of the mean force constraint in that work leads to unsatisfactory prediction in the 3d situation although the 2d prediction is very accurate the present work shows that a different mean strain constraint chosen for the force distribution can yield the correct prediction when compared to computer simulation results in both 2d and 3d furthermore new simulation results are used to verify the interpretation that the shape of the force distribution measured by a parameter in the predicted force distribution is a structure sensitive term

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

scaling and correlation in financial time series we discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of nancial time series i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to very drastic events such as the crash of 19 october 1987 sometimes referred to as black monday the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as 8 orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by 3 orders of magnitude from 1 min up to approximately 10 days ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law persisting for several months iii volatility and trading activity we quantify the relation between trading activity measured by the number of transactions nt and the price change gt for a given stock over a time interval t t t we nd that nt displays long range power law correlations in time which leads to the interpretation that the long range correlations previously found for jgtj are connected to those of nt c 2000 elsevier science b v all rights reserved

measuring nonextensitivity parameters in a turbulent couette taylor flow we investigate probability density functions of velocity differences at different distances r measured in a couette taylor flow for a range of reynolds numbers re there is good agreement with the predictions of a theoretical model based on nonextensive statistical mechanics where the entropies are nonadditive for independent subsystems we extract the scale dependent nonextensitivity parameter q r mathrm re from the laboratory data

asymptotic behavior of a stationary silo with absorbing walls we study the nearest neighbors one dimensional uniform q model of force fluctuations in bead packs 1 a stochastic model to simulate the stress of granular media in two dimensional silos the vertical coordinate plays the role of time and the horizontal coordinate the role of space the process is a discrete time markov process with state space 1 n at each layer time the weight supported by each grain is a random variable of mean one its own weight plus the sum of random fractions of the weights supported by the nearest neighboring grains at the previous layer the fraction of the weight given to the right neighbor of the successive layer is a uniform random variable in 0 1 independent of everything the remaining weight is given to the left neighbor in the boundaries a uniform fraction of the weight leans on the wall of the silo this corresponds to absorbing boundary conditions for this model we show that there exists a unique invariant measure the mean weight at site i under the invariant measure is i n 1 i we prove that its variance is frac 1 2 i n 1 i 2 o n3 and the covariances between grains i j are of order o n3 moreover as n the law under the invariant measure of the weights divided by n2 around site integer part of rn r 0 1 converges to a product of gamma distributions with parameters 2 and 2 r 1 r 1 sum of two exponentials of mean r 1 r 2 liu et al 2 proved that for a silo with infinitely many weightless grains any product of gamma distributions with parameters 2 and 2 with 0 are invariant our result shows that as the silo grows the model selects exactly one of these gamma s at each macroscopic place

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

nonlinear scattering based imaging in elastic media theory theorems and imaging conditions with the more widespread introduction of multicomponent recording devices in land and marine ocean bottom seismic acquisition elastic imaging may become mainstream in coming years we have derived new nonlinear elastic imaging conditions a correlation type representation theorem for perturbed elastic media commonly used in seismic interferometry to explain how a scattered wave response between two receivers sources may be predicted given a boundary of sources receivers can be considered as a starting point for the derivation here we use this theorem to derive and interpret imaging conditions for elastic migration by wavefield extrapolation e g elastic reverse time migration some approximations lead to a known heuristically derived imaging condition that crosscorrelates p and s wave potentials that are separated in the subsurface after full wavefield extrapolation this formal connection reveals that the nonapproximated correlation type representation theorem can be interpreted as a nonlinear imaging condition that accounts also for multiply scattered and multiply converted waves properly focusing such energy at each image point we present a synthetic data example using either an ideal acquisition on a full closed boundary or a real partial boundary seismic exploration survey and we demonstrate the importance of nonlinearities in pure and converted mode imaging in pp imaging they result in better illumination and artifact reduction whereas in ps imaging they show how zero time lag and zero space lag crosscorrelation imaging conditions are not ideal for imaging of converted mode waves because no conversion arises from zero offset experiments

herd behaviors in the stock and foreign exchange markets the herd behavior of returns for the won dollar exchange rate and the korean stock price index kospi is analyzed in korean financial markets it is reported that the probability distribution p r of returns r for three types of herding parameter satisfies the power law behavior p r r with the exponents 2 2 the won dollar exchange rate and 2 4 the kospi when the herding parameter h satisfies h 2 33 the crash regime in which p r increases with the increasing r appears the active state of the transaction exists to decrease for h 2 33 especially we find that the distribution of normalized returns shows a crossover to a gaussian distribution when the time step t 252 is used our results will also be compared to the other well known analyses

