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the propagation of cosmic rays ( crs ) through turbulent magnetic fields is effectively described as a diffusion process . this approach provides an excellent description of the observed spectra and chemical abundances of galactic crs up the cr knee at a few pev . the arrival direction of crs is mostly isotropized in this process and the location of the sources is hidden . the first order correction to the isotropic distribution is a small dipole anisotropy that scales with the local cr gradient according to fick s law . the phase and strength of this dipole is expected to be a combined effect of the relative motion of the solar system with respect to the frame where crs are isotropic  @xcite , the density gradient of crs , the strength of diffusion , and the strength and orientation of regular magnetic fields  @xcite . various cr observatories have revealed anisotropies in the arrival directions of tev  pev galactic crs on large and small angular scales  @xcite . in this letter we will focus on small - scale anisotropies that have been studied via a power spectrum analysis by icecube  @xcite and hawc  @xcite ( see fig .  [ fig2 ] ) . @xcite have argued that small - scale anisotropies can naturally emerge as a next - to - leading order correction of the diffusive approximation if one takes into account the local scattering of crs within the diffusive scattering length . @xcite showed that under certain simplifying assumptions these small - scale fluctuations necessarily arise from the application of liouville s theorem . assuming a simple hierarchical evolution equation of the multipoles predicts a power spectrum that matches well the observations . fluctuations of the arrival direction of crs induced by local turbulent fields are not directly accessible in standard diffusion theory . this becomes obvious if one considers the angular power spectrum of the cr phase - space distribution ( psd ) @xmath0 defined as @xmath1 where @xmath2 denotes a unit vector of @xmath3 and @xmath4 are legendre polynomials of degree @xmath5 . here and in the following we use the abbreviation @xmath6 for the local psd . in general , the small - scale magnetic field that determines the psd is unknown . therefore , the psd can only be predicted as an average over a statistical ensemble of random small - scale magnetic fields , usually characterized by its power spectrum . in ergodic systems , the average over turbulent magnetic field configurations , _ i.e. _ ,  the _ ensemble _ average denoted in the following by @xmath7 , is equivalent to a time - average . however , the ensemble average of @xmath8 is proportional to @xmath9 , which is _ larger _ than the product of the ensemble averages , _ i.e. _ ,  the standard solution of diffusion theory . on large timescales , the motion of crs in turbulent magnetic fields can be effectively approximated as a diffusion process . the angular - integrated psd @xmath10 then satisfies a diffusion equation , @xmath11 , where the diffusion tensor @xmath12 can be expressed as @xmath13 here , @xmath14 denotes a unit vector along the regular magnetic field . in general , the ( scattering ) rates @xmath15 , @xmath16 and @xmath17 are independent of one another . the presence of a small local gradient @xmath18 induces a dipole anisotropy in the quasi - stationary solution as a result of fick s law , @xmath19 . therefore , standard diffusion theory predicts that the dipole strength of the relative intensity @xmath20 is given as @xmath21 we will see in the following that this relation becomes modified once we consider corrections of the psd product in the ensemble average . this will also introduce multipoles at small angular scales , which can be related to properties of relative diffusion . we now study the effect of small local fluctuations of the psd around the ensemble average , @xmath22 . according to liouville s theorem we can relate the local ( _ i.e. _ ,  @xmath23 ) psd @xmath6 to the contribution backtracked along cr trajectories to an arbitrary time , @xmath24\nabla n\,,\end{gathered}\ ] ] where @xmath25 and @xmath26 denote the local cr density and gradient and @xmath27 and @xmath28 are the position and momentum of a cr ( that is at position @xmath29 and @xmath30 at time @xmath31 ) at backtracking time @xmath32 . now , in the limit of large @xmath32 the last term in eq . ( [ eq : backtrack ] ) is dominated by the third term scaling with the position of the particle . also , for two momenta @xmath33 we can assume that the ensemble average of fluctuations are uncorrelated , @xmath34 , for sufficiently large backtracking times when the cr trajectories eventually separate . in the degenerate case @xmath35 the two backtracked cr trajectories stay correlated over arbitrarily long backtracking times . it will be sufficient to assume that @xmath36 remains finite . we can then express the multipole spectrum of the ensemble - averaged relative intensity as the limit @xmath37 @xmath38 being shorthand for @xmath39 . note that the @xmath40 multipole spectrum is generated through _ relative _ diffusion : it can be easily seen that the sum over _ all _ ensemble - averaged multipoles of the relative intensity can be expressed via the symmetric part of the diffusion tensor @xmath41 in the limit of large backtracking times @xmath32 , @xmath42 on the other hand , the average monopole contribution in this limit can be expressed as @xmath43 where the symmetric part of the relative diffusion tensor is defined as @xmath44 with @xmath45 . therefore , the sum of multipoles @xmath40 is related to the relative diffusion tensor . for uncorrelated particle trajectories , this expression reduces to the normal diffusion tensor . however , particle trajectories with a small relative opening angle will follow similar trajectories and the relative contribution ( [ eq : krel ] ) remains small over long timescales . note that the multipoles in eq . ( [ eq : cell ] ) are expected to be finite in the limit of large backtracking times since particle trajectories with arbitrarily small opening angles will eventually become uncorrelated , @xmath46 .    ) for a cr gradient parallel ( solid lines ) and perpendicular ( dotted lines ) to the regular magnetic field and the 3d turbulence model discussed in the main text . we show results in terms of the dipole @xmath47 ( black ) , monopole @xmath48 ( blue ) , and medium-@xmath5 multipoles ( green ) . we also show the asymptotic noise level ( [ eq : noise ] ) ( red ) and the dipole prediction ( [ eq : standardc1 ] ) of standard diffusion ( magenta ) evaluated by the replacement @xmath49 in eq . ( [ eq : cell ] ) . ] in the following , we will study the development of small - scale anisotropies via numerical simulations ( see also @xcite ) . we follow the approach of @xcite and define a three - dimensional ( 3d ) turbulent magnetic field as the sum @xmath50 with @xmath51 random phases @xmath52 and wave vectors @xmath53 with 3d random orientations , on top of a regular field @xmath54 . the wave vector amplitudes @xmath55 range from @xmath56 to @xmath57 with equal logarithmic steps . the vectors @xmath58 have 3d random orientations subject to the conditions @xmath59 and @xmath60 with a coherence scale @xmath61 . we assume a kolmogorov - type phenomenology with @xmath62 and the strength of the turbulence relative to the regular magnetic field is fixed via a parameter @xmath63 and the normalization of the spatial average @xmath64 .    for the galactic environment we expect a regular magnetic fields of the order of @xmath65 g and a turbulent component with @xmath66 and a coherence scale @xmath67  pc . the larmor radius of 1 - 10  tev crs is then of the order of @xmath68 corresponding to a few hundred au . a numerical study of this configuration with the methods described above is very time - consuming since the asymptotic behavior relevant for the multipole configuration is reached at a very late stage . instead , here we will study a simple turbulent configuration that allows us to reach the necessary statistics to study the predicted effect qualitatively . we choose @xmath69 with @xmath70 and @xmath71 with @xmath72 and fix the rigidity of the crs to @xmath73 . we sample over @xmath74 different turbulent magnetic field configurations . for each field configuration we uniformly sample @xmath75 cr orientations @xmath76 following the _ healpix _ parametrization ( @xmath77 )  @xcite , which we backtrack through the static magnetic field @xmath78 to find the initial position @xmath27 . figure  [ fig1 ] shows the power spectrum at different backtracking times determined via a multipole expansion of the relative intensity sky maps using the _ healpix _ utilities . we show the cases of cr gradients parallel ( solid lines ) and perpendicular ( dotted lines ) to @xmath54 . the black lines show the evolution of the ensemble - averaged dipole induced by the cr gradient . the magenta lines correspond to the standard dipole anisotropy defined via eq . ( [ eq : standardc1 ] ) evaluated by the replacement @xmath49 in eq . ( [ eq : cell ] ) . as expected , this dipole estimate is smaller than the ensemble average . note that for the case of perpendicular diffusion ( dotted lines ) the difference in the asymptotic values is about one order of magnitude . the green lines show the multipole power for @xmath79 . we also show the expected monopole component ( [ eq : c0 ] ) as blue lines . at large times , the high-@xmath5 components in the multipole expansion are dominated by noise , due to the finite number of trajectories . the noise level can be estimated for large backtracking times @xmath32 as pixel shot noise @xmath80 this is indicated as red lines in fig . [ fig1 ] and clearly influences the level of high-@xmath5 multipoles ( green lines ) in the map . ( normalized to @xmath81 ) for the parallel ( filled red diamonds ) and perpendicular ( filled green triangles ) cr gradient for the same simulations as shown in fig . [ fig1 ] at a backtracking time @xmath82 . we also show the icecube  @xcite ( blue open circles ) and the hawc  @xcite ( magenta open squares ) power spectra renormalized by a factor of @xmath83 for better comparison . the different noise level of the data is indicated as dotted lines . the three gray lines correspond to the prediction of a relative scattering term @xmath84 in eq . ( [ eq : finalcl ] ) for three different values of @xmath85 . ] the best estimator for the true power spectrum is then @xmath86 and the variance ( excluding cosmic variance ) is given by @xmath87 ; see,_e.g . _ , @xcite . in fig . [ fig2 ] we show the estimators @xmath88 normalized to the dipole @xmath81 at a backtracking time @xmath89 , corresponding to the timescale of the transition into the diffusion regime ( cf .  