the scientific process asks questions of the universe .much of the education of a scientist consists of learning , painstakingly , to ask those questions .if the question is ill - posed , the answer will be ambiguous or even unintelligible ; even if it is well - posed , the question needs to be the one whose answer is pertinent to whatever problem it is we wish to solve .simulations are one good route to finding out about the clarity and worth of our questions , since variables can be easily controlled and extraneousness determined .  quantum imformation is the systematic use of the fundamental quantum mechanical nature of the universe to do calculations that are either very difficult or even impossible to do with a classical computer . exactly what this allows us to do , and how we can make use of that nature , turns out to be a very difficult problem .the quantum system itself contains that information .but how do we exploit those abilities ?how do we even find out what they are ?  in this paperwe use a quantum neural network ( qnn ) , which necessarily complex - valued , to calculate the essential attributes of the quantum information .while this is done in simulation , the method , as outlined , could be implemented experimentally to create a quantum computer , programmed through neural network training , to calculate its own entanglement and phase .determination of entanglement is one very good example .this is an important question because entanglement is what allows us to quantum computations ; it is an essentially question since no classical system can have quantum correlations ; thus , it is an excellent testbed for this general problem . the quantum system which is ourquantum computer `` knows '' what its entanglement is , but how do we ask the question ? and how do we know if we have the right answer ?  given the density matrix , entanglement can actually be calculated for the two - qubit system . for definiteness , we choose the `` entanglement of formation ''  , which is the number of pure singlets necessary to create a given entangled state . of course, we may not even know the density matrix .we could determine the density matrix through quantum tomography , but the number of measurements necessary goes like , where is the number of qubits .thus , as the system grows this method becomes prohibitive .is there instead a relatively cheap way to determine the entanglement ?a number of researchers ( see , _ e.g. _ , ) have devised single measurement `` entanglement witnesses '' , to indicate the presence or absence of entanglement .an entanglement witness , , is a hermitian operator such that for a density matrix representing a fully separable ( i.e. , unentangled ) state .most of them are measures of `` closeness '' to a single , entangled state , and fail for other kinds of entangled states .this is easiest to understand by considering the so - called `` bell '' states : for , this is a commonly used basis for the two - qubit system , the `` bell basis '' ( it is an orthonormal set ) .each of these states is fully entangled , ( i.e. , each could be created using exactly one singlet ) , independent of the angle .a witness can be devised that measures closeness to any one of these states .for example , the mintert witness is given by , where , is the projector onto the antisymmetric states , and is the projector onto the symmetric states . as proposed this witness is set up to work well for states close to the singlet ( antisymmetric ) state , , so it does not detect entanglement in either or .worse , the witness increasingly fails to detect entanglement in as increases .see figure [ mintert1fig ] , which shows ( set for closeness to the epr singlet , ) , calculated for each of the bell states as functions of their respective .entanglement is indicated by negativity of the result .the straight line showing the constant result of zero shows that this witness predicts no entanglement for either or .the two oscillatory curves correspond to the and states . at becomes the other ; thus , at that point , is predicted to lose entanglement and to become entangled .  , set for closeness to , tested on each of the bell basis states as functions of phase offset .both and are calculated to have zero entanglement , while and oscillate , showing entanglement for half the range of .note that the answer is that each state is fully entangled for all values ( i.e. , the witness should be negative everywhere . )this oscillation is what we are calling `` anomalous oscillation '' in the entanglement witness .[ mintert1fig],width=240 ]  witnesses can be `` reset '' so that they will work for other members of the bell basis ; for this witness , all that is necessary is to change the projection operators .see figure [ mintert2fig ] , which shows results of this procedure : all the predictions now lie atop each other ( as they must do by symmetry . )so , if you know which entangled state your unknown state is close to , you can use a witness which will correctly predict its entanglement .however this creates an obvious dilemma : _ you must have knowledge of the state before you do the measurement to gain knowledge of the state . _  , set for closeness to each of the bell basis states ( at ) in turn , and tested as functions of phase offset . allshow exactly the same oscillatory behavior .again , the correct answer is that each state is fully entangled for all values ( the witness should be negative everywhere . )[ mintert2fig],width=240 ]  all single measurement witnesses must show this anomalous oscillation , shown in figures [ mintert1fig ] and [ mintert2fig ] .clearly , it would be of value to devise a corrector to this oscillation , so that the witness would then correctly predict that all the states of equation 1 are fully entangled . with such a method, we could thereby determine whether a general , unknown , input state was entangled , and perhaps even the degree of entanglement .we propose here such a method , using a quantum neural network .in previous work we have shown that a quantum system can be trained to deal with both parts of the problem : that it can be trained to output an entanglement indicator or a phase indicator .briefly , the details are as follows .  for the 2-qubit system ,we write for the hamiltonian : where are the pauli operators corresponding to each of the two qubits , a and b , and are the tunneling amplitudes , and are the biases , and the qubit - qubit coupling .