average stresses and force fluctuations in noncohesive granular materials a lattice model is presented for investigating the fluctuations in static granular materials under gravitationally induced stress the model is similar in spirit to the scalar q model of coppersmith et al phys rev e 53 4673 1996 but ensures balance of all components of forces and torques at each site the geometric randomness in real granular materials is modeled by choosing random variables at each site consistent with the assumption of cohesionless grains configurations of the model can be generated rapidly allowing the statistical study of relatively large systems for a two dimensional 2d system with rough walls the model generates configurations consistent with continuum theories for the average stresses unlike the q model without requiring the assumption of a constitutive relation for a 2d system with periodic boundary conditions the model generates single grain force distributions similar to those obtained from the q model with a singular distribution of q s

universal renormalization group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics we uncover the dynamics at the chaos threshold m of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period doubling cascade that occurs for m m we corroborate this structure analytically via the feigenbaum renormalization group rg transformation and find that the sensitivity to initial conditions has precisely the form of a q exponential of which we determine the q index and the q generalized lyapunov coefficientl q our results are an unequivocal validation of the applicability of the nonextensive generalization of boltzmann gibbs statistical mechanics to critical points of nonlinear maps critical points of nonlinear maps offer a suitable playground for testing the validity of the nonextensive generalization of the boltzmann gibbs bg statistical mechanics proposed by tsallis over a decade ago 1 2 here we describe universal properties related to the dynamics of iterates at the onset of chaos in unimodal maps 3 which provide a literal confirmation of the generalized nonextensive theory to this end we employ the celebrated one dimensional logistic map f m x 512muxu 2 21 x 1 and the properties of its renormalization group rg fixed point to present evidence of previously unexposed scaling properties at the onset of chaos m5m at this state the most prominent of the map critical points the trajectories of the iterates exhibit an intricate structure that we describe and show is governed by the feigenbaum rg transformation 3 the domain of validity of bg statistical mechanics has been implicitly challenged by the proposal of its nonextensive generalization subsequent studies have offered experimental and numerical evidence that point out both the inadequacy of the standard bg statistics and the plausible competence of the generalized theory in describing various types of phenomena and systems this theoretical development represents an exceptional event in the long and trustworthy history of bg statistical mechanics however it is still in the process of being converted into a rigorously corroborated and fully understood fact the suggested circumstances under which the generalized theory is believed to be applicable at least with regards to nonlinear dynamical systems are those associated to a phase space with power law sensitivity to initial conditions to the consequent vanishing of the largest lyapunov exponent and to a fractal or multifractal geometrical structure 2 here we show that our results for the dynamics at the onset of chaos in unimodal maps constitute an unequivocal proof of the universal validity of the nonextensive statistics at such critical points in fact at the chaos threshold as well as at other critical points of the map the lyapunov exponent l 1 vanishes and the sensitivity to initial conditions j t for large iteration time t ceases to obey exponential behavior exhibiting instead a power law behavior 4 in order to describe the dynamics at such critical points the q exponential expression

a nonextensive maximum entropy approach to a family of nonlinear reaction diffusion equations we consider a monoparametric family of reaction diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time dependent particular solutions of the tsallis maximum entropy maxent form the evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically a simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q maxent solutions when the reaction term vanishes the time dependent distributions studied here reduce to the previously known tsallis maxent solutions for the nonlinear diffusion equation

conditional q entropies and quantum separability a numerical exploration we revisit the relationship between quantum separability and the sign of the relative q entropies of composite quantum systems the q entropies depend on the density matrix eigenvalues pi through the quantity q ipqi renyi s and tsallis measures constitute particular instances of these entropies we perform a systematic numerical survey of the space of mixed states of two qubit systems in order to determine as a function of the degree of mixture and for different values of the entropic parameter q the volume in state space occupied by those states characterized by positive values of the relative entropy similar calculations are performed for qubit qutrit systems and for composite systems described by hilbert spaces of larger dimensionality we pay particular attention to the limit case q our numerical results indicate that as the dimensionalities of both subsystems increase typical mixed states of composite quantum systems tend as far as their relative q entropies are concerned to behave in a classical way

