You are an expert at summarizing long articles. Proceed to summarize the following text: 
materials placed in a close vicinity to each other modify the modes of the electromagnetic field . this results in a change of the vacuum energy , which eventually manifests itself as a net force known as the casimir force @xcite . the casimir force has been the subject of a number of experimental investigations at object separations ranging from tens of nanometers to a few micrometers . starting with the experiments by lamoreaux @xcite and mohideen @xcite , the casimir effect has experienced an enormous increase in experimental activities in recent years @xcite . theoretical approaches to the casimir force are usually built on an effective medium description of the interacting materials . within such an approximation , the local details of the materials microstructure are neglected and the objects are described by macroscopic , spatially - independent dielectric constants . while the effective medium description is in general quite satisfactory for describing dense materials that indeed look homogenous at the typical scales of the casimir force , this is not necessarily the case for strongly heterogeneous ( `` disordered '' ) media that are made of many constituting elements ( `` scatterers '' ) well separated from one another . examples of such heterogeneous systems include nanoporous materials @xcite , clouds of cold atoms @xcite and , in a slightly different context , corrugated surfaces @xcite .    from a theoretical viewpoint , interaction phenomena involving strongly heterogeneous materials have been little studied . seminal works on that subject considered the thermal casimir interaction between slabs made of piled layers separated from random distances ( one - dimensional disorder ) @xcite . the question of disorder was also addressed recently @xcite in the context of the casimir - polder ( cp ) interaction @xcite between a sphere and a plate @xcite . in a recent work finally , the cp interaction between a dielectric sphere ( or an atom ) and a three - dimensional disordered dielectric material was also investigated @xcite . this is the scenario we consider in the present paper . when a probe sphere or an atom interacts with a spatially heterogeneous material such as a semi - infinite disordered medium , the cp potential naturally fluctuates in space . in other words , the casimir interaction depends on the specific statistical realization of the disorder . a shared conclusion of refs . @xcite is that when the two objects are far enough from each other , precisely when the distance between them is large compared to the typical separation between two heterogeneities , the value of the casimir potential from a realization to another is well captured by its configuration average , which coincides with the prediction of the effective medium prescription . in strong contrast , at smaller distances fluctuations of the potential become larger than its mean , which is consequently no longer representative . in practice , this conclusion is crucial for measurements of quantum reflection @xcite , and more generally for any measurement of the casimir force involving heterogeneous materials .    in our previous work @xcite , we developed an exact mathematical treatment of the fluctuations of the cp interaction between a dielectric sphere and a dilute disordered dielectric medium , and applied it to the calculation of the mean value of the cp potential and of its standard deviation . in this paper , we consider the same geometry ( recalled in sec . [ framework ] ) , for which we perform extensive numerical simulations of the cp potential . the results of these simulations confirm the predictions of @xcite ( sec . [ numerics_sec ] ) , and additionally allow us to compute the full probability distribution of the cp potential which , for a given distribution of the scatterers , does not depend on the microscopic properties of the latter . in a second time ( sec . [ theory_sec ] ) , we present a simple statistical model based on a pairwise summation of the individual contributions of the scatterers , and confront it with the simulations . concluding remarks are collected in sec . [ conclusion_sec ] . we address the cp interaction between a probe dielectric sphere ( placed in vacuum ) of static polarizability @xmath0 ( here and in the following , polarizabilities are expressed in si units divided by @xmath1 ) and a semi - infinite , three - dimensional disordered medium consisting of a collection of many scatterers , as illustrated in fig . [ scheme ] . we denote by @xmath2 the distance between the sphere and the surface of the disordered medium . for definiteness , in this paper we restrict our discussion to the retarded regime of the casimir interaction where @xmath2 much exceeds the resonance wavelength @xmath3 of the probe sphere ( the treatment