( generalization to an n - qubit system is possible but will not be considered here . )the time evolution of the system is given by the schrdinger equation :  \label{schr}\ ] ] where is the density matrix and is the hamiltonian .the parameters determine the time evolution of the system in the sense that , if one or more of them is changed , the way a given state will evolve in time will also change .this is the basis for using our quantum system as a neural network .the role of the `` weights '' of the network is played by the parameters of the hamiltonian , , all of which we take to be experimentally adjustable as functions of time ( see , _ e.g. _ , , for the case of squid charge qubits . ) by adjusting the parameters using a neural learning algorithm we can train the system to evolve in time to a set of chosen target outputs at the final time , in response to a corresponding ( one - to - one ) set of given inputs . because the time evolution is quantum mechanical ( and , we assume , coherent ) , a quantum mechanical function can be mapped to an observable of the system s final state , a measurement made at the final time .the time evolution of the quantum system is calculated by integrating the schrdinger equation numerically in matlab simulink , using ode4 ( runge - kutta ) , with a fixed integration step size of 0.05 ns .the system was initialized in each input state in the training set , in turn , then allowed to evolve for 190 ns .a measurement is then made at the final time ; this is the `` output '' of the network .an error , , is calculated , and the parameters are adjusted slightly to reduce the error .this is repeated for each pair multiple times until the calculation converges on parameters that work well for the entire training set .complete details , including a derivation of the quantum dynamic learning paradigm using backpropagation in time , are given in and in .for the entanglement indicator , we chose as our output the qubit - qubit correlation function evaluated at the final time , , and trained the indicator using a training set of four ( one fully entangled state , two product ( unentangled ) states , and one partially entangled state . )that is , the input , output ) pairs are :    with prepared input states at zero time , and corresponding targets , given in table [ enttraining ] , which also shows the trained values and the entanglement of formation , calculated using the analytic formula for comparison .note that the qnn indicator systematically underestimates for partially entangled states ; this is because we found through simulation that the net naturally trained to the target value of 0.44 .see for details .that is , we seek here not exact with ( in which case we would train the state to a target value of 0.55 ) , but a robust and internally self consistent measure , which we would hope would track well with an analytic measure like .  .training data for qnn entanglement witness . [ cols="<,<,<,<",options="header " , ]   , , and , as functions of time , as trained for the phase offsets , and plotted with curvefits to single oscillatory functions , as explained in the text .on the scale of the graph the curvefits are indistinguishable from the plotted points of the trained values .[ paramphasefitfig],width=240 ]  agreement is good even with the fitted functions in place of the trained ones ; and it is certainly easier to use a fixed analytic function than a large collection of discrete data samples . for testing on more general states than those of equation 7 , though , we needed to expand the target function .  because quantum computational power comes from entanglement , it is not surprising that the ability of the net to determine a relative phase between two basis states depends on there being entanglement between those states , and , indeed , the size of the signal diminishes with decreasing entanglement . for the bell states with unequal magnitude , , we found that if we used the target function we got excellent results .note that , the coefficient in front of the term , is the concurrence for this state - a monotonic measure , like , of the entanglement .testing on states of the form was equally good when we again adjusted for the diminished amount of entanglement in the target function .in fact , the parameter functions we found through training only on the eleven states of equation 7 gave us all three of the phases , with symmetrically adjusted outputs and target functions ( e.g. , to find we used the output . )the phase information seems inextricably linked with the entanglement information . for the simple case of a fully entangled state , either ] , our phase indicator returns the cosine squared ( of or , respectively ) for the projections , so the procedure is straightforward : using two copies of a given state , measure the phase offset on one copy , then apply a rotation operator to the other , and measure the entanglement .results are shown in figure [ fixbellfig ] . ] states .[ fixbellfig],width=240 ]  the linkage does happen in both directions , though . if we examine the anomalous oscillation in , e.g. , figure [ qnnbellfig ] , it is also remarkably close to a cosine squared .a good approximation is shown in figure [ qnn_model_approximation_jan12fig ] .results are exactly similar for the  ] ( and symmetrical equivalents , such as  ] , as a function of and , as calculated by the qnn , and compared with a curvefit to ( the partially transparent grid , superimposed . ) [ qnn_model_approximation_jan12fig],width=240 ]   ] with a high degree of probability , our method would allow for good reproducible determination of the entanglement .of course this is not a complete solution even to the measurement of the entanglement of a two qubit system .that is , our results in the previous section do not transform figure [ cxcoeffig ] completely into figure [ realfig ] , as we have allowed only one of the coefficients at a time to be complex .we consider this work to be only proof - of - concept : though our phase indicator does give us all relative phases , we need something more to solve this problem completely , for three reasons .first , because each measurement we make on a quantum system destroys all other information , so that we would need , minimum , four copies of our input state .with so many measurements necessary , it seems it might be easier simply to determine the state completely and then to calculate or whatever we wish to know , analytically .second , inversion of the phase indicator target functions is increasingly problematic for the general input state . and third, it is not true that if we successfully unwind all the phases we will be able to recover the entanglement of the original state .consider the entanglement of formation of the `` flat state '' ( equal magnitudes of all of the charge basis states  ] , and since is a monotonic bijective function of the concurrence , we can easily see that ( since is maximum ) whenever the difference in phase between the `` bell part '' and the `` epr part '' is .that is , if the `` bell - type '' entanglement and the `` epr - type '' entanglement are exactly in phase , the superposition is a product state , and , thus , has zero entanglement ; if they are exactly out of phase , the density matrix can not be factorized , and , thus , is entangled .this type of phase dependence actually causes entanglement , so , if we wish to determine the entanglement for a general input state , we will have to do more than simply determine all the phases and unwind them .it is not yet completely clear how much of this behavior the trained qnn net has already captured , since for real coefficients , including negative ones , we get good results ( figure [ realfig ] ) .further work is needed , and is ongoing .we have not considered here the multiple qubit systems that are of most interest to practical quantum computing ; however , it is worth considering that a neural approach has natural advantages for scale - up .in fact , our entanglement indicator did easily generalize from two to three to four to five qubits .we have also not considered mixed states , though our entanglement indicator did work also for these types of states , which gives us hope that this , larger , work may be easily extended to those also .behrman and j.e .steck , _ a quantum neural network computes its own relative phase _ , in _ proceedings of the ieee symposium on computational intelligence 2013 _ ( singapore , april 15 - 19 , 2013 .