most physicists agree that some form of new physics ( np ) must exist beyond the standard model(sm)we simply do nt know what it is yet .though there are many prejudices based on preconceived ideas about the form this np may take , it will be up to experiments at future colliders , the lhc and ilc , to reveal its true nature . while we are all familiar with the list of these theoretical possibilities one must keep in mind that nature may prove to be more creative than we are and that something completely unexpected may be discovered .after all , we certainly have not yet explored more than a fraction of the theory landscape .one now well - known possibility is that extra spatial dimensions will begin to show themselves at or near the tev scale .only a few years ago not many of us would have thought this even a remote possibility , yet the discovery of extra dimensions(eds ) would produce a fundamental change in how we view the universe .the study of the physics of tev - scale eds that has taken place over the past few years has its origins in the ground breaking work of arkani - hamed , dimopoulos and dvali(add) . since that time eds has evolved from a single idea to a new paradigm with many authors employing eds as a tool to address the large number of outstanding issues that remain unanswerable in the sm context .this in turn has lead to other phenomenological implications which should be testable at colliders and elsewhere . a ( very ) incomplete list of some of these ideasincludes , ,  * addressing the hierarchy problem * producing electroweak symmetry breaking without a higgs boson * the generation of the ordinary fermion and neutrino mass hierarchy , the ckm matrix and new sources of cp violation * tev scale grand unification or unification without susy while suppressing proton decay * new dark matter candidates and a new cosmological perspective * black hole production at future colliders as a window on quantum gravity  this list hardly does justice to the wide range of issues that have been considered in the ed context .clearly a discussion of all these ideas is beyond the scope of the present work and only some of them will be briefly considered in the text which follows .however it is clear from this list that ed ideas have found their way into essentially every area of interest in high energy physics which certainly makes them worthy of detailed study .  of course for many the real reason to study extra dimensionsis that they are fun to think about and almost always lead to surprising and unanticipated results .most analyses of eds are within the context of quantum field theory . can we learn anything about eds from ` classical ' considerations and some general principles without going into the complexities of field theory ?consider a massless particle moving in 5d ` cartesian ' co - ordinates and assume that 5d lorentz invariance holds .then the square of the 5d momentum for this particle is given by *p* where is the 5d metric tensor ( , defined by the invariant interval ) , is the usual particle energy , * p* is the square of the particle 3-momentum and is its momentum along the 5th dimension .( note that the indices run over all 5d .we will sometimes denote the 5th dimension as and sometimes just as . )the ` zero ' in the equality above arises from the fact that the particle is assumed to be massless in 5d .note that a priori we do not know the sign of the metric tensor for the 5th dimension but as we will now see some basic physics dictates a preference ; the choice of the + ( - ) sign corresponds to either a time- or space - like ed .we can re - write the equation above in a more traditional form as *p* and we recall , for all the familiar particles we know of which satisfy 4d lorentz invariance , that , which is just the square of the particle mass .( note that greek indices will be assumed to run only over 4d here . )notice if we choose a time - like extra dimension that the sign of the square of the mass of the particle will appear to be _ negative _ , , the particle is a _tachyon_. tachyons are well known to be very dangerous in most models , even classically , as they can cause severe causality problems something we d like to avoid in any theory provided they interact with sm particles .this implies that we should pick the space - like solution .generally , it turns out that to avoid tachyons appearing in our ed theory we must always choose eds to be space - like and therefore we assume there will always be only one time dimension even though we could all use some extra time ..  now lets think about a real massless scalar field in a flat 5d - space ( assuming a space - like ed ! ) which is a solution of the 5d klein - gordon equation : ( , where here represents the extra dimension .we can do a fast and dirty trick by performing something like separation of variables , , take and plug it into the klein - gordon equation above giving us .now we note that if , we obtain a set of equations that appears like which looks like an infinite set equations for a collection of distinct 4d scalar fields with masses .this bunch of states with different masses is called a kaluza - klein(kk ) tower .note that we labeled the states by a set of integers ( ) so that the levels are discrete ; we could just as well have replaced the sum by an integral and treat as a continuous variable .the difference between these two possibilities and the link to the nature of the 5d space and the associated boundary conditions will be made clear below .now we have pulled a bit of a fast one in performing this quick and dirty analysis so let us return and do a somewhat better job ; we will still assume , however , that is a distinct integer label for reasons to be clarified below .let us start from the action ( , the integral of the lagrangian ) for the massless 5d scalar assuming with for now arbitrary :    now we recall and substitute the decomposition as above .then the integrand of the action becomes a double sum proportional to $ ] .this appears to be a mess but we can ` diagonalize ' this equation in a few steps . first if we choose to orthonormalize the such that    then the kinetic term of the ( the first one in the bracket above ) reduces to a single sum after integration becoming simply .this is essentially just sum of the kinetic terms for a set of distinct 4d scalars . to handle the second term in the brackets we integrate by - parts andnote that _ if _ we take the boundary conditions to be and also require that as above , we can integrate the entire action over and obtain an effective 4d theory  \,,\ ] ] which is just the sum of the actions of the independent 4d scalars labeled by with masses , , the kk tower states .one sees that in this derivation it was important for the above boundary conditions(bcs ) to hold in order to obtain this result .it is important to note that in fact the various masses we observe in 4d correspond to ( apparently ) quantized values of for the different .the fields can be thought of as the wave functions of the various kk states in the 5th dimension and in this simple , flat 5d scenario are simple harmonic functions : .what are the ? what are and ?to say more we must discuss bcs a bit further . in thinking about bcs in this kind of modelit is good to recall one s experience with one - dimensional quantum mechanics(qm ) that we all learned ( too ) many years ago .first recall the schrdinger equation for a free particle moving along the direction .it has the same form as eq.4 above and since the direction is infinite , , _ noncompact _ , the solution is just where is the particle momentum which can take on an infinite set of continuous values .we say that in this case the momentum is not quantized and this is due to the fact that the space is noncompact .now let us consider a slightly different problem , a particle in a box , , a situation where the potential is zero for but is infinite elsewhere so that the wavefunction vanishes outside this region .since the physical region is of a finite size it is called _compact_. we know that the solution inside the box takes the same general form as does the case of a free particle or above but it must also vanish at the boundaries .these bcs tell us and so that the solutions actually takes the form and that the momenta are quantized , , with . clearly these two situations are completely analogous to having a 5th dimension which is either infinite ( , noncompact ) or finite ( , compact ) in size ._ under almost all circumstances _ we assume that extra dimensions are compact . for a flat 5th dimension of length analysis above tells us that the kk masses are given by , , the masses are large if the size of the extra dimension is small .perhaps it is natural to think that the reason that we have not seen eds is that they are very small and the corresponding kk states are then too massive to be produced at colliders .in fact , the observation of kk excitations is the hallmark of eds .it is interesting to observe that there are no solutions in the ` particle in a box ' example corresponding to massless particles , , those with , the so - called zero modes .there are other sorts of bcs that can be important . in qm we also examined the case of a particle moving on a circle where angular momentum is quantized .in the 5d case we can imagine that the 5th dimension is curled up into a circle , one dimensional sphere of radius that the points and are the same , , we have periodic boundary conditions . herethe kk masses are given by so that we still see the correlation between the kk masses and the inverse size of the extra dimensions but the solutions take the form of with .note that here a massless mode does exist due to the periodicity of the bcs .we can change this solution slightly by imagining defining a parity operation on the interval which maps .there are now 2 special points on this interval , call the fixed points , which are left invariant by this operation when combined with the translation and the periodicity property ; these are the points .the eigenfunctions of our wave equation must now respect the discrete , , parity symmetry so that our solutions are _ either _ -even , or -odd , .note that only -even states have a zero - mode amongst them .this geometry is called and is the simplest example of an _ orbifold _ , a manifold with a discrete symmetry that identifies different points in the