the diffusion of atoms and molecules on surfaces is of importance in a wide range of research fields and applications , consequently , a wide range of dedicated experimental and theoretical methods have been developed over the years .one of these techniques is quasi elastic helium atom scattering ( qhas ) .this method which has received a significant boost with the availability of the helium spin echo ( hse ) apparatus , provides a unique opportunity to follow atomic scale motions on time scales of pico to nano - seconds . with new datacomes the need for new or improved models to interpret the data and extract the underlying physical properties of the surface system .one commonly used interpretation model is based on a 2d langevin simulation which allows the extraction of a potential energy surface term and a friction parameter from the experimental data. one obvious drawback of this model is that the complex dynamics of the substrate atoms are not explicitly treated and correlations between the motion of the adsorbate and substrate particles can not be accounted for .another approach , which to the best of our knowledge was only applied twice to interpret qhas measurements , is using molecular dynamics ( md ) simulations . in these md simulations , the motion of the surface atoms and their interaction with their neighbors are explicitly calculated and correlation effects between the motion of the adsorbate and the substrate atoms are inherently included . on the down side ,these explicit md simulations are computationally expensive . in this manuscriptwe describe a numerical study which aims to quantify the differences between these two approaches and probe the validity of using the simpler and less time consuming langevin approach .this comparison is performed by calculating the observables with both simulations under similar conditions .the paper is organized in the following manner , we start by introducing some useful relations and definitions , then we describe the two numerical simulations and explain how they were tuned to simulate similar surface systems and finally we present the results of the comparison .surface diffusion is a general process which describes the motion of particles ranging from atoms to macro molecules which are confined to a surface . for most systems surface diffusionis essentially a classical process - with hydrogen diffusion at low temperatures being an example of an exception . for molecular adsorbatesa simple and sometimes sufficiently good description can be obtained by ignoring the internal degrees of freedom , although it should be noted that these degrees of freedom can play an important role in some systems .there are several physical properties which are accessible to experiments and can be used to characterize surface motion , particularly popular choices are the the tracer diffusion coefficient in the case of isolated diffusion and the chemical diffusion coefficient for the case of collective motion .another way of characterizing motion is using pair correlation functions which , unlike the diffusion coefficients mentioned above , contain a full statistical description of the motion and its underlying mechanism .the pair correlation function we use can be written as a sum of the self correlation and the distinct correlation functions , defined by   these correlation functions can be interpreted as a measure of the probability of finding a particle at location at time , given that the same or a different particle was at the origin at time .the self correlation function describes the complete dynamics of an individual adsorbate .it is also the dominant contribution for dilute adsorbate coverages . in this workwe will focus on the zero coverage limit , neglecting the contribution from the distinct correlation function .one advantage of using these pair correlation functions is their close relation to quantities which can be measured in experiments .in particular fourier transforming to the momentum domain gives , the self part of the intermediate scattering function ( isf ) where is proportional to the momentum exchange parallel to the surface in a scattering experiment . within the kinematic scattering approximationit can be shown that the hse technique mentioned above measures this quantity directly , where the experimentally accessible values range between 0.01 to a few inverse angstroms and the times , , range from 0.1ps to a few nano seconds . using the same scattering approximationit can be shown that for a time of flight helium atom scattering apparatus , the observable quantity is the temporal fourier transform of known as the dynamic structure factor ( dsf ) , , where is the energy exchange during the scattering event .  generally speaking , when random motion takes place on a surface decays with time reflecting the loss of correlation of the position of the surface particles , a decaying isf corresponds to a peak in the dsf which is centered around and has a width which is inversely proportional to the decay rate .this peak is called the quasi - elastic peak ( qep ) and its width , , is often termed the quasi - elastic broadening . whether one measures the isf ( using hse ) or the dsf ( using time of flight helium scattering ) , an analytical or numerical model is needed to extract the physical properties of the surface dynamics from the data .one particularly useful analytical model for surface diffusion of adsorbates , which can be used to calculate the quasi - elastic broadening , is the jump diffusion model which was first derived by chudley and elliott for neutron scattering measurements of bulk diffusion . inthis model vibrational motions within the adsorption sites ( intra - cell motions ) are ignored and the inter - cell jumps between one adsorption site and another are assumed to be instantaneous .generally speaking , the chudley elliot model is suited for systems where i ) the energy barrier for diffusion is large compared with the thermal energy ii ) the adsorption sites form a bravais lattice and iii ) the adsorbate coverage is sufficiently small to not be affected by the presence of other adsorbates , i.e. we can ignore the influence of the distinct pair correlation function .the results of this simple model are an isf which decays exponentially with time , which corresponds to a dsf which contains a lorentzian peak centered at with a finite width , . for the case of the chudley - elliot model the dependence of the quasi elastic broadening on the momentum transferis given by    where the sum is over of a discrete set of possible jump vectors connecting two adsorption sites and is the jump rate for a particular jump vector .( [ eq : jump qhas-1 ] ) contains all the information about the jump diffusion process , hence , the experimental isf allows us to extract the different jump rates .furthermore , if we make the assumption that the jump rate is of the form we can find the potential barrier the adsorbate has to overcome by finding the temperature dependence of the coefficients .an analytic approach in general and eq .