( institute of electrical and electronics engineers ) .behrman and j.e .steck , _ dynamic learning of pairwise and three - way entanglement _, in _ proceedings of the third world congress on nature and biologically inspired computing ( nabic 2011 ) _( salamanca , spain , october 19 - 21 , 2011 .( institute of electrical and electronics engineers ) .yann le cun , _ a theoretical framework for back - propagation _ in _ proc .1998 connectionist models summer school , _d. touretzky , g. hinton , and t. sejnowski , eds . ,morgan kaufmann , ( san mateo ) , pp .21 - 28 ( 1988 ) .behrman earned an sc.b from brown university in mathematics , and an m.s . in chemistry and ph.d . in physics from university of illinois at urbana - champaign , where she performed research using feynman path integrals to study the dynamics of tunneling systems in condensed phases at finite temperatures .she did postdoctoral work at suny stony brook , and worked for four years at nys college of ceramics at alfred university , doing research in calcium aluminosilicate glasses , chalcogenide glasses , viscosity of glass melts , and high temperature ceramic superconductors .she is currently professor of mathematics and of physics at wichita state university where she has taught for twenty - two years .she teaches undergraduate and graduate courses in physics at all levels , including quantum mechanics , statistical mechanics , and mathematical methods .her work at wichita state includes inorganic buckyballs and buckytubes , reaction intermediates , and defects in radiation - damaged materials as well as quantum neural computing .bonde earned b.s .degrees in aerospace engineering , physics ( honors ) , and mathematics ( honors ) from wichita state university , where , in addition to quantum neural networks , he also did research with the pierre auger array .he is currently a ph.d .graduate student at the university of texas at arlington , working on the atlas experiment .steck has a b.s . and m.s .in aerospace engineering and a ph.d . in mechanical engineering from the university of missouri - rolla where he performed research in the use of finite element methods in mechanics , fluids and aero - acoustics .he has done postdoctoral work in artificial neural networks as control systems for the intelligent systems center at the university of missouri - rolla , and worked for two years at mcdonnell aircraft company in the aerodynamics department doing flight dynamics support for flight simulation , wind tunnel and flight testing of the av-8b aircraft .he is currently professor of aerospace engineering at wichita state university where he has taught for seventeen years .he teaches undergraduate and graduate courses in flight dynamics and controls , artificial neural networks , and computational methods .his current work includes : intelligent adaptive control systems for general aviation aircraft , integrated aircraft structural health monitoring , quantum neural computing , optical aircraft ice detection , optical neural computing , and the use of artificial neural networks for system modeling and control .behrman will earn a b.a . in physics ( honors ) from harvard university in may 2013 .in addition to work in quantum neural computing , she has also done research in the history of physics , under the direction of h. georgi of harvard ; simulation and analysis of cosmic ray showers , under the direction of p. horowitz of harvard , and simulation and experimental work on disordered ino at the superconductor - insulator transition , under the direction of d. shahar of the weizmann institute .

Abstract: entanglement of a quantum system depends upon relative phase in complicated ways , which no single measurement can reflect . because of this , entanglement witnesses are necessarily limited in applicability and/or utility . we propose here a solution to the problem using quantum neural networks . a quantum system contains the information of its entanglement ; thus , if we are clever , we can extract that information efficiently . as proof of concept , we show how this can be done for the case of pure states of a two - qubit system , using an entanglement indicator corrected for the anomalous phase oscillation . both the entanglement indicator and the phase correction are calculated by the quantum system itself acting as a neural network .   shell : bare demo of ieeetran.cls for journals


rocks , especially igneous rocks , behave as semiconductors under certain conditions .although the magnetic fields produced by small semiconductors are often negligible , semiconductors on geophysical scales may produce significant magnetic fields .this is of particular interest since these fields can be observed at the earth s surface and they seem to indicate that rock is being stressed deep in the crust .ultra - low frequency ( ulf ) electromagnetic emissions have been observed prior to earthquakes , possibly resulting from electric currents flowing deep in the crust .low frequency fluctuations in the earth s magnetic field have been reported in the literature since at least 1635 . increased levels of magnetic fluctuationshave been repeatedly observed prior to earthquakes since at least 1964 , but these transient phenomena have received relatively little attention and , as such , they are not yet fully understood.  observed pre - earthquake electromagnetic waves typically have frequencies between 0.01hz and 20hz , possibly owing to the fact that only low - frequency components may traverse tens of kilometers through the rock column .they have been independently observed prior to many earthquakes over the past 50 years . during the weeks leading up to the m=5.4 alum rock earthquake of oct .30 , 2007 , a magnetometer located about 2 km from the epicenter recorded unusual non - alternating magnetic pulses , reaching amplitudes up to 30 nt .the incidence of these pulses increased as the day of the earthquake approached .a pair of magnetometer stations in peru recently recorded similar unipolar pulses prior to several medium - sized earthquakes , and triangulating the source of these pulses revealed the location of subsequent earthquake epicenters .the unipolar magnetic pulses observed prior to earthquakes have a characteristic shape similar to a gaussian that is attenuated over time .the unipolar nature of the magnetic pulses is somewhat unusual and bears resemblance to pulses produced by lightning and other electrical breakdown phenomena .however , the duration of many pre - earthquake pulses exceeds several seconds , much longer than any lightning strike .moreover , triangulation of such pulses near lima , peru revealed that strong pulses originated almost exclusively from locations within a few kilometers of future earthquake epicenters .  