of the opposite limit @xmath4 is analogous ) . scatterers are also modeled by dielectric spheres of size @xmath5 and of static polarizability @xmath6 . throughout the paper , we assume that they are uniformly distributed in space with density @xmath7 , and we consider the case of a dilute disordered medium , such that the average distance between the scattering spheres is larger than their size , @xmath8 . this is the typical regime where the statistical fluctuations of the cp potential are the largest @xcite . in the opposite limit @xmath9 of scatterers very close to each other , the approach developed below does not apply but we expect the statistics of the casimir - polder potential to be approximately captured by a model where the atom interacts with a rough surface @xcite . ( color online ) we consider the casimir - polder interaction between a dielectric sphere ( placed in vacuum ) and a semi - infinite disordered medium . the disordered medium consists of a collection of dielectric spheres ( size @xmath5 , density @xmath7 ) whose positions are uniformly distributed in space . ]    in @xcite , the question of fluctuations in the limit @xmath10 was tackled with the help of a statistical description of the disordered material , in which the cp potential @xmath11 becomes a random variable . its mean , @xmath12 , and its variance , @xmath13 , were calculated from an exact treatment of radiation - matter interaction , based on the scattering approach to casimir forces @xcite combined with a diagrammatic description of radiation scattering off the disordered medium @xcite . in the limit @xmath14 , the following expression for the mean was found @xcite : @xmath15 where @xmath16 is the casimir potential between the probe sphere and a perfect mirror . as announced , the result ( [ ubar_theo ] ) coincides with the prediction of an effective medium description where the probe sphere interacts with an homogeneous surface of relative permittivity @xmath17 . the amplitude of fluctuations , quantified by the ratio @xmath18 of the standard deviation of the cp potential and its mean , was found to be ( for @xmath14 ) : @xmath19 with @xmath20 . equation ( [ gamma_theo ] ) indicates that @xmath12 , i.e. the prediction of the effective medium theory , is well representative of @xmath11 only when @xmath21 . at smaller scales , @xmath18 becomes larger than unity and @xmath12 no longer provides a trustful estimation of the interaction . we now propose to investigate the statistical properties of the cp potential from exact numerical simulations in the geometry of fig . [ scheme ] . for this purpose , we proceed as follows . we generate an ensemble of @xmath22 dielectric spheres of radius @xmath5 and frequency - dependent permittivity @xmath23 , uniformly distributed in a cube of side @xmath24 . this system constitutes a disordered medium of average density @xmath25 . an additional probe sphere is placed above this cube , at a distance @xmath2 to the center of one face , as in fig . [ scheme ] . denoting by @xmath26 the total , internal casimir energy between the @xmath27 spheres  @xcite , the cp interaction @xmath11 is by definition @xmath28 the strategy thus consists in calculating the interaction energy as the difference between the internal energies of @xmath27 and @xmath22 spheres . within the scattering formalism @xcite , the total casimir energy between @xmath22 spheres is given by  @xcite @xmath29 @xmath30 is a block - square matrix of dimension @xmath22 with the following structure : @xmath31 the diagonal blocks of @xmath30 are the inverse of the spheres reflection operators @xmath32 . the @xmath33 off - diagonal block of @xmath30 contains the translation operator @xmath34 , which relates an outgoing spherical wave centered on @xmath35 to an incoming spherical wave centered on @xmath36 @xcite . finally , @xmath37 is the block - diagonal matrix @xmath38 . for the simulations , we express the scattering and translation operators @xmath39 and @xmath34 in a basis of spherical vector waves @xmath40 , with @xmath41 , @xmath42 and @xmath43 . in this basis , the matrix elements of @xmath39 are given by the standard mie scattering amplitudes @xcite . we compute these amplitudes without any approximation , taking into account the full multipole expansion . finally , we evaluate the matrix elements of @xmath34 using the formalism of ref . @xcite .      