( [ eq : jump qhas-1 ] ) in particular , provides significant insights into the dynamics when analyzing experimental data ( e.g. ) . on the other hand using equation [ eq : jump qhas-1 ] , which treats the surface as a discrete set of point - like adsorption sites that an isolated point - like particle jumps between , is typically restricted to relatively simple surface dynamics systems where this ideal - jump model is valid . as mentioned above ,an alternative approach which has been extensively used in the last few decades is to numerically calculate the trajectories using a set of parameters which describe the various interactions , extract observables such as the dsf or isf from the trajectories and by comparison with the experimental observables improve the interaction parameters until a good fit is obtained . for practical reasons the comparison is typically performed on 1d quantities such as the dependence of the quasi - elastic broadening as function of momentum transfer , temperature or coverage rather than direct comparison of the two dimensional dsf or isf functions .the two numerical approaches we compare in this work are molecular dynamics ( md ) and 2d langevin simulations .the first , provides an explicit treatment of the interactions between all the particles , whereas the second provides fast computation and a simple separation of the static and dynamic interactions characterizing the surface and has been heavily used to interpret quasi - elastic helium scattering experiments .md simulations include the degrees of freedom of both the adsorbate and substrate atoms . in order to mimic experimental systems in a realistic way, complex many - body interactions can be used .however , the purpose of this work is to study how well langevin simulations can reproduce the explicit approach of md modeling .this can be studied with particularly simple interactions for the md simulation ; pair - wise harmonic interaction between the substrate atoms and a morse potential between the adsorbate and the substrate atoms .the parameters of these two interaction models and the mass of the particles were initially chosen to resemble an experimentally relevant situation - the motion of a sodium atom ( mass = 23 amu ) on a flat ( 001 ) copper surface .the na / cu(001 ) system has been extensively studied experimentally , both in the regime of low coverage where the sodium atom can be assumed to move as an isolated adsorbate and at higher coverages where correlated motion effects dominate .furthermore , this system represents a rare case where a md simulation has been used to interpret quasi - elastic helium scattering measurements , which conveniently supplies us with a set of parameters for both the harmonic and the morse potentials .  for the harmonic term , , a single force constant between nearest neighbohrs of shown to provide a good description of the copper bulk phonons .as mentioned above , the surface - adsorbate potential was modeled with a morse like potential where is the distance between the jth substrate atoms and the adsorbate and the sum runs over all the substrate atoms .the following values were used for the parameters which reproduce the experimental measurements of the adsorption height and vibrational frequencies .the geometry we used included a copper solid consisting of 7 layers of lattice cells with a total of 896 atoms .periodic boundary conditions were imposed parallel to the surface . the bottom layer of the slab was frozen to simulate the rest of bulk layers and to fix the center of mass in its place .the substrate atoms at were placed in their bulk equilibrium positions , and were given a random initial velocity using a maxwell - boltzmann distribution .the atoms were then allowed to relax , after this relaxation period a single adsorbate atom was added on the surface and the system was allowed to relax again to the desired temperature .the simulation was carried in the micro canonical ensemble .the newtonian equations of motion were solved using beeman s algorithm for both substrate atoms and the single adsorbate .a popular approach for interpreting quasi - elastic helium scattering experiments is using a 2d langevin simulation .when inter - adsorbate interactions can be ignored ( the zero coverage limit ) , the force is given by where is the adsorbate s 2d momentum .the equation includes a constant potential energy surface ( pes ) term which is the potential the adsorbate experiences when the surface atoms are at their equilibrium points ( see sec .( [ sec : constructing - a - comparable ] ) for a description of the procedure used to obtain the pes ) .the two terms which replace the explicit treatment of the dynamic interaction between the adsorbate and substrate atoms are a dissipation term which leads to energy losses and a random fluctuating force ( typically chosen as a white noise force ) which allows energy to be supplied to the adsorbate .these two terms are not independent , as they are related through the fluctuation dissipation theorem it should be noted that if the issue of restricted computational time is ignored it would be preferable to extend the simulation to include a three dimensional motion of the adsorbate .however , since the height of an adsorbate above the surface is typically restricted to a very narrow range ( fractions of an angstrom ) and correspondingly the vibrational period perpendicular to the surface is about an order of magnitude faster than the motion of the adsorbate parallel to the surface , the effect of the vertical motion is typically assumed to be averaged out and a 2d langevin approach is used for analysis . as this is the most frequently used approach we chose to use it in our comparison .when fitting experimental data with a langevin simulation ( in the zero coverage limit ) , the free parameters are those used to define the pes and the friction parameter .the parameters of the pes provide important insight into the average interaction between an adsorbate and the surface , they can be used to compare the energy of multiple adsorption sites and provide an important benchmark for density functional theory calculations .the friction parameter reflects the atomic - scale energy transfer mechanism and plays an important role in a wide range of research fields and applications .since measurements of atomic scale friction of isloated adsorbates are scarse , the ability to extract such values from langevin analysis of quasi - elastic helium scattering measurements is particularly important .the current theoretical understanding of surface friction is rather limited , it is however custom to separate the frictional coupling into two main contributions , namely , electronic and phononic friction . within the langevin approach ,the friction coupling is a fitting parameter and its value reflects the total friction regardless of its origin , this is in contrast with md simulations where the friction is not a parameter , rather it is a result of the explicit interactions between atoms .consequently , since typically the interactions calculated in md simulations are between the ions , the only friction mechanism which is simulated is phononic friction and systems where electronic friction is important can not be accurately studied with simple md simulations of the type described above .   as mentioned above , in a langevin simulation the energy transfer due to the substrate motion or other mechanismsis accounted for using a damping term and a fluctuating force term , both of which are determined by a single friction parameter , . in this work , the friction parameter is used as an adjustable fitting parameter .the langevin simulation also requires the adiabatic interaction potential , i.e. the pes . when analyzing experimental data , the pesis derived using adjustable fitting parameters , however in this study , our goal is to perform a relevant comparison with a specific md model . in order to do this we chose the pes to be the time averaged potential in the md simulation .the procedure for deriving the pes is the following :  1 .generate a 2d grid above the periodic unit of the substrate top layer .2 .  for every point in the grid ,fix the lateral coordinates of the adsorbate , leaving all other degrees of freedom free .allow the system to relax to the equilibrium geometry . in the explicit md simulation , where the force acting on atom along its free coordinate in the system and is the velocity .if this product is positive then the atoms are moved according to the force , if it is negative the velocity is set to zero .the quenching procedure described in the previous stage continues until the change in the system s total energy between time steps drops to a negligible level .the forces acting on the atoms are the same forces used in the explicit md simulation described above .the value of the pes at the grid point is set to be the potential energy of the entire system - contributions from the adsorbate - substrate interaction as well as interaction between the substrate s atoms .using this procedure and the interaction parameters mentioned earlier ( for the harmonic and morse interactions ) the potential difference between a local minimum and saddle point of the pes was found to be 75 mev , this energy barrier corresponds to values in the range 2.9 - 6.2 for the temperatures this study was performed at ( from 300k down to 140k ) ] .the same pes was used for all adsorbate masses in this work .both of the simulations used in this work generate trajectories of the adsorbate . from each trajectorywe can construct the isf of the adsorbate using eq .