to model the electromagnetic phenomena associated with volumes of rock , we solve a three - dimensional drift - diffusion model of a semiconductor and calculate the magnetic fields induced by its electric currents .the model is seen to describe transient low - frequency unipolar magnetic pulses .we will show that unipolar pulses can emerge simply from the electrical drift and random diffusion of charge carriers in a semiconducting volume of rock .there are several reasons why this is a plausible mechanism for the observed pulses .large electrical currents are known to accompany earthquakes , occasionally so large that luminous effects known as earthquake lights become apparent .there is experimental evidence indicating that , during stressing , electrons and holes become activated in igneous rocks that subsequently behave as semiconductors .one proposed source of charge carriers in rock is the break - up of peroxy defects as a result of the increase in tectonic stresses .the oxygen sublattice of a wide variety of silicate minerals can form peroxy defects that act as sources of electron / hole pairs , causing these minerals to exhibit semiconductivity . once activated , highly mobile electronic charge carriers diffuse through the minerals .peroxy defects are point defects , typically introduced through the incorporation of into nominally anhydrous minerals that crystallize in -laden magmas or recrystallize in high - temperature -laden environments .the incorporation of into oxides and silicates leads to pairs that subsequently undergo redox conversion .the two of the pairs combine to form , and the ions bind to form a peroxy bond .the formation of these peroxy bonds has been extensively studied in laboratory experiments and treated by computational chemistry .  when peroxy bonds are energized via stresses in the rock or by heat, they may produce electron - hole pairs .the peroxy bond breaks , forming a transient state with two unpaired electrons .this is followed by a fully dissociated state in which a hole is free to move through the crystal structure .a neighboring oxygen atom donates an electron and becomes a hole , as its valence shell becomes deficient by one electron .the donated electron becomes trapped near the broken peroxy bond in a new state whose energy level is slightly below the upper edge of the valence band . in terms of the valence state ,the neighboring oxygen atom , which was previously in an state , becomes .this oxygen anion in the 1- state is effectively a positive hole with an incomplete valence shell and could also be regarded as an unstable oxygen radical .holes are capable of propagating through the oxygen lattice , exchanging valence electrons by a phonon - assisted vacancy hopping mechanism .this process effectively constitutes a diffusion of holes through a lattice of atoms .the trapped electrons are immobile but participate through recombination and electrostatic interactions .the drift - diffusion equations are the most frequently used model for semiconductor physics , and perform well on scales greater than about 0.5 micrometers .they describe current in terms of charge carrier concentrations and an electrostatic field , and this determines the change in charge carrier concentrations via continuity of the current density .the drift - diffusion equations are :   here , , , , , and are defined on a domain , where is a subset of a 3-dimensional space .the functions and are concentrations of electron and hole charges , respectively , and is the charge of any dopant ions that are present . is the recombination / generation rate of electrons and holes .the third equation is poisson s law of electrostatics whose solution describes the electric potential . is the electric permittivity .the constants and are the mobilities of electrons and holes , respectively , ( not to be confused with the magnetic permeability or ) and and are the corresponding diffusion coefficients . in the particular instance of the model under consideration , and are approximately zero due to electrons becoming trapped in the valence band .maxwell s equations describe propagation at the speed of light , much faster than the charge carriers diffusing in a typical semiconductor . rather than modeling propagation on two very different time scales , we make use of the quasi - static ( magnetostatic ) approximation , assuming that currents do not alternate rapidly or approach the speed of light .specifically , the maxwell displacement current appearing in ampere s law is assumed to be negligible : .this assumption is implicit in the drift - diffusion model due to its use of poisson s equation for the static electrical potential .the electric current density acts as the source of a magnetic field .it may be expressed as the sum of a drift term , involving the electric field , and a diffusive term , involving the concentration gradient .the rate of change of the concentration then becomes a continuity equation that is a function of current density . explicitly separating the current and continuity equations facilitates coupling to the magnetic field . in this form ,the current densities are :   the continuity equations that describe the change in electron and hole concentrations are , then :   the current densities and are summed to obtain the total current density that acts as a source for the magnetic field . in a magnetostatic approximation , the solution to the magnetic field on a domain may be efficiently computed by solving a set of poisson equations for the magnetic vector potential . in this case , however , we calculate the field at an arbitrary point in space , which could be outside the domain .we apply the biot - savart law to obtain the magnetic field at the point :   here , is the magnitude of the vector from to and is the magnetic permeability .the velocities of the holes are not sufficiently large for the lorentz force to be significantly influenced by magnetic fields so we do not conisder the effect of the magnetic field on the charge carriers .the drift - diffusion equations are solved by expressing the partial differential equations as a system of ordinary differential equations for the time derivatives and . a finite - difference approximation to this systemis then integrated using a fourth - order runge - kutta scheme ( rk4 ) .poisson s equation is solved separately at each timestep using successive over - relaxation ( sor ) with an adaptive relaxation parameter and open boundary conditions . for the other pdes ,the dirichlet boundary conditions and are applied to the boundary of a grid of uniformly spaced points representing the , , and coordinates over which functions are evaluated .all spatial partial derivatives ( , , , ) of the current and continuity equations are approximated using a fourth - order central difference approximation .  