manifold , here and .the orbifold is of particular interest in model building as we will see below . by the way , we note that all of the bcs of interest to us above are such as to satisfy the conditions following from eq.3 .so far we have seen that a 5d scalar field decomposes into a tower of 4d scalars when going from the 5d to 4d framework .what about other 5d fields ? in a way we have encountered a somewhat similar question when we first learned special relativity , , howdo 3-vectors and scalars get embedded into 4d fields ? just as a 4-vector contains a 3-vector and a 3-scalar , one finds that , , a 5d massless gauge field ( which has 3 polarization states ! ) contains two kk towers , one corresponding to a 4d gauge field ( with 2 polarization states ) and the other to a 4d scalar field .in fact in ( 4+n)-dimensions a gauge field will decompose into a 4d gauge kk tower plus distinct scalar towers . at this point a subtlety exists .we know from our previous discussion that kk tower states above the zero mode are massive. how can there be massive 4d gauge fields with only 2 transverse polarization states ?consider for simplicity the 5d case .it turns out that we can make a gauge transformation to eliminate the scalar kk tower fields and have them ` eaten ' by the gauge field in a manner similar to the higgs - goldstone mechanism becoming their longitudinal components . in a way this is a geometric goldstone mechanism .the scalar kk tower is then identified to be just the set of goldstone bosons eaten by the gauge fields to acquire masses .thus in the _ unitary _ or physical gauge the massless 5d gauge field becomes a massive tower of 4d gauge fields . in ( 4+n)-dimensions only one linear combination of the scalars is eaten so that a massless ( 4+n)-dimensional gauge field produces a massive tower of 4d gauge fields together with scalar towers in the unitary gauge .to see how this 5d decomposition for gauge fields works in practice consider a massless 5d gauge field with the 5th dimension compactified on . in the first step in the kk decompositionone can show that the 2-component vector kk s are -even with 5d wavefunctions like , whereas the kk scalars are -odd with wavefunctions like . note the absence of an scalar mode due to the orbifold symmetry .once we employ the kk version of the higgs - goldstone trick all the gauge kk tower fields become massive 3-component gauge fields with , having eaten their scalar partners .however , the zero mode remains massless , , there was no goldstone boson for it to eat . in 4d the masslessness of the zero mode tells us that gauge invariance has not been broken .we see that orbifold bcs are useful at generating massless modes and maintaining gauge invariance .one can translate these orbifold bcs for the gauge fields into the simple relations for all and where ` ' stands for a boundary .interestingly the physics changes completely if we change the bcs in this case . instead of an orbifold , consider compactifying on a line segment or interval , , and taking as bcs at and at . hereone now finds that and no massless zero mode exists . after the higgs - goldstone trick _ all _the 4d gauge kk tower fields become massive ( leaving no remaining scalars ) with implying that gauge invariance is now broken . herewe see a simple example that demonstrates that we can use bcs to break gauge symmetries .the possibility that such techniques can be successfully employed to break the symmetries of the sm without the introduction of fundamental higgs fields has been recently extensively discussed in the literature .how do other higher dimensional fields decompose ? as the simplest example consider the gravitational field in 5d , represented as by symmetric tensor which decomposes as when going to 4d ; as before is a kk tower index .if we compactify on to keep the zero mode massless ( and which we ll identify as the ordinary graviton of general relativity ) then for all of the and fields get eaten to generate a massive kk graviton tower with fields that have 5 polarization states ( as they should ) .for there is no to eat due to the orbifold symmetry yet a massless scalar remains in addition to the massless graviton .this field is the _ radion _ , which corresponds to an fluctuation of the size of the extra dimension , and whose vacuum value must be stabilized by other new physics to keep radius the extra dimension stable .the physical spectrum is then just the radion and the graviton kk tower in this case which , as we will see below , corresponds to what happens in the rs model .  in terms of manifolds on which to compactify higher dimensional fields it is easy to imagine that as we go to higher and higher dimensions the types of manifolds and the complexity of the possible orbifold symmetries grows rapidly .typical manifolds that are most commonly considered are torii , , which are simply products of circles , and spheres , .let us now turn to a few important representative ed models .the large extra dimensions scenario of arkani - hamed , dimopoulos and dvali(add) was proposed as a potential solution to the hierarchy problem , , the question of why the ( reduced ) planck scale , gev , is so much larger than the weak scale tev .add propose that we ( and all other sm particles ! ) live on an assumed to be rigid 4d hypersurface ( sometimes called a wall or brane ) .on the other hand gravity is allowed to propagate in a ( 4+n)-dimensional ` bulk ' which is , , an , .this brane is conveniently located at the origin in the eds , , * y*=0 .gauss law then tells us that the planck scale we measure in 4d , , is related to the true ( 4+n)-dimensional fundamental scale , , that appears in the higher dimensional action , via the relation where is the volume of the -dimensional compactified space . ( sometimes denoted as in the literature ) can be thought of as the _ true _ planck scale since it appears in the higher dimensional action which is assumed to describe general relativity in ( 4+n)-dimensions . is is possible that could be a few tev thus ` eliminating ' the hierarchy problem ?could we have been fooled in our extrapolation of the behavior of gravity from what we know up to the tev scale and beyond and that gravity , becomes strong at and not at ?to get an idea whether this scenario can work at all we need to get some idea about the size of , the volume of the compactified space . as a simple example , andthe one most often considered in the literature , imagine that this space is a torus all of whose radii are equal to .then it is easy to see that ; knowing and assuming a few tev we can estimate the value of . before we do thiswe need to think about how gravity behaves in eds.   plane excluded by table top searches for deviations from newtonian gravity from adelberger .the add prediction with and tev is also shown.,width=453 ]  if one considers two masses separated by a distance in -dimensions the force of attraction now depends on the relative magnitudes of and the compactification radius . to see thisfirst imagine ; in this case the extra dimensions are essentially invisible and to all appearances the space looks to be 4d .then we know that thanks to newton. however , in the opposite limit the effects of being in a full ( 4+n)-dimensional space will become obvious ; we do nt even realize that the ed is compactified .either via gauss law or by recalling the nature of the solution to the inhomogeneous laplace s equation in eds one finds that now .clearly one will start to see significant deviations from newtonian gravity once so that can not be very large .let s assume ; then we can solve the equation above and obtain m. this is a scale of order the earth - sun distance over which we know newton s law holds very well ; thus is excluded .fortunately the size of decreases rapidly as increases ; for one obtains m which is close to the limit of current table top experimental searches for deviations from newton s law of gravity .these are summarized in fig .[ fig1 ] from the work of adelberger .note that these deviations from newtonian gravity are conventionally parameterized by adding a yukawa - type interaction of relative strength and scale length to the usual newtonian potential . in the figurethe deviations expected in the scenario are shown assuming tev ; the bounds from the data tell us that a few tev in this case .  if is further increased becomes way too small to probe for direct deviations from the law .it is interesting to note that for , which is already being constrained by table top measurements , ev telling us that we have not probed gravity directly beyond energy scales of this magnitude .this ignorance is rather amazing but it is what allows the large parameter space in which the add model successfully functions . from this discussionit appears that the add model will work as long as with being somewhat close to the boundary of the excluded regime . as we will see below the naive caseis highly disfavored by other measurements though larger values of are much more weakly constrained .how large can be ?if we believe in superstring theory at high scales then we can expect that or 7 .a priori , however , there is no reason not to consider larger values in a bottom - up approach .it is curious to note that when one has which is perhaps what we might expect based on naturalness assumptions ; for any or less , a few tev .this point is important for several reasons .( ) one may ask why we required that the sm fields remain on the brane .if the sm or any part thereof were in the bulk , those fields would have kk towers associated with them . since the masses of these kk fields would be of order as discussed above _ and_ we have not experimentally observed any kks of the sm particles at collider so far , we must have gev or so . for any it is clear that this condition can not be satisfied .so if we believe in strings the sm fields must remain on the wall .