( [ eq : isf self ] ) . figure ( [ fig : isf ] ) shows an example of the isf from a 10 nanosecond trajectory calculated by the langevin simulation .the isf contains 2 main features with different characteristic time scales i ) a slow decay of the isf which takes place over tens of pico seconds and ii ) a rapid initial decay and an oscillatory pattern , which can be seen more clearly in the inset in figure ( [ fig : isf ] ) which depicts the isf at short times .both of the features mentioned above are characteristic of surface diffusion systems and have been seen in experimental and theoretical work .example of an isf function .the inset shows the isf at short times , where a combination of decaying oscillations as well as a decaying exponential can be seen .their origin is explained in the text.,title="fig:",width=650 ] +  example of a dsf function calculated from the isf shown in figure [ fig : isf ] , focusing on the two peaks which are located at the origin .the much sharper qep dominates the dsf , whereas the underlying broad qeb can be seen more clearly in the inset plot.,width=650 ]  the slow exponential decay is due to intercell diffusion i.e. transitions between local minimum in a corrugated potential as was discussed in section [ sec : basic - definitions - and ] .the quasi - elastic broadening , ( or the decay rate of the isf ) , , and its dependence on and t can be related to the dynamics either using simple analytical theory ( e.g. equation [ eq : jump qhas-1 ] ) or as will be demonstrated below using more detailed numerical models . in the following section we will use to compare the diffusion process calculated by the two numerical simulations .the oscillation and decay seen at short time scales , is related to the motion within the adsorption site .the oscillation period reflects the vibrational motion of the adsorbate whereas the fast decay reflects the loss of phase coherency due to the random nature of this motion , a process sometimes referred to as intra - cell diffusion . in the dsfthis intra - cell motion appears as three peaks , two inelastic peaks located at the energy gain / loss values which correspond to vibrational frequency and one additional peak centered at , as shown in figure ( [ fig : dsf ] ) .the width of all three peaks is related to the rapid decay due to the phase loss of the intra - cell motion mentioned above . since this decay is typically much faster than that due to the inter - cell motion , the widths of all three peaks are substantially larger than that of the qep ( i.e. the quasi - elastic broadening ) . in this work we will refer to the intra - cell motion contribution centered at as the quasi - elastic base ( qeb ) to differentiate it from the much sharper qep which is also centered at . as mentioned above , in many cases the time scale of the intercell diffusion is much slower compared to the intra - cell one , and the differentiation of the different contributions mentioned above is valid . in section [ sub : estimating - the - friction ] we will make use of this separation scheme in order to extract values for the frictional coupling within the adsorption site .as mentioned above , under many circumstances , including the conditions encountered in this work , inter - cell motion leads to an exponentially decaying isf equivalent to a lorentzian qep in the dsf . under these conditions , the quasi - elastic broadening , which can be extracted either from the decay rate of the isf or from the width of the qep peak in the dsf , can be used to characterize the inter - cell motion from both experiments and theory .a method which allowed us to reliably extract , , from the calculated isf is to delay the fitting procedure to times which are sufficiently long to avoid mixing the contributions of the intra - cell motion mentioned above .. $ ] was fitted to a single exponential , with being advanced in time at each iteration until the decay times between successive iterations differed by less than 1% ]  we start with the case of an adsorbate with a mass of 23 amu , representing the na / cu(001 ) system mentioned earlier .figure [ fig : comparison optimal friction 23 amu ] shows a comparison of calculated using the two simulation approaches .the left panel shows an example for calculations performed at 160k , the md results are shown using the black dot symbols , where as langevin results using different friction values in the range 0.44thz-0.72thz are plotted with coloured symbols according to the legend .one immediate feature which can be seen for both simulations is the oscillatory nature of the width of the qep as function of the momentum transfer value .this is a characteristic feature of jump diffusion as can be seen from the chudley elliot equation ( [ eq : jump qhas-1 ] ) .a second observation which can be made is that the langevin simulation can reproduce the md result quite well if the friction parameter is set to a value of 0.56thz , can be expressed with the dimensionless quantity . ], we will refer to the friction parameter which provides the best fit as the `` optimal friction value '' , .this particular value is consistent with the results obtained in the past when analyzing experimental measurements of na / cu(001 ) with langevin simulations . for lower friction valueswe observe a slower jump rate due to weak coupling between the substrate and adsorbate , while for higher friction values the shape of the curve is narrower , indicating the dominance of single jumps ( equation ( [ eq : jump qhas-1 ] ) reverts to a single oscillating term when only nearest neighbor jumps take place ) .a ) quasi - elastic broadening , , calculated along the ( 1,1,0 ) crystal azimuth for a 23 amu adsorbate at 160k .md results are shown with full black circles alongside with the results of langevin simulations with different friction parameters as indicated in the legend .b ) comparison between the calculated by the md ( black circles ) and those calculated by the langevin using the optimal friction values indicated in the legend alongside the relevant surface temperature ., width=650 ]  the right panel of figure [ fig : comparison optimal friction 23 amu ] shows the same comparison in the temperature range 140k-300k .for each temperature we plot the md calculation together with the langevin simulation obtained using the optimal friction values , , i.e. the friction values which gave the minimal standard deviation between the two curves .again one can see that the langevin simulation can reproduce the md values quite well , however , the optimal friction parameters ( indicated in the legend ) are not identical for the different temperatures , instead there is a subtle but clear trend where the friction parameter , , increases with the temperature , i.e. the langevin simulation we used can not exactly reproduce the md results if a single temperature independent friction value is used .   