at each timestep ,the electric potential is determined by solving poisson s equation , starting the sor iteration with the electric potential from the previous timestep . using the electric potential and the charge carrier concentrations , the components of the current vector fields and are evaluated . from and ,the continuity equations are integrated , yielding the concentrations of the charge carriers at the next timestep .the magnetic field is evaluated by applying a discretized biot - savart law to the currents . is calculated at each timestep but since the result does not affect the dynamics , it may be evaluated at a single point .since holes are mobile and electrons are immobile , diffusion separates the two species , creating an electric current that acts as an electromagnet . the boundary of a region of activated charge behaves , essentially , like the p - n junction of a diode .since only holes may flow out of this volume , the initial current diffusing across the boundary is unidirectional , corresponding to forward bias in the diode .however , after a delay period during which recombination reduces the diffusive current , the diode switches to reverse bias .the p - n junction capacitance generates a reverse recovery current and potentially a reverse - bias electrical breakdown .this current of holes flows back into the source volume , producing a magnetic pulse that is opposite in polarity and potentially much stronger than the initial magnetic field .we use the semiconductor model to calculate an example of a unipolar magnetic pulse .the electric permittivity and magnetic permeability are estimated based on the static properties of mgo ( and ) and a temperature of . since electrons are trapped and immobile in broken peroxy bonds , and are set to .the mobility and diffusion constant of holes were roughly estimated based on experimental data from a haynes - shockley - style experiment whereby a rapid pressure impulse to the center of a gabbro tile injected holes that diffused and drifted away from their source .the parameters used are m ( vs) and m s , comparable to their values in pure undoped silicon .charge generation is not explicitly considered in this calculation , and a pre - existing excess concentration of c m of both electrons and holes exists as an initial condition .these dissociated charges are initially present only at points within a ball of radius m. their recombination rate is proportional to the product of electron and hole concentrations , . the attenuation of the magnetic field as it passes through the earth is not considered , nor are effects associated with the surface of the earth .calculated and observed magnetic pulses are illustrated in figure 1 .the first figure shows the value of the x - component of the magnetic field as a function of time , measured 10 km directly above the center of the simulated volume .the amplitude , frequency , and shape of the pulse are similar to pulses that have been observed before earthquakes . for comparison, the second figure shows several magnetic pulses observed prior to an earthquake near lima , peru .these pulses were measured over a period of several days by a pair of magnetometers approximately 25 km away , and the locations of their sources were triangulated .the sources were clustered within a few kilometers of the epicenter of an earthquake that occurred two weeks after the onset of the pulses .this analysis has been performed prior to several moderate earthquakes near lima , with similar results .when a volume of rock is stressed , excess holes and electrons are injected .the mobile holes begin to diffuse out of the source volume , while electrons are trapped within the source volume and undergo recombination with the holes that have not diffused out .the flux of holes leaving the source effectively creates a p - n diode : the source volume becomes an n - type semiconductor and the surrounding rock becomes p - type .a depletion region forms between the two layers of the p - n junction and the p - n double layer screens electric fields outside its immediate vicinity .after charge injection , a diffusive current of holes flows as a result of the concentration gradient across the source boundary .this corresponds to a forward bias state of the diode , dominated by diffusion capacitance rather than junction capacitance .this current creates a transient magnetic field . as the hole concentration gradient decreases , the diffusive current and the magnetic field decay .after holes have diffused outward , creating p - type and n - type regions , a junction capacitance results from layers of positive and negative charge separated by a depletion region at the junction .after a delay period , the diode effectively switches to a reverse bias state .electron - hole recombination consumes the holes remaining within the source volume , leaving mostly electrons inside .the junction capacitance causes a transient reverse recovery current , and , if the potential drop across the depletion region is sufficiently strong , reverse - bias electrical breakdown may occur .coulomb attraction pulls the holes back into the source volume , and this reverse current of holes can create transient magnetic pulses similar to those observed before earthquakes .  a distinctive form and the ability to pass through the earth at ultra - low frequencies make magnetic pulses a compelling tool for the observation of pre - seismic shifts in the stress level of rocks that are otherwise inaccessible due to depth . by triangulating the source of these magnetic pulses , the increased buildup of stress around future earthquake epicenters may be identified weeks in advance of seismicity .in addition to unipolar pulses , other types of electromagnetic precursors might be predicted from the semiconductor model .oscillatory ulf fields , for example , have been observed immediately preceding earthquake activity .the positive hole semiconductor theory modeled here seeks to unify a wide range of electromagnetic phenomena associated with seismic activity .the direct coupling of semiconductor drift - diffusion currents and electromagnetism produces a model consistent with observations of pre - seismic magnetic pulses .this suggests that pre - earthquake ulf activity may be the result of geophysical semiconductor processes .the authors would like to thank tom bleier and clark dunson of quakefinder for informative discussions .this research was funded in part by nasa earth surface and interior grant nnx12al71 g ( dr . john labrecque ) .       in this context, ultra - low refers to electromagnetic waves having frequencies from millihertz to a few hertz , in contrast to the international telecommunications union ( itu ) definition of ultra - low , which would correspond to waves having frequencies of 300hz-3khz .

Abstract: a semiconductor model of rocks is shown to describe unipolar magnetic pulses , a phenomenon that has been observed prior to earthquakes . these pulses are observable because their extremely long wavelength allows them to pass through the earth s crust . interestingly , the source of these pulses may be triangulated to pinpoint locations where stress is building deep within the crust . we couple a semiconductor drift - diffusion model to a magnetic field in order to describe the electromagnetic effects associated with electrical currents flowing within rocks . the resulting system of equations is solved numerically and it is seen that a volume of rock may act as a diode that produces transient currents when it switches bias . these unidirectional currents are expected to produce transient unipolar magnetic pulses similar in form , amplitude , and duration to those observed before earthquakes , and this suggests that the pulses could be the result of geophysical semiconductor processes .


