research on financial markets using methods and concepts developed in physics has increased considerably over the last decade .various kinds of stylized facts or empirical laws of markets have been discovered from high precision market data of gigantic size .the next goal of this econophysics study is to attempt to establish the reasons for these empirical findings .just as with the boyle - charles macroscopic law which can be derived from a simple microscopic ideal - gas model , we hope to construct a simple microscopic model of a market that can reproduce major empirical findings . by relating macroscopic market behavior to microscopic dealers actions , we may find a pathway to control the markets , so as to avert bubbles and crashes , which occasionally cause problems in the market .the study of modeling dealers action is carried out with so - called agent - based models .this approach is supported not only by economists but also by information scientists and physicists .agent - based models can in practice reproduce dealers actions in the market and they can also reflect empirical laws of markets to some extent .however , agent models generally include a huge number of parameters , and it has proved difficult to understand the relation between the parameters of the model and resulting market behavior .  in order to find relationships between the parameters of dealers actions and market behavior , we have already introduced a kind of minimal model of an agent - based market which consists of dealers with simple deterministic time evolution rules . with this model , we successfully reproduced most of the basic empirical laws using a minimal number of parameters , and found that there are only three important effects needed to reproduce the empirical laws .the first effect is the compromise pricing of both buyers and sellers , who tend to allow the particular transaction price they have in mind to approach the current market price in order to make a deal . from this effect , transactions occur spontaneously in the market and the price rises and falls almost randomly .the second effect is the self - modulation of transaction intervals , that is , the rate of a dealer s clock depends on the latest moving average value of transaction intervals .when market activity becomes high , dealers accelerate their transaction rates , and by this effect we can reproduce empirical statistical properties of transaction intervals which deviate from a simple poisson process .the third is the trend - follow effect , that is , dealers forecast upcoming prices using the latest market trend which is defined by a moving average of price changes . this forecasting effect makes the price change distribution follow a power law quite similar to that of the real market .  in this paperwe first introduce a stochastic version of the dealer model which is even simpler than the above ( deterministic ) model . in the case of the deterministic dealer model we needed at least three dealers to reproduce market properties ; however , in the present stochastic model we require only two .the advantages of this stochastic model are not only its simplicity but also its solvability by analytical calculation . in the usual agent - based approaches intensive numerical simulation is the only way to obtain results ; in such cases exact or strict results are rarely obtained .based on this stochastic dealer model and its variants we can derive the major empirical results mentioned above , that have already been obtained by simulation of the deterministic dealer model by theoretical analysis .  apart from agent - based modeling , the standard way to model markets is by utilizing random walks .it is now widely known that bachelier introduced a random walk model for market prices five years earlier than einstein s random walk model of brownian motion .work on portfolio theory , option price formulation and the arch and garch models , which has led to nobel prizes for their developers , are all based on random walk models .  recently , one of the authors ( m.t . )has introduced a new type of extended random walk model of the market , the so - called puck model , in which a random walker moves according to a deforming potential force , the center point of which is given by the moving average of the random walker s traces . by using this generalization ,all major empirical laws can be established ; moreover , dynamical behaviors such as bubbles , crashes and inflations can also be described as following from special cases of the market potential force .the arch model can also be derived as a special limiting case of this extended random walk model .considering the wide applicability of the puck model has led to an open question concerning the origin of market potential forces .this question has been partially answered by using the deterministic dealer model . here , we are able to provide quantitative answers by using the stochastic dealer model .  in the next sectionwe introduce our stochastic dealer models step by step in sequential subsections .the third section is devoted to the relationship with the puck model , in which we will see how dealers actions produce a market s potential force in a quantitative discussion .the final section contains a summary .in this section we introduce three stochastic dealer models , model-1 , model-2 and model-3 .model-1 is the simplest market model in which the framework of the stochastic dealer model is introduced .then , we note two empirical properties which model-1 can not reflect . in model-2 and model-3we introduce two additional effects respectively to deal with these difficulties . after combining these revisions , the stochastic dealer model fully reflects all major empirical laws of markets .firstly , we assume an artificial market consisting of only two dealers who are offering both buying and selling prices .the buying price , or bid price , is the current maximum price at which the dealer wants to buy . the selling price , or ask price , is the minimum price at which he will sell . for each dealerthe ask price is always higher than the bid price , because they want some margin , and the difference between these prices is called the spread , which is assumed to be a constant , , in this model .we define the -th dealer s mid - price at time , , as the average of his bid and ask prices .when is less than , these dealers do not transact as their transaction conditions are not fulfilled ( fig.[fig : transaction2.eps]a ) .in such a case dealers are assumed to change their prices randomly according to the following rule . here , is a random noise for the -th dealer , and is a constant parameter .then , the distance between and is checked .if it is greater than or equal to , then one dealer s bid price is higher than the other s ask price , and a transaction occurs ( fig.[fig : transaction2.eps]c ) . in such a case a unit volume deal is assumed to be made , with the market price given by the averaged price of the two dealers mid - prices .after this transaction their mid - prices are assumed to shift to the market price .these processes are repeated again and again and the time proceeds in unit of .it should be noted that there is a possibility that this model produces a negative value for market price .in such a case the step width of the price change , , should depend on the market price , such that the value of is proportional to the market price to avoid crossing the origin . here , we pay attention only to the case that the price fluctuation level is much smaller than the market price and for simplicity , we assume that is constant .-th dealer s mid - price is denoted by .( a ) : in this situation no transaction occurs .( b ) : the dealers prices follow random walks .( c ) : when the distance between and is greater than or equal to , a transaction occurs and the market price is defined by the averaged price of the two mid - prices .( d ) : after this transaction both dealers mid - prices move to the market price .these processes are repeated ., width=302 ]  for convenience of analysis we define another unit of time called the tick time , denoted by , which takes an integer value incremented at each occurrence of a transaction .accordingly , denotes the market price at tick time , and the -th transaction interval , , is defined by the time difference between the -th and -th transactions .  in fig.[fig :ts - pi ] , we plot an example of resulting market prices and corresponding transaction intervals . in the sub - windows of these figureswe also plot the probability density function of price changes and transaction intervals both on a semi - log scale .it is clear that the tail parts of both of these distributions are well characterized by exponential laws . as for the intervals, this result implies that the occurrence of transactions of this model is approximated by a poisson process .it is interesting that the price change distribution of this simplest model follows an exponential distribution except around , instead of gaussian distribution ., , and .,title="fig:",width=302 ] , , and .,title="fig:",width=302 ]  we can explain the functional form of the tails of these distributions as follows .we define the difference of the dealers prices by , then the condition for the occurrence of a transaction is given by , and we also define the mass center by the average of the two mid - prices , . as the mass center at the time of transaction gives the market price ,market price statistics can be calculated from the information about for times .namely , is given by which is defined by , where is the previous transaction time .these two variables define a two dimensional random walk with absorbing walls at and as shown in fig.[fig : footprint_d - dg_model-1.eps ] .the stochastic dynamics is described by the following set of equations ( [ eq:2d - brownian_m1 ] ) .[ eq:2d - brownian_m1 ]   when the random walker reaches one of the absorbing walls a transaction occurs , and by the transaction rule of model-1 that the prices of the two dealers are then set to the market price , the random walker goes to the origin , and a new random walk begins .in this 2-dimensional formulation the transaction interval , , is given by the survival time , that is , the time the random walker starting from the origin takes to reach one of the absorbing walls .similarly , the market price change , , is given by the random walker s location on the axis .the particle starts from the origin and continues a random walk until it touches one of the horizontal walls , meaning that a market transaction occurs .the transaction interval is given by the survival time of this random walker , and the market price change is given by the distance along the axis from the origin to the position of the particle on the absorbing wall . , width=302 ]  in fig.3 an example of a random walk is shown for better understanding of this mapping of model-1 to a 2-dimensional random walk . here ,the horizontal axis is the -axis and the vertical one is the -axis . for theoretical analysiswe consider a continuum limit such that the mesh sizes of space and time go to zero . under the condition , we find that the probability density of the particle in the plane at time is described by the following diffusion equation . here , is equivalent to the diffusion coefficient of this 2-dimensional random walk .the initial condition is the delta function , and the boundary condition is given by the absorbing walls on the axis , while there is no boundary in the direction .this diffusion equation is solved exactly as follows , here , .we obtain the distributions of transaction intervals and price changes by calculating distributions of survival times and absorbed points from ( [ eq : solution_u ] ) .[ eq : pdf - i - adp ] [ eq : q1q2 ]   in the case of large values of and in eq.([eq : q1q2 ] ) , these summations are dominated by the term of .so , the functions and can be approximated as  [ allequations ]   from these results we can derive the exponential laws of interval distributions and the price change distributions already seen in fig.[fig : ts - pi ] .it is confirmed that the theoretical values of the decay constants and , fit well with the numerical results .higher order moments of the distributions of transaction intervals and price changes are also obtained exactly from eq.([eq : pdf - i - adp ] ) .the -th moments of and are calculated as follows .[ allequations ]   here is the gamma function and are the euler numbers appearing in the expansion of ; . is the dirichlet beta function defined as . using this resultwe can calculate means and variances of transaction intervals and volatilities with results as shown in table [ tb : ave - vari ] ..exact solutions for the means and variances of transaction intervals and the absolute value of price changes in model-1 . is catalan s constant ; . [ cols="^,^,^",options="header " , ]   in this subsection we focus on the statistical differences between transaction intervals of real markets and those of model-1 . in real dollar - yen exchange market data provided by ebs for six years from 2000 to 2005 , we find that the transaction intervals exhibit a circadian pattern even for foreign exchange markets which are open continuously as shown in fig.[fig : avent_15min_ydt_00 - 05.eps ] for the dollar - yen market .as can be clearly seen , large numbers of transactions occur during office hours of tokyo , london and new york and the transaction density is least a little before the open of the tokyo offices .in addition to this 24-hour pattern , there are fluctuations with much shorter time scales as typically shown in fig.[fig : price_20050616_2.eps ] . in this figure transaction eventsare shown by bars at the top and corresponding dollar - yen rates are plotted in the lower section . herethe window size is ten minutes and we can find places where bars tend to cluster , marked as `` dense '' , and others where bars are `` sparse '' .the distribution of these intervals is clearly seen not to correspond to the simple theoretical model of a poisson process .the clustering properties of transactions are known to be well modeled by a self - modulation process introduced by the authors , which is described as follows . where is the time scale of self - modulation , and  and are independent noises , and is the number of transactions occurring within seconds .this process is a modified poisson process whose mean value is given by the moving average of transaction intervals over the past seconds .as a result , there is a greater tendency to cluster and the so - called 1/f fluctuation is realized in general .model-2 is designed to satisfy the real interval property by applying the self - modulation process .we estimate the distribution of from real data by using the following relation for the transaction intervals , .  here the typical value of is 150 seconds .it is confirmed from dollar - yen rate data that the distribution of follows an exponential distribution in general with mean value of unity .this exponential distribution is favorable for our model construction as model-1 automatically produces the exponential interval distribution . as we can control the speed of transaction intervals by controlling the speed of diffusion , we obtain a revised model , model-2 , by modifying the constant parameter in eq.([eq : model-1 ] ) , which is directly related to the diffusion coefficient , making it a time - dependent parameter as follows .  where in eq.([eq : c(t ) ] ) , is a mean of transaction intervals shown in table [ tb : ave - vari ] in the case of and . is the moving average of transaction intervals averaged over the latest seconds defined as . in this equation, is the transaction interval that is the -th tick earlier than the -th tick . is the number of transactions within seconds from time . if , we set .it is known from eq.([eq : c(t ) ] ) that for larger over , the value of is larger , that is , dealers tend to make larger changes their prices to effect more rapid transactions when transaction intervals become shorter in the market . by this effect , eq.([eq:2d - brownian_m1 ] ) of model-1 is modified to  [ eq:2d - brownian_m2 ]   examples of random walk traces are shown in fig.[fig : footprint_d - dg_model-2.eps ] . as known from this figure the initial condition and the boundary conditions are the same ; however , the step size changes for each random walk following the self - modulation formulation . by this effect the transaction intervals tend to form clusters as shown in fig.[fig : s - i - model1-model2.eps ] .   for model-1 ,the step size depends on past transaction intervals.,width=302 ]    in order to make the interval distribution fit well with that of the real dollar - yen market we introduce two thresholds for the value of .when , we set , and when , we set . these restrictions are needed to prevent intervals from converging to zero , or from diverging to infinity . with these minor revisions an example time sequence produced by model-2is plotted together with one produced by model-1 in fig.[fig : s - i - model1-model2.eps ] .comparing these two sequences , we observe that model-2 can reproduce the clustering property quite well .moreover , the lower interval sequence looks similar to the real sequence shown in fig.[fig : price_20050616_2.eps ] . actually , the distribution of transaction intervals arising from model-2 is now very close to that of the actual interval distribution as shown in fig.[fig : cdf_i_model2.eps ] .  , , and .,title="fig:",width=302 ] , , and .,title="fig:",width=226 ]   in this subsectionwe shift our attention from transaction intervals to price changes . we know that the price change distribution of model-1 is characterized by an exponential distribution while that of the real market is often characterized by a power law .it has been established that such power law distributions can be derived by introducing the effect of trend - following prediction .this effect can be introduced to our stochastic dealer model by simply adding a further term , , which is defined as follows .  where here is the price change at the -th tick .the new term , , is a kind of moving average of price changes for ticks with weights that decay linearly .the parameter in ( [ eq : model3 - 1 ] ) is an important parameter that governs the dealers strategy .a dealer with positive is a trend - follower who predicts upcoming market prices proportional to the latest price slope . on the other hand , a dealer with a negative called a contrarian who forecasts that upcoming market prices will go against the trend and that the present market price is close to a local maximum or minimum .  adding this effect , the equations ( [ eq:2d - brownian_m1 ] ) in model-1 are modified as  [ eq:2d - brownian_m3 ]   in the 2-dimensional random walk representation the initial conditions and the boundary conditions are invariant, however , we have a horizontal flow proportional to as shown in fig.[fig : footprint_d - dg_model-3.eps ] .the existence of this flow implies that the distance of the absorption point from the origin is greater than for the original model-1 . as the vertical motions are completely identical the transaction interval is also identical ; however , the absorbed point on the horizontal axis is shifted by .the strength of the flow depends on the parameter and the latest price changes .   - space for model-3 .two traces are plotted for comparison with the results of model-1 : the dotted line is for model-1 and the solid line is for model-3 , both produced using the same random number generator ., width=302 ]  in this revised model the transaction intervals are identical to those of model-1 because eq.([eq:2d - brownian_m3 - 1 ] ) is the same as eq.([eq:2d - brownian_m1 - 1 ] ) , while the market price change is described by the following equation . here , the first term of the right hand side is the distance covered by the flow , and gives the intensity of the flow , is the transaction interval .the second term is identical to the price change of model-1 . from the results already obtained for model-1it is clear that both and are random variables characterized by exponential functions , so eq.([eq : dp - model3 ] ) follows a random multiplicative process .we know that a time series which is produced by a random multiplicative process generally follows a power law if the process satisfies a stationary condition . in particular , in the case that and and are independent in eq.([eq : dp - model3 ] ) , we have an exact solution . in this model-3 , and are not statistically independent and so we do not have an exact solution ; however , if the effect of is negligibly small , then we can approximate the result using the solution for the independent case .we have the exponent for the power law of cumulative price change distribution as follows . here is the -th order moment of , so we can apply eq.([eq : moment - i ] ) to eq.([eq : beta-1 ] ) . as a result , we have it is known that empirical values of the power exponent of the cumulative price change distribution are around in the actual market , so we set , and in eq.([eq : beta-2 ] ). then we have .we can reproduce the power law of price change distribution with exponent as represented in fig.[fig : cdf_adp_model3.eps ] .other parameters are taken to be , and .we have seen that the stochastic dealer models can reproduce important empirical features of markets . in this sectionwe examine the relation to the market potential model called puck . in our previous work, we showed that the deterministic dealer model can reproduce market potentials confirmed by numerical simulation , and the essence for reconstruction of the potential is found to be the dealers forecasting effect using moving averages .we now present an analysis based on the present stochastic dealer model .it is easy to confirm that price changes produced by model-3 yield non - trivial market potential functions as shown in fig.[fig : potential_model3.eps ] . in this figurethe parameter in eq.([eq : model3 - 1 ] ) is changed .we set the parameter during the period from n=1 to 1000 ticks , that is , the dealers are contrarians who predict that the future price will move against the latest trend .the parameter in the period n=1001 to 2000 ticks , that is , the dealers are simple random walkers. and during 2001 to 3000 ticks , that is , the dealers are trend - followers predicting that near future prices will be proportional to a moving average of price changes .we can clearly observe a stable , a flat and an unstable potential function , respectively , as expected . during 3001 to 4000, we set when the average of the past price changes , , is greater than or equal to zero , and we set when . in such an asymmetric case , we can find an asymmetric potential as shown in fig.[fig : potential_model3.eps]d ; in that case the market price increases nearly linearly on a large scale .it is apparent that the market potential function and the dealers forecasting effect are also deeply related in this stochastic model .the puck model is formulated as follows : here is the noise reduction price introduced by ohnisi et .al . , also referred to as the optimal moving average price , is the potential function defined by eq.([eq : pm - p ] ) , is an uncorrelated noise term , and is the simple moving average over ticks : .if the market potential is asymmetric as in fig.[fig : potential_model3.eps](d ) , we define the potential function for and respectively by using quadratic functions . in the case of symmetric potential as fig.[fig : potential_model3.eps]abc , we can transform ( ) to the following equation :  is defined by eq.( ) .we note that eq.([eq : dp - model3 ] ) in model-3 and eq.([eq : pm-2 ] ) have the same form of a linear stochastic equation , so the statistical property is independent of the noise property . comparing the coefficients of in eq.([eq : dp - model3 ] ) and eq.([eq : pm-2 ] ), we have the simple relation : by taking the average over tick times , we have where denotes the average of over tick time .this equation implies that the market is unstable ( ) when dealers are trend followers ( ) while the market is stable when dealers are contrarians ( ) .this result is consistent with our previous simulation results using the deterministic dealer model .  in the puck model , when the potential is of quadratic type such as in fig.[fig : potential_model3.eps]abc , the diffusion coefficient of market prices , , is theoretically calculated as a function of the potential coefficient . by introducing the newly derived relation , eq.([eq : sigma - b ] ) , into the formula , we have a theoretical evaluation of the price diffusion coefficient for our model-3 . from this relation we find that in the range of the value of is finite , and we can expect the market price to follow a normal random walk over a long time scale .when the above formula is meaningless ; however , the actual market price moves nearly monotonically or even exponentially as occurs for bubbles in real markets .actually we can generate a bubble - like phenomenon by setting as shown in fig.[fig : bubble_yahoo.eps ] . here , the time evolution is well approximated by an exponential function as predicted by the puck model .we do not investigate this phenomenon further in this paper , but it should be stressed that our model can describe not only normal states of markets , but also abnormal states such as bubbles and crashes where prices move monotonically , by tuning our model s parameters .we have introduced a new stochastic dealer model which consists of only two dealers , and showed that basic empirical laws of financial markets are well reflected in terms of transaction intervals and price changes . in model-1 ,both dealers change their prices randomly , so fluctuations of transaction intervals and price changes are also random .we calculated these statistical properties exactly .the occurrence of transactions is well approximated by a poisson process , and the price change distribution is well described by an exponential distribution .  in order to make our model more realistic the following two feedback effects were introduced .one was the feedback effect of transaction intervals , and the other was the feedback effect of price changes , both caused by the dealers observations of the latest market status .as a result , the basic model s random noise properties were modulated and the artificial market reproduced both the distributions of transaction intervals and price changes to follow long - tailed distributions which are quite similar to those of real markets .  moreover , we established that the dealers action of prediction by using a moving average of past price changes generates market potentials in the puck formulation , and we derived a simple theoretical relation between the stochastic dealer model and the puck model .namely , we found the relation between the microscopic dealers strategy and the macroscopic market s stability as defined by the puck model .as an application of the relation to the puck model , we checked the condition when market prices in our stochastic dealer model exhibit a bubble - like phenomenon in which price motion is approximated by an exponential function rather than a random walk .this transition from a random walk phase to an exponential growth phase is considered to be quite useful for discussions concerning how to realize a stable market .we expect that our models can be used as a base for market experiments .for example , we may be able to observe the effect of governmental intervention by introducing a third dealer who only buys dollars over a given period . from the viewpoint of numerical simulationit is quite easy to increase the number of dealers , each of whom will have his own strategy .so we can construct any experimental market by adding or subtracting specific dealers , and observe the change in the market s macroscopic behavior as a result .for example , we may be able to find a way to avoid market crashes by introducing a specially designed dealer who acts to stabilize the market .this kind of market experiment may contribute to future attempts at real market stabilization .p. gopikrishnan , m. meyer , l. a. n. amaral , v. plerou and h. e. stanley , eur .j. b 3 , 139 - 140 , ( 1998 ) .y. liu , p. gopikrishnan , p. cizeau , m. meyer , c .- k .peng , and h. e. stanley , phys .e 60 , 1390 - 1400 ( 1999 ) .t. ohnishi , h. takayasu , t. ito , y. hashimoto , t. watanabe , m. takayasu " journal of economic interaction and coordination , vol . 3 , number 1 , 99 - 106 ( 2008 ) .t. lux and m. marchesi , nature 397 , 498 - 500 ( 1999 ) .

Abstract: we introduce solvable stochastic dealer models , which can reproduce basic empirical laws of financial markets such as the power law of price change . starting from the simplest model that is almost equivalent to a poisson random noise generator , the model becomes fairly realistic by adding only two effects , the self - modulation of transaction intervals and a forecasting tendency , which uses a moving average of the latest market price changes . based on the present microscopic model of markets , we find a quantitative relation with market potential forces , which has recently been discovered in the study of market price modeling based on random walks .


ground - based helioseismic observations ( _ e.g. _ gong : harvey _ et al . _ , 1996 ; bison : broomhall _ et al . _ , 2009 ) and space - based ones ( _ e.g. _ mdi : scherrer _ et al . _ , 1995 ; golf : gabriel _ et al . _ , 1995 ; or hmi : scherrer _ et al ._ , 2012 ) , have allowed us to derive a good description of the dynamics of the solar interior ( _ e.g. _ thompson _ et al . _ , 2003 ,garca _ et al . _ , 2007 , eff - darwich _ et al . _ , 2002 , 2008 , howe , 2009 ) .helioseismic inferences have confirmed that the differential rotation observed at the surface persists throughout the convection zone .the outer radiative zone ( ) appears to rotate approximately as a solid body at an almost constant rate ( nhz ) , whereas it is not possible to rule out a different rotation rate for the innermost core ( ) . at the base of the convection zone , a shear layer known as the tachocline separates the region of differential rotation throughout the convection zone from the one with rigid rotation in the radiative zone .finally , there is a subsurface shear layer between the fastest - rotating layer , located at about , and the surface .of course , this rotation profile is not constant ; the time - varying component of the rotation displays clear variations near the surface ( known as the torsional oscillations ) , while we see hints of variations at the base of the convection zone , both being likely related to the driving mechanisms of the solar - activity cycle .our understanding of the dynamics of the solar interior has undoubtedly improved ; however we still need to constrain the rotation profile near the core and fully analyze the nature of the torsional oscillations .we still do not know how thin the tachocline really is and what is keeping it this way .understanding the tachocline should help discern if there is a fossil magnetic field in the radiative zone that prevents the spread of the tachocline ( zahn _ et al ._ , 2007 ) , or an oscillating magnetic field ( forgcs - dajka and petrovay , 2001 ) .no purely fluid - dynamics mechanism can explain the tachocline , resulting in a compelling argument for the presence of a strong magnetic field ( gough and mcintyre , 1998 ) .the proper knowledge of the relationship between the solar dynamics and its structure is not important only in order to understand the present conditions of the sun , but also to understand the temporal evolution of our star and other solar - like stars .it is usually assumed that the main characteristics of the dynamics of the sun were established during its contraction phase ( turck - chize _ et al ._ , 2010 ) , hence the sun was not a rapid rotator when it entered the zero age main sequence ( zams ) .the transport of momentum during the contraction phase might have been carried out by a magnetic field in the core and the diffusion of this field flattened the rotation profile in the rest of the radiative zone ( duez _ et al ._ , 2010 ) . in any case, theories about the mechanisms that drive the solar rotation and its spatial and temporal variations remain to be tightly constrained by improved helioseismic inversion results .better rotation profiles mean not only improved inversion methodologies but improved estimates of rotational - frequency splittings .we present here results derived using an improved inversion methodology that i ) adjusts the inversion grid ( over both depth and latitude ) based on the data set and its precision , and ii ) solves the inversion problem iteratively .but first we review recent developments in global mode characterization ( korzennik , 2008 ) that allowed us to infer with better confidence the internal - rotation rate and its time varying patterns .we describe in detail the inversion methodology and show the resulting profiles .we have used rotational frequency splittings determined from fitting data acquired with five different instruments .two are ground based : the _ birmingham solar oscillations network _ ( bison ) and the _ global oscillation network group _ ( gong ) and three are on - board spacecraft : the _ global oscillations at low frequencies _ ( golf ) and the _ michelson doppler imager _ ( mdi ) on board the _ solar and heliospheric observatory _ ( soho ) , and the _ helioseismic and magnetic imager _ ( hmi ) on board the _ solar dynamics observatory _ ( sdo ) . for all but the last instrument , the available data sets span well over a decade of observations .table [ tab : datasets ] summarizes what data , from which instrument , and for what time span are included in this study ..list of instruments and time span from which data sets were used in the work presented here . [ cols="<,^",options="header " , ]   plane .the color represents how often a mode is fitted , with red indicating all the time ( ) , green 90% , etc .the top panels correspond to mdi improved symmetric and asymmetric fit and gong pipeline fit .the bottom panels correspond to our alternative fitting methodology , for , , , -day long time series ( left to right ) . ]the starting point of all helioseismic , linear rotational inversion methodologies is the functional form of the perturbation in frequency [ induced by the rotation of the sun , :   the perturbation in frequency [ with the observational error [ that corresponds to the rotational component of the frequency splittings , is given by the integral of the product of a sensitivity function , or kernel [ with the rotation rate [ ] over the radius [ and the co - latitude [ . the kernels [ are known functions of the solar model.  equation ( [ eq : equation1 ] ) defines a classical inverse problem for the sun s rotation . the inversion of this set of integral equations one for each measured  allows us to infer the rotation rate profile as a function of radius and latitude from a set of observed rotational frequency splittings ( hereafter referred as splittings ) .our inversion method requires the discretization of the integral relation to be inverted . in our case ,equation ( [ eq : equation1 ] ) is transformed into a matrix relation where is the data vector , with elements and dimension , is the solution vector to be determined at model grid points , is the matrix with the kernels of dimension , and is the vector containing the corresponding observational uncertainties . the number and location of the model grid nodes are calculated according to the effective spatial resolution of the inverted data set .such a procedure produces a non - equally spaced ( _ i.e. _  unstructured ) mesh distribution .a complete description and examples of the gridding methodology can be found in eff - darwich and prez - hernndez ( 1997 ) and eff - darwich _ et al . _( 2010 ) .the resulting unstructured grid is used to compute the matrix in equation ( [ eq : equation2 ] ) .that equation is then solved with a modified version of the iterative method developed by starostenko and zavorotko ( 1996 ) .this approach calculates according to the following algorithm : where is the iteration index .the diagonal matrices and are calculated from the summation of columns and rows of matrix , respectively . for each iteration ,values for the error propagation and data misfit , , are calculated .helioseismology , as a tool to infer the properties of the solar interior , is based on the fact that different mode sets are sensitive to different layers of the sun .hence , by combining these mode sets , it is possible to derive the structure and dynamics of the solar interior .however , these sets are not homogeneous and the number and quality of the modes that are sensitive to the solar radiative interior is significantly lower than those sensitive to the convective zone and the surface layers ( as shown in figure [ fig:0 ] ) . therefore , the dispersion and the level of uncertainties of the modes that are sensitive to the core are the largest for the entire data set .another problem arises as we look closely at the uncertainties ( see figure [ fig:0 ] ) : the error level as a function of radius is not strictly monotonic . for a given inner turning radius ,the scatter of the errors is rather large , and is primarily the consequence of the reduced accuracy of estimates at high frequencies .-day long time series ) as a function of the ratio , a proxy for the inner turning radius . for illustrative purposes ,the approximate extent of the solar core , radiative zone , tachocline and convective zone are represented .right panel : as in the left panel , but for the observational uncertainties of sectoral frequency splittings ., title="fig : " ] -day long time series ) as a function of the ratio , a proxy for the inner turning radius . for illustrative purposes ,the approximate extent of the solar core , radiative zone , tachocline and convective zone are represented .right panel : as in the left panel , but for the observational uncertainties of sectoral frequency splittings ., title="fig : " ]  figure [ fig:17 ] shows how consistent both the range in degree and frequency are when the alternative fitting technique developed for this work is used on data from different instruments .by contrast , the mode sets obtained by the team pipelines , for both gong and mdi , differ significantly , especially for their frequency spans . the consistency of our fitting technique is shown in figure [ fig:15 ] : this figure shows how both the uncertainties and the data dispersion are reduced when the length of the time series analyzed is increased .the improvement is less apparent for the modes that are more sensitive to the solar core and in the data sets corresponding to shorter time - series .however , in the case of the -day long mode set , the uncertainties for the modes sensitive to the core are reduced by a factor of four relative to the mode sets obtained from shorter time series .ratio , a proxy for the inner turning radius , for modes obtained by fitting mdi , gong -day long time - series and the mdi and gong team pipelines .right panel : as in the left panel , but for the observational uncertainties ., title="fig : " ]  ratio , a proxy for the inner turning radius , for modes obtained by fitting mdi , gong -day long time - series and the mdi and gong team pipelines .right panel : as in the left panel , but for the observational uncertainties ., title="fig : " ]   ratio , a proxy for the inner turning radius , for modes obtained by fitting mdi , , , , , and -day long time - series .right panel : as in the left panel , but for the observational uncertainties ., title="fig : " ]  ratio , a proxy for the inner turning radius , for modes obtained by fitting mdi , , , , , and -day long time - series .right panel : as in the left panel , but for the observational uncertainties ., title="fig : " ]  hence , uncertainties of the data sensitive to the solar core rotation decrease when longer time - series and better fitting technique are used .the level of uncertainties that we need to reach to counteract the low sensitivity of the modes to these regions is illustrated in figure [ fig:3 ] , with test profiles .two sets are presented : i ) one set where the radiative zone is rotating rigidly , at a rate of nhz , and below at rates of , , and nhz ; ii ) the other set where the radiative zone is also rotating rigidly at a rate of nhz , and where below the rates are again set to , , and nhz . out of these six test profiles ,only one is substantially and significantly different from the frequency - averaged rotational splittings ( _ i.e. _  averaged over frequencies in the 1.1 to 3.3 mhz range ) derived from our peak - fitting methodology ( mdi and gong days ) , the mdi pipeline ( using a 2088-day long time series ) , golf , and bison data sets .modes ( mdi -day long time series ) .the vertical colored lines represent the frequency - averaged ( 1.1to 3.3 mhz ) rotational splittings derived from our peak - fitting methodology ( mdi and gong days ) , the mdi pipeline ( using a 2088-day long time series ) , golf and bison.the red dashed lines represent the average values of the theoretical rotational splittings , if the radiative zone were rotating rigidly at a rate of nhz , with rates below of , , and nhz ( a , b and c respectively ) .the blue dashed lines represent the average values of the theoretical rotational splittings , with rates below of , , and nhz ( d , e and f respectively ) . ]  the diagnostic potential of the new global - mode fitting technique when combined with the improved inversion methodology is illustrated in figures [ fig:27 ] , [ fig:28 ] , and [ fig:30 ] , where we present the time - averaged rotation profiles of the sun from the surface down to that were calculated , using either mdi or gong , and , , , , , and -day long time - series .inversions using recent hmi data ( and -day long ) are also presented , although they are not yet comparable to either mdi or gong results , since the amount of hmi observations is still significantly smaller .both mdi and gong inversions give similar results , with the largest discrepancies at high latitudes and below .the most significant difference between the inversions obtained by the same instrument is the reduction of the uncertainties , in particular random noise , when the length of the time - series used to fit is increased .this reduction is particularly important in the inner radiative core .all results are compatible with a radiative zone rotating rigidly at a rate of approximately 431 nhz ; however , it is not possible to disregard a faster or slower rotator below ( _ i.e. _  up to 600 or down to 300 nhz ) .although the radiative zone seems to rotate rigidly , there is a consistent and systematic dip in the rotation profile located at approximately and in latitude .this dip is seen in both mdi and gong results , notwithstanding the actual length of the fitted time series .this result is intriguing , particularly if we analyze the time evolution of the dip , for both mdi and gong derived profiles , as shown in figure [ fig:35 ] .it was not possible to include the and -day long results , since the quality of the inverted profiles at that depth and latitude is too low .therefore , we used the -day long data , since its precision and temporal resolution allow us to carry out a temporal evolution analysis with adequate quality of the resulting profiles .although the dip is certainly at the limit of the resolution of the data and the inversion method , there is a systematic temporal change of the dip .this variation is not found at other latitudes .  , , , , , and -day long time - series .black , red , green , dark - blue , and light - blue lines correspond to the rotational rate at different latitudes , namely , and , respectively .vertical lines represent the error bars for the rotational rate at the equator . ]  , , , , , and -day long time - series .black , red , green , dark - blue , and light - blue lines correspond to the rotational rate at different latitudes , namely , and , respectively .vertical lines represent the error bars for the rotational rate at the equator ]   and -day long sets .black , red , green , dark - blue , and light - blue lines correspond to the rotational rate at different latitudes , namely , and , respectively .vertical lines represent the error bars for the rotational rate at the equator . ]  , where nhz , at a depth of and for different latitudes obtained from the inversion of the different mdi -day long data sets .right panel : as in the left panel , but for gong data ., title="fig : " ] , where nhz , at a depth of and for different latitudes obtained from the inversion of the different mdi -day long data sets .right panel : as in the left panel , but for gong data ., title="fig : " ]  the consequences of using different peak fitting techniques on the inversion results is illustrated in figure [ fig:29 ] .that figure shows the time - averaged rotational rates obtained using mdi and gong -day long alternative fitting method and gong and mdi respective project pipelines .the lengths of the fitted time - series are comparable , however the spherical harmonic degree and frequency ranges of the fitted mode sets differ significantly .in particular , the mode sets obtained by the project pipelines result in rotational profiles that significantly disagree in the spatial extent of the optimal inversion grid and in the inverted rotation rates at high latitudes and in the radiative zone .the mode sets obtained through the alternate technique devised for this work are , in contrast , homogeneous , even though data from different instruments were fitted .hence systematic differences introduced by different fitting techniques and different mode sets are greatly reduced .  ,gong -day long , gong pipeline , and mdi pipeline .black , red , green , dark - blue , and light - blue lines correspond to the rotational rate at different latitudes , namely , and , respectively .vertical lines represent the error bars for the rotational rate at the equator . ]we have fitted one solar cycle of mdi and gong data and the latest hmi data using a new fitting methodology .this method fits individual multiplets , an asymmetric mode profile , incorporates all known instrumental distortion , uses our best estimate of the leakage matrix , and uses an optimal sine multi - tapered spectral estimator . it was applied to time series of varying lengths to study the effect of trading off precision for temporal resolution in the inversion results . on the other hand , the improved inversion method that we used is one that estimates the optimal inversion model grid based on the extent of the mode set ( over spherical harmonic degree and frequency ) and the data uncertainties .our results are summarized in figure [ fig:18 ] , where we present the rotational profiles obtained from inverting frequency splitting derived from fitting time series spanning an entire solar cycle , cycle 23 , for both gong and mdi observations .these profiles are our best inferences of the rotation in the radiative region , to date .both results are compatible with a radiative zone rotating rigidly at a rate of approximately 431 nhz ; however , it is not possible to disregard a faster or slower rotator below ( _ i.e. _  up to 600 or down to 300 nhz ) .although the radiative zone seems to rotate rigidly , there is a consistent and systematic dip in the rotation profile located at around and of latitude .this dip appears to evolve with time , although this last result has to be confirmed when additional time series covering cycle 24 become available .( left panel ) and gong -day long ( right panel ) .the rotational rates at different latitudes , from the equator to at steps of , are represented by colored lines .vertical lines represent the error bars for the rotational rate at the equator.,title="fig : " ]  ( left panel ) and gong -day long ( right panel ) .the rotational rates at different latitudes , from the equator to at steps of , are represented by colored lines .vertical lines represent the error bars for the rotational rate at the equator.,title="fig : " ]  anderson , e.r . , duvall , t.l.jr . , jefferies , s.m . : 1990 , _ astrophys . j. _ * 364 * , 699 , ads : .broomhall , a.m. , chaplin , w.j . ,davies , g.r . , elsworth , y. , fletcher , s.t . , hale , s.j . , miller , b. , new , r. : 2009 , _ mon ._ * 396 * , l100 , ads : , doi : .scherrer , p.h . ,schou , j. , bush , r.i . ,kosovichev , a.g . ,bogart , r.s . ,hoeksema , j.t . ,liu , y. , duvall , t.l . , zhao , j. , title , a.m. , schrijver , c.j . ,tarbell , t.d . ,tomczyk , s. , 2012 , _ solar phys ._ * 365 * , 207 , ads : , doi : .

Abstract: the dynamics of the solar radiative interior are still poorly constrained by comparison to the convective zone . this disparity is even more marked when we attempt to derive meaningful temporal variations . many data sets contain a small number of modes that are sensitive to the inner layers of the sun , but we found that the estimates of their uncertainties are often inaccurate . as a result , these data sets allow us to obtain , at best , a low resolution estimate of the solar core rotation rate down to approximately . we present inferences based on mode determination resulting from an alternate peak - fitting methodology aimed at increasing the amount of observed modes that are sensitive to the radiative zone , while special care was taken in the determination of their uncertainties . this methodology has been applied to mdi and gong data , for the whole solar cycle 23 , and to the newly available hmi data . the numerical inversions of all these data sets result in the best inferences to date of the rotation in the radiative region . these results and the method used to obtain them are discussed . the resulting profiles are shown and analyzed , and the significance of the detected changes discussed .


this paper deals with the modeling of behavioral - social crowd dynamics , where this term is used to indicate that laws of classical mechanics can be substantially modified by individual behaviors and strategies developed by living entities .namely , the individuals in the crowd , are viewed as self - propelled micro - systems , who can modify their collective behaviors according to their walking strategies .these micro - scale subsystems are featured by an heterogeneously distributed ability to express their walking strategy related to the interactions with the other entities .the paper aims at presenting a unified approach , to modeling and simulation , based on the methods of the kinetic theory and stochastic games which , according to the authors bias , offer an appropriate framework to capture the greatest part of the complexity features of the systems under consideration .classical methods of the kinetic theory for molecular fluids can not be straightforwardly applied . indeed ,conservation of number of particles can be claimed in the case of self propelled particles , but not conservation of momentum and energy .the modeling involves several technical difficulties .understanding these difficulties is a necessary preliminary step to develop a successful approach .the interested reader can take advantage of some review papers , which enlighten different aspects of the modeling of crowd dynamics . restricting our attention to the most recent literature , the survey paper presents andcritically analyzes a variety of crowd models at all representation scales , from the microscopic ( individual ) to macroscopic ( hydrodynamical ) . a mathematical approach to the modeling of swarm dynamics , which has some intersection with crowd dynamics ,is proposed in .paper is proposed for crowd dynamics in unbounded domain with homogeneous distribution of walking ability of pedestrians .the present paper aims at including more general features of behavioral - social dynamics , as well as interactions with walls .moreover , the approach of uses discrete velocity variables , while a continuous variable is used in this present paper .this choice allows to overcome the uncertainty problem induced by the choice , up to now heuristic , of the number of velocity module and directions .the contents of the paper are as follows :  section 2 defines an assessment of the basic principles of behavioral - social dynamics of a large systems of interacting self - propelled particles with focus on crowd dynamics .the analysis is related to the complexity features of these specific systems referring to a scaling and representation , which anticipates the mathematical kinetic theory for active particles .the microscopic state of the latter , which constitute the micro - scale system , whose micro - state includes not only mechanical variables , typically position and velocity , but also an additional variable , called activity , suitable to model their strategy .section 3 develops the concept of behavioral social dynamics as a new science of mechanics , where interactions depend on the behaviors of the entities of the crowd , which is viewed as a living system .a general structure is used toward the derivation of specific models .the approach consists in representing the state of the system by a probability distribution over the micro - scale state of pedestrians and in deriving a general framework to describe the time and space dynamics of such distribution by a balance of particles in the elementary volume of the space of the micro - states .interactions , which are non local and nonlinearly additive , are modeled by theoretical tools of stochastic game theory .section 4 takes advantage of the framework presented in the preceding section to derive models which include also the case of individuals which change the rules of their participation to the dynamics .the whole section is in three parts .first the dynamics is modeled in unbounded domains .then it is shown how the decision process by which individuals select their trajectories include the actions to avoid walls and obstacles .finally , it is shown their social behavior evolves in time and space focusing on panic conditions .section 5 presents some simulations with to test the predictive ability of the model .particles methods ( monte carlo type ) are used as this approach appears to be well consistent with the general mathematical structure proposed in section 3 .two types of simulations are developed , the first one to show how the individuals moving in opposite directions interact , while the second one aims at understanding how panic conditions can affect the evacuation dynamics .paper is an important reference to define specific objective of simulations .section 6 presents a critical analysis and indicates how the approach can be further developed to include additional aspects of the dynamics such as modeling interactions which induce large deviations in the social behavior of the crowd .this section also outlines some research perspectives focusing on a systems approach to crowd dynamics in complex environments .this section provides an heuristic description of behavioral dynamics of pedestrian crowds to be transferred into appropriate mathematical structures . of the mesoscopic ( kinetic ) theories , where the micro - state of pedestrians is given by position and velocity , while the system is represented a probability distribution function over such state .mathematical models describe the time evolution of this function in the phase space by means of integro - differential equations .the micro - scale description , where pedestrians are identified singularly , is delivered by position , velocity , and additional variables suitable to retain features of pedestrians viewed as living entities .let us now understand what is a crowd and which are the most important features of a crowd to be taken into account in the modeling approach .first a definition of the crowd needs to be given .the following one ( due to helbing and johanson ) appears to be particularly well focused :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ agglomeration of many people in the same area at the same time .the density of the crowd is assumed to be high enough to cause continuous interactions with or reaction to other individuals . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  according to the authors bias , only three important specific features of the crowd are selected among various ones .subsequently , a critical analysis of the existing literature will focus on topics to be considered open , and to be treated in the next sections ..1 cm strategy : pedestrians have the ability to express walking strategies based on interactions with other individuals and with the surrounding environment .the latter includes vocal and visual signaling which addresses them toward optimal and safe ways , including evacuation paths ._ the modeling of pedestrians strategy should include several features , for instance trend toward the exit or a meeting point , following or avoiding streams and clusters , avoiding overcrowding in the proximity of walls , clustering of individuals with similar activity , avoiding individuals with different activity , and possibly others_.  .1 cm heterogeneity : the behavior of pedestrians is heterogeneously distributed due both to different psychologic attitudes and mobility abilities from individuals with handicap to high level walking ability .in addition , a crowd might need to to be split into different groups related to different strategies , e.g. reaching different objectives , or even an internal hierarchy , which induces different interaction rules ._ heterogeneity can refer to social behaviors , for instance aggressiveness in a crowd where two groups contrast each other or panicking behaviors .indeed , crowds can lose , in panic conditions , optimal strategies. this feature can be important in extreme situations when sudden dangers can induce the said conditions_.  .1 cm interactions : interactions involve both mechanical and social - behavioral features , and are nonlocal as individuals communicate and develop a visual activity at a distance ; these are nonlinearly additive as the strategy developed by a pedestrian is a nonlinear combination of different stimuli , while mechanics induces social exchanges and these modify the walking dynamics ._ interactions with the external environment where the pedestrians move , namely different geometrical and environmental contexts , say corridors , rooms , stairs , sudden changes of directions , luminosity conditions , and many others , can have an important effect on the dynamics as the interaction rules , which also depend on the quality of the environment_.  .1 cm  although only three general features have been selected , one can rapidly verify that the existing literature does not yet provides an exhaustive answer to all of them .in fact , recent papers , e.g. , only treat some of the aforesaid topics , while the interplay between mechanics and social behaviors deserves further attention .the meso - scale approach , introduced in and further developed in , shows how pedestrians , viewed as heterogeneous entities chose , by a decision process modeled by tools of game theory , between different trends from avoiding walls to reach prescribed exits .it is an approach quite different from that developed at the microscopic scale , for instance by the social force model , which is based on the assumption that pedestrians are subject to an acceleration to be related to social interactions within the crowd .however , models at the microscopic scale can contribute not only to understand how individual behaviors can be described by equations , but also to model interactions . a useful example is given by the paper by faure and maury , which provides a detailed analysis of the granular dynamics based on different aspects of attraction and repulsion between pedestrians .the next sections present a model , which should at least partially , include the predictive ability of the aforesaid features .the derivation will be followed by a critical analysis the said ability also referred to validation issues .this critical analysis can take advantage of the literature in the field , which already offers valuable contributions on theoretical models and interpretation of empirical data , among others , .finally , let us mention the strategic motivation to study anomalous behaviors and , in particular , the onset and propagation of panic conditions .this study is motivated by the related safety problems , see for a deeper understanding of a psycho - mechanical study of this extreme phenomenon and for a detailed analysis of evacuation times and the impact that panic conditions can have on it .indeed , the study of this feature is an important issue of this present paper .let us consider a large heterogeneous system of pedestrians moving in two dimensional domains .the mathematical approach to modeling can not relay , as observed in  , on the deterministic causality principles typical of classical mechanics . in fact , pedestrians develop a their own dynamics based on an individual interpretation of that of the other individuals .namely , they have a strategy , which is heterogeneously distributed and which depends on several factors to be included in the modeling approach .these reasonings lead to introduce the concept of behavioral - social dynamics as a new science of mechanics , where interactions depend on the behaviors of the component of the crowd viewed as a living system .this section proposes a general structure to be used toward the derivation of specific models .+ bearing this in mind , let us anticipate some terminology and some preliminary ideas of the approach that will be developed hereinafter .  *the modeling approach proposed in this paper is based on suitable developments of the so - called kinetic theory for active particles , which applies to large systems of interacting entities  . hence the meso - scale representation is chosen . *pedestrians , namely the micro - system , are viewed as active particles , that have the ability of expressing a their own strategy , called activity .this ability can differ for different groups in the same crowd , being understood that the activity is heterogeneously distributed .* the overall state of the system is described by a probability distribution over the variable deemed to define the physical state of the said entities , while interactions are modeled by theoretical tools of game theory  .a general overview of this approach is presented in the survey paper  , while applications to model crowd dynamics and social systems are proposed in  . *the overall system is subdivided into groups of pedestrians who share common features .the partition follows the hallmarks of a systems theory of social systems introduced in and followed in .* active particles can communicate and develop a social dynamics , as they learn from interactions and accordingly modify both strategy and dynamical rules followed in the movement .the output is a collective behavior which can be observed in the whole .modeling is developed in three steps , each of them presented in the next subsections , namely representation of the system , modeling of micro - scale interactions , and derivation of a mathematical structure consistent with the complexity paradigms of behavioral dynamics .this last subsection also proposes a concise critical analysis .let us now consider a crowd which can be subdivided into different groups distinguished either by different walking objectives and/or abilities .pedestrians move in the whole plane , but at an initial time are located in a domain .each group , called functional subsystem , has a different walking purpose , for instance either following a certain direction or reaching a meeting point .the microscopic state of pedestrians , viewed as active particles , is defined by position , velocity , and activity .dynamics in two space dimensions is considered , while polar coordinates are used for the velocity variable , namely , where is the velocity modulus and denotes the velocity direction .moreover , we assume that each group , called functional subsystem , can be featured by a different group s strategy , ability to express it is modeled by a variable  ] , where the lower value denotes the worse environmental conditions that prevents the movement , while the best one .dimensionless , or normalized , quantities are used by referring the components and to the length that corresponds to the largest dimension of , while the velocity modulus is divided by the maximum admissible velocity , which can be reached by a fast pedestrian in free flow conditions ; is the dimensionless time variable obtained referring the real time to a suitable critical time identified by the ratio between and .the mesoscopic ( kinetic ) representation of the overall system is delivered by the statistical distribution at time , over the micro - scale state : , \quad \theta \in [ -1,1 ) \quad u\in [ 0,1],\ ] ] for each functional subsystem labeled by , and denoted by the acronym -fs ..1 cm * remark 3.1 . * _ functional subsystems can be identified by different features , for instance trend to different exits or directions , different roles in the crowd such as hierarchical behaviors .examples will be given in the applications proposed in the following ._ .1 cm  if is locally integrable then is the ( expected ) infinitesimal number of pedestrians who , at time , have a micro - state comprised in the elementary volume of the space of the micro - states of each -fs .the following compact expression , with , is occasionally used in the following so that , while the are divided by , which which defines the maximal full packing density of pedestrians .although their state is represented by a point , pedestrians are supposed to have a finite dimension .macroscopic observable quantities can be obtained , under suitable integrability assumptions , by weighted moments of the distribution functions : while analogous calculations lead to the flux global quantities are obtained by summing over the index labeling the functional subsystems .for instance , mean velocity and velocity variance are computed as follows : and   an additional quantity to be taken into account in the modeling of interactions is the _ perceived density _ along the direction .according to  , this quantity is defined as follows : = \rho + \frac{\partial_\theta \rho}{\sqrt{1 + ( \partial_\theta \rho)^2}}\,\big[(1- \rho)\ , h(\partial_\theta \rho ) + \rho \ , h(- \partial_\theta \rho)\big]\va\ ] ] where denotes the derivative along the direction , while is the heaviside function , , and .therefore , positive gradients increase the perceived density up to the limit , while negative gradients decrease it down to the limit in a way that    as already mentioned , interactions may depend not only on the micro - state of the interacting particles ( pedestrians ) , but also on their distribution functions .when only the first type of interaction appears , one talks about linear interactions , otherwise when also second type occur , the concept of nonlinear interactions needs to be used .while linearity involves only independent variables , i.e. , micro - state variables , nonlinearity involves the dependent variables , i.e. , the distribution function and/or its moments . in the following , round and square parenthesis distinguish , respectively , the former and latter interactions .the modeling of interactions corresponds to a decision process for each particle related to the micro - state , and distribution function , of the particle and of all those in its interaction domain . for each functional subsystem ,three types of particles are involved in the process at each time : the test particle with micro - state and distribution function ; the field particle , in , with micro - state and distribution function ; and candidate particle , in , with micro - state and distribution function .the candidate particle can acquire , in probability , the micro - state of the test particle after interaction with the field particles , while the test particle loses its state in the interaction with the field particles .the test particle is representative , for each functional subsystem , of the whole system of particles .interactions of test and candidate particles with field particles , can be modeled by the following quantities : interaction domain , interaction rate , and transition probability density .these quantities can depend , as already mentioned , on the micro - state and distribution function of the interacting particles , as well as on the quality of the environment .moreover , they refer to interactions involving all functional subsystems .the expression of these terms is reported in the following , where the term -particle is used to denote a pedestrian belonging to the -th functional subsystem .* interaction domain : active particles have an interaction domain to be related to their visibility domain which can be defined as an arc of circle , with radius and , symmetric with respect to the velocity direction being defined by the visibility angles and . for practical applications andaccording to the normalization , one can assume  ] .analogous notation is used for a test particle in and a field particle in .+ .2 cm * transition probability density : interactions can be modeled by the probability density ({\boldsymbol{w } } _ * \to { \boldsymbol{w}}|{\boldsymbol{w } } _ * , { \boldsymbol{w}}^*;\alpha) ] , where correspond the worse conditions preventing the dynamics , while to the best ones , which allow a rapid dynamics .this type of nonlinearity will be discussed later . _.1 cm   the mathematical framework consists in an integro - differential equation suitable to describe the time dynamics of the distribution functions , which can be obtained by a balance of particles in the elementary volume of the space of the micro - states .this conservation law writes : where the inlet and outlet fluxes are caused by interactions .this equality corresponds to the following structure : (t , \bx , { \boldsymbol{w } } ; \alpha ) = \big(g_i -  l_i \big)[f](t , \bx , { \boldsymbol{w } } ; \alpha),\ ] ] where the various terms can be formally expressed , consistently with the definition of , and . the formal result , in the case of individuals who do not move from one functional subsystem to the other ,is as follows : ({\boldsymbol{w}}_*,{\boldsymbol{w}}^ * ; \alpha )  \ , \ca_{ik}[{f } ] ( { \boldsymbol{w } } _ * \to { \boldsymbol{w}}|{\boldsymbol{w } } _ * , { \boldsymbol{w}}^ * , u _ * ;  \alpha ) \\ \label{eq : gain }   & \times &  f_i(t , \bx , { \boldsymbol{w } } _ * ) f_k(t , \bx^ * , { \boldsymbol{w}}^*)\,d{\boldsymbol{w}}_*\,d{\boldsymbol{w}}^ * \ , d\bx^ * ,  \end{aligned}\ ] ] and ({\boldsymbol{w}}_*,{\boldsymbol{w}}^ * ; \alpha)\ , f_k(t , \bx^ * , { \boldsymbol{w}}^*)\ , d{\boldsymbol{w}}^ * \ , d\bx^*.\ ] ]  when the aforesaid crossing over functional subsystems , induced by change of behaviors , are allowed , the interaction term of the structure modifies as follows : ({\boldsymbol{w}}_*,{\boldsymbol{w}}^ * ) \ , \ca_{hk}^i[{f } ] ( { \boldsymbol{w } } _ * \to { \boldsymbol{w}}|{\boldsymbol{w } } _ * , { \boldsymbol{w}}^ * , u _ * ) \nonumber \\ \label{eq : gain2 }  & \times &  f_h(t , \bx , { \boldsymbol{w } } _ * ) f_k(t , \bx^ * , { \boldsymbol{w}}^*)\,d{\boldsymbol{w}}_*\,d{\boldsymbol{w}}^ * \ , d\bx^ * ,  \end{aligned}\ ] ] and ({\boldsymbol{w}}_*,{\boldsymbol{w}}^*)\ , f_k(t , \bx^ * , { \boldsymbol{w}}^*)\ , d{\boldsymbol{w}}^ * \ , d\bx^*.\ ] ]  finally , it can be briefly shown that this structure is consistent with the paradigms presented in section 2 .the ability of pedestrians to express walking strategies based on interactions with other individuals is modeled by the transition probability density , while the heterogeneous distribution of the said strategy ( behavior ) corresponding both to different psychologic attitudes and mobility abilities is taken into account by the use of a probability distribution over the activity variable .interactions have been assumed to be nonlocal and nonlinearly additive as the strategy developed by a pedestrian is a nonlinear combination of different stimuli generated by interactions with other pedestrians and with the external environment .the hallmarks to derive models by the kinetic theory for active particles are reported in , where the hallmarks proposed in this present paper can be followed . in detail , first a general structure suitable to capture the main complexity features of living systems is derived and subsequently such a structure is implemented by models of interactions at the micro - scale .this approach aims at overcoming the lack of first principles that govern the living matter .specific models are derived in this section within the framework proposed in the preceding one .the approach first refers to eqs .( 3.7),(3.8),(3.9 ) focusing to well defined case studies , where active particles do not move from one subsystem to the other ; subsequently , it is shown how models can be generalized to include also the aforesaid transition as by eqs .( 3.7),(3.10),(3.11 ) .the next two subsections are devoted to this objective , while the third one presents a critical analysis focused on the modeling of panic conditions , and more in general anomalous behaviors .let us now consider the modeling of crowd dynamics when pedestrians have a well defined strategic objective and do not cross from one functional subsystem to the other . in this specific case, the activity variable corresponds to the walking ability and the derivation of models needs defining the terms and , which describe interactions at the micro - scale .first some phenomenological assumptions on the qualitative micro - scale dynamics are made and subsequently these are transferred into models to be implemented into the mathematical structure . in detail :  1 .the encounter rate corresponds to the frequency by which each pedestrian develops contacts with the other ones in the visibility zone .+ .2 cm 2 .interactions modify the dynamics of pedestrians who first might change the direction of movements , and subsequently adapt their velocity to the local density conditions .+ .2 cm 3 .three types of stimuli contribute to the modification of walking direction : ( i ) desire to reach a well defined target , namely a direction or a meeting point ; ( ii ) attraction toward the mean stream ; ( iii ) attempt to avoid overcrowded areas .+ .2 cm 4 .pedestrians moving from one direction to the other , adapt their velocity to the local new density conditions , namely they decrease speed for increasing density and increase it for decreasing density .+ .2 cm 5 .the dynamics is more rapid in amply and high quality areas .moreover rapidity is heterogeneously distributed and increases for high values of the activity variable ..  interactions also modify the activity variable according to a social dynamics based on attraction and/or repulsion of social behaviors .bearing all above in mind , let us transfer this qualitative description into mathematical models of a micro - scale dynamics . according to these phenomenological assumptions , the gain and loss terms in the kinetic equation , eqs .  and respectively ,simplify to : ({\boldsymbol{w } } _ * ; \alpha ) \ ;    \ca_{i}[f ] ( { \boldsymbol{w } } _ * \to { \boldsymbol{w}}|{\boldsymbol{w } } _ * , u _ * ;  \alpha ) f_i(t , \bx , { \boldsymbol{w}}_*)\,d{\boldsymbol{w } } _ * , \\ \label{eq : loss_simplified } l_i & = & \int_{\omega \times d_{\boldsymbol{w } } } \eta_{i}[{f}]({\boldsymbol{w } } _ * ; \alpha)\,d{\boldsymbol{w } } _ * \ ; f_i(t , \bx , { \boldsymbol{w}}).\end{aligned}\ ] ]  .1 cm * modeling the encounter rate , . * in general , the encounter rate can depend on the micro - state and the meso - state of the interacting individuals viewed as active particles .a rule generally adopted is that the frequency of interactions depends on the number of particles in the interaction domain .however , the conjecture that individuals consider only a fixed number of individuals to develop the decision process on their walking strategy was posed in  .see also the general formalization of  and the computational hints in  .therefore , if the visibility allows to capture this critical density , a reasonable assumption is simply .however , lack of visibility or presence of obstacles can reduce this encounter rate ..1 cm * modeling the transition probability density , . *let us consider a candidate pedestrian located in with velocity direction and modulus , and let us define the following unit vectors :  * directed to a prescribed meeting point or to a walking direction ; .2 cm * (\bx) ] directed along the direction of lowest density gradient ;  where both and are computed within the visibility zone .pedestrians in addition to the trend to , which depends on the position only , develop a decision process between the attraction by the stream , namely , and the search of a low density path , where both unit vectors depend on and on through velocity and density , respectively .paper  suggests that a preferred direction can be heuristically chosen .accepting this hint , the following model of preferred direction is proposed : }         { \left\| ( 1-\rho ) { \boldsymbol{\nu}_i^{(t)}}+ \rho \left [ \beta { \boldsymbol{\nu}_i^{(s)}}+ ( 1-\beta ) { \boldsymbol{\nu}^{(v)}}\right ] \right\|},\ ] ] where is a parameter which models the sensitivity to the stream with respect to the search of vacuum ..1 cm * remark 4.1 . *_ if the crowd has not a well defined trend to a prescribed direction or to a meeting point , then the dynamics is simply ruled by the remaining two trends and the preferred direction is : _  .1 cm * remark 4.3 . * _ a correspondence can be easily identified , for practical simulations , between the preferred direction and the preferred angle of motion , , i.e. , ._  .1 cm all tools have now been defined to model the decision process leading to the transition probability density , , which is defined as follows (t , \bx , { \boldsymbol{w } } ; \alpha ) = \cd [ \rho ] ( u _ * \to u ) \cc [ \rho ] ( v _ * \to v ) \cb_i [ \rho,\mathbf{q}_i ] ( \theta _ * \to \theta).\ ] ]  more in detail :  1 .the candidate pedestrian in with velocity direction changes in probability into depending on the direction , on the local density , activity , and quality of the environment ; + .2 cm 2 .  after havingchanged velocity direction the velocity is modified by increasing ( decreasing ) its modulus depending on whether the perceived density is lower ( higher ) than the previous one ; + .2 cm 3 .finally , the candidate pedestrian modifies the activity by social communications .according to the aforesaid phenomenological description , the transition probability density for angles is assumed to vary linearly from to , i.e. , \left(\theta _ * \to \theta=\left(\theta _ * + \delta\theta \right)\;\mbox{mod}\;2\pi\right ) = 2 \frac{\phi_\theta-\psi_{i,\theta}}{\delta \theta_{i , max}^2 } |\delta\theta| + \frac{1+\psi_{i,\theta}-\phi_\theta}{|\delta \theta_{i , max}|},\ ] ] where  ] with and the preferred velocity is given by .an explicit definition of and is reported in fig .[ fig : p_transizione ] . in eq . , and give the negative and positive contributions to the trend to , respectively , finally let us consider the third step consisting in modeling how interactions modify the activity variable by increasing or decreasing it , but supposing that each pedestrian keeps one s own strategy without transition into another functional subsystem .the simplest assumption is that given a random initial condition of the distribution over such variable , then this is not modified by interactions .this assumption correspond to the following expression of the transition probability density    when dealing with dynamics in a bounded domain , nonlocal interactions occur with walls , obstacles , and one or more exists .the strategy pursued in our modeling approach is that the aforementioned interactions , which replace the classical local boundary conditions , define a new preferred walking direction which is determined by a two - steps procedure . as a first step , the candidate pedestrian changes in probability the direction of motion and the velocity modulus by following the same rules described in the preceding subsection , eqs .  and , respectively . as a second step , by keeping the same velocity modulus , the direction of motionis further changed so as to account for the presence of solid walls .accordingly , the probability transition density modifies to  (t , \bx , { \boldsymbol{w } } ; \alpha ) = \cb^{(2)}(\theta^{(1 ) } \to \theta ) \cd [ \rho ] ( u _ * \to u ) \cc [ \rho ] ( v _ * \to v ) \cb_i^{(1 ) } [ \rho,\mathbf{q}_i ] ( \theta _ * \to \theta^{(1)}).\ ] ]  where is defined by eq . , with in place of , and reads    in eq ., the angle of motion gives the direction of the velocity when the latter is rotated so as to reduce its normal component linearly with the distance from the wall .more specifically , after the first step , the velocity components normal and tangential to the wall are given , respectively , by    where is the velocity of the pedestrian after the decisional process that weigths target , stream and vacuum effects , and are the projection operators in the directions normal and tangential to the wall . due to the interaction with the wall , in a second step ,the pedestrian velocity components are modified as follows  ^{1/2}\end{aligned}\ ] ]  eq .shows the the velocity component normal to the wall decreases linearly approching the wall and becomes nought in the limiting case of a pedestrian in contact with the wall .the tangential component , as given by eq ., is obtained by requiring that the modulus of the velocity is constant .the angle which enters in eq ., is thus given by     the mathematical model proposed in the preceding subsections is such that pedestrians equally share the different trends , however already paper  observed that the presence of panic increases their attraction toward the stream effect .namely , pedestrians try excessively to do what the others do and neglect the search of less congested areas .this feature can be modeled by the parameter in eq . . however , the attraction toward the stream is not the only effect of panic , which increases the walking ability up to a certain extent . summarizing , increasing the level of panic indices the following :  * attraction to what the others do against the search of less crowded areas .+ .2 cm * increasing of the ability to express the walking ability .if the crowd is in a sufficiently small domain , one can suppose that at a certain critical time the aforesaid effects are homogeneously captured by the whole population .therefore a simplest approach consists in introducing a parameter suitable to define the level of panic , or other anomalous behaviors , by $ ] , where corresponds to normal conditions , while to the highest admissible level .then , the transition of the parameter of the model can be defined as follows : .\nonumber\end{aligned}\ ] ]  on the other hand , in case of overcrowding in a large environment , one can figure out situations where panic , or any anomalous behavior , is localized in a small area and is transported in the whole domain . for instance :   the modeling approach can be developed by adding to the set , which defines the micro - state so that the distribution functions are , while the transition probability density is factorized as follows : ( { \boldsymbol{w } } _ * \to { \boldsymbol{w } } ) =  \cc [ \rho ] ( v _ * \to v ) \cb [ \rho ] ( \theta _ * \to \theta){\mathcal{e}}[\rho ] ( p _ * \to p ) \cd [ \rho ] ( u _ * \to u ) , \ ] ] where now .the modeling of can be based on a consensus and learning type games , see section 3 of , while the following adjustment of eq .( 4.21) can be used againthis section develops some computation and simulations to test the capability of the model to depict emerging behaviors which are observed in reality and are of interest in the evacuation dynamics .more specifically , we consider the problem of two groups of people walking in opposite directions in a crowded street  .the analysis focuses in particular on the role of the parameter which is descriptive of the presence of panic conditions .it is well understood that this case study do not cover the whole varieties of dynamics to be considered .however , it leads to a number of dynamical behaviors worth to be studied . in subsection[ sub : particle_method ] , we give a a brief overview of the numerical method adopted in the present study to solve eq . .the simulations results are then presented and discussed in subsection [ sub : numerical_results ] .obtaining numerical solutions of eq .  is a challenging task because the unknown function depends , in principle , on seven variables .moreover , the computation of the interaction term on the right hand side requires the approximate evaluation of a multidimensional integral .numerical methods for solving similar kinetic equations in rarefied gas dynamics studies can be roughly divided into three groups :  * particle methods * semi - regular methods * regular methods  particle methods are by far the most popular and widely used simulation methods  .the basic idea is to represent the distribution function by a number of mathematical particles which move in the computational domain and interact according to stochastic rules derived from the kinetic equation .macroscopic flow properties are obtained by means of weighted averages of the particle properties .methods in groups ( b ) and ( c ) adopt similar strategies in discretizing the distribution function on a regular grid in the phase space and in using finite difference schemes to approximate the streaming term on the l.h.s of eq . . however , they differ in the way the interaction integral is evaluated . in semi - regular methods ,the interaction integral is computed by monte carlo or quasi monte carlo quadrature methods whereas deterministic integration schemes are used in regular methods . + kinetic equations for pedestrian dynamics are usually solved by means of regular methods of solution  . in the present work, however , eq .  is solved by using a nanbu - like particle scheme .compared with different methods of solutions , the proposed approach provides some advantages such as the possibilities to easily deal with complex geometries as well as to account for a sophisticated behavioral decisional process .   in the following ,simulations are carried out for two flows of pedestrians moving in opposite directions in a narrow street .the geometry of the problem is displayed in fig .[ fig : config ] .initially , the two groups of pedestrians are uniformly distributed in a rectangular room of dimension which is open on the left and on the right sides .periodic boundary conditions are assumed in the transversal direction . + the objectives of the simulations are the following :  1 .  describing the segregation of pedestrians into lanes of uniform walking direction andassess the influence of the parameter and of the density on the shape of the pattern .study of the evacuation dynamics for the two crowds .we first show that the model predicts the spontaneous formation of parallel lanes .the transient from disorder to order can be quantitatively assess by computing the band index , which measures the segregation of opposite flow directions  . in the present work it may be generalized to the expression    .4  pedestrians . left panel : temporal evolution of the band index .right panels : contour plots of the pedestrian density at time ( upper panel ) and ( lower panel).,title="fig : " ]  .6  pedestrians . left panel : temporal evolution of the band index .right panels : contour plots of the pedestrian density at time ( upper panel ) and ( lower panel).,title="fig : " ]  .4  pedestrians . left panel : temporal evolution of the band index .right panels : contour plots of the pedestrian density at time ( upper panel ) and ( lower panel).,title="fig : " ]  .6  pedestrians . left panel : temporal evolution of the band index .right panels : contour plots of the pedestrian density at time ( upper panel ) and ( lower panel).,title="fig : " ]  according to its definition , for mixed counterflows and for a perfect segregation of the opposite flows . in a low density crowd of pedestrians ,[ fig : segregation_rho10]a shows that the band index is only slightly different from zero .indeed pedestrians randomly fill the domain and no segregation takes place as shown in fig .[ fig : segregation_rho10]b which reports snapshots of pedestrian density at time ( upper panel ) and ( lower panel ) . figs .[ fig : segregation_rho150]a and [ fig : segregation_rho150]b show the same results but for a dense crowd of pedestrians . in this case, the emergence of spatial segregation is apparent , namely the different groups of pedestrians form in line and file alternatively .it is worth pointing out that , unlike most of the previous studies on the subject , no specific repulsion forces have been assumed between pedestrians belonging to different populations .we then compute the ratio between the pedestrian mean flow rate through a section of the street and the number of pedestrian as a function of the number of pedestrian for a different sensitivity to the stream with respect to the search of vacuum . below a threshold value , ,the reduced mean flow rate is almost constant for any value of , whereas , as the number of pedestrian increases , it reduces .this is not unexpected since in the behavioral decisional process of pedestrians the weight of the target direction decreases with the mean crowd density , .when , which corresponds to , the stream and vacuum effects become dominant thus leading to a loss of efficiency in reaching the target .  .the sequential steps followed in this paper to pursue the objective of developing an approach to behavioral social dynamics can be listed as follows :  1 .definition of the features of behavioral dynamics ; + .1 cm 2 .derivation of a general structure suitable to offer the conceptual basis for the derivation of models ; + .1 cm 3 .modeling interactions at the microscopic scale to implement the aforesaid structure and derive specific models in unbounded and bounded domains ;  this process has been focused on a detailed analysis of the influence of panic conditions on the overall dynamics , which included some aspects of behavioral dynamics by the activity variable assumed to be heterogeneously distributed over individuals .it has been shown how this distribution evolves in time and space due to interactions .however , the approach does not yet tackle a challenging problem , which consists in modeling the dynamics of a crowd subdivided into different functional subsystems behaving with different features and purposes . in particular , the dynamics should predict transitions from one subsystem to the other .it is worth mentioning that this type of investigation is motivated by security problems . as an example, one can consider a crowd of individuals in a public demonstration to support or make opposition to political issue .these individuals can be subdivided into a large group manifest correctly their position , while a small group are rioters .their number can grow in time due to interactions which might persuade the other part of the crowd to join them .similarly a small group of security forces might react to provocations out of the settle protocol , but their number might also grow due to interactions , which cause an excess of reactions . the modeling can take advantage of the more general structure given in eqs .( 3.7 ) , ( 3.10 ) , ( 3.11 ) , as well as of the literature on social dynamics .these brief perspective ideas guide a research program already in progress .ballerini m. , cabibbo n. , candelier r. , cavagna a. , cisbani e. , giardina i. , lecomte v. , orlandi a. , parisi g. , procaccini a. , viale m. , and zdravkovic v. , interaction ruling animal collective behavior depends on topological rather than metric distance : evidence from a field study , _ proc .* 105(4 ) * ( 2008 ) , 12321237 .moussad m. , helbing d. , garnier s. , johansson a. , combe m. , and theraulaz g. , experimental study of the behavioural mechanisms underlying self - organization in human crowds , _ proc .b _ , * 276 * ( 2009 ) , 27552762 .

Abstract: this paper presents a new approach to behavioral - social dynamics of pedestrian crowds by suitable development of methods of the kinetic theory . it is shown how heterogeneous individual behaviors can modify the collective dynamics , as well as how local unusual behaviors can propagate in the crowd . the main feature of this approach is a detailed analysis of the interactions between dynamics and social behaviors . + self - propelled particles , scaling , nonlinear interactions , crowd dynamics , kinetic theory , active particles .