( ) since the gravitons _ are _ bulk fields their kk masses are given by where is a integer labeling the kk momenta for the ed . as we noted above , for not too large values of these massesare generally very small compared to the 1 tev or even 100 gev scale .this will have important phenomenological implications below .if or 7 in string theory why do nt we just assume this value ?consider a small variation on a theme .so far we have assumed that all of the ed compactification radii are equal ; this need not be the case .assume there are eds but let of them have radius and of them radius .then from above we have now imagine that ; then we d have instead , it would _ appear _ that we really only have large dimensions .the keyword here is ` large ' , the dimensions are actually ` small ' of order tev and not far from the fundamental scale in size .thus there could be , , 7 eds as suggested by strings but only 4 of them are large . if any sm field lived _ only _ in these small eds that would be ( at least superficially ) experimentally acceptable since their kk masses would be tev and as of yet these kks would be unobserved at colliders . since the sm fields can live in these tev size eds we will see below this the possibility is quite popular for model building purposes .to proceed further we need to know what these kk graviton states do , , how they interact with the sm fields on the wall .a derivation of the feynman rules for the add model is beyond the scope of the present talk but can be found in ref. with some elementary applications discussed in ref. .a glance at the feynman rules tells us several things : ( ) all of the states in the graviton kk tower couple to sm matter on the wall with the same strength as the ordinary zero - mode graviton , , to lowest order in the coupling where are the kk graviton fields in the unitary gauge and is the stress - energy tensor of the sm wall fields .( ) since there are at least 2 eds we might expect that the vector fields ( or some remaining combination of them ) , where , would couple to the sm particles .it turns out that such couplings are absent by symmetry arguments since the sm wall resides at * y*=0 .( ) similarly we might expect that some combination of the scalar fields to couple to the sm . here in factone kk tower of scalars does couple to .however , since for massless particles ( except for anomalies ) this coupling is rather small for most sm fields except for top quarks and massive gauge bosons . thus , under most circumstances , these scalar contributions to various processes are rather small .( ) though each of the are rather weakly coupled there are a _ lot _ of them and their density of states is closely packed .this is very important when performing sums over the graviton kk tower as we will see below .how would add eds appear at colliders ?essentially , there are two important signatures for add eds and there has been an enormous amount of work on the phenomenology of the add model in the literature [ for a review see ] .the first signature is the emission of graviton kk tower states during the collision of two sm particles .consider , , either the collision of to make a gluon or to make a photon and during either process have the sm fields emit a tower kk graviton states. note that since each of the graviton kks is very weakly coupled this cross section is quite small for any given kk state .also , once emitted , the graviton interacts so weakly it will not scatter or decay in the detector and will thus appear only as missing energy .now apart from their individual masses all graviton kk states will yield the same cross section as far as this final state is concerned , , jet or photon plus missing energy ; thus we should sum up all the contributions of the kk states that are kinematically accessible .for example , at an collider this means we sum over the contributions of all kk states with masses less than .this is a _ lot _ of states and , since these states are closely packed , we can replace the sum with an integral over the appropriate density of states .while _ no one _ kk graviton state yields a large cross section the resulting sum over so many kks does yield a potentially large rate for either of the processes above .these rates only depends on the values of and . as reviewed by landsberg in these proceedings, both the tevatron and lepii have looked for such signatures with no luck and have placed bounds on the add model parameters .clearly it is up to searches at future colliders to find these signals if they exist .[ fig2 ] from vacavant and hinchliffe shows the missing spectrum at the lhc assuming an integrated luminosity of 100 fb for the process jet plus missing energy in the sm and the excess induced by add graviton emission assuming different values of and .once the rather large sm backgrounds are well understood this excess will be clearly visible .the more difficult question to address is whether such an excess if observed at the lhc would naturally be interpreted as arising from add eds as other new physics can lead to the same apparent final state .signature at the lhc assuming an integrated luminosity of 100 fb from ref .both the sm backgrounds and the signal excesses from graviton emission in the add model are shown . here and .,width=377 ]  at the ilc , the backgrounds for the photon plus missing energy process are far simpler and better understood , essentially arising from the final state .these backgrounds can be measured directly by modifying the electron ( and positron ) beam polarization(s ) since the intermediate state gives the dominant contribution to this process .measuring the excess event cross section at two different center of mass energies allows us to determine both and as shown in fig .[ fig3 ] from the tesla tdr .if fitting the data taken at different center of mass energies results in a poor using these parameters we will know that the photon plus missing energy excess is due to some other new physics source and not to add eds .   plus missing energy final state at the ilc in the add model as a function of for various from ref. normalized to a common value at gev . combining measurements at two distinct values of one can extract both the values of and for the add model.,width=377 ]  these is another way to see , at least indirectly , the effect of graviton kks in the add model : gravitons can be exchanged between colliding sm particles .this means that processes such as or can proceed through graviton kk tower exchange as well as through the usual sm fields . as before ,the amplitude for one kk intermediate state is quite tiny but we must again sum over all their exchanges ( of which there are very many ) thus obtaining a potentially large result . unlike the case of graviton emission where the kk sum was cut off by the kinematics here there is no obvious cutoff and , in principle , the kk sum should include all the tower states .one problem with this is that this kk sum is divergent once as is the case here .( in fact the sum is log divergent for and power law divergent for larger . )the conventional approach to this problem is to remember that once we pass the mass scale the gravitons in the add model become strongly coupled and we can no longer rely on perturbation theory so perhaps we should cut off the sum near .there are several ways to implement this in detail described in the literature . in all casesthe effect of graviton exchange is to produce a set of dimension-8 operators containing sm fields , , in the notation of hewett where is the cutoff scale , and are the stress energy tensors for the sm fields in the initial and final state , respectively .this is just a contact interaction albeit of dimension-8 and with an unconventional tensor structure owing to the spin-2 nature of the gravitons being exchanged .graviton exchange contributions to sm processes can lead to substantial deviations from conventional expectations ; fig [ fig4 ] shows the effects of graviton kk exchange on the process at the ilc .note that the differential cross section as well as the left - right polarization asymmetry , , are both altered from the usual sm predictions .can the effects of graviton exchange be uniquely identified , , separated from other new physics which induces contact interaction - like effects , such as exchange ? this has been addressed by several groups of authors .for example , by taking moments of the angular distributions and employing polarized beams it is possible to uniquely identify the spin-2 nature of the graviton kk exchange up to tev at a tev ilc with an integrated luminosity of 1 ab .this is about half of the discovery reach at the ilc for add eds : for reasonable luminosities .if both beams could be polarized this could be improved somewhat by also employing transverse polarization asymmetries .it is possible to constrain the add model in other ways , , the emission of add kk gravitons can be constrained by astrophysical processes as reviewed in ref. .these essentially disfavor values of less than several hundred tev for but yield significantly weaker bounds as increases .  before turning to a different modellet us briefly discuss the dirty little secret of the add model .the purpose of this model was to eliminate the hierarchy problem , , remove the large ratio between the weak scale and the true fundamental scale , hence the requirement that a few tev .however , if we look carefully we see that this large ratio has been eliminated in terms of another large ratio , , , which for smallish is a very large number as large as the hierarchy we wanted to avoid .thus we see that add really only trades one large ratio for another and does not really eliminate the hierarchy problem .the next model we will discuss does a much better job in that regard .the warped extra dimensions scenario was created by randall and sundrum(rs) and is quite different and more flexible than the add model .the basic rs model assumes the existence of only one ed which is compactified on the now - familiar orbifold discussed above . in this setupthere are two branes , one at ( called the planck brane ) while the other is at ( called the tev or sm brane ) which are the two orbifold fixed points .what makes this model special is the metric : where is the usual minkowski metric and is some a priori unknown function .this type of geometry is called ` non - factorizable ' because the metric of the 4d subspace is . in the simplest version of the rs modelit is assumed , like in the add case , that the sm fields live on the so - called tev brane while gravity lives everywhere . unlike in the add case , however , there is a ` cosmological ' constant in the 5d bulk and both branes have distinct tensions .solving the 5d einstein s equations provides a unique solution for these quantities and also determines that , where is a dimensionful parameter .a basic assumption of this model is that there are no large mass hierarchies present so that very roughly we expect that , the 5d fundamental or planck scale .in fact , once we solve einstein s equations and plug the solutions back into the original action and integrate over we find that as we will see below the _ warp factor _ will be a very small quantity which implies that and have essentially comparable magnitudes following from the assumption that no hierarchies exist .if we calculate the ricci curvature invariant for this 5d space we find it is a constant , , and thus is a measure of the constant curvature of this space . a space with constant negative curvature is called an anti - desitter space andso this 5d version is called . due to the presence of the exponential warp factor this spaceis also called a _ warped _ space . on the other hand ,a space with a constant metric is called ` flat ' ; the add model is an example of flat eds when compactified on as has been assumed here .