making use of the approach described in sec . [ methodology ] , we compute the cp potential @xmath11 between a dielectric sphere of radius @xmath44 nm and a disordered medium consisting of @xmath45 other , identical spheres with the same radius @xmath5 , uniformly distributed in a cube of side @xmath46 m . for these parameters , the disordered medium is dilute , @xmath47 , and we are effectively describing the geometry of a semi - infinite bulk system as long as @xmath48 . for definiteness , we give to all the spheres the permittivity of silicon : @xmath49\omega_0 ^ 2/(\omega^2-\omega_0 ^ 2+i\gamma\omega)$ ] , where @xmath50 with @xmath51 nm , @xmath52 and @xmath53 @xcite . figure [ averagez_cp ] displays the absolute value of the disorder - averaged cp potential , @xmath12 , computed with these parameters , for several values of @xmath2 ( red dots ) . each dot is obtained by generating thousands ( typically between @xmath54 and @xmath55 depending on @xmath2 ) of disorder realizations , computing @xmath11 for each of them and finally averaging the results . ( color online ) absolute value of the mean cp potential between a dielectric sphere and a semi - infinite disordered medium , as a function @xmath2 . dots are the results of exact numerical simulations and the dashed line is the theoretical prediction ( [ ubar_theo ] ) . ]    in fig . [ averagez_cp ] , we also show the theoretical prediction ( [ ubar_theo ] ) ( dashed line ) , which is in very good agreement with the numerics . the small disagreement visible at large @xmath2 stems from deviations to the geometry of the semi - infinite medium : when @xmath2 becomes of the order of @xmath56 , the probe sphere starts to be sensitive to the boundaries of the system , and a cross - over toward the sphere - cube geometry is expected . we also show in fig . [ varz_cp ] the standard deviation of the cp potential relative to its mean , @xmath18 , as a function of @xmath57 . red dots are the numerical results , and the dashed curve is the theoretical prediction ( [ gamma_theo ] ) . ( color online ) relative fluctuations of the cp potential as a function of @xmath57 . dots are the results of exact numerical simulations and the dashed line is the theoretical prediction ( [ gamma_theo ] ) . ] the agreement between theory and numerics is very good , up to small finite - size effects at large @xmath2 . as was pointed out in @xcite , the fluctuations of @xmath11 become significant at distances @xmath58 , when @xmath18 becomes larger than unity , see eq . ( [ gamma_theo ] ) and fig . [ varz_cp ] . this suggests that at small distances , the mean @xmath12 is no longer representative of @xmath11 . in order to confirm this picture , we compute the full probability distribution function @xmath59 of the cp potential normalized to its mean , @xmath60 , by constructing histograms of the numerical data . in fig . [ scattered_u ] , we show as an example a scatter plot of the data obtained for 5500 disorder realizations , for @xmath61 . the associated histogram @xmath59 is displayed in fig . [ ps_fig ] , together with the histograms corresponding to three other values of the parameter @xmath57 . ( color online ) scatter plot of the cp potential normalized to its mean value , @xmath60 , computed for 5500 disorder realizations , for @xmath61 . ] ( color online ) probability distribution function @xmath62 , for four increasing values of @xmath57 . histograms ( vertical lines ) are the results of exact numerical simulations , and solid curves are the theoretical prediction , eq . ( [ ps_general ] ) . the lower - right panel also displays as a dashed curve the gaussian distribution expected in the limit of very large distances , eq . ( [ gaussian_ps ] ) . ]    a quick look at the distributions in fig . [ ps_fig ] forthwith confirms the property already outlined by the analysis of @xmath18 : as the sphere gets closer to the disordered medium , the distribution function becomes more and more peaked around a value @xmath63 , corresponding to a cp potential much smaller than its mean . in other words , @xmath11 is no longer a self - averaging quantity . only when @xmath64 does the maximum of the distribution approaches @xmath65 . such a phenomenon was previously observed in the context of the interaction between plates with one - dimensional disorder @xcite . we see here that it is a quite general property , not restricted to one - dimensional systems . to conclude our numerical study , we address the question of the sensitivity of @xmath59 with respect to changes in the microscopic properties of the scatterers . for this purpose , we perform additional numerical simulations involving scatterers with a different radius @xmath66 nm and made of a different material with frequency - independent permittivity @xmath67 . we also set the radius of the probe sphere to @xmath66 nm , but keep the same value of the permittivity of silicon as in the previous section . ( color online ) numerical probability distribution @xmath59 for two different sets of parameters of the scattering spheres , for @xmath68 ( histograms , vertical lines ) . no visible difference is seen between the two histograms . this is confirmed by the theoretical prediction ( solid curve ) , eq . ( [ ps_general ] ) , which depends on the single parameter @xmath57 . ]    the distribution @xmath59 obtained for these new parameters is shown in the