in the previous section we saw that we can find a good agreement between the quasi - elastic broadenings , , calculated by the two numerical models with only one free parameter , - the frictional couplinghowever , in order to optimize the fit we had to slightly adjust the friction according to the temperature . during the last two decades various systems have been measured using qhas , most of which were analyzed using langevin simulations where a single , temperature independent , friction coefficient was assumed .if the temperature dependence of the friction is significant for some of these systems , the analysis method which was applied in the past to extract an activation energy for these systems , resulted in a small but systematic error which needs to be taken into account . in order to study and understand this apparent temperature dependence we performed further calculations for heavier adsorbates , as this allows us to change the strength of the frictional coupling while leaving the inter - atomic forces unchanged .figure [ fig : optimal friction vs temperature all masses ] shows the friction values which give the best quasi - elastic broadening match between the two simulations at different temperatures for 100 amu and 200 amu adsorbates .the resolution of the friction parameter is and for the 100 and 200 amu masses respectively .  from top to bottom :the optimal friction as a function of temperature for 23 , 100 and 200 amu adsorbates respectively .results for the `` straight '' ( over the bridge site ) crystal azimuth ( 110 ) are plotted in green and results for the `` diagonal '' ( over the top ) azimuth ( 100 ) in red .a linear form was fitted for both crystal azimuths .note that the relative change of the optimal friction parameter in the temperature range 150 - 300k can be as large as 100% for the heavier adsorbates we simulated ., width=377 ]  two main observations can be made when comparing the results of the different masses : i ) the friction values needed to fit the two simulations are significantly reduced for heavier adsorbates .this is the expected trend , since heavier adsorbates have a lower vibration frequency and are expected to have a weaker coupling to the substrate .ii ) the need to adjust the friction parameter according to the temperature in order to get an agreement between the two simulations is more pronounced for the heavier adsorbates .thus , this temperature dependent friction which is rather subtle for the 23 amu adsorbate , and would have a small effect on the interpretation of experimental data , becomes a more significant effect for heavier adsorbates .we have shown above , that in order to mimic the md results using a langevin simulation we need to allow the friction parameter to increase with temperature .one explanation for this is that by changing the friction we simply make use of our only free parameter to compensate for the fact that the langevin simulation can not exactly mimic the md results , either due to the fundamental differences between the two , or due to our particular choice for the pes . on the other hand ,since the friction is not an explicit parameter in the md simulation , another possibility is that the friction coupling changes with temperature in the md simulation and that the comparison with the langevin simulation is revealing this trend . in order to try and differentiate between these possibilitieswe have attempted to extract an effective `` friction parameter '' from the md simulation and study its temperature dependence .we achieve this by extracting the width of the quasi - elastic base ( qeb ) , as mentioned in section ( [ sec : basic - definitions - and ] ) .the width of the qeb is governed by the dephasing rate of the intracell motion , i.e. it is related to the friction coupling within the adsorption site , a relation which has been shown , both analytically and numerically .in fact , if one looks at the lowest order of the analytically derived expression for the dsf , the half width of the qeb ( in the angular frequency domain ) is simply equal to  .while the qeb width will undoubtedly be related to the frictional coupling , the accuracy of the simple relation between the two mentioned above is unknown . in particular ,the analytical relation is valid within certain approximations .furthermore , the friction in the langevin simulation reflects the average energy exchange rate , both within and outside the adsorption site , whereas the qeb is only related to the intracell motion within the adsorption site , hence the two properties are obviously not identical . in order to validate our approach , we first start by applying this method on the dsf calculated by langevin . since langevin simulations include an explicit friction value , our ability to reproduce this value from the qeb acts as a self consistency check for our method .  in order to assist the fitting procedure andseparate any contributions from intercell diffusion , the dsf calculations were performed along the straight azimuth for , conditions under which the qep has a negligible width due to the jump diffusion process ( minima values of in eq .[ eq : jump qhas-1 ] ) .. , where is the frequency resolution of the calculated dsf .this range was chosen to eliminate the qep contribution which manifests itself in the dsf as a single data point at .the fitting range extended to a frequency which provided enough data points for the fit , yet avoided contribution from the lorentzian centered about the vibration frequency ] the fit for the langevin data is shown in figure ( [ fig : langevin qeb ] ) .at each temperature , the dsf corresponds to a calculation using the optimal friction value from figure ( [ fig : optimal friction vs temperature all masses ] ) .the inset in figure ( [ fig : langevin qeb ] ) shows the friction values extracted from the qeb width ( blue circles ) versus the friction parameters used in the simulation ( denoted .overall , the two values are very close , with the qeb underestimating the friction parameters by 15%-20% , similar calculation for higher masses ( 100 amu and 200 amu , data not shown ) produce even smaller deviations between the two .thus , we conclude that the qeb width provides a reasonable way to estimate the friction within the accuracy stated above .fitting the qeb peak calculated by the langevin simulations for the different temperatures .the full symbols are the dsf values calculated by the langevin simulations , performed using friction parameters .the solid lines show the lorentzian fit described in the text which was used to extract the qeb width .the inset compares with the friction values extracted from the width of the lorentzian peak which best fitted the qeb . , width=650 ]  next , we applied the same procedure on the md data in order to extract effective friction values and check how they change with temperature .figure [ fig : qeb md ] shows the qeb peaks calculated for the different temperatures and different adsorbate masses , and a lorentzian peak fit ( full lines ) to the qeb .first we note , that the lorentzian fit to the qeb peak is not quite as good as it was for the spectra calculated by the langevin simulation , mostly due to low frequency peaks related to the surface vibrations and an incomplete subtraction of the qep peak ( assumed to be a delta function at the diffraction condition ) .nevertheless , we see that the values extracted from the lorentzian width are quite close to the langevin friction values which were obtained by fitting the curves ( ) .the inset depicts the comparison between and qeb widths ( extracted from the md simulation ) for the different temperatures and adosrbate masses .an obvious feature which can be seen from these graphs is that for all three masses the qeb widths extracted from the md calculations increase as function of temperature , more or less following the trend of .consequently , we conclude that the need to increase as function of t when trying to reproduce the md results with the langevin simulation , represents a property of the frictional coupling of the md simulation , which is then revealed in the comparison with the langevin simulations . in other words, the fact we had to increase with temperature in order to mimic the md results with the langevin code , does not indicate a discrepancy between the two simulations .  