thermodynamics of relation based systems with applications in econophysics sociophysics and music abstract a methodology was developed to analyze relation based systems evolving in time by using the fundamental concepts of thermodynamics the behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors or pathways connecting subsequent values which characterize an event such as the index values in stock markets a system behaves in a rigid elastic way to an external effect and resists permanent deformation or it behaves in a viscous or soft way and deforms in an irreversible way it was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation based systems gunduz 2009 5 gunduz and gunduz 2010 6 in this research the work associated with elastic component and heat associated with viscous component were discussed and elaborated in a simple two subsequent pathway system in a scattering diagram the first vector represents the cause and the second the effect by using work and heat energy relations that involve force and also storage and loss modulus terms respectively one can calculate the energy involved in relation based systems the modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector once work like and heat like terms were determined the internal energy is also easily found from their summation the parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system three cases i the behavior of the nasdaq 100 index ii a social revolt and iii the structure of a melody were analyzed for their work like heat like and torque like energies in the course of their evolution nasdaq 100 exhibits highly dissipative behavior and its work terms are very small but heat terms are of large magnitude its internal energy highly fluctuates in time in the social revolt studied work and heat terms are of comparable magnitude the melody depicts highly organized structure and usually has larger work terms than heat terms but at some intervals heat terms burst out and attain very large magnitudes torque terms reach high values when the system is recovering from a minimum value

transmission of information and herd behavior an application to financial markets we propose a model for stochastic formation of opinion clusters modeled by an evolving network and herd behavior to account for the observed fat tail distribution in returns of financial price data the only parameter of the model is h the rate of information dispersion per trade which is a measure of herding behavior for h below a critical h the system displays a power law distribution of the returns with exponential cutoff however for hg h an increase in the probability of large returns is found and may be associated with the occurrence of large crashes

tsallis distributions and 1 f noise from nonlinear stochastic differential equations probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems in this article we unite modeling of such distributions with the model of widespread 1 f noise we propose a class of nonlinear stochastic differential equations giving both the q exponential or q gaussian distributions of signal intensity revealing long range correlations and 1 f behavior of the power spectral density the superstatistical framework to get 1 f noise with q exponential and q gaussian distributions of the signal intensity is proposed as well

scaling in the norwegian stock market the main objective of this paper is to investigate the validity of the much used assumptions that stock market returns follow a random walk and are normally distributed for this purpose the concepts of chaos theory and fractals are applied two independent models are used to examine price variations in the norwegian and us stock markets the first model used is the range over standard deviation or r s statistic which tests for persistence or antipersistence in the time series both the norwegian and us stock markets show significant persistence caused by long run memory components in the series in addition an average non periodic cycle of four years is found for the us stock market these results are not consistent with the random walk assumption the second model investigates the distributional scaling behaviour of the high frequency price variations in the norwegian stock market the results show a remarkable constant scaling behaviour between different time intervals this means that there is no intrinsic time scale for the dynamics of stock price variations the relationship can be expressed through a scaling exponent describing the development of the distributions as the time scale changes this description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour the empirical distributions of high frequency price variations for the norwegian stock market is then compared to the levy stable distribution with the relevant scaling exponent found by using the r s and distributional scaling analysis good agreement is found between the levy profile and the empirical distribution for price variations less than 6 standard deviations covering almost three orders of magnitude in the data for probabilities larger than 6 standard deviations there seem to be an exponential fall off from the levy profile in the tails which indicates that the second moment may be finite

two dimensional maps at the edge of chaos numerical results for the henon map the mixing properties or sensitivity to initial conditions of the two dimensional henon map have been explored numerically at the edge of chaos three independent methods which have been developed and used so far for one dimensional maps have been used to accomplish this task these methods are i the measure of the divergence of initially nearby orbits ii analysis of the multifractal spectrum and iii computation of nonextensive entropy increase rates the results obtained closely agree with those of the one dimensional cases and constitute a verification of this scenario in two dimensional maps this obviously makes the idea of weak chaos even more robust

itineration of the internet over nonequilibrium stationary states in tsallis statistics the cumulative probability distribution of sparseness time interval in the internet is studied by the method of data analysis round trip time between a local host and a destination host through ten odd routers is measured using the ping command i e doing an echo experiment the data are found to be well described by q exponential distributions which maximize the tsallis entropy indexed by q less or larger than unity showing a scale invariant feature of the system the network is observed to itinerate over a series of the nonequilibrium stationary states characterized by tsallis statistics

cancellation of spurious arrivals in green s function extraction and the generalized optical theorem the extraction of the green s function by cross correlation of waves recorded at two receivers nowadays finds much application we show that for an arbitrary small scatterer the cross terms of scattered waves give an unphysical wave with an arrival time that is independent of the source position this constitutes an apparent inconsistency because theory predicts that such spurious arrivals do not arise after integration over a complete source aperture this puzzling inconsistency can be resolved for an arbitrary scatterer by integrating the contribution of all sources in the stationary phase approximation to show that the stationary phase contributions to the source integral cancel the spurious arrival by virtue of the generalized optical theorem this work constitutes an alternative derivation of this theorem when the source aperture is incomplete the spurious arrival is not canceled and could be misinterpreted to be part of the green s function we give an example of how spurious arrivals provide information about the medium complementary to that given by the direct and scattered waves the spurious waves can thus potentially be used to better constrain the medium