right panel of fig . [ universality_fig ] , for @xmath68 . for comparison , we also display the distribution @xmath59 computed with the parameters of the previous section . no visible difference is seen between the two histograms , which indicates that in the dilute limit @xmath59 is in fact a function of the parameter @xmath57 only . in particular , the parameter @xmath69 is irrelevant . this could have been anticipated since in the limit @xmath10 of independent scatterers , @xmath69 enters both @xmath70 and @xmath71 within the same prefactor , which thus cancels out when considering the ratio @xmath60 ( see sec . [ theory_sec ] for a general proof ) . this property is in particular fulfilled by the second moment of the distribution , @xmath72 , see eq . ( [ gamma_theo ] ) . we now develop a simplified statistical description of the cp interaction between a dielectric sphere and a disordered bulk medium , based on a pairwise summation ( pws ) approximation @xcite . this approximation describes the total cp interaction @xmath11 as a sum of the pair interaction @xmath73 between the probe sphere and each of the @xmath22 scatterers . it has to be distinguished from the perturbative expansion , as it can in principle be used for non perturbative pair interactions . in the problem studied in this paper however , the validity of the two approximations is a consequence of the same assumption of a dilute disorder ( @xmath10 ) .      as in the numerical simulations ( sec . [ numerics_sec ] ) , we consider a situation where the distance @xmath2 much exceeds the sphere resonance wavelength @xmath3 ( in the opposite limit @xmath4 , the reasoning follows exactly the same lines ) . consequently , the interaction potential between the probe sphere and a scatterer located at distance @xmath74 takes the simple form @xmath75 , where @xmath76 is a constant characteristic of the microscopic properties of the two interacting objects . let us consider a small spherical cap of volume @xmath77 containing @xmath78 scatterers , as illustrated in the left panel of fig . [ parametrization ] . the elementary cp interaction between them and the probe sphere is @xmath79 . as in the simulations , we assume the positions of the scatterers to follow a uniform distribution , so the random variable @xmath78 is poisson distributed . the @xmath80 cumulant of @xmath78 , @xmath81 , thus fulfills : @xmath82 where @xmath83 is the average density of scatterers . within the pws approximation , the cumulant of the elementary cp potential @xmath84 involving @xmath78 scatterers reads : @xmath85 where we have used the property @xmath86 in the first equality , and eq . ( [ kmdn ] ) in the second . the cumulant of the total cp potential between the probe sphere and the @xmath22 scatterers is finally obtained by using the parametrization @xmath87 , see fig . [ parametrization ] , and integrating over @xmath74 from @xmath88 to @xmath89 . ( color online ) left : parametrization of the geometry for the statistical approach of sec . right : typical ( rare ) disorder realization contributing to the lifshitz tail : the atom is above a large region free of scatterers , of size @xmath90 . ] this yields @xmath91 from which various statistical properties can be deduced , as we now discuss .      as a preliminary application of our statistical approach , we propose to re - derive the predictions ( [ ubar_theo ] ) and ( [ gamma_theo ] ) , previously obtained from an exact treatment of the radiation - matter interaction . for simplicity we assume the scatterers to be spheres of static polarizability @xmath0 , identical to the probe sphere . the coefficient @xmath76 then describes the large - distance interaction between two identical spheres . it can be readily evaluated , for instance from the casimir - polder law @xcite for the energy between two atoms @xcite : @xmath92    the mean @xmath12 is by definition the first - order cumulant : @xmath93 , where we have used eq . ( [ cumulants ] ) for @xmath94 . combining this expression with eq . ( [ gammasept ] ) , we recover eq . ( [ ubar_theo ] ) . the variance @xmath13 is , on the other hand , given by the second - order cumulant , @xmath95 . taking the ratio with @xmath96 , we find @xmath97 which is nothing but eq . ( [ gamma_theo ] ) , with an analytic expression of the coefficient @xmath98 . let us now derive the probability distribution function @xmath62 that has been studied numerically in sec . [ ps_numerics ] . @xmath59 is given by the inverse laplace transform @xmath99 where @xmath100 is greater than the real part of all singularities of @xmath101 . @xmath102 is the cumulant generating function of @xmath103 , and can be expressed as a power series of the cumulants ( [ cumulants ] ) : @xmath104 making use of eq . ( [ cumulants ] ) and of the relation @xmath105 obtained above , we find : @xmath106,\ ] ] where we have introduced @xmath107 and where @xmath108 is the lower incomplete gamma function . since @xmath101 has no singularities in the complex plane , we can set @xmath109 in eq . ( [ lmoinsun ] ) . furthermore , we have the property @xmath110 , such that after the substitution @xmath111 , eq . ( [ lmoinsun ] ) simplifies to @xmath112dx.