the famous thomson problem  is to find , for an arbitrary natural number , a configuration of classical electrons on the unit sphere , , that minimizes the coulomb energy .there is no general theoretical solution to this problem .the apparent obstacle is strong evidence suggesting that the ground state for the coulomb potential in two dimensions has a hexagonal structure .the sphere , however , can not be tiled exclusively with hexagons . if one places points numbered on the sphere , and divides the sphere into voronoi cells centered at each of the points , then the euler characteristic of the sphere ensures that where is the number of sides of the voronoi cell associated with the point .one can see examples of these non - hexagonal voronoi cells , which are commonly referred to as defects or scars , in figure [ fig : sample ] .finding the energy minimizing configuration will likely require finding the right defect structure .many numerical techniques that aim to identify minimal energy configurations rely on gradient information and tend to find configurations that are stable , but not minimal .these stable configurations also have a local hexagonal structure , but differ from one another ( and presumably the energy minimizing configuration ) largely in location and structure of defects .   the configuration with the lowest observed energy for and points generated by wales , mckay and altschuler  .each point is depicted as a dot on the surface of the sphere surrounded by its voronoi cell , which we computed with qhull  . in the image on the left the five sided cells are gray ( red in the online version ) ,the six sided cells are light gray ( green in the online version ) and the seven sided cells are dark gray ( blue in the online version ) .the image on the right shows the same configuration in the same orientation , but with the voronoi cells colored by point energy .( in the online version blue indicates the lowest point energies , green average point energies and red , the highest . ) note that the fluctuations in point energy extend out from the defects into the surrounding `` hexagonal sea''.,title="fig:",height=264 ]  the configuration with the lowest observed energy for and points generated by wales , mckay and altschuler  .each point is depicted as a dot on the surface of the sphere surrounded by its voronoi cell , which we computed with qhull  . in the image on the leftthe five sided cells are gray ( red in the online version ) , the six sided cells are light gray ( green in the online version ) and the seven sided cells are dark gray ( blue in the online version ) .the image on the right shows the same configuration in the same orientation , but with the voronoi cells colored by point energy .( in the online version blue indicates the lowest point energies , green average point energies and red , the highest . ) note that the fluctuations in point energy extend out from the defects into the surrounding `` hexagonal sea''.,title="fig:",height=264 ]  a natural question to ask is : how much does the energy change as the structure and location of defects changes ?because stable configurations differ from the minimal configuration in location and structure of defects , a related question is : how much does the average energy of stable configurations differ from the true minimal energy ?we answer this empirically by developing a large library of stable configurations and comparing the resulting average energy of the stable configurations with the lowest observed energy .minimal energy is often approximated in an asymptotic expansion , in , and we compare the difference between the average and lowest observed energy with the terms in these asymptotic expansions .that is , we empirically identify the terms in the asymptotic expansion that approximate the lowest observed energy , but not the average energy .we believe that these terms likely reflect characteristics of defects .this work has value in several ways .first , because the energy of any configuration of points on the sphere is an upper bound for the minimal energy , these results provide a lower bound for the difference between the average energy of stable configurations and the minimal energy .second , there are methods that have a controllable error bound for quickly approximating the pairwise energy , most notably the fast multipole method  . forsuch approximations the results in this paper will help select the error bound necessary to distinguish stable configurations from minimal configurations .finally , this work suggests which terms in the asymptotic expansion will require an understanding of defect structure .the rest of the paper is organized as follows : section [ sec : background ] review some of the relevant work .section [ sec : genconf ] describes our method for generating stable configurations , and reports properties of these stable configurations .section [ sec : minest ] compares theory and conjecture with minimal observed energy and reports the observed asymptotic differences between the average energy of stable configurations and minimal observed energy .additionally , we examine and extend some conjectures regarding the second order term for the thomson problem . in section [ sec : conclusion ] we summarize our results .some of the earliest computational work on the thomson problem was done by erber and hockney  where they rely on optimization techniques to search for minimal energy configurations .rakhmanov , saff and zhou  presented a comprehensive search for the minimal energies for up to for the logarithmic as well as coulomb energies .morris , deavon and ho  used a genetic algorithm in an effort to avoid becoming trapped in stable non - minimal configurations . an important effort to constructively generate candidate minimal energy configurations came from altschuler , williams , ratner , tipton , stong , dowla and wooten  , where the authors of that paper identified configurations with twelve point defects and high symmetry .these configurations were later shown not to be minimal by prez - garrido , dodgson , moore , ortuno and diaz - sanchez  and prez - garrido , dodgson , moore  .these authors found that , as increased , the defects were not point defects , but had considerable structure such as those in figure [ fig : sample ] .efforts to understand and characterize this structure , as well as find minimal energy configurations , include the work of wales and ulker  and wales , mckay and altschuler  .the results of the experiments described in these two publications are collected in the cambridge cluster database  , and provide , to our knowledge , the lowest observed energies for the thomson problem .bowick , cacciuto , nelson and travesset  use a continuum elasticity model to describe the interaction of defects . in these works the empirical evidence is that configurations with low energy consist of a `` hexagonal sea '' with complex defects at the vertices of an icosahedron inscribed in .theoretical examinations of the thomson problem provide valuable insights and language for the problem , and we review some of the relevant theory here .let denote a set of distinct points in .we consider the following discrete energy of  where is the function given by and where is the euclidean norm inherited from .note that many papers on this topic report an energy where the second sum is over leading to a factor of two difference in our values for energy .the functions , , are the riesz potentials , which are a natural generalization of the coulomb potential . the questions in which we are interested apply to riesz potentials in general , and we present results for the riesz potentials corresponding to , , , and .we denote the point ( -)energy of the point in by for any compact set of hausdorff dimension , the lower semi - continuity of ensures that there is at least one configuration contained in , which we denote , that satisfies that is to say , there is at least one energy - minimizing configuration , , and the minimal -point -energy is denoted . in this setting onecan search for an expansion of the minimal energy as a function of of the form in certain cases , e.g. and , this expansion will also include logarithmic terms .  