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

finite time singularity signature of hyperinflation abstract we present a novel analysis extending the recent work of mizuno et al physica a 308 2002 411 on the hyperinflations of germany 1920 1 1 1923 11 1 hungary 1945 4 30 1946 7 15 brazil 1969 1994 israel 1969 1985 nicaragua 1969 1991 peru 1969 1990 and bolivia 1969 1985 on the basis of a generalization of cagan s model of inflation based on the mechanism of inflationary expectation of positive feedbacks between realized growth rate and people s expected growth rate we find that hyperinflations can be characterized by a power law singularity culminating at a critical time t c mizuno et al s double exponential function can be seen as a discrete time step approximation of our more general non linear ode formulation of the price dynamics which exhibits a finite time singular behavior this extension of cagan s model which makes natural the appearance of a critical time t c has the advantage of providing a well defined end of the clearly unsustainable hyperinflation regime we find an excellent and reliable agreement between theory and data for germany hungary peru and bolivia for brazil israel and nicaragua the super exponential growth seems to be already contaminated significantly by the existence of a cross over to a stationary regime

generalized relative entropy in functional magnetic resonance imaging abstract the generalized kullback leibler distance d q q is the tsallis parameter is shown to be an useful measure for analysis of functional magnetic resonance imaging fmri data series this generalized form of entropy is used to evaluate the distance between the probability functions p 1 and p 2 of the signal levels related to periods of stimulus and non stimulus in event related fmri experiments the probability densities of the mean distance d q averaged over the n epochs of the entire experiment are obtained through numerical simulations for different values of signal to noise ratio snr and found to be fitted very well by gamma distributions 2 0 0008 for small values of n n 30 these distributions allow us to determine the sensitivity and specificity of the method by construction of the receiver operating characteristic roc curves the performance of the method is also investigated in terms of the parameters q and l number of signal levels and our results indicate that the optimum choice is q 0 8 and l 3 the entropic index q is found to exert control on both sensitivity and specificity of the method as q q 0 is raised sensitivity increases but specificity decreases finally the method is applied in the analysis of a real event related fmri motor stimulus experiment and the resulting maps show activation in primary and secondary motor brain areas

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

pdf of velocity fluctuation in turbulence by a statistics based on generalized entropy an analytical formula for the probability density function pdf of the velocity fluctuation in fully developed turbulence is derived non perturbatively by assuming that its underlying statistics is the one based on the generalized measures of entropy the renyi entropy or the tsallis entropy the parameters appearing in the pdf including the index q in the generalized measures are determined self consistently with the help of the observed value of the intermittency exponent the derived pdf explains quite well the experimentally observed density functions

on the closed form solutions for non extensive value at risk abstract we examine possible closed form solutions for the cumulative distribution function for systems where the probability density function can be adequately described by the generalized non extensive statistics framework application to financial time series as a possible value at risk technique indicates reasonable agreement with the data under consideration including all possible extremes and asymmetries of the returns numerical results to illustrate the efficiency of the method are presented

convergence to the critical attractor of dissipative maps log periodic oscillations fractality and nonextensivity for a family of logisticlike maps we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space we found that the phase space volume occupied by the ensemble w t depicts a power law decay with log periodic oscillations reflecting the multifractal character of the critical attractor we explore the parametric dependence of the power law exponent and the amplitude of the log periodic oscillations with the attractor s fractal dimension governed by the inflection of the map near its extremal point further we investigate the temporal evolution of w t for the circle map whose critical attractor is dense in this case we found w t to exhibit a rich pattern with a slow logarithmic decay of the lower bounds these results are discussed in the context of nonextensive tsallis entropies

theoretical estimates for the largest lyapunov exponent of many particle systems the largest lyapunov exponent of an ergodic hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space for an n particle hamiltonian system with a smooth hamiltonian of the type p 2 v q the evolution of tangent vectors is governed by the hessian matrix v of the potential ergodicity implies that the lyapunov exponent is independent of initial conditions on the energy shell which can then be chosen randomly according to the microcanonical distribution in this way a stochastic process v t is defined and the evolution equation for tangent vectors can now be seen as a stochastic differential equation an equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cumulant expansion we show that if cumulants higher than the second one are discarded the lyapunov exponent can be obtained by diagonalizing a small dimension matrix that in some cases can be as small as 3 3 in all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process we discuss the connection between our approach and an alternative theory the so called geometric method