\ ] ] inserting eq . ( [ phit ] ) into this relation , we finally obtain    @xmath113 \right\}d\tau.\ ] ]    distributions @xmath59 obtained from numerical evaluation of eq . ( [ ps_general ] ) are displayed in fig . [ ps_fig ] as solid curves on top of the numerical results of sec . [ numerics_sec ] . the agreement is excellent for all the values of @xmath57 . furthermore , we notice that eq . ( [ ps_general ] ) confirms the conclusion drawn from the simulations in sec . [ universality ] : @xmath59 depends on the parameter @xmath57 only , being completely independent of the microscopic details of the probe sphere and of the scatterers .      although the distribution @xmath59 , eq . ( [ ps_general ] ) , has no evident analytic expression , several simple asymptotic limits can be readily examined .    _ large-@xmath103 limit _ we describe the limit of large @xmath103 by expanding the term inside the square brackets in eq . ( [ ps_general ] ) up to second order in @xmath114 . since essentially the values of @xmath115 such that @xmath116 contribute to the integral , this expansion is a good approximation provided @xmath117 . it results in a gaussian integral which is straightforwardly performed to give : @xmath118.\ ] ] making use of eq . ( [ gamma_theo_pws ] ) , we rewrite eq . ( [ gaussian1 ] ) as @xmath119.\ ] ] at large @xmath103 , @xmath59 is thus simply a ( normalized ) gaussian distribution of mean @xmath71 and variance @xmath120 . the expansion used to derive eq . ( [ gaussian_ps ] ) being valid as long as @xmath121 , the gaussian shape is a very good approximation of the whole distribution when @xmath122 , which is a direct consequence of the weakness of fluctuations at large distances . we expect this gaussian distribution to be universal at large distances , as a consequence of the central - limit theorem ( many scatterers contribute to @xmath11 when @xmath122 ) , regardless the nature of the disordered medium . this conclusion is supported by similar predictions previously made in the context of the thermal casimir effect in one - dimensional disordered media @xcite , as well as in recent studies of the cp interaction involving quasi two - dimensional disordered metals @xcite . for comparison , we show eq . ( [ gaussian_ps ] ) in the lower - right panel of fig . [ ps_fig ] as a dashed curve , for @xmath123 . note that in the chosen geometry this limit is difficult to reach in the numerical simulations , because it requires the generation of a significant number of scattering spheres in order to satisfy the condition @xmath124 , while maintaining @xmath125 to avoid finite - size effects . when @xmath126 , eq . ( [ gaussian_ps ] ) still holds but only in the very far tail of the distribution , @xmath127 . the physical reason for which we recover a gaussian tail even for small values of @xmath57 is the following . large values of @xmath103 correspond to particular disorder realizations @xmath128 for which the casimir potential is very large , i.e. for which the density of scatterers @xmath129 underneath the probe atom is very high . thus , for these specific disorder realizations the condition @xmath130 is effectively fulfilled . _ moderate values of @xmath103 _     ( color online ) distribution @xmath59 for @xmath131 ( solid red curve ) , together with the approximate form ( [ ps_moderate ] ) at moderate @xmath103 ( dashed curve ) . ]    to describe the limit @xmath132 , we expand the term inside the square brackets in eq . ( [ ps_general ] ) for large @xmath115 : @xmath133\right\}.\end{aligned}\ ] ] we then perform a wick rotation @xmath134 , and express both exponential terms @xmath135 and @xmath136 as power series . this gives @xmath137^n \left[\pi n z^3\gamma(5/7)\left(-x\right)^{2/7}\right]^m.\end{aligned}\ ] ] if we additionally assume @xmath138 , the terms @xmath139 and @xmath140 give the leading contribution to @xmath59 ( the term @xmath141 is purely real and does not contribute ) . keeping only these two terms and computing the remaining integral , we find @xmath142.\ ] ] equation ( [ ps_moderate ] ) holds for @xmath143 . it is shown in fig . [ ps_moderates ] as a dashed curve for @xmath131 , together with the exact distribution calculated from eq . ( [ ps_general ] ) ( solid red curve ) .    _ small-@xmath103 limit _ we finally consider the low - potential tail @xmath144 of @xmath59 . in order to find an asymptotic expansion in that limit , we come back to eq . ( [ ps_smallnzc ] ) and apply the change of variables @xmath145 . this yields @xmath146\right\}.