in the general case where is any dimensional compact set and , plya and szegestablish the first order term  by connecting the asymptotic behavior of the discrete minimal energy with a continuum problem . specifically , let denote the positive borel measures supported on , and denote the borel probability measures supported on .one may interpret as a continuous charge distribution and consider the energy functional defined for any , by analogous to the discrete point energy , , the potential due to at a point , is there is a unique energy - minimizing measure so that ( cf . also gtz  provides a proof of a key step without using standard fourier techniques . )further , for all with the possible exception of a set that supports no measures of finite energy ( cf .* theorem 2.4 ) ) .roughly speaking equation  asserts that the potential is constant in regions where there is charge .the essence of the proof is that , if this were not the case , energy could be decreased by moving charge from regions of high potential to regions of low potential .the celebrated transfinite diameter result of plya and szeg relates the continuous and discrete problems as follows ( also cf . ) : for any continuous function and any sequence of energy - minimizing configurations , and for this range of the discrete minimal energy configurations are converging in the weak - star topology of measures to .the minimal energy grows as , where the coefficient is given by .the proof of these results indicates that the first order approximation of the minimal energy is determined by the global distribution of points within energy minimizing configurations .kuijlaars and saff have shown  that the second order term on the sphere in the expansion  grows as and the , still to be proven , coefficient is conjectured to depend on the presumed local hexagonal structure .  if , then for all , ( cf .8) ) and other techniques are required to estimate growth in minimal energy .hardin and saff  and borodachov , hardin and saff  show that when has certain smoothness properties and where is the dimensional hausdorff measure restricted to , is a constant that depends only on and and not the underlying set , and is the closed unit ball in .these results demonstrate that for the asymptotic distribution of points in energy - minimizing configurations is uniform .furthermore , the minimal -point energy grows at a rate exceeding and is determined largely by the local structure of the energy minimizing configurations .indeed for the case , numerical evidence supports the conjecture that is given by a hexagonal zeta function evaluated at , i.e. the sum of the reciprocal non - zero distances in the hexagonal lattice raised to the power .brauchart , hardin and saff present a summary of theory and conjecture regarding minimal energy configurations on the sphere  .to generate candidate configurations we begin with a random , well - separated , initial configuration of points on and alternate between the polak - ribire variant of conjugate gradient ( cf . ) with a line minimization of the energy , and an exact newton s method to find a root of the gradient . to solve the linear system arising in newton s method we use lapack  .we use a direct evaluation of the energy sum , given in equation  omitting obvious duplicate calculations , which involves terms , the smallest of which is , while can grow into the hundreds of millions for some values of and considered . to control the numerical round - off errorassociated with adding two numbers whose ratio is far from unity ( cf . for relevant work on this problem ) we logarithmically bin our summands . by only adding summands in the same bin , we bound the ratio of any two intermediate summands to be added .the final sum is computed by iterating over our bins in increasing magnitude and summing their contents .  for we ran thousands of trials . for we ran tens to hundreds of trials .we report lowest observed energies on the sphere only for those where the cambridge cluster database provides a configuration with which we can initialize our solver .the above optimization process leads to a candidate configuration , which we assume is close enough to a true stable configuration so that the linear approximation about for the gradient is reasonable . here is the gradient of the energy with respect to the free parameters that define and is the hessian represented in the same coordinates .were the hessian invertible this would lead to the bound where is the smallest eigenvalue of the hessian , is the unnormalized two - norm of the parameters defining the argument , and is the associated operator two - norm .our choice of coordinates leads to three degrees of freedom corresponding to rigid motions of the sphere and so the smallest three eigenvalues of the hessian are zero .we assume a rotation and reflection of so that the difference between and and does not reflect these rigid motions .we let denote the fourth lowest eigenvalue , then we have the bound we desire that our reasoning is that the free parameters are the polar and azimuthal angles , and , on the unit sphere , changes in position are always bounded from above by changes in angle .the above bound will ensure that no point in is further from its corresponding point in the true stable state by more than the arbitrary bound of one ten - thousandth of the minimum separation in .this is ensured if where , again , we used lapack to compute .we reiterate that these estimates hinge on the assumption that the gradient at the true stable state is well approximated by a linear expansion of the gradient about the observed state .we keep candidate configurations if equation  holds or if the configuration possesses the lowest observed energy .  note that equation  is quite stringent . as increases, the minimum pairwise separation between points goes as .in addition we have bounded from above the infinity - norm with the unnormalized two - norm .such a bound is tight only when all the components but one are zero .this condition was relaxed for , where we simply required that all but lowest three eigenvalues be positive .   in figure[ fig : frac_avg ] one can see the average fraction of points that have six - sided voronoi cells . for each and ,these data are obtained by computing this fraction per configuration , and then averaging over all the observed configurations and weighting by the number of times the configuration occurred .this is the same averaging method we use when computing the average energy of stable configurations .as one can see this average fraction is better than percent for , supporting the claim that stable configurations are largely hexagonal .  as a point of comparison ,we ve also computed this fraction for the configurations that have the lowest observed energy .this is shown in figure [ fig : frac_min ] .one important feature of this plot is that the configurations with the lowest observed energy have far more non - six - sided voronoi cells than the minimum allowed if no voronoi cell has fewer than five sides .if one further assumes that no voronoi cell has more than seven sides , then the number of voronoi cells with other than six sides must be even .this corroborates previous observation that as increases , the defects cease to be single points and develop structure .this plot shows the fraction of points that have six - sided voronoi cells for and for the values of in which we are interested .the inset plot provides more detail for .in addition we ve plotted the upper bound for this fraction assuming that no voronoi cell has fewer than five sides .each data point is averaged over all the configurations , weighted by number of occurrences , for the specified and .,height=340 ]   as in figure [ fig : frac_avg ] we ve plotted the fraction of points that have six - sided voronoi cells . hereeach data point corresponds to the configuration with the lowest observed energy . in the outer plot the solid line is the upper bound on this fraction , while the dashed line indicates what this fraction would be if there were defects each consisting of a three