from top to bottom , results of the fit to a single lorentzian for 23 , 100 and 200 amu adsorbate .the inset shows the qeb widths extracted from md data simulations at different temperatures alongside the optimal friction values , , used by the langevin simulation to fit the md curves .,title="fig:",width=377 ] +we have compared two different numerical approaches for interpreting adsorbate diffusion on a solid substrate , namely , md and langevin simulations .a major difference between these two approaches is the substitution of the dynamic substrate which is explicitly simulated by the md code , with a friction damping term and a stochastic force in the langevin simulation . since this substitution can not accurately account for correlations between the relative motion of the substrate and adsorbate atoms which takes place in the md , a certain discrepancy in the simulated dynamics is anticipated .for example , a substrate phonon creates a time dependent distortion of the potential energy surface on which the adsorbate moves , hence one could expect that the rate of single jumps and longer jumps would be affected by the frequency and amplitude of the substrate vibrations . while it is obvious that such correlations will take place to some degree , a quantitative assessment of the discrepancies was missing in the literature , and it was unclear whether they are sufficiently large to affect the interpretation of realistic ( noisy ) experimental data .the observables we chose to compare are the isf and dsf correlation functions , focusing on the width of the quasi - elastic peak , , in particular .the dependence of the quasi - elastic peak width on the momentum transfer and sample temperature provides a sensitive measure of the motion rate and mechanism , and is also accessible to helium scattering experiments .the comparison we performed showed that for the particular systems we simulated , the two simulations can produce very similar observables , using the friction parameter as the only free parameter used to fit the two .thus , within the conditions we simulated , correlation effects do not seem to lead to any noticeable discrepancies between the two simulations , and the langevin simulations provides a good approach for simulating the surface dynsmics .we did notice that the optimal friction values which we obtained from fitting the two simulations increased slightly with temperature , an effect which was more significant for adsorbates with a higher mass .one possible interpretation of this observation is that the need to increase the friction parameter of the langevin simulation at higher temperatures is an indication of a discrepancy between the two numerical approaches ( i.e. we are compensating for fundamental differences between the two simulations by adjusting the fit parameter ) .another interpretation is that the frictional coupling rate is increasing with temperature in the md simulation and that the comparison with the langevin simulation ( which produces an optimal friction parameter ) is simply revealing this fact .we used the quasi - elastic base width as a method to estimate the frictional coupling from the dephasing rate of the motion within the adsorption site and extract effective friction values from the md data .our results show that the effective friction values extracted using this method , are in close proximity to those used to fit the langevin simulation .furthermore , the effective friction values also show an increase with temperature supporting the second interpretation mentioned above , i.e the frictional coupling increases with temperature in the md simulation and the need to adjust the friction parameter of the langevin simulation to fit the two simulations does not indicate a discrepancy between the two numerical approaches .in conclusion , when using the particular interaction models mentioned above , adsorbate masses ranging from 23 to 200 amu , and temperatures within the range of 140k to 300k , we do not observe significant differences between the langevin and md simulations .thus , even if differences exist , they are subtle and should not affect the analysis of experimental data with similar or larger noise levels. an explanation for this lack of discrepancy , might be that the relatively fast time scales which characterize the substrate motion lead to an averaging effect which reduces the importance of explicit correlations and allows us to treat the interaction as a sum of a static interaction ( pes ) and a stochastic force with a good accuracy .it is also worth noting , that in the past when applying langevin simulations for data analysis , it was assumed that the friction is independent of surface temperature .while the particular trend we observed in the md simulation reflects our choice of model for simulating the substrate ( harmonic potential ) and adsorbate ( morse potential ) and is not directly related to other systems and interaction models , it is worth remembering that the friction might change as function of temperature also in other systems .if this temperature dependence is not negligible and is not taken into account , systematic errors might be produced when extracting physical properties from the simulations , in particular the energy barrier for diffusion deduced from arrhenius graphs .finally , we assume that there will be other systems and conditions under which correlations will produce noticeable effects , however , these will probably require substantially different time scales ( faster adsorbate motion or slower substrate motions ) and it should be interesting to study such systems in the future .the authors would like to thank prof .erio tossati for valuable scientific discussions .this work was supported by the israeli science foundation ( grant no .2011185 ) and the european research council under the european unions seventh framework program ( fp/2007- 2013)/ erc grant 307267 .peter fouquet , mark r. johnson , holly hedgeland , andrew p. jardine , john ellis , and william allison .molecular dynamics simulations of the diffusion of benzene sub - monolayer films on graphite basal plane surfaces . , 47(11):26272639 , 2009 .a. p. jardine , e. y. m. lee , d. j. ward , g. alexandrowicz , h. hedgeland , w. allison , j. ellis , and e. pollak .determination of the quantum contribution to the activated motion of hydrogen on a metal surface : h / pt(111 ) ., 105(13):136101 , 2010 .a. p. graham , f. hofmann , j. p. toennies , l. y. chen , and s. c. ying .experimental and theoretical investigation of the microscopic vibrational and diffusional dynamics of sodium atoms on a cu(001 ) surface . , 56(16):10567 , october 1997 .gil alexandrowicz , pepijn r. kole , everett y. m. lee , holly hedgeland , riccardo ferrando , andrew p. jardine , william allison , and john ellis .observation of uncorrelated microscopic motion in a strongly interacting adsorbate system ., 130(21):67896794 , may 2008 .a p jardine , h hedgeland , d ward , y xiaoqing , w allison , j ellis , and g alexandrowicz . probing molecule surface interactions through ultra - fast adsorbatedynamics : propane / pt(111 ) . , 10(12):125026 , 2008 .

Abstract: diffusion studies of adsorbates moving on a surface are often analyzed using 2d langevin simulations . these simulations are computationally cheap and offer valuable insight into the dynamics , however , they simplify the complex interactions between the substrate and adsorbate atoms , neglecting correlations in the motion of the two species . the effect of this simplification on the accuracy of observables extracted using langevin simulations was previously unquantified . here we report a numerical study aimed at assessing the validity of this approach . we compared experimentally accessible observables which were calculated using a langevin simulation with those obtained from explicit molecular dynamics simulations . our results show that within the range of parameters we explored langevin simulations provide a good alternative for calculating the diffusion procress , i.e. the effect of correlations is too small to be observed within the numerical accuracy of this study and most likely would not have a significant effect on the interpretation of experimental data . our comparison of the two numerical approaches also demonstrates the effect temperature dependent friction has on the calculated observables , illustrating the importance of accounting for such a temperature dependence when interpreting experimental data .