( the spaces are flat since one can make a conformal transformation to a metric with constant coefficients . ) before going further we note that if the scale of curvature is too small , , if the inverse radius of curvature becomes larger than the 5d planck scale , then quantum gravity effects can dominate our discussion and our whole scenario may break down since we are studying the model in its ` classical ' , , non - quantum limit .this essentially means that we must require which implies a bound that or so , which is not much of a hierarchy .now for the magic of the rs model . in fitting in with the rs philosophy it will be assumed that all dimensionful parameters in the action will have their mass scale set by so that there is no fine - tuning. however , the warp factor rescales them as one moves about in so that , in particular , all masses will appear to be of order the tev scale on the sm brane , , to us .this means that if there is some mass parameter , , in the action which is order , we on tev brane will measure it to be reduced by the warp factor , , .note that if ( a small hierarchy ) this exponential suppression reduces a mass of order gev to only 1 tev. thus the ratio of the weak scale to is explained through an exponential factor and no large ratios appear anywhere else in the model .it has been shown by goldberger and wise that values of are indeed natural and can be provided by a stable configuration .hence we have obtained a true solution to the hierarchy problem .   in the rs model from ref .the different curves correspond to various choices of and as described in the text.the ever widening resonances correspond to increasing the value of .,width=302 ]   in the rs model from ref .the different curves correspond to various choices of and as described in the text.the ever widening resonances correspond to increasing the value of .,width=302 ]  how does this ` warping ' really work ?let s do a simple example by considering the action for the higgs field on the tev brane : where is the determinant of the metric tensor , is the usual quartic coupling and is the higgs vev , which , keeping with the rs philosophy , is assumed to be of order . now and so that we can trivially integrate over due to the delta - function .this yields now to get a canonically normalized higgs field ( one with no extra constants in front of the kinetic term ) we rescale the field by letting which now gives where we have contracted the indices using .we see that the vev that we observe on the sm brane is _ not _ but which is of order the tev scale .thus the higgs gets a tev scale vev even though the parameters we started with in the action are all of order !this warping effect is a general result of the rs model .  .the present anticipated parameter space is inside the triangular shaped region.,width=340 ]  what do the gravitons look like in this model ?even though gravitons are spin-2 it turns out that their masses and wave functions are identical to the case of a scalar field in the rs bulk which is far simpler to analyze .let us return to the klein - gordon equation above but now in the case of curved space ; one obtains ` separation of variables ' via the kk decomposition then yields    which reduces to the result above for a space of zero constant curvature , , when .the solutions to this equation for the wave functions yield linear combinations of the bessel functions and not sines and cosines as in the flat space case and the masses of the kk states are given by where the are roots of . heresince is a few hundred gev at most , we see that the kk graviton masses are of a similar magnitude with comparable , but unequal , spacing , , the kk gravitons have approximately weak / tev scale masses .this is quite different than in the add model .returning to the 5d einstein action we can insert the wavefunctions for the kk states and determine how they couple to sm fields on the tev brane ; one finds where is of order tev .here we see that the ordinary graviton zero mode couples as in the add model as it should but all the higher kk modes have couplings that are exponentially larger due to the common warp factor .we thus have weak scale graviton kks with weak scale couplings that should be produced as spin-2 _ resonances _ at colliders . due to the universality of gravity thesekk graviton resonances should be observable in many processes . there are no table top or astrophysical constraints on this scenario unlike in the add model .it is interesting to note that this model also has only 2 free parameters which we can take to be the mass of the lightest kk excitation , , and the ratio ; given these parameters as input all other masses and couplings can be determined . as we will see essentially controls the kk resonance width for a fixed value of the resonance mass .the rs model in its simplest form is thus highly predictive .      at this point onemay wonder why in the rs model the zero - mode coupling is so weak while the couplings of all the other kk tower states are so much stronger .the strength of the graviton kk coupling to any sm state on the tev brane is proportional to the value of its 5d wavefunction at . in the flat space cases discussed above the typical wavefunctions for kk gravitonswere and so took on essentially the same value at the location of the sm fields for all . herethe relevant combinations of the bessel functions behave quite differently when , , for the zero mode , versus the case when takes on a non - zero value as it does for the kk excitations . for the zero modethe 5d wavefunction is highly peaked near the planck brane and so its value is very small near the tev brane where we are ; the opposite is true for the other kk states .thus it is the strong peaking of the graviton wavefunctons that determine the strength of the gravitational interactions of the kk states with us .what will these graviton kk states look like at a collider ?[ fig5 ] shows the production of graviton resonances at the lhc in the drell - yan channel and in -pairs at the ilc for different values of and .note that the width of the resonance grows as so that the resonance appears rather like a spike when this ratio is small .also note that due to the nonrenormalizable coupling of the graviton kk states the resonance width also grows as as we go up the kk tower .hence heavier states are rather wide ; for any reasonable fixed value of at some point as one goes up the kk tower one reaches states which are quite wide with signaling the existence of the strongly interaction sector of the theory . examining the parameter space and making some simple assumptions one sees that the lhc has a very good chance of covering all of it once 100 fb of integrated luminosity have been accumulated .part of the present constraints on rs follow from the requirement that , as discussed above , and how large we are willing to let be before we start worrying about fine tuning again . given these considerationswe see that the lhc has excellent rs parameter space coverage as seen in fig .[ fig6 ] .once we discover a new resonance at the lhc or ilc we d like to know whether or not it is a graviton kk state .the first thing to do is to determine the spin of the state ; fig .[ fig7 ] shows that differentiating spin-2 from other possibilities is relatively straightforward at either machine . to truly identify these spin-2 resonances as gravitons , however , we need to demonstrate that they couple universally as expected from general relativity . the only way to do this is to measure the various branching fractions and this is most easily done at the ilc .[ fig8 ] shows the expected branching fractions for a graviton kk as a function of its mass assuming only decays to sm particles with a higgs mass of 120 gev .one unique test is based on the fact that both of which can be easily measured at either collider ., , , , , and , respectively.,width=321 ]  before concluding this section we should note that the simple rs model scenario is barely the tip of the iceberg and has been extended in many ways to help with various model building efforts . a few possibilities that have been considered ( with limited references ! )are  * extend to 3 or more branes * extend to 6 or more dimensions * put the sm gauge fields and fermions in the bulk with or without localized brane term interactions .this is very active are of current research .let us return to the case where we again have one flat tev - scale ed as we might have in add with different compactification radii as discussed above .then we can imagine putting all of the sm fields into this part of the bulk so that they have kk excitations ; this is the universal extra dimensions(ued ) scenario of appelquist , cheng and dobrescu . for simplicity we can think of this one ed as just as we are familiar with from above .the kk masses of the various fields at tree level are then given by where are the various sm particle masses and is the usual compactification radius .note that even if is only 300 - 500 gev there will be a lot of degeneracy among the kk levels .the details of the kk mass spectrum are clearly very important for understanding the phenomenology of this model , , the width for is very sensitive to the details of the various masses .thus the radiative corrections to the naive mass formula above need to be considered ; these were calculated by cheng , matchev and schmaltz and they move the spectrum around by a substantial amount .these radiative corrections arise from two sources : ( ) bulk terms which are finite and ( ) terms induced at the orbifold fixed points which depend logarithmically on a cutoff .the size of such terms are set in the model by the assumption that this non - normalizable theory is replaced by some other new physics above the scale so that the magnitude of the divergent terms are chosen to vanish at the cutoff .