\end{aligned}\ ] ] in the limit of very small @xmath103 , we are thus led to evaluate @xmath147,\ ] ] where @xmath148 and @xmath149 . equation ( [ i_def ] ) naturally calls for the method of steepest descent . there are five saddle points , solutions of @xmath150 . only one of them , denoted by @xmath151 , turns out to give a nonzero contribution to @xmath59 : @xmath152^{1/4}e^{-i\pi/14}.\ ] ] using cauchy s theorem , we then deform the path of integration to a path coinciding with the path of steepest descent in the vicinity of @xmath151 . this is achieved by expanding @xmath153 up to second order around @xmath154 and performing the change of variables @xmath155 , leaving us with a gaussian integral whose evaluation leads to : @xmath156,\end{aligned}\ ] ] with prefactors @xmath157 and @xmath158 . the asymptotic form ( [ lifshitz_tail ] ) is completely analogous to the so - called lifshitz tail that describes the band edge of the density of states of disordered conductors in solid - state physics @xcite . physically , it can be understood from the following qualitative argument . low values of @xmath103 are achieved for rare disorder realizations where the probe sphere stands above a large region free of scatterers , as illustrated in the right scheme of fig . [ parametrization ] . since the distribution of scatterers is poissonian ( sec . [ theory_sec ] ) , the distance between the events of this poisson process follows an exponential distribution . consequently , the probability to find a large region of size @xmath159 free of scatterers is @xmath160 , where @xmath161 is a numerical constant . in such configuration , the casimir potential felt by the atom can be estimated as ( see fig . [ parametrization ] ) : @xmath162 to leading order in @xmath163 . on the other hand , we have seen in sec . [ mean_var_pws ] that the average casimir potential is @xmath164 . therefore , for the rare disorder realization displayed in fig . [ parametrization ] , we have @xmath165 , such that @xmath166\nonumber\\ & = & \exp\left(-\dfrac{cnz^3}{s^{3/4}}\right),\end{aligned}\ ] ] which is nothing but the asymptotic form ( [ lifshitz_tail ] ) . in this paper , we have developed a statistical description of casimir - polder potentials from both a numerical and an analytical perspective . this approach is well suited for describing the casimir interaction between a simple dielectric object and a strongly heterogenous medium made of a large number of independent constituting elements . it can be readily extended to other geometries and to heterogeneous media characterized by a more complex statistics involving , for instance , a non - uniform or polydisperse distribution of scatterers .    as a first extension of our work , it would be interesting to investigate deviations to the dilute limit @xmath8 where the scatterers can no longer be systematically considered independent . we expect these deviations to primarily affect the far tails of the distribution @xmath59 . a second open question concerns the change in the statistics of the casimir - polder potential when the host ( homogeneous ) medium has a dielectric constant differing from unity . this problem is more difficult to treat since it now involves a surface , which implies multiple reflections inside and outside the medium . the presence of a host medium may therefore strongly affect the distribution @xmath59 [ in the obvious limit where the dielectric constant of the host medium goes to infinity , one recovers a perfectly reflecting interface and @xmath59 should tend to the dirac function @xmath167 .    in practice , the distribution @xmath59 could be experimentally accessed either by moving the sphere over a static disordered medium to record different disorder distributions , or by taking advantage of a brownian motion of the scatterers if the measurement process is fast enough . indeed , in that case different disorder realizations can be obtained by detecting the casimir force and then letting the scatterers move before carrying out the next measurement . if the measurement process is slow , the effect of the motion of the scatterers is to average the casimir potential , giving him a value well approximated by eq . ( [ ubar_theo ] ) since doppler shifts have a negligible effect at thermal velocities @xcite .
we explore the statistical properties of the casimir - polder potential between a dielectric sphere and a three - dimensional heterogeneous medium , by means of extensive numerical simulations based on the scattering theory of casimir forces . 
 the simulations allow us to confirm recent predictions for the mean and standard deviation of the casimir potential , and give us access to its full distribution function in the limit of a dilute distribution of heterogeneities . 
 these predictions are compared with a simple statistical model based on a pairwise summation of the individual contributions of the constituting elements of the medium .