fig . [ fig1 ] ) . given the finite number of trajectories , it might be difficult to disentangle noise variance and cosmic variance in the simulations ; we hence limit ourselves to the former only . similar to the simulation , the sensitivity of experimental data to high-@xmath5 multipoles is limited to shot noise indicated as dotted lines in the plot . overall , the simulation agrees well with the icecube and hawc data . for an isotropic gaussian field , deviations at multipoles of @xmath90 can be due to cosmic variance with @xmath91 . furthermore , the limited field of view of the observatories introduces additional systematic errors , in particular for low-@xmath5 multipoles , which are not included in the error bars of fig . in the following , we derive the asymptotic form of the power spectrum via a microscopic description of relative diffusion . as before , we split the psd into the average and fluctuating parts , @xmath92 . the ensemble - averaged boltzmann equation can be written as  @xcite @xmath93 here we have introduced the angular momentum operator @xmath94 and the rotation vectors @xmath95 and @xmath96 .    the rhs of eq . ( [ eq : genboltzmann ] ) encodes the particle scattering in the random fields . the influence of the turbulence is typically approximated as a friction term introduced by @xcite ( bgk ) . in its original version this term drives the ensemble - averaged distribution @xmath97 to its isotropic mean @xmath25 with an effective relaxation rate @xmath98 , _ i.e. _ ,  @xmath99 . the parameters @xmath15 , @xmath16 , and @xmath17 are the effective parallel , perpendicular , and axial scattering rates , and can be related to the bgk rate @xmath98 via @xmath100 , @xmath101 , and @xmath102 . substituting into eq . ( [ eq : k ] ) we find that the standard @xmath103 and @xmath104 ( [ eq : standardc1 ] ) scale as @xmath105 and @xmath106 , respectively , similar to the sum ( [ eq : id1 ] ) in the relative diffusion approach . we can generalize the bgk approximation of the friction term to higher multipoles if we consider the effect of turbulence on the particle momenta as a brownian motion on a sphere  @xcite . for a distribution @xmath107 on a sphere the brownian diffusion of momenta is described via @xmath108 . in particular , if initially the configuration is localized , @xmath109 , then the solution at later times is given as @xmath110 . this reduces to a two - dimensional gaussian distribution of the opening angle @xmath111 defined via @xmath112 with a width @xmath113 . therefore , for the rhs of eq . ( [ eq : genboltzmann ] ) we make the ansatz @xmath114 which is a natural generalization of the bgk approach . note that this simple treatment of the scattering term is only approximate . depending on the strength and coherence length of the turbulent magnetic field , the effective relaxation can in general be anisotropic even for a fully isotropic turbulence . however , in our case we are interested in the qualitative behavior and development of high-@xmath5 multipoles , and the first order bgk approximation seems appropriate . now , for the asymptotic solution of @xmath8 at large backtracking times we have to look for stationary solutions of the equation @xmath115 as before , we make the ansatz @xmath116 with the stationary solution @xmath117 . assuming that the spatial dependence of @xmath118 is small compared to @xmath119 we can replace the gradient term in eq . ( [ eq : clevolution ] ) via @xmath120 . on the other hand , the second and third terms in eq . ( [ eq : clevolution ] ) are now non - trivial terms in the evolution that will lead to mixing and damping of multipoles . analogous to the bgk - type approximation  @xcite of eq . ( [ eq : bgk ] ) , we make the ansatz , @xmath121\langle f_1f_2\rangle\,,\end{gathered}\ ] ] with @xmath122 and @xmath123 . the relative and correlated scattering rates , @xmath124 and @xmath125 , respectively , depend on the relative distance of the trajectories at early times , which in turn depends on the trajectories opening angle . note that eq . ( [ eq : bgk2 ] ) is necessarily symmetric under the interchange of @xmath126 as well as @xmath127 and is the most general linear approximation of all possible scalar combinations of @xmath128 and @xmath129 . note that as in the original bgk ansatz , we do not account for the possibility of anisotropic scattering in turbulent fields . we now look for the power spectrum of a stationary solution of eq . ( [ eq : clevolution ] ) using the approximation ( [ eq : bgk2 ] ) . the product @xmath130 can be expanded into eigenstates of @xmath131 and @xmath132 and the power spectrum integral ( [ eq : celldef ] ) corresponds to projections onto the state @xmath133 with @xmath134 . the rotation term corresponding to the regular magnetic field in eq . ( [ eq : clevolution ] ) does not contribute . since the relaxation rates @xmath135 and @xmath136 in the approximation ( [ eq : bgk2 ] ) are assumed to depend only on @xmath137 the relative rotation proportional to @xmath138 is the only non - vanishing term .    