in order to restore a corrupted image , one needs a model of how uncorrupted ( i.e. natural ) images appear . in the markov random field bayesian paradigm for image restoration ,natural images are modeled via an _image prior_. this is a probabilistic model that encodes how natural images behave _ locally _ ( in the vicinity of every pixel ) .an inference algorithm is then used to restore the image , whose aim is to find a consensus among all vicinities on which global solution is most compatible with a natural image ( while still similar to the corrupted image a corrupted tree should be restored as an uncorrupted tree , not as an uncorrupted house ) .the simplest example of an image prior is perhaps the pairwise model presented in  which simply expresses that neighboring pixels are likely to share similar gray - levels .however , it is easy to see that such a model fails to capture a great deal of important information about natural images .for example , ` edges ' are highly penalized by such a prior , and it is unable to encode any information about ` texture ' . only by using higher - order priors will one be able to capture this important information .an example of a high - order prior is the _ field of experts _ model , which is parameterized as the product of a selection of filters ( or ` experts ' ) .each of these filters is typically a patch of or pixels , resulting in a or -dimensional prior respectively. unfortunately , such a high - dimensional prior limits the practicality of many inference algorithms .even though it may be possible to use smaller ( for example , ) patches , we are still limited by the number of gray - levels used to properly represent natural images ( typically ) .updating a single pixel using a gibbs sampler ( for example ) requires us to consider all possible gray - levels .since gibbs samplers may typically take hundreds ( or thousands ) of iterations to converge , they are simply impractical in this setting .  while belief - propagation techniques tend to converge in fewer iterations , they are often equally impractical .since adjacent cliques may share as many as nodes ( using a model ) , the size of the messages passed between them may be as large as .even if we only use cliques , the size of our messages may still be as large as , which remains impractical for many purposes .although inpainting has been previously approached using belief - propagation techniques in , they do not deal with ( or larger ) models , and their approach can therefore only capture limited textural information .  to avoid the above problems ,image restoration is typically performed using gradient - ascent , thereby eliminating the need to deal with many discrete gray - levels , and avoiding expensive sampling .while gradient - based approaches are generally considered to be fast , they may still require several thousand iterations in order to converge , and even then will converge only to a _ local _ optimum .  in this paper, we propose a method that gives us the best of both worlds : we manage to render belief - propagation practical using a high - order ( ) model , and use it for the task of image inpainting . by using a nonparametric prior, we avoid the need to discretize images , resulting in much smaller messages being passed between cliques .our experiments show that belief - propagation techniques are able to produce competitive results after only a single iteration , rendering them faster than many gradient - based approaches , while retaining similar visual quality of the restoration .in this section , we define the markov random field ( mrf ) image prior to be used in our model .although we shall not present any significant results in terms of _ learning _ the prior , we have nevertheless made a number of modifications to the ` standard ' image prior in order to render inference tractable .the hammersley - clifford theorem states that the joint probability distribution of a markov random field with clique set ( assuming maximal cliques ) is given by ( where is the set of variables in belonging to the clique ; is a normalization constant ) .when dealing with images , the are often assumed to be homogeneous , meaning that the prior can be defined entirely in terms of a single potential function , . in the field of experts model , this potential function is assumed to take the form of a _ product of experts _ , in which each ` expert ' is the response of the image patch \{ } to a particular filter \{}. that is , the potential function takes the form ( where the s are simply weighting coefficients controlling the importances of the filters ) . specifically , each expert is assumed to take the form of a student s t - distribution , namely although use contrastive divergence learning to select the filters and alphas , it has been shown that the filters can more easily be selected using principal component analysis ( pca ) .this leaves only the problem of learning the alphas , which we shall deal with in section [ sec : approx ] .inference in the mrf setting can be formulated as a message passing problem .two common message passing algorithms exist , namely the _ junction - tree algorithm _ , and _ loopy belief - propagation _ . in our case , which algorithm should be applied depends upon the ` shape ' of the region being inpainted. we will give only a brief overview of these algorithms in order to explain why it is infeasible to apply them directly when using the above prior .a more complete specification is given in ; similar ideas are also used in an image inpainting setting in .  belief - propagation algorithms work by having cliques pass ` messages ' to other cliques which share one or more nodes in common .if we denote by the intersection of the two cliques and , and denote by the neighbors of ( i.e. those cliques which share one or more nodes with ) , then the message , , sent from to is given by that is , the outgoing message from to is defined as the product of the local potential \{ } with the incoming messages from all neighbors _ except _ , marginalized over the variables in but not in .once all messages have been sent , the final distribution of \{ } is given by even when using the model , evaluating requires us to consider possible gray - level combinations .although it may be possible to approximate the marginal being computed in equation ( [ eq : message ] ) without computing explicitly , the message itself still contains elements .this problem is dealt with in by using a _ factor - graph _ , which requires only that one dimensional marginals are computed ; however the running time of their method is still linear in the number of gray - levels , in addition to the fact that the factor - graph fails to fully capture the conditional independencies implied by the model .  as a result, we seek in such a form that the sum in equation ( [ eq : message ] ) may be replaced by an integral . in ,the authors defined such a model in which the potential function takes the form of a gaussian mixture , that is , with taking the form ( this is sometimes known as a _ gaussian random field _ ) . unfortunately , the method they use to learn these mixtures appears to be applicable only to low - order models ( the largest mixture models they learn are 3-dimensional ) .  in the remainder of this paper, we will show that the experts \{ } can be approximated as a gaussian mixture , resulting in a high - order model which closely matches the one given in equations ( [ eq : pot1 ] ) and ( [ eq : pot2 ] ) .we will describe belief - propagation in this setting , and show how this approach can be used for fast image inpainting .in , the authors showed that the filters \{ } in equation ( [ eq : pot1 ] ) can be learned by performing a pca on a collection of natural image patches . herewe follow this idea . to learn our filters ( for a model ), we randomly cropped 50,000 patches from images in the berkeley segmentation database , and used their principal components as our filters .it was found in that the first component of such a pca always corresponds to a uniform gray patch , which should be ignored in order to obtain a model invariant to intensity .hence we only use the _ last three _ filters for our model .the resulting patches appear to make sense visually , and are shown in figure [ fig : filters ] .this technique can also be used to learn or models ( resulting in 8 or 24 filters , respectively ) , which are not shown .model.,title="fig : " ]  model.,title="fig : " ]  model.,title="fig : " ]  this model requires also that we learn the ` importances ' of each filter . from equation ( [ eq : pot2 ] ) , it can be seen that the simply control the shape ( or ` peakedness ' ) of the student s t - distribution . rather than try to learn the explicitly, we will learn them implicitly through our approximation .  