[ fig9 ] shows how these radiative corrections shift the tree level spectrum in this case .   the spectrum of the first kk level at ( a ) tree level and ( b ) one - loop , for gev , , gev , , and assuming vanishing boundary terms at the cut - off scale from cheng , matchev and schmaltz.,title="fig : " ]  the spectrum of the first kk level at ( a ) tree level and ( b ) one - loop , for gev , , gev , , and assuming vanishing boundary terms at the cut - off scale from cheng , matchev and schmaltz.,title="fig : " ]  in a theory with only periodic bcs , such as one compactified on , the value(s ) of the momenta in the extra dimensions are conserved in any process . in the case of ued , the effect of radiative corrections inducing boundary terms at the fixed points is to violate this conservation law .instead one finds a discrete conservation law , called kk - parity , which takes on a value with being the kk level .the effect of this conservation law is quite restrictive when considering the properties of the first kk states but is useful in helping the model avoid known collider and electroweak constraints .clearly the kks can only be pair produced and the lightest kk particle , the lkp , is stable and can be a dark matter candidate for in the 500 - 1000 gev range ; this sounds a lot like supersymmetry with -parity conservation .in fact the kk spectrum after radiative corrections looks a lot like a typical susy spectrum .is it possible that the two models could be confused ? how can they be distinguished at a collider ?clearly a major difference between susy and ued is that the new heavy particles have different spins in the two cases .determining the spin of a new particle which is pair produced at the lhc is very difficult but has been discussed ; this is a much simpler task at the ilc as new charged spin-0 ( susy ) and spin- ( ued ) states have completely different threshold behaviors as discussed in ref  .another possibility , provided is not too large , is the single production of the states in the ued framework .no particles analogous to these exist in the susy case so if observed would uniquely point to the ued scenario .the subject of eds has become a huge research area over the last six years and we have hardly scratched the surface in the present discussion .as one can see there are at present an immense number of ideas and models floating around connected to eds and we certainly can expect there to be many more in the future .eds can lead to a wide range of new phenomena ( dark matter , collider signatures , ) that will be sought over the coming decade .of course , only experiment can tell us if eds have anything to do with reality and , if they do exist , what their nature may be .the discovery of eds will certainly radically alter our view of the universe on the very small and very large scales .the author would like to thank hooman davoudiasl , joanne hewett , ben lillie , frank petriello and jim wells for collaboration on research on eds over the last 6 years .work supported by department of energy contract de - ac02 - 76sf00515 .# 1 # 2 # 3 mod .phys . lett . *# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 nucl . phys . *# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . lett . * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . rep . *# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . rev . *# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys .# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 rev .phys . * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 nuc .# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 z. phys . * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 e. phys .j. * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 int . j. mod . phys . *# 1 * , # 2 ( # 3 ) # 1 # 2 # 3 j. high en . phys . * # 1 * , # 2 ( # 3 )   n. arkani - hamed , s. dimopoulos and g. r. dvali , phys . rev .d * 59 * , 086004 ( 1999 ) [ arxiv : hep - ph/9807344 ] and phys .b * 429 * , 263 ( 1998 ) [ arxiv : hep - ph/9803315 ] ; 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Abstract: extra dimensions provide a new window on a number of problems faced by the standard model . the following provides an introduction to this very broad subject aimed at experimental graduate students and post - docs based on a lecture given at the 2004 slac summer institute .


the plasma dynamics is governed by infinite - range interactions , i.e. coulomb potentials , and thus behaves differently than ordinary molecular gases . at the kinetic level , among various plasma models , the _ vlasov - poisson _ ( vp ) equations and _ fokker - planck - landau _ ( fpl ) equations are the most representative ones describing , respectively , collisionless and collisional plasma systems .  the vp system is a nonlinear kinetic system modeling the transport of charged particles in a collisionless plasma , under the effect of a self - consistent electrostatic field and possibly an externally supplied field .the electrostatic potential is coupled through poisson equation . some natural plasma , as for example solar wind , behaves as collisionless , since the mean free path of a particle traveling from the sun to the earth is of the order of sun - earth distance . because of its comparative simplicity , numerical schemes for vp equations have been not only thoroughly explored but also well developed .one can refer to , for example .the collisionless vp system exhibits a variety of dynamical phenomena .for example , the well - known filamentation ( filaments in phase space and steep gradients in ) due to its dispersive nature and landau damping mechanism for near equilibrium states satisfying some conditions .readers can refer to for more physical insights .if collisions are taken into account , particles are scattered and things could be different . to our best knowledge , there is rare work on such models .thus , we expect to study the numerical behaviors of the inhomogeneous fpl system for multiple species .the transport of probability density for the particle species is given by subject to some initial and boundary conditions on .here , is the distribution for species , the term is a nonlinear , nonlocal operator in divergence form and models the pair collisions ( e.g. electron - electron , ion - ion , electron - ion , etc . ) and are the coupling parameters . in our present work ,we take to be collision frequency with the knudsen number .the case corresponds to the vlasov - poisson system .the force field only depends on time and space position and can be external or self - consistent .if it is self - consistent , it corresponds to the electrostatic force , where is the charge and is the self - consistent electrostatic field obtained from the poisson equation for charges subject to some boundary condition on .the fpl transport equation is used to model long - range coulomb interactions between charged particles ( e.g binary collisions occurring in a plasma ) .it is of primary importance in modeling evolution of collisional plasma and actually a rather realistic model especially when the magnetic field is very weak .the fpl transport equation can be derived from the general boltzmann transport equation by taking the so - called binary grazing collision limit , i.e collisions that only result in very small deflections of particle trajectories , as is the case for coulomb potentials with rutherford scattering .the original derivation is due to landau .it is also often called fokker - planck - landau equation in plasma physics due to the independent derivation in the fokker - planck form in .  with the general non - isotropic landau collision operator , the inhomogeneous fpl model gains huge difficulties to handle , both analytically and numerically .the main factors generating such difficulties are the nonlinearity , non - locality and diffusive nature with high dimensionality .unlike other kinetic models , for example boltzmann equations , where some non - deterministic methods ( dsmc ) have been successfully applied , the infinite - range potential interactions greatly limit the applications of these type of monte carlo methods .many have tried to develop efficient deterministic solvers for the inhomogeneous fpl equations . however , due to the computational complexity mentioned above , they have turned to some simplified versions of this problem . among them , the space homogeneous fpl equations in the isotropic case were study in , 1d fokker - planck type operator , the cylindrically symmetric problem in , as well as very recent work in for a conservative scheme for a multispecies system of fpl equations .l. pareschi et al .proposed a spectral method to solve fpl equations , by taking truncated fourier series and extending solutions by periodicity .this method was not intended to preserve the natural collision invariants , so , as a consequence they introduced unphysical binary collisions .it can not avoid _ aliasing effects _ , which will be present whenever a vanishing function is approximated by a periodic one .later , filbet and pareschi applied the spectral method to study inhomogeneous fpl with 1d in space and 2d in velocity .the pure transport equations was further splitted and a finite volume scheme was used .then , crouseilles and filbet proposed a solver for inhomogeneous fpl with 1d in space and 3d in velocity , where the pure transport part was treated with a finite volume scheme and the landau operator was approximated by averaging of uncentered finite difference operators .however , the solver in only preserved mass and energy at the discrete level ( for the uncentered finite difference approximate landau operator ) , under some symmetry assumptions on the initial datum .  in this work ,we follow a regular time - splitting scheme , splitting the original inhomogeneous fpl equation into a pure transport problem , i.e vlasov - poisson equation for advection and a homogeneous fpl equation for collisions .these two subproblems can be treated with completely different schemes . for the vp problem, we apply the rkdg method with a piecewise polynomial basis subspace covering all collision invariants , which can be proved to conserve mass , momentum and kinetic energy up to some boundary effects that disappear if the domain is taken large enough .while for the homogeneous fpl equation , different than the one in , we extend the spectral method first introduced in for the nonlinear boltzmann transport equation and propose a conservative spectral method for homogeneous fpl equation , by first extending the solution by zero , representing the collision integral through choosing fourier modes as the test functions in the weak form and enforcing conservation routines . since two completely different numerical scheme are applied separately , our challenge is not only to link two different meshes and at the same time , but also to keep the conserved quantities . we have designed a new conservation correction process such that , after projecting the conservative spectral solution onto the dg mesh , the conserved moments are transferred to the dg solution as well .this work is based on a section of thesis dissertation of the first author under the direction of the second author of this manuscript .the fpl operator models binary collisions in a system of single- or multi - species and reads with the nonnegative and symmetric projection matrix where is the identity matrix ; is the orthogonal projection upon the space orthogonal to .it s semi positive definite with eigenvalues 0,1,1 .the constant is a positive(a value related to the logarithm of the dimensionless debye radius of screening of the coulomb potential in plasma ) . for simplicity, we take in the following .  the inverse - power laws has .similar to boltzmann equations , different categorizes hard potentials for , maxwellian molecules for and soft potentials for . but here , we only focus on most interesting case , corresponding to coulomb interactions .  when , the operator will be a nonlinear ( bilinear ) integro - differential operator in divergence form . here andin the following , when talking about single - species distributions , we will drop the subscript for simplicity .the strong form of this nonlinear partial integrodifferential equation is where the collision kernel is of the form   the fpl operator , as a limit of the boltzmann collision operator , possesses a similar conservation laws and decay of entropy(-theorem ) . that is if and only if corresponding to the conservation of mass ( charge ) , momentum and kinetic energy . as in the previous chapters , we call the test functions _ collision invariants_.  in addition , for any , if set , one can show the following dissipation of entropy which also implied the equilibrium states given by the maxwellian distribution where is the boltzmann constant .the local dependence of is from the mass , the mean velocity and the kinetic temperature , given by   when , the operator models collisions between two different species .it is essentially a linear operator and the treatment will be similar and sometimes even much simpler compared with the fully nonlinear one ( [ q_fpl ] ). we will consider different problems associated to different forms of the operator in the following sections .the main challenges come from the high dimensionality , nonlinearity , diffusive nature , the conservation properties , positivity , etc , which require very careful design of the numerical scheme .we divide and conquer starting from a time splitting method . for zero force field , i.e , the time - splitting is an efficient and reliable way for conquering inhomogeneous problems ; however , we will employ the time - splitting to non - zero force field as well and show that it also works .we discretize time , where is the time step size .denote . in a time interval ] with defined in ( [ eqn : fft_requirement ] ) , and the first integral in the above last formula is zero if .apparantly , it can be computed in , through fft .let be the total number of fourier modes and be the inverse discrete fourier transform of in ( [ qhat_analytical ] ) and be the distribution vector at current time step .the conservation routine here is actually following a similar argument as for the conservative dgfem solver for nonlinear bte .if we use some quadrature , say trapezoidal rule , to evaluate the integrals over the whole velocity domain , then here the constrained matrix , analogous to ( [ constraintc ] ) , is of size and defined as where is the -th integration weight of the quadrature rule ( say , trapezoidal rule ) .following a same derivations , we obtain a conservation correction for the original collisional operator .\mathbf{q}\ , .\ ] ]  then , the semi - discrete problem ( with conservation ) is    the high dimensionality and nonlinearity would make an implicit iterative time discretization really expensive .thus , an explicit method is preferred .due to the diffusive nature of the collision operator , a stiff problem has to be solved , and thus the corresponding stability condition forces the time step to be on the order of the square of the velocity step .we will show this property in the following .the original proof is due to and can easily extend to our spectral method .what we need to solve is the following problem where with defined in ( [ qhat_analytical ] ) .  in practice ,we employ a fourth - order explicit runge - kutta scheme that achieves high temporal accuracy and at the same time does not ruin the spectral accuracy . since the runge - kutta method is just a convex combination of first order euler scheme , we only need to consider the first order euler scheme where the superscript denotes the mode value at the -th time step .the linear stability theory tells us the stability condition is determined by the eigenvalues of the jacobian .we need to find an upper bound on the ( negative ) eigenvalues , such that .then , we have the following proposition  for the first order euler scheme , the time step should satisfy the following stability condition , where is the lateral size of the fixed velocity domain , is the knudesen number and constant only depends on the norm of the current solution .we rewrite ( [ qhat_analytical ] ) into two convolution forms with , being defined component - wisely , the convolutions in ( [ eqn : qhat_2conv ] ) will be evaluated by the trapezoidal quadrature rule , with the fourier nodes being the quadrature points .that is , \ , , \ ] ] where is the step size in fourier space as determined by ( [ eqn : fft_requirement ] ) , and are quadrature weights .  according to ,the time step should satisfy where is the lipschitz norm of .this can be found through estimating the upper bound on the jacobian \\ & \leq \frac{c}{\varepsilon l_{v}}|\widehat{f}^{n}(0)|l^{2}_{\xi } \\ & \leq \frac{c}{\varepsilon l_{v}}\|f\|_{l^{1}(\mathbb{r}^{d})}\frac{1}{(\delta v)^{2}}\ , , \end{split}\ ] ] where the fft relationship ( [ eqn : fft_requirement ] ) is applied , and it is not hard to observe the following uniform bound estimates   therefore , the time step has to satisfy the stability condition   this proves ( [ eqn : stability_homog_fpl ] ) . in practice , we employ a fourth - order explicit runge - kutta scheme and the conservation routine should be performed ad every intermediate step .recall our discretization of time , where is the time step size .denote by the distribution vector at time step . in a time interval ] , 1d in and 3d in .denote by and the regular partitions of and , respectively , with with , , and .then , defines a partition of .denote by and be set of edges of and , respectively .then , the edges of will be .in addition , with and being the interior and boundary edges , respectively .same for variable velocity domain .the mesh size , with and .next , we define the following approximation space .( note that we only have 1d in ) : and where denotes the space of polynomials of total degree at most on some element , while the space of polynomials of degree in each variable on . has number of degrees of freedom , while has degrees of freedom ( here ) .since basis polynomials are piecewise defined over each element , we need to introduce the concepts of jumps and averages . for any test function ( or , ) , define , . for any edge , which is actually one end point of intervals , and any edge , with as the outward unit normal to ,the jump across and are defined as {x_{i}}=(\phi_{h})^{+}_{i-1/2,v}-(\phi_{h})^{-}_{i-1/2,v } , \quad [ \phi_{h}]_{v}=(\phi_{h})^{+}_{x , k_{v}}\mathbf{n}^{+}_{v } + ( \phi_{h})^{-}_{x , k_{v}}\mathbf{n}^{-}_{v}\ , .\ ] ]  and the averages are   here and below , we denote by the discrete electric field computed from the poisson s equation . with proper partitioning, we can assume each direction of has a single sign .the dg scheme for the nonlinear vp equation is described as follows .we seek an approximation solution ( or , ) , such that , for any test function ( or , ) , where here , is the multi - index , corresponding to the three directions of .the following upwinding fluxes ( the trace at the element interfaces ) are used ,   and   the electric field is solved from the poisson s equation , as is used in . in the one - dimensional case , the exact solution of the poisson s equation can be obtained through the classical representation of green s function , if we enforce the periodicity condition , where , , and   the above semi - dg problem ( [ semidg_vp ] ) can be solved by coupling with a suitable time discretization , e.g. total variation diminishing ( tvd ) runge - kutta method .the third order tvd - rk method for evolving is implemented as where are also obtained through the exact representation ( [ exacte_vp ] ) .readers can refer to for a detailed introduction to tvd runge - kutta methods .this completes the rkdg scheme for nonlinear vp problem .we propose to apply basis function , as is proposed in study of vlasov - maxwell equations in , hoping that the rkdg scheme can well preserve mass , momentum and energy .  * remark . *  _ to ensure a positive dg solution , many authors have successfully applied positivity - preserving limiters in the intermediate time steps. please refer to for full descriptions and applications .we summarize the scheme here .for each intermediate step of runge - kutta method , _  * on each mesh element , compute , where , and are sets of gauss quadrature points and sets of gauss - lobatto quadrature points . *compute with the average over element and .* update .the above limiter adjusts the function to be positive while preserving the cell average .thus , application of such positivity - preserving limiter still achieves conservation of total mass , yet however will deteriorate the conservation of energy .this limiter maybe added when necessary , but for the time being , we would like to highlight the conservation of all desired moments .   a piecewise polynomial approximation subspace containing all collision invariants will be applied .we will show , the total mass ( charge ) and momentum is conserved , up to some boundary effects ; as for the total energy , the variation relies on the approximation accuracy of the solution together with the projection error of the potential . also , the approximate solution is stable .the following propositions are extensions of some related results studied in in higher dimensions .the approximate solution ( or , ) for semi - dg problem ( [ semidg_vp ] ) satisfies with   and with here , boundary effects and are negligible if is large enough or equal zero if assume a compact support in the velocity space for .take , then {x}ds_{x}dv - \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\widehat{e_{h}g_{h}}[1]_{v_{1}}ds_{v}dx \\ & = \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon^{b}_{v}}(\widehat{e_{h}g_{h}})_{x , n_{v}+\frac{1}{2}}ds_{v}dx - \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon^{b}_{v}}(\widehat{e_{h}g_{h}})_{x,\frac{1}{2}}ds_{v}dx\ , .\end{split}\ ] ] where considering the periodicity in .take , then {x}ds_{x}dv -\int_{\mathcal{t}^{x}_{h}}\int_{\mathcal{t}^{v}_{h}}e_{h}g_{h}dxdv - \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\widehat{e_{h}g_{h}}[v_{1}]_{v_{1}}ds_{v}dx\ , .\ ] ] the first term above is zero due to the periodic boundary conditions ; the third term is the boundary effect same as above ; let s only look at the second term . thanks to the exact solver for poisson equation ( [ eqn : exactpoisson_phi ] ) and ( [ exacte_vp ] ) ,   the cases for and follow a same way .the total energy of the approximate solution ( or , ) for semi - dg problem ( [ semidg_vp ] ) satisfies with where is the projection of onto ( or , ) and on all interfaces of ( such that is continuous ) .take , then {x}ds_{x}dv - \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\widehat{e_{h}g_{h}}\frac{1}{2}[|v|^{2}]_{v_{1}}ds_{v}dx  \\ & -\int_{\mathcal{t}_{h}}e_{h}g_{h}\partial_{v_1}\varphi_{h}dxdv\ , .\end{split}\ ] ] the first term above is zero due to the periodicity ; the second term is the boundary effect , which is zero if we assume the solution if compactly supported in .  on the other hand ,noticing again the exact poisson solver ( [ eqn : exactpoisson_phi ] ) and ( [ exacte_vp ] ) , which gives ( [ eqn : vp_totene ] ) .if take ( or , ) , then we obtain , which is also valid for the exact solution . the exact solution also obviously conserves total energy , which implies .thus , .this proposition means the variation of total energy relies on the numerical error of and projection error . if the poisson equation is not solved by exact formula but instead through a local dg method , then with special choice of flux , the total energy on the discrete level is proven to be conserved , see in the case of periodic boundary condition with a large enough domain depending on the initial data . buthere , we focus on the inhomogeneous model coupled with the landau collision operator , thus the exact poisson solver is preferred without many extra efforts . actually , when a relatively fine dg mesh is applied , the variations on total energy are negligible .the approximate solution ( or , ) for semi - dg problem ( [ semidg_vp ] ) decays enstrophy with ^{2}_{x}ds_{x}dv -\frac{1}{2}\int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v } } |e_{h}|[g_{h}]^{2}_{v_{1}}ds_{v}dx\ , .\ ] ]  take , then {x}ds_{x}dv \\ & -\int_{\mathcal{t}_{h}}e_{h}g_{h}(g_{h})_{v_{1}}dxdv - \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\widehat{e_{h}g_{h}}[g_{h}]_{v_{1}}ds_{v}dx  \\ & : = a_{1 } + a_{2}\ , , \end{split}\ ] ] where , noticing the definition of upwinding flux {x}\ , , \ ] ] we have , {x}ds_{x}dv \\ & = \int_{\mathcal{t}_{h}}v_{1}\left(\frac{g^{2}_{h}}{2 } \right)_{x}dxdv + \int_{\mathcal{t}^{v}_{h}}\int_{\varepsilon_{x}}\widehat{v_1 g_{h}}[g_{h}]_{x}ds_{x}dv \\ & = -\int_{\mathcal{t}^{v}_{h}}\int_{\varepsilon_{x}}\frac{1}{2}[v_{1}g^{2}_{h}]_{x}ds_{x}dv + \int_{\mathcal{t}^{v}_{h}}\int_{\varepsilon_{x}}\widehat{v_1 g_{h}}[g_{h}]_{x}ds_{x}dv \\ & = \int_{\mathcal{t}^{v}_{h}}\int_{\varepsilon_{x}}\left(-\frac{1}{2}[v_{1}g^{2}_{h}]_{x } + \{v_{1}g_{h}\}_{x}[g_{h}]_{x}-\frac{|v_{1}|}{2}[g_{h}]^{2}_{x } \right)ds_{x}dv \\ & = -\frac{1}{2}\int_{\mathcal{t}^{v}_{h}}\int_{\varepsilon_{x}}|v_{1}|[g_{h}]^{2}_{x}ds_{x}dv\ , , \end{split}\ ] ] and similarly {v_{1}}ds_{v}dx \\ & = -\int_{\mathcal{t}_{h}}e_{h}\left(\frac{g^{2}_{h}}{2}\right)_{v_{1}}dxdv- \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\widehat{e_{h}g_{h}}[g_{h}]_{v_{1}}ds_{v}dx \\ & = \int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}\left(\frac{1}{2}[e_{h}g^{2}_{h}]_{v_{1 } } - \{e_{h}g_{h}\}_{v_{1}}[g_{h}]_{v_{1}}-\frac{|e_{h}|}{2}[g_{h}]^{2}_{v_{1}}\right)ds_{v}dx \\ & = -\frac{1}{2}\int_{\mathcal{t}^{x}_{h}}\int_{\varepsilon_{v}}|e_{h}|[g_{h}]^{2}_{v_{1}}ds_{v}dx\ , .\end{split}\ ] ] so , .so far , we have solved two subproblems separately : vlasov - poisson equation and homogeneous fokker - planck - landau equation .the next step is to link them together , i.e project the fourier series solution of the homogeneous fpl equation onto the dg mesh .if denote by the fourier series solution of the homogeneous fpl equation at the -th time step , and ( or , ) the projection , then according to the time splitting scheme , the initial condition for -st vlasov - poisson problem ( [ eqn : timesplitting_vp ] ) is   during this linking process , conservation of desired moments will be lost if the conservation routine correct the collision operator on discrete level of fourier modes , as in section [ sec : conserveroutine_fpl ] .thus , such a concern inspires us to develop a new conservation routine , which conserves moments fully on level of piecewise polynomials in the dg space ( or , ) .this is shown in the following .take for example . after having , the will be approximated by a partial sum of fourier series , where are the spectral modes , is the multi - index .let be the total number of discretizations in the velocity space , i.e. the total number of fourier modes .we will find corrected mode coefficients , such that is independent of the quadrature rules . here are the collision invariants .plugging this back into gives constraints on the corrected mode coefficients .if denote by the vector of mode coefficients , , the real and imaginary parts , respectively , then where the constraint matrices , , are the real and imaginary parts of the following where .indeed ,   the conservation correction is found by solving the following constrained optimization problem : find ^{t}\in \mathbb{r}^{2m} ] .  following a same derivation ,we obtain the conservative correction \widehat{\mathbf{q}}_{o}\ , , \ ] ] where is a identity matrix .thus , in the temporal evolution , the above _ conserve _ ( [ correctedq ] ) and _ reconstruct _ ( [ q_fs ] ) routines have to be implemented at every intermediate step of runge - kutta schemes .  * remark . *_ we expect the whole discrete scheme to be stable and also to be able to construct a priori error estimates .these two goals will be done in a future project . _one common feature for nearly all realistic kinetic models is the high dimensionality .plus the higher than linear complexity , it addresses the importance of implementations of parallel computing .  for rkdg schemes for vp problem ,the parallelization becomes more natural due to the locality of basis functions .once all the nodes can access to the information from previous time step , the evolution of each grid point is done independently without communications across computing nodes .after evolution is done for the current time step for all nodes , the information will be gathered together and redistributed to all computing nodes in the community .we will use message passing interface ( mpi ) to distribute the velocity grid points .  for the spectral solver for homogeneous fpl equation , at each time step , a single grid point only `` sees '' the particles at the same spatial grid point , through the collision term .since collisions involve all participating particles and take most of the computation time , to avoid large amount of communicating latency , we restrict all of the information needed for the current time step on the same computing node , and thus only distribute spatial grid points across the computing node community .  to further parallelize the computation, we realize that , for each phase velocity grid point , the computation of is a weighted sum over all phase velocities , with no information interrupted by other grid points s .similar features also apply to the integrations in rkdg method for the vp problem .thus , the work load will be further shared using openmp .as the majority of computations occur in the collision steps , the computational time consumed in collisions will dominate .since all information needed for collisions will be kept on the same computing node and only spatial grid points are distributed , an almost linear strong scaling efficiency would be expected .we run tests on a typical linear landau damping problem for the landau - poisson system , and record the time consumed for one single time step in table [ table : para_inhomofpl ] .this example is associated with the one in figure [ fig : damping_k05 ] .tests run on xeon intel 3.33ghz westmere processors ( on cluster lonestar - tacc ) ..the wall clock time for one single time step of a typical linear landau damping problem . [ cols="^,^,^",options="header " , ]    this example is to validate our conservative solver for in the homogeneous setting .we test our scheme to a sum of two gaussians in 3d velocity space , to compute the evolution of entropy and moments and thus verify its validity , \ , , \ ] ] with parameter and .we select domain ^{3} ] .such an initial state has been chosen by many authors , see for instance , as a benchmark problem for studying damping properties .  to study linear damping , we have to make the amplitude small enough , e.g. , to restrict the problem under linear regimes .to well capture the landau damping , the velocity domain must be large enough .it has to be larger than the phase velocity , where is the frequency approximated by  here , we select .the classical landau theory tells that the square root of the electrostatic energy is expected to decay exponentially with frequency . we will plot the evolution of logarithm of square root of the electrostatic energy andcompute its numerical damping rate .  according to , the theoretical damping rate can be estimated as where is the damping rate for collisionless plasma and is the correction " for collisional case . with denoting the collision frequency . and , is estimated by however , as pointed out in , ( [ eqn : dampingrate0 ] ) is more accurate when wave number is large ; so , for small wave numbers , more accurate estimate is available in  and frequency   we will test with initials ( [ eqn : init_lineardamping ] ) for both collisionless and collisional cases .we assume and , and fix wave number .since here amplitude is small enough , the model can be seen in its linear regime and we can compare the numerical damping results against theoretical predictions ( [ eqn : dampingrate ] ) .our numerical results recovered the exponential damping behaviors and show that the damping is stronger if collisions are taking effects . and the damping rate increases with larger wave number . in collisionless case , i.e , when , formula ( [ eqn : dampingrate0 ] ) gives an estimation which agrees well with our numerical result in figure [ fig : damping_k05 ] ; but for , formula ( [ eqn : dampingrate1 ] ) gives a more accurate estimate ( formula ( [ eqn : dampingrate0 ] ) gives ) .in collisional case ,e.g , theoretically estimated damping rate for is , while for is .also , from the damping result , we know larger collision frequency impose a stronger damping .[ fig : damping_k05 ]  : ( left ) , ( right),width=283 ]  : ( left ) , ( right),width=283 ]  [ fig : damping_k03 ]  : ( left ) , ( right),width=283 ]  : ( left ) , ( right),width=283 ]   the linear theory regarding plasmas has been relatively well developed ( though still many problems remain unsolved ) .however , the nonlinear phenomena of plasma is much less understood . from last section , we know as long as the wave amplitude is small enough , a well - developed linear theory is valid .nevertheless , when the wave amplitude gets larger , many waves in experiments can be no longer described by the linear theory . some of them are even not trackable through analysis .one example would be electron trapping " phenomena .it occurs with the nonlinear landau damping of the waves .since the particles travel relative to the wave , a large electric potential together with collisions will trap the electrons in a potential well of the wave .the trapped electrons will be bounced back and forth in the well , causing fluctuating amplitudes of the wave .thus , one can not always expect an exponential damping as in the linear case .  in order to capture the electron trapping , we extract the contours of the following marginal distribution   in phase space , will form peaks whenever there is a potential through .trapped electrons will move in closed orbits in phase space , since the contours are also the electron trajectories .please refer to for more explanations .  in this section , we will study the nonlinear damping with the following initial wave \times \mathbb{r}^{3}\ , , \ ] ] for a relatively large amplitude such that it is no longer in the linear regime . here , we choose the maxwellian figure [ fig : nonlindamping ] shows the nonlinear damping results for , , and a large enough velocity domain , with different collision frequencies .we choose mesh elements on -direction , mesh elements on each direction of velocity for the rkdg vp problem , and fourier nodes for the spectral method .we can see the electric energy , in all cases , decreases exponentially at first . in the collisionless regime ,the electric energy then starts to oscillate around a constant , which agrees well with the known property . with collision ,the oscillations are weaken . in particular , with the presence of stronger collisions , the amplitude of electric energy will start to form an exponential decay again .although a relatively large amplitude is imposed and moderate resolution of mesh is applied , we still obtain good preservation of the total energy , which is even better conserved that .see figure [ fig : vartotene_nonlindamping ] on variations of total energy during the whole process of simulation .   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ,width=340 ]  figure [ fig : eletrapping ] shows the electron trapping effects for much larger amplitude , , and .we choose , and .collision effects range from weak to strong , that is , .one can observe that , without collisions , the much more electrons are trapped in the potential through . while with collisions get stronger , less and lesselectrons are trapped and a stationary state is reached early .   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]   for ( left ) , ( middle ) and ( right ) ]this is of primary importance for studying nonlinear effects of plasmas in future . in this section, we consider a plasma with fixed ion background and only consider the electron - electron collisions .we will study how well the above time - splitting and conservative linking process work , by initializing with a non - isotropic two - stream flow . where is the amplitude of the perturbation and the wave number , and \ , , \ ] ] with parameter and .we would like it to be far from the linear regime , so a relatively large perturbation is considered , with .a large enough velocity domain is selected .we choose mesh elements on -direction , mesh elements on each direction of velocity for the rkdg vp problem , and fourier nodes for the spectral method .relatively stronger collision effects are considered by taking a relatively large collision frequency ( relatively small knudesen number ) .results are also compared to collisionless case , i.e. . in figure ( [ fig : inhomolp_2stream_kineticene ] ) , ( [ fig : inhomolp_2stream_elecene ] ) and ( [ fig : inhomolp_2stream_varene ] ) , the total energy initially comes from both the kinetic and electrostatic energy , but with time forwarding , the electrostatic energy decays with oscillations down to zero and the total energy at the end all comes from pure kinetic motions , which means the system has reached at its global equilibrium . during the whole process , the total energy is well preserved only with negligible variations .in addition , from figure ( [ fig : inhomolp_2stream_kineticene ] ) and figure ( [ fig : inhomolp_2stream_elecene ] ) one can observe that , since the landau operator is essentially a diffusive operator , the oscillations generated by coupling with the poisson equations damps with collisions , and thus the state reaches stationary in a much earlier stage .we studied the inhomogeneous fpl equations , coupled with poisson equations which governing the self - consistent electric field .the complicated inhomogeneous problem is split into two subproblems , by time - splitting scheme .we applied two different methods for treating the pure transport vlasov problem and the pure collisional homogeneous fpl equation .the former is solved by rkdg method , which has achieved its success in many other kinetic problems ; while the latter is treated using conservative spectral method .the conservative spectral method was well developed for solving boltzmann equations and we extended the method to the fpl problems and apply to study the multi - component plasma .the temperature relaxation of the multi - component plasma is studied both analytically and numerically . to link the two different methods , or computing grids, we developed a new conservation routine which can guarantee no loss of moments when projecting the fourier solution onto dg meshes .all desired moments are preserved only with error of dg approximations .the whole scheme has been applied to study the well - known landau damping problems , whose results agree well with theoretical estimates , and to two stream flows .  the project is implemented with parallelization , hybrid mpi and openmp .  in the future, we would like to speed up the collision and conservation processes and meanwhile increase the gridding resolution , such that the current solver can tackle , in real world of collisional plasma , more challenging problems which is tough to be treated numerically .for example , we would like to apply non - periodic boundary conditons on the poisson equations and thus to study more nonlinear effects , for instance plasma sheath problems , which is of primary importance for aerospace engineering .00 a.v . bobylev , s.a . karpov and i.f .potapenko , , , american institute of physics , 9 - 13 july , 2012 .e. gabriel , g.e .fagg , g. bosilca , t. angskun , j. j. dongarra , j. m. squyres and v. sahay , p. kambadur , b. barrett , a. lumsdaine , r. h. castain , d. j. daniel , r. l. graham , t. s. woodall , , , 97 - 104 , september , 2004 , budapest , hungary .

Abstract: we have developed a deterministic conservative solver for the inhomogeneous fokker - planck - landau equation coupled with poisson equation , which is a rather realistic and primary model for collisional plasmas . two subproblems , i.e vlasov - poisson problem and homogeneous landau problem , are obtained through time - splitting methods , and treated separately by runge - kutta discontinuous galerkin method and conservative spectral method , respectively . to ensure conservation when projecting between the two different computing grids , a special conservation routine is designed to link the two solutions to these two subproblems . the entire numerical scheme is implemented with parallelization with hybrid mpi and openmp . numerical experiments are provided to study linear and nonlinear landau damping problems and two - stream flow problem as well .   inhomogeneous fokker - planck - landau equation , discontinuous galerkin , conservative spectral methods , collisional plasma , landau damping