for identical momenta and rigidities , the relative diffusion should vanish completely . one way to see this is by considering a toy model where the initial configuration of crs is homogeneous  @xcite . from this setup one can infer that the psd at any time obeys @xmath139 . therefore , in this case we must also have @xmath140 . since this is independent of the initial configuration we must therefore conclude that @xmath141 . a functional dependence can be obtained from a geometrical argument : two particles emitted locally under a small opening angle will separate with a reduced relative velocity @xmath142 as long as both particles are experiencing a strongly correlated magnetic field . this leads to the ansatz @xmath143 with @xmath144 . to allow for a weaker or stronger @xmath145-dependence , we have generalized this form to @xmath143 , with @xmath146 .    under these assumptions and approximations , the stationary average @xmath147 spectrum then obeys the set of equations ( see appendix  [ app1 ] ) @xmath148 with the effective dipole source term @xmath149 . since the rhs of the previous equation is an expansion in terms of legendre polynomials , we can easily derive the stationary spectrum as @xmath150 with the assumed @xmath143 , this simplifies to @xmath151 in the limit @xmath152 , @xmath153 , and so @xmath154 . for @xmath155 , this is close to the @xmath156 behavior exhibited by the data ( cf . [ fig2 ] ) . figure  [ fig2 ] also shows examples of the power spectrum ( [ eq : finalcl ] ) for three models @xmath157 with @xmath158 , @xmath159 , and @xmath160 . while our ansatz @xmath143 is phenomenological in nature , a microscopic derivation of @xmath135 , in quasi - linear theory , will be presented in future work . we have discussed the appearance of small - scale anisotropy from the propagation of crs in local turbulent magnetic fields . we have shown that this effect can be understood as a phenomenon of relative diffusion , where crs with similar arrival directions experience correlated motion in the local magnetic environment . our numerical simulation showed that the ensemble - averaged dipoles can be significantly larger than those expected from standard diffusion theory . in addition , the formalism predicts significant power in the ensemble - averaged small - scale multipoles . we have also provided an analytic framework that can describe the multipole power spectrum via an effective relative relaxation rate of multipole components in the evolution of the ensemble - averaged psd products . this formalism is a natural extension of the well - known bgk approach . assuming that the relative relaxation rate has a simple power - law dependence on the relative opening angle of cr arrival directions , we arrive at a power - law spectrum very similar to that obtained from simulations and the observed cr data .    _ acknowledgements._the authors would like to thank paolo desiati , luke drury , gwenael giacinti , alex lazarian , vanessa lpez - barquero , martin pohl , and stefan westerhoff for comments . we thank dan fiorino and marcos santander for discussions and providing the data points of fig . is supported by the national science foundation under grants opp-0236449 and phy-0236449 . p.m.  is supported by doe contract de - ac02 - 76sf00515 and a kipac kavli fellowship . the decomposition of the psd @xmath161 into spherical harmonics @xmath162 corresponds to a projection onto eigenfunctions @xmath163 of the angular momentum operator @xmath164 . the products of psds that appear in the definition of the power spectrum in eq . ( [ eq : celldef ] ) are then eigenfunctions of @xmath165 that can be decomposed into eigenfunctions @xmath166 of the total angular momentum operator @xmath167 . as noted earlier , the integral over arrival directions in eq . ( [ eq : celldef ] ) corresponds to a projection of @xmath168 onto states with @xmath133 and @xmath134 and @xmath169 , which can be seen from the expansion into spherical harmonics @xmath170 following from the clebsch - gordan coefficients @xmath171 now , the bgk ansatz ( [ eq : bgk2 ] ) assumes that the scattering rates @xmath172 and @xmath173 are only functions of @xmath174 and can thus be expanded into a sum over states with @xmath133 and @xmath134 . the projection of eq . ( [ eq : bgk2 ] ) onto @xmath175 can thus only receive contributions from the projection of the product of psds onto @xmath133 and @xmath134 states , @xmath176 finally , the stationary solution ( [ eq : stationary ] ) has the form @xmath177 and hence @xmath178 . from this we immediately arrive at eq . ( [ eq : finalcl ] ) .
the arrival directions of multi - tev cosmic rays show significant anisotropies at small angular scales . it has been argued that this small - scale structure can naturally arise from cosmic ray scattering in _ local _ turbulent magnetic fields that distort a global dipole anisotropy set by diffusion . 
 we study this effect in terms of the power spectrum of cosmic ray arrival directions and show that the strength of small - scale anisotropies is related to properties of relative diffusion . 
 we provide a formalism for how these power spectra can be inferred from simulations and motivate a simple analytic extension of the ensemble - averaged diffusion equation that can account for the effect .