in order to approximate the experts , we first calculated the inner products for a random selection of 5,000 image patches ( again cropped from the berkeley segmentation database ) .a normal probability plot of the data ( against the first filter ) is shown in figure [ fig : normplot ] this plot reveals that the data is more heavily tailed than would be suggested by a normal distribution , indicating that the student s t - distribution may indeed be valid .however , rather than assume that this data is generated according to a student s t - distribution , we simply tried to approximate it directly using a mixture of gaussians .    in order to estimate the distribution governing this data , we used the expectation - maximization ( em ) algorithm , assuming that the set of inner products for each filter was generated by a mixture of three gaussians .all of our parameters to be learned \{ } were initialized by using a k - means clustering on the original inner products .we used this approach to learn a separate mixture model for each expert .this algorithm produces an approximation of the form the alpha terms are no longer relevant the ` shape ' of the distribution is implicitly controlled by the other parameters .however , the expression in equation ( [ eq : approximation ] ) is not yet in the same form as equation ( [ eq : mog ] ) .hence we need to solve the system that is , we are trying to solve for ( a matrix ) and ( a vector ) , in terms of ( a vector ) and ( a scalar ) .it is not difficult to see that the only solution for is \label{eq : covinv}\ ] ] ( where is the size of the filter  in our case , ) . alternately , there are infinite solutions for . one obvious solution is however , we found for all of our filters that , meaning that this solution would be highly unstable . a more stable solution ( which we used ) is given by   our potential function is now of the form in order to expand the above product , we use the following result about the product of gaussian distributions : given gaussians ( with means , and covariances ) , the covariance of the product \{ } is given by ( although each is singular in our case , their sum is not ) .the mean of the product \{ } is given by the corresponding beta term for the product is just . in our casethis results in a final approximation which is a mixture of gaussians .in order to perform belief - propagation , we must first be able to express equations ( [ eq : message ] ) and ( [ eq : messageproduct ] ) in terms of the gaussian mixtures we have defined . in our setting , the sum in equation ( [ eq : message ] ) becomes an integral , resulting in the new equation we have already suggested how to perform the above multiplication in equations ( [ eq : prod1 ] ) and ( [ eq : prod2 ] ) .the only difference in this case is that the mixtures for each message may contain fewer variables ( and smaller covariance matrices ) than the local distribution \{}. in such a case , the inverse covariance matrices for each message \{ } are simply assumed to be zero in all missing variables .  to compute the marginal distribution of a gaussian mixture with mean and covariance matrix ( i.e. the integral in equation ( [ eq : newmessage ] ) ), we simply take the elements of and corresponding to the variables whose marginals we want .the importances for each gaussian in the mixture remain the same .of course , when we compute the products in equation ( [ eq : newmessage ] ) , we produce a model with an exponentially increasing number of gaussians . as a simple solution to this problem, we restrict the maximum number of gaussians to a certain limit ( see section [ sec : results ] ) , by including only those with the highest importances .when solving an inpainting problem , we only wish to treat some of the variables in each clique as unknowns ( for example , the ` scratched ' sections ) .hence the potential function for these cliques should be conditioned upon the ` observed ' regions of the image .suppose that for a clique we have unknowns , and observed variables ( i.e.  ) .then we may partition the mean and covariance matrix ( for a particular gaussian in the mixture ) as and .\ ] ] the mean of the conditional distribution \{ } is now given by and the covariance matrix \{ } is given by   finally , once all messages have been propagated , we are able to compute the marginal distribution for a given node ( or pixel , belonging to clique ) by marginalizing ( equation ( [ eq : messageproduct ] ) ) in terms of that node . in order to estimate the ` most likely ' configuration for this pixel , we simply consider each of the 256 possible gray - levels .as we mentioned in section [ sec : bprop ] , the two propagation techniques we will deal with are the junction - tree algorithm and loopy belief - propagation . although we will not cover these in great detail ( see for a more complete exposition ) , we will explain the differences between the two in terms of image inpainting .  both algorithms work by passing messages between those cliques with non - empty intersection .however , when using the junction - tree algorithm , we connect only enough cliques to form a maximal spanning tree . now, suppose that two cliques and have intersection .if each clique along the path between them also contains , we say that this spanning tree obeys the ` junction - tree property ' .if this property holds , it can be proven that exact inference is possible ( subject only to the approximations used by our gaussian model ) , and requires that messages be passed only for a single iteration .technically , the graphs which obey this property are the so - called__triangulated _ _ or _ chordal _ graphs .the tractability of exact inference in these graphs depends on their _ tree - width _ : graphs that are more ` tree - like ' are better suited to efficient exact inference .see for details.  if this property _ does nt _ hold , then we may resort to using loopy belief - propagation , in which case we simply connect _ all _ cliques with non - empty intersection .there is no longer any message passing order for which equation ( [ eq : newmessage ] ) is well defined ( i.e. we must have a criterion to initialize some messages , and the common choice is to assume they have a uniform distribution ) , meaning that messages must be passed for many iterations in the hope that they will converge .figure [ fig : animal ] shows an inpainting problem for which a junction - tree exists , and two problems for which one does not ( assuming -pixel cliques ) . since the regions being inpainted are usually thin lines ( or ` scratches ' ) , we may often observe graphs which do in fact obey the junction - tree property in practice .the graphs formed from the white pixels in the other two images do not.,title="fig : " ]  model ) .the graphs formed from the white pixels in the other two images do not.,title="fig : " ]  model ) .the graphs formed from the white pixels in the other two images do not.,title="fig : " ]  fortunately , we found that even in those cases where no junction - tree existed , loopy belief - propagation tended to converge in very few iterations .although there are few theoretical results to justify this behavior , loopy - belief propagation typically converges quickly in those cases where the graph _ almost _ forms a tree ( as is usually the case for the regions being inpainted ) .in order to perform image inpainting , we used a high - level ( python ) implementation of the junction - tree algorithm and loopy belief - propagation , which is capable of constructing markov random fields with any topology . despite being written in a high - level language ,our implementation is able to inpaint images within a reasonably short period of time .since it is difficult to assess the quality of our results visually , we have reported both the peak signal - to - noise ratio ( psnr ) , and the structured similarity ( ssim ) .we were not able to directly compare our psnr results to those in , since they only presented and models .while it is certainly true that their model produces a _ much _ higher psnr than our technique ( e.g. a psnr of for the image in figure [ fig : text ] ) , its execution time is simply impractical .fortunately , it is still possible to measure approximate execution times of a model using their approach , even without reporting psnrs .these results are presented in the next section .  while it is true that the difference between the two models being compared makes meaningful comparison difficult , it is most important to note that there is little _ visual _ difference between the two models . in the next section, we will show that our model is faster than a similar model using gradient - ascent it is the combination of these two results which we believe makes our technique viable .figure [ fig : text ] shows a corrupted image from which we want to remove the text .the image has been inpainted using a model containing only a single gaussian ( although the learned mixtures contained three gaussians see below ) .after a single iteration , most of the text has been removed , and after two iterations it is almost completely gone .although the current state - of - the - art inpainting techniques produce superior results in terms of psnr , they give similar visual results and take several thousand iterations to converge , compared to ours which takes only two ( no further improvement was observed after a third iteration ) .+  +  ' '' ''   +  +   figure [ fig : results ] compares models of various sizes , varying both the number of gaussians used to approximate each mixture , as well as the maximum number of gaussians allowed during the inference stage .the same results are summarized in table [ tab : results ] .model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ] model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ] +  ' '' ''   +  model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ] model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ] +  model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ] model , after three iterations .see table [ tab : results ] for more detail.,title="fig : " ]  .comparison of inpainting performance for several models . herewe vary the number of gaussians used to compute the initial mixture , as well as the maximum number of gaussians allowed during propagation . [ cols="<,<,<,<,<",options="header " , ]   as a simple experiment , we timed these operations in matlab ( using random matrices and vectors ) .we found that computing 69948 inner products was approximately 10 times faster than computing the matrix operations shown in table [ tab : runtime ] .this leads us to believe that a low - level implementation of our belief - propagation algorithms may be significantly faster even than the results we have shown .our results have shown than even a model is able to produce very satisfactory inpainting performance .we believe that even this small model is able to capture much of the important information about natural images . while higher - order models exist , the improvements appear to be quite incremental , despite a significant increase in their execution time .while it is certainly the case that our results fall short of the state - of - the - art in terms of psnr , the differences are difficult to distinguish visually .it is therefore pleasing that we are able to produce competitive results within only a short period of time .we have not yet fully explored the possibility of using the junction - tree algorithm to inpaint images .unfortunately , determining whether a graph obeys the junction - tree property ( see section [ sec : propmethods ] ) is very expensive , meaning we simply used loopy belief - propagation in all cases , without even performing this test .however , there are many cases in which we can be _ sure _ that a junction - tree exists for example , if the inpainting region is a scratch which is only one or two pixels wide . in such cases ,optimal results can be produced after only a single iteration , which would render our algorithm several times faster again .  in spite of this, we found that loopy belief - propagation tended to converge in very few iterations .while we believe it helped that the regions we are inpainting appear to be fairly ` tree - like ' , there is very little theory to support this claim . on the other hand, loopy belief - propagation often converges far slower when dealing with large regions , meaning that we can inpaint a ` scratch ' much faster than a ` coffee stain ' .we have also not considered the possibility that the corrupted pixels may contain some information about the original image .many gradient - ascent approaches implicitly exploit this possibility by initializing their algorithms using the corrupted pixels .if the restored image is ` close to ' the corrupted image , this can result in faster convergence .our approach is also able to deal with this possibility by augmenting the graphical model with an observation layer with the respective noise model for the damaged pixels .in this paper , we have developed a model for inpainting images quickly using belief - propagation .while image inpainting has previously been performed using low - order models by belief - propagation , and high - order models by gradient - ascent , we have presented new methods which manage to exploit the benefits of both , while avoiding their shortcomings .we have shown these algorithms to give satisfactory visual results and to be faster than existing gradient - based techniques , even in spite of our high - level implementation .anat levin , assaf zomet , and yair weiss .learning how to inpaint from global image statistics . in _iccv 03 : proceedings of the ninth ieee international conference on computer vision _, page 305 , washington , dc , usa , 2003 .ieee computer society .j. b. macqueen .some methods of classification and analysis of multivariate observations . in _ proceedings of the fifth berkeley symposium on mathemtical statistics and probability _ , pages 281297 , 1967 .d. martin , c. fowlkes , d. tal , and j. malik .a database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics . in _ proc .8th intl conf . computer vision _, volume 2 , pages 416423 , july 2001 .julian j. mcauley , tibrio s. caetano , alex j. smola , and matthias o. franz .learning high - order mrf priors of color images . in _icml 06 : proceedings of the 23rd international conference on machine learning _ , pages 617624 , new york , ny , usa , 2006 .acm press .

Abstract: in this paper , we use belief - propagation techniques to develop fast algorithms for image inpainting . unlike traditional gradient - based approaches , which may require many iterations to converge , our techniques achieve competitive results after only a few iterations . on the other hand , while belief - propagation techniques are often unable to deal with high - order models due to the explosion in the size of messages , we avoid this problem by approximating our high - order prior model using a gaussian mixture . by using such an approximation , we are able to inpaint images quickly while at the same time retaining good visual results .


standard models of stock - market fluctuations predict a normal ( gaussian ) distribution for stock price log - returns . however , the empirical distribution exhibits significant kurtosis with a greater probability mass in its tails and centre of the distribution . in quantitative finance 2002 ( ) , drgulescu and yakovenko ( dy ) proposed an analytical formula for the probability density function ( _ pdf _ ) of stock price log - returns , based on the heston model  a geometric brownian motion for the log - returns time series coupled with a stochastic mean - reverting volatility .the resulting _ pdf _ is claimed to outperform the gaussian on a large scale of time lags ( and days ) .+   this model starts with the usual assumption that the price follows a geometric brownian motion described by the following stochastic differential equation   [ cols= " < , < " , ]   +the dy analytical formula is quite exciting , and would validate the heston model , if the resulting expected distribution was similar to the empirical one .our method of pre - processing the data is different from the one suggested by drgulescu and yakovenko , and , we believe , more truly captures the market .we find that the dy formula consistently outperforms the gaussian in terms of best fit ( the error between the model and the observed data ) , but a higher complexity is the price to pay : the number of parameters is greater in the heston model , which enables the fitted _ pdf _ to exhibit kurtosis and fat tails when the empirical dataset does , but also penalizes it when it does nt .hence , in terms of goodness - of - fit ( the trade - off between best fit and complexity ) , the gaussian is preferable for low frequencies ( days ) , since the empirical dataset is then quite smooth and does not exhibit kurtosis . for medium frequencies ( days ) ,both models are accepted at a .05 confidence level . finally , for high frequencies , although it performs better than the gaussian , the dy formula is still rejected ( and days ) , mainly because of the extremely high kurtosis of the observed data .

Abstract: an analytical formula for the probability distribution of stock - market returns , derived from the heston model assuming a mean - reverting stochastic volatility , was recently proposed by drgulescu and yakovenko in quantitative finance 2002 . while replicating their results , we found two significant weaknesses in their method to pre - process the data , which cast a shadow over the effective goodness - of - fit of the model . we propose a new method , more truly capturing the market , and perform a kolmogorov - smirnov test and a test on the resulting probability distribution . the results raise some significant questions for large time lags  to days where the smoothness of the data does not require such a complex model ; nevertheless , we also provide some statistical evidence in favour of the heston model for small time lags  and days compared with the traditional gaussian model assuming constant volatility .   heston model , dy formula , mean - reverting stochastic volatility , goodness - of - fit