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

statistical properties of share volume traded in financial markets we quantitatively investigate the ideas behind the often expressed adage it takes volume to move stock prices and study the statistical properties of the number of shares traded q deltat for a given stock in a fixed time interval deltat we analyze transaction data for the largest 1000 stocks for the two year period 1994 95 using a database that records every transaction for all securities in three major us stock markets we find that the distribution p q deltat displays a power law decay and that the time correlations in q deltat display long range persistence further we investigate the relation between q deltat and the number of transactions n deltat in a time interval deltat and find that the long range correlations in q deltat are largely due to those of n deltat our results are consistent with the interpretation that the large equal time correlation previously found between q deltat and the absolute value of price change g deltat related to volatility are largely due to n deltat

the reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem by analyzing correlation type reciprocity theorems for wavefields in perturbed media it is shown that the correlation type reciprocity theorem for the scattered field is the progenitor of the generalized optical theorem this reciprocity theorem in contrast to the generalized optical theorem allows for inhomogeneous background properties and does not make use of a far field condition this theorem specializes to the generalized optical theorem when considering a finite size scatterer embedded in a homogeneous background medium and when utilizing the far field condition moreover it is shown that the reciprocity theorem for the scattered field is responsible for the cancellation of non physical spurious arrivals in seismic interferometry and as such provides the mathematical description of such arrivals even though here only acoustic waves are treated the presented treatment is not limited to such wavefields and can be generalized to general wavefields therefore this work provides the framework for deriving equivalents of the generalized optical theorem for general wavefields

multifractal features of financial markets abstract we study the tick dynamical behavior of three assets in financial markets the kospi the won dollar and yen dollar exchange rates using the rescaled range r s analysis the multifractal hurst exponents with long run memory effect can be obtained from those assets and we discuss whether there exists the crossover or not for the hurst exponents at characteristic time scales particularly we find that the probability distribution of returns approaches to a lorentz distribution different from gaussian properties

scaling laws for the largest lyapunov exponent in long range systems a random matrix approach we investigate the laws that rule the behavior of the largest lyapunov exponent lle in many particle systems with long range interactions we consider as a representative system the so called hamiltonian ensuremath alpha ensuremath mathrm xy model where the adjustable parameter ensuremath alpha controls the range of the interactions of n ferromagnetic spins in a lattice of dimension d in previous work the dependence of the lle with the system size n for sufficiently high energies was established through numerical simulations in the thermodynamic limit the lle becomes constant for ensuremath alpha gd whereas it decays as an inverse power law of n for ensuremath alpha ld a recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results here we show that the scaling behavior can also be explained by a random matrix approach in which the tangent mappings that define the lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble

fractional brownian motion fractional gaussian noise and tsallis permutation entropy abstract in this work we analyze two important stochastic processes the fractional brownian motion and fractional gaussian noise within the framework of the tsallis permutation entropy this entropic measure evaluated after using the bandt pompe method to extract the associated probability distribution is shown to be a powerful tool to characterize fractal stochastic processes it allows for a better discrimination of the processes than the shannon counterpart for appropriate ranges of values of the entropic index moreover we find the optimum value of this entropic index for the stochastic processes under study

risk attitudes and decision weights to accommodate the observed pattern of risk aversion and risk seeking as well as common violations of expected utility e g the certainty effect we introduce and characterize a weighting function according to which an event has greater impact when it turns impossibility into possibility or possibility into certainty than when it merely makes a possibility more or less likely we show how to compare such weighting functions of different individuals with respect to the degree of departure from expected utility and we present a method for comparing an individual s weighting functions for risk and for uncertainty

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

source receiver reverse time imaging of dual source vector acoustic seismic data abstractnovel technologies in seismic data acquisition allow for recording full vector acoustic va data pointwise recordings of pressure and its multicomponent gradient excited by pressure only as well as dipole gradient sources building on recent connections between imaging and seismic interferometry we present a wave equation based nonlinear reverse time imaging approach that takes full advantage of dual source multicomponent data the method s formulation relies on source receiver scattering reciprocity thus making proper use of va fields in the wavefield extrapolation and imaging condition steps in a self consistent manner the va imaging method is capable of simultaneously focusing energy from all in and outgoing waves the receiver side up and downgoing receiver ghosts fields are handled by the va receiver extrapolation whereas source side in and outgoing source ghosts arrivals are accounted for when combining dual source data at the imaging condition additionally va imaging handl

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators