You are an expert at summarizing long articles. Proceed to summarize the following text: 
many complex systems in various areas of science exhibit a spatio - temporal dynamics that is inhomogeneous and can be effectively described by a superposition of several statistics on different scales , in short a superstatistics @xcite . the superstatistics notion was introduced in @xcite , in the mean time many applications for a variety of complex systems have been pointed out @xcite . essential for this approach is the existence of sufficient time scale separation between two relevant dynamics within the complex system . there is an intensive parameter @xmath1 that fluctuates on a much larger time scale than the typical relaxation time of the local dynamics . in a thermodynamic setting , @xmath1 can be interpreted as a local inverse temperature of the system , but much broader interpretations are possible . the stationary distributions of superstatistical systems , obtained by averaging over all @xmath1 , typically exhibit non - gaussian behavior with fat tails , which can be a power law , or a stretched exponential , or other functional forms as well @xcite . in general , the superstatistical parameter @xmath1 need not to be an inverse temperature but can be an effective parameter in a stochastic differential equation , a volatility in finance , or just a local variance parameter extracted from some experimental time series . many applications have been recently reported , for example in hydrodynamic turbulence @xcite , for defect turbulence @xcite , for cosmic rays @xcite and other scattering processes in high energy physics @xcite , solar flares @xcite , share price fluctuations @xcite , random matrix theory @xcite , random networks @xcite , multiplicative - noise stochastic processes @xcite , wind velocity fluctuations @xcite , hydro - climatic fluctuations @xcite , the statistics of train departure delays @xcite and models of the metastatic cascade in cancerous systems @xcite . on the theoretical side , there have been recent efforts to formulate maximum entropy principles for superstatistical systems @xcite .    in this paper we provide an overview over some recent developments in the area of superstatistics . three examples of recent applications are discussed in somewhat more detail : the statistics of lagrangian turbulence , the statistics of train departure delays , and the survival statistics of cancer patients . in all cases the superstatistical model predictions are in very good agreement with real data . we also comment on recent theoretical approaches to develop generalized maximum entropy principles for superstatistical systems . in generalized versions of statistical mechanics one starts from more general entropic measures than the boltzmann - gibbs shannon entropy . a well - known example is the @xmath2-entropy @xcite @xmath3 but other forms are possible as well ( see , e.g. , @xcite for a recent review ) . the @xmath4 are the probabilities of the microstates @xmath5 , and @xmath2 is a real number , the entropic index . the ordinary shannon entropy is contained as the special case @xmath6 :    @xmath7    extremizing @xmath8 subject to suitable constraints yields more general canonical ensembles , where the probability to observe a microstate with energy @xmath9 is given by    @xmath10    one obtains a kind of power - law boltzmann factor , of the so - called @xmath2-exponential form . the important question is what could be a physical ( non - equilibrium mechanism ) to obtain such distributions . the reason could indeed be a driven nonequilibrium situation with local fluctuations of the environment . this is the situation where the superstatistics concept enters . our starting point is the following well - known formula    @xmath11    where @xmath12 is the @xmath0 ( or @xmath13 ) probability distribution . we see that averaged _ ordinary _ boltzmann factors with @xmath0 distributed @xmath1 yield a _ generalized _ boltzmann factor of @xmath2-exponential form . the physical interpretation is that tsallis type of statistical mechanics is relevant for _ nonequilibrium _ systems with temperature fluctuations . this approach was made popular by two prls in 2000/2001 @xcite , which used the @xmath0-distribution for @xmath14 . general @xmath14 were then discussed in @xcite .    in @xcite it was suggested to construct a dynamical realization of @xmath2-statistics in terms of e.g. a linear langevin equation    @xmath15    with fluctuating parameters @xmath16 . here @xmath17 denotes gaussian white noise . the parameters @xmath18 are supposed to fluctuate on a much larger time scale than the velocity @xmath19 . one can think of a brownian particle that moves through spatial cells with different local @xmath20 in each cell ( a nonequilibrium situation ) . assume the probability distribution of @xmath1 in the various cells is a @xmath0-distribution of degree @xmath21 : @xmath22 then the conditional probability given some fixed @xmath1 in a given cell is gaussian , @xmath23 , the joint probability is @xmath24 and the marginal probability is @xmath25 . integration yields @xmath26 i.e. we obtain power - law boltzmann factors with @xmath27 , @xmath28 , and @xmath29 . here @xmath30 is the average of @xmath1 . the idea of superstatistics is to generalize this example to much broader systems . for example , @xmath1 need not be an inverse temperature but can in principle be any intensive parameter . most importantly , one can generalize to _ general probability densities @xmath14 _ and _ general hamiltonians_. in all cases one obtains a superposition of two different statistics : that of @xmath1 and that of ordinary statistical mechanics . superstatistics hence describes complex nonequilibrium systems with spatio - temporal fluctuations of an intensive parameter on a large scale . the _ effective _ boltzmann factors @xmath31 for such systems are given by    @xmath32    some recent theoretical developments of the superstatistics concept include the following :    * can prove a superstatistical generalization of fluctuation theorems @xcite * can develop a variational principle for the large - energy asymptotics of general superstatistics @xcite ( depending on @xmath14 , one can get not only power laws for large @xmath33 but e.g. also stretched exponentials ) * can formally define generalized entropies for general superstatistics @xcite * can study microcanonical superstatistics ( related to a mixture of @xmath2-values ) @xcite * can prove a superstatistical version of a central limit theorem leading to @xmath2-statistics @xcite * can relate it to fractional reaction equations @xcite * can consider superstatistical random matrix theory @xcite * can apply superstatistical techniques to networks @xcite * can define superstatistical path integrals @xcite * can do superstatistical time series analysis @xcite    ... and some more practical applications :    * can apply superstatistical methods to analyze the statistics of 3d hydrodynamic turbulence @xcite * can apply it to atmospheric turbulence ( wind velocity fluctuations @xcite ) * can apply superstatistical methods to finance and economics @xcite * can apply it to blinking quantum dots @xcite * can apply it to cosmic ray statistics @xcite * can apply it to various scattering processes in particle physics @xcite * can apply it to hydroclimatic fluctuations @xcite * can apply it to train delay statistics @xcite * can consider medical applications @xcite while in principle any @xmath14 is possible in the superstatistics approach , in practice one usually observes only a few relevant distributions . these are the @xmath0 , inverse @xmath0 and lognormal distribution . in other words , in typical complex systems with time scale separation one usually observes 3 physically relevant universality classes @xcite    * \(a ) @xmath0-superstatistics ( @xmath34 tsallis statistics ) * \(b ) inverse @xmath0-superstatistics * \(c ) lognormal superstatistics    what could be a plausible reason for this ? consider , e.g. , case ( a ) . assume there are many microscopic random variables @xmath35 , @xmath36 , contributing to @xmath1 in an additive way . for large @xmath37 , the sum @xmath38 will approach a gaussian random variable @xmath39 due to the ( ordinary ) central limit theorem . there can be @xmath21 gaussian random variables @xmath40 of the same variance due to various relevant degrees of freedom in the system . @xmath1 should be positive , hence the simplest way to get such a positive @xmath1 is to square the gaussian random variables and sum them up . as a result , @xmath41 is @xmath0-distributed with degree @xmath21 , @xmath42 where @xmath43 is the average of @xmath1 . \(b ) the same considerations can be applied if the temperature @xmath44 rather than @xmath1 itself is the sum of several squared gaussian random variables arising out of many microscopic degrees of freedom @xmath35 . the resulting @xmath14 is the inverse @xmath0-distribution : @xmath45 it generates superstatistical distributions @xmath46 that decay as @xmath47 for large @xmath33 . \(c ) @xmath1 may be generated by multiplicative random processes . consider a local cascade random variable @xmath48 , where @xmath37 is the number of cascade steps and the @xmath35 are positive microscopic random variables . by the ( ordinary ) central limit theorem , for large @xmath37 the random variable @xmath49 becomes gaussian for large @xmath37 . hence @xmath39 is log - normally distributed . in general there may be @xmath21 such product contributions to @xmath1 , i.e. , @xmath50 . then @xmath51 is a sum of gaussian random variables , hence it is gaussian as well . thus @xmath1 is log - normally distributed , i.e. , @xmath52 where @xmath53 and @xmath54 are suitable parameters . we will now discuss examples of the three different superstatistical universality classes . our first example is the departure delay statistics on the british rail network . clearly , at the various stations there are sometimes train departure delays of length @xmath55 . the 0th - order model for the waiting time would be a poisson process which predicts that the waiting time distribution until the train finally departs is @xmath56 , where @xmath1 is some parameter . but this does not agree with the actually observed data @xcite . a much better fit is given by a @xmath2-exponential , see fig .  1 .    ) .,width=302 ]    what may cause this power law that fits the data ? the idea is that there are fluctuations in the parameter @xmath1 as well . these fluctuations describe large - scale temporal or spatial variations of the british rail network environment . examples of causes of these @xmath1-fluctuations :    * begin of the holiday season with lots of passengers * problem with the track * bad weather conditions * extreme events such as derailments , industrial action , terror alerts , etc .    as a result , the long - term distribution of train delays is then a mixture of exponential distributions where the parameter @xmath1 fluctuates :    @xmath57    for a @xmath0-distributed @xmath1 with @xmath21 degrees of freedom one obtains    @xmath58    where @xmath59 and @xmath60 . the model discussed in @xcite generates @xmath2-exponential distributions of train delays by a simple mechanism , namely a @xmath0-distributed parameter @xmath1 of the local poisson process . this is an example for @xmath0 superstatistics . our next example is an application in turbulence . consider a single tracer particle advected by a fully developed turbulent flow . for a while it will see regions of strong turbulent activity , then move on to calmer regions , just to continue in yet another region of strong activity , and so on . this is a superstatistical dynamics , and in fact superstatistical models of turbulence have been very successful in recent years @xcite . the typical shape of a trajectory of such a tracer particle is plotted in fig .  2 . this is lagrangian turbulence in contrast to eulerian turbulence , meaning that one is following a single particle in the flow . in particular , one is interested in velocity differences @xmath61 of the particle on a small time scale @xmath62 . for @xmath63 this velocity difference becomes the local acceleration @xmath64 . a superstatistical lagrangian model for 3-dim velocity differences of the tracer particle has been developed in @xcite . one simply looks at a superstatistical langevin equation of the form    @xmath65    here @xmath66 and @xmath67 are constants . note that the term proportional to @xmath67 introduces some rotational movement of the particle , mimicking the vortices in the flow . the noise strength @xmath68 and the unit vector @xmath69 evolve stochastically on a large time scale @xmath70 and @xmath71 , respectively , thus obtaining a superstatistical dynamics . @xmath70 is of the same order of magnitude as the integral time scale @xmath72 , whereas @xmath73 is of the same order of magnitude as the kolmogorov time scale @xmath74 . one can show that the reynolds number @xmath75 is basically given by the time scale ratio @xmath76 . the time scale @xmath77 describes the average life time of a region of given vorticity surrounding the test particle .    in this superstatistical turbulence model one defines the parameter @xmath1 to be @xmath78 , but it does _ not _ have the meaning of a physical inverse temperature in the flow . rather , one has @xmath79 , where @xmath80 is the kinematic viscosity and @xmath81 is the average energy dissipation , which is known to fluctuate in turbulent flows . in fact , kolmogorov s theory of 1961 suggests a lognormal distribution for @xmath82 , which automatically leads us to lognormal superstatistics : it is reasonable to assume that the probability density of the stochastic process @xmath83 is approximately a lognormal distribution @xmath84 for very small @xmath62 the 1d acceleration component of the particle is given by @xmath85 and one gets out of the model the 1-point distribution @xmath86 this prediction agrees very well with experimentally measured data of the acceleration statistics , which exhibits very pronounced ( non - gaussian ) tails , see fig .  3 for an example .    ) ( see @xcite for more details).,width=302 ]    it is interesting to see that our 3-dimensional superstatistical model predicts the existence of correlations between the acceleration components . for example , the acceleration @xmath87 in @xmath88 direction is not statistically independent of the acceleration @xmath89 in @xmath90-direction . we may study the ratio @xmath91 of the joint probability @xmath92 to the 1-point probabilities @xmath93 and @xmath94 . for independent acceleration components this ratio would always be given by @xmath95 . however , our 3-dimensional superstatistical model yields prediction @xmath96 this is a very general formula , valid for any superstatistics , for example also tsallis statistics , obtained when @xmath14 is the @xmath0-distribution . the trivial result @xmath95 is obtained only for @xmath97 , i.e.  no fluctuations in the parameter @xmath1 at all . 4 shows @xmath91 as predicted by lognormal superstatistics :    ) , @xmath14 being the lognormal distribution.,width=302 ]    the shape of this is very similar to experimental measurements @xcite . our final example of application of superstatistics is for a completely different area : medicine . we will look at cell migration processes describing the metastatic cascade of cancerous cells in the body @xcite . there are various pathways in which cancerous cells can migrate : via the blood system , the lymphatic system , and so on . the diffusion constants for these various pathways are different . in this way superstatistics enters , describing different diffusion speeds for different pathways ( see fig . 5 ) . but there is another important issue here : when looking at a large ensemble of patients then the spread of cancerous cells can be very different from patient to patient . for some patients the cancer spreads in a very aggressive way , whereas for others it is much slower and less aggressive . so superstatistics also arises from the fact that all patients are different .    a superstatistical model of metastasis and cancer survival has been developed in @xcite . details are described in that paper . here we just mention the final result that comes out of the model : one obtains the following prediction for the probability density function of survival time @xmath55 of a patient that is diagnosed with cancer at @xmath98 : @xmath99 or @xmath100 , \label{eq7}\end{aligned}\ ] ] where @xmath101 is the modified bessel function . note that this is inverse @xmath0 superstatistics . the role of the parameter @xmath1 is now played by the parameter @xmath102 , which in a sense describes how aggressively the cancer propagates .    the above formula based on inverse @xmath0 superstatistics is in good agreement with real data of the survival statistics of breast cancer patients in the us . the superstatistical formula fits the observed distribution very well , both in a linear and logarithmic plot ( see fig.6 ) . one remark is at order . when looking at the relevant time scales one should keep in mind that the data shown are survival distributions _ conditioned on the fact that death occurs due to cancer_. many patients , in particular if they are diagnosed at an early stage , will live a long happy life and die from something else than cancer . these cases are _ not _ included in the data .    , both in a linear and double logarithmic plot . only patients that die from cancer are included in the statistics . the solid line is the superstatistical model prediction @xcite . , title="fig:",width=302 ] , both in a linear and double logarithmic plot . only patients that die from cancer are included in the statistics . the solid line is the superstatistical model prediction @xcite . , title="fig:",width=302 ] we finish this article by briefly mentioning some other recent interesting theoretical developments . one major theoretical concern is that a priori the superstatistical distribution @xmath14 can be anything . but perhaps one should single out the really relevant distributions @xmath14 by a least biased guess , given some constraints on the complex system under consideration . this program has been developed in some recent papers @xcite . there are some ambiguities which constraints should be implemented , and how . a very general formalism is presented in @xcite , which contains previous work @xcite as special cases . the three important universality classes discussed above , namely @xmath0 superstatistics , inverse @xmath0 superstatistics and lognormal superstatistics are contained as special cases in the formalism of @xcite . in principle , once a suitable generalized maximum entropy principle has been formulated for superstatistical systems , one can proceed to a generalized thermodynamic description , get a generalized equation of state , and so on . there is still a lot of scope of future research to develop the most suitable formalism . but the general tendency seems to be to apply maximum entropy principles and least biased guesses to nonequilibrium situations as well . in fact , jaynes @xcite always thought this is possible . another interesting development is what one could call a superstatistical path integral . these are just ordinary path integrals but with an additional integration over a parameter @xmath1 that make the wiener process become something more complicated , due to large - scale fluctuations of its diffusion constant . jizba et al . investigate under which conditions one obtains a markov process again @xcite . it seems some distributions @xmath14 are distinguished as making the superstatistical process simpler than others , preserving markovian - like properties . these types of superstatistical path integral processes have applications in finance , and possibly also in quantum field theory and elementary particle physics .    in high energy physics , many of the power laws observed for differential cross sections and energy spectra in high energy scattering processes can also be explained using superstatistical models @xcite . the key point here is to extend the hagedorn theory @xcite to a superstatistical one which properly takes into account temperature fluctuations @xcite . superstatistical techniques have also been recently used to describe the space - time foam in string theory @xcite . superstatistics ( a statistics of a statistics ) provides a physical reason why more general types of boltzmann factors ( e.g.  @xmath2-exponentials or other functional forms ) are relevant for _ nonequilibrium _ systems with suitable fluctuations of an intensive parameter . let us summarize :    * there is evidence for three major physically relevant universality classes : @xmath0-superstatistics @xmath34 tsallis statistics , inverse @xmath0-superstatistics , and lognormal superstatistics . these arise as universal limit statistics for many different systems . * superstatistical techniques can be successfully applied to a variety of complex systems with time scale separation . * the train delays on the british railway network are an example of @xmath0 superstatistics = tsallis statistics @xcite . * a superstatistical model of _ lagrangian turbulence _ @xcite is in excellent agreement with the experimental data for probability densities , correlations between components , decay of correlations , lagrangian scaling exponents , etc . this is an example of lognormal superstatistics @xcite . * cancer survival statistics is described by inverse @xmath0 superstatistics @xcite . * the long - term aim is to find a good thermodynamic description for general superstatistical systems . a generalized maximum entropy principle may help to achieve this goal . 99 c. beck and e.g.d . cohen , physica a * 322 * , 267 ( 2003 ) c. beck , e.g.d . cohen , and h.l . swinney , phys . e * 72 * , 056133 ( 2005 ) c. beck and e.g.d . cohen , physica a * 344 * , 393 ( 2004 ) h. touchette and c. beck , phys . e * 71 * , 016131 ( 2005 ) c. tsallis and a.m.c . souza , phys . e * 67 * , 026106 ( 2003 ) p. jizba , h. kleinert , phys . e * 78 * , 031122 ( 2008 ) c. vignat , a. plastino and a.r . plastino , cond - mat/0505580 c. vignat , a. plastino , arxiv 0706.0151 p .- h . chavanis , physica a * 359 * , 177 ( 2006 ) g. wilk and z. wlodarczyk , phys . lett . * 84 * , 2770 ( 2000 ) c. beck , phys . * 87 * , 180601 ( 2001 ) k.  e. daniels , c. beck , and e. bodenschatz , physica d * 193 * , 208 ( 2004 ) c. beck , physica a * 331 * , 173 ( 2004 )    m. baiesi , m. paczuski and a.l . stella , phys . lett . * 96 * , 051103 ( 2006 ) y. ohtaki and h.h . hasegawa , cond - mat/0312568 a.y . abul - magd , physica a * 361 * , 41 ( 2006 ) s. rizzo and a. rapisarda , aip conf . proc . * 742 * , 176 ( 2004 ) ( cond - mat/0406684 ) t. laubrich , f. ghasemi , j. peinke , h. kantz , arxiv:0811.3337 a. porporato , g. vico , and p.a . fay , geophys . res . lett . * 33 * , l15402 ( 2006 ) a. reynolds , phys . * 91 * , 084503 ( 2003 ) h. aoyama et al . , arxiv:0805.2792 e. van der straeten , c. beck , arxiv:0901.2271 a.y . abul - magd , g. akemann , p. vivo , arxiv 0811.1992 c. beck , europhys . lett . * 64 * , 151 ( 2003 ) c. beck , phys . * 98 * , 064502 ( 2007 ) g. wilk , z. wlodarczyk , arxiv:0810.2939 c. beck , arxiv:0902.2459 m. ausloos and k. ivanova , phys . e * 68 * , 046122 ( 2003 ) j .- p . bouchard and m. potters , _ theory of financial risk and derivative pricing _ , cambridge university press , cambridge ( 2003 )    a.y . abul - magd , b. dietz , t. friedrich , a. richer , phys . e * 77 * , 046202 ( 2008 ) s. abe and s. thurner , phys . e * 72 * , 036102 ( 2005 ) slvio m. duarte queirs , braz . * 38 * , 203 ( 2008 ) k. briggs , c. beck , physica a * 378 * , 498 ( 2007 ) l. leon chen , c. beck , physica a * 387 * , 3162 ( 2008 ) s. abe , c. beck and g. d. cohen , phys . e * 76 * , 031102 ( 2007 ) g. e. crooks , phys . e * 75 * , 041119 ( 2007 ) j. naudts , aip conference proceedings * 965 * , 84 ( 2007 )    e. van der straeten , c. beck , phys . e * 78 * , 051101 ( 2008 ) c. tsallis , j. stat . phys . * 52 * , 479 ( 1988 ) c. tsallis , r.s . mendes , a.r . plastino , physica a * 261 * , 534 ( 1998 ) c. beck , arxiv:0902:1235 , to appear in contemporary physics ( 2009 ) a.m. mathai and h.j . haubold , physica a * 375 * , 110 ( 2007 )        r. d. rosenkrantz , _ e.t . jaynes : papers on probability , statistics and statistical physics , _ kluwer ( 1989 ) r. hagedorn , suppl . nuovo cim . * 3 * , 147 ( 1965 ) c. beck , physica a * 286 * , 164 ( 2000 ) n.e . mavromatos and s. sarkar , arxiv:0812.3952
we provide an overview on superstatistical techniques applied to complex systems with time scale separation . 
 three examples of recent applications are dealt with in somewhat more detail : the statistics of small - scale velocity differences in lagrangian turbulence experiments , train delay statistics on the british rail network , and survival statistics of cancer patients once diagnosed with cancer . 
 these examples correspond to three different universality classes : lognormal superstatistics , @xmath0-superstatistics and inverse @xmath0 superstatistics .