points with five , seven and five sided voronoi cells respectively . in the insetone sees lines corresponding to non six - sided voronoi cells.,height=340 ]in this section we compare theory and conjecture for the minimal -point energy with the lowest observed -point energy . in the case we extend a conjecture for the second order term on to certain smooth manifolds .we report the asymptotics of the difference between the average and minimal observed energies and compare this difference with terms in the asymptotic expansion .like all computational works of this type , we have no assurances that the lowest available energies are indeed minimal .systematic errors of this type would cause us to underestimate the difference between the average and the minimal energies . consequently our results that indicate that a term in the asymptotic expansion does not describe both the average and minimal energy should be trusted more than results indicate that a term does describe both the average and minimal energies .we shall use the following notation : is the lowest _ observed _ minimal -point -energy on a set . is the difference between the minimal -point -energy on and an -term asymptotic expansion of the minimal -energy , while is the difference between the lowest _ observed _ energy and the -term expansion .this is the thomson problem , and the leading order term in the asymptotic expansion of the minimal energy follows from the transfinite diameter result in equation , i.e. for a set of dimension it is . for the spherea simple calculation shows that .we now review an existing conjecture for the second order term on , and show how it may be generalized for compact -manifold .a trivial representation of the first order term and the correction for a set is we shall consider the case that is absolutely continuous with respect to , the support of is all of , and equation  holds for the entire support of , that is for all .these assumptions are satisfied for .the potential is linear in and so , with our assumptions , we may write equation  as   the above equation is exact regardless of where on we choose to evaluate the potential .however , choosing to evaluate the potential at the points that form a minimal -point configuration suggests one way to express the correction : the point energy for should be corrected by subtracting the potential at due to times the equilibrium measure and adding the energy due to the presence of the other discrete points . in broader termsthe point at sees other discrete points , not a smoothed out average density .  for the point, the correction given by equation  may be written as two terms , which we refer to as `` near '' and `` far '' contributions .  this decomposition is motivated by the reasoning presented by kuijlaars and saff ( * ? ? ?* section 2 ) , namely that the second order correction for is determined by the local structure .where kuijlaars and saff use a cutoff at radius , we use an exponential damping that allows use of the poisson summation formula and ewald type arguments for the case .we fix small enough so that changes on a scale much larger than , and we consider large enough so that the nearest neighbor distance is much smaller than .then for most we can expect a local hexagonal structure around and so we consider the following estimate for the near term in equation :  here is the hexagonal lattice of unit spacing , is the hexagonal lattice where the generating vectors have been scaled by , and is the area of the fundamental cell of the scaled lattice .finally , denotes integration with respect to area .the essential statement of the approximation in equation  is that , for most points in a configuration with low energy , the energy due to neighboring points is well approximated by the energy due to the neighboring points in an appropriately scaled hexagonal lattice , and that the density represented by equilibrium measure changes little on the scale of nearest neighbor separation .this assumption is qualitatively supported by figure [ fig : sample ] where most points are surrounded by a local hexagonal structure .we compute the sum over a lattice that is scaled by , which is intended to reflect the local point density of the energy minimizing configuration near the point .for the case , is independent of . to generalize to an arbitrary 2-manifoldone may estimate as follows : let be the nearest - neighbor spacing .assume that for large , hence small , the voronoi cells within are all hexagonal and of the same size .this gives here indicates the number of points in the following set . is the area of a hexagon of inner radius , which is .the second estimate follows from the weak - star convergence of the discrete minimal energy points to the equilibrium measure and the assumption that is -almost clopen. then , for sufficiently high ,   dividing by gives , for sufficiently large as decreases to zero , the right hand side tends toward the radon - nikodm derivative of with respect to and we have that the nearest neighbor spacing , and the appropriate scaling for the lattice at , is given by with some substitutions , the limit as grows to infinity of may be expressed as we evaluate this limit ( omitting the factor ) in the appendix as and denote its value as .discarding the far piece in equation , assuming a local hexagonal structure , and replacing the outer sum with an integral on the right hand side of equation  gives the following conjecture .[ conj : simple ] let be a compact -manifold where is absolutely continuous with respect to , where the support of is all of , and where for all .then where and where is the unit hexagonal lattice .conjecture [ conj : simple ] follows from a number of simplifying and possibly unnecessary assumptions .a broader conjecture that is closer in form to conjecture 2 given by kuijlaars and saff  is  [ conj : ks - ext ] let be a compact -manifold , , and absolutely continuous with respect to , then where   here is the analytic extension of the riemann zeta function and is the dirichlet l - function given by conjectures [ conj : simple ] and [ conj : ks - ext ] both predict for the coefficient of the term on , and are in good agreement with energies on the sphere .we now consider two additional numerical tests of these conjectures . in the first testwe shall look at the torus using a modest data set of low energy configurations .however , we also need an approximation of , and we turn to the work of brauchart , hardin and saff on sets of revolution  . in that workthe authors begin with the fact that for sets of revolution , the equilibrium measure must be invariant under revolution .they develop a lower dimensional minimization problem on the set , which when rotated , gives .while the theory does not address the case , we use their theory as a recipe to approximate numerically and present the results in table [ table : energycompare ] .we denote the torus of major radius and minor radius by .landkof  provides the following formula for the energy of the equilibrium measure on the torus : ,\ ] ] where and where and are legendre functions of the first and second kind .we use the gnu scientific library  to evaluate the legendre functions in the above sum . in table [ table : energycompare ] we see good agreement between the energies that result from extending the work in  to and the energies given by . because the equilibrium measure is the unique measure that minimizes the energy ,we conclude that the measure generated by applying the theory in  to the torus for generates a reasonable approximation of the equilibrium measure on the torus .further , our numerical experiments show that the support of the equilibrium measure is . in figure[ fig : secondterm ] we plot the difference between the observed minimal energy and the first order term , i.e. .we also plot the conjectured value for the term using our numerical approximation of .the agreement suggests that conjectures [ conj : simple ] and [ conj : ks - ext ] appears to hold for the torus ..[table : energycompare ] a comparison of the energy of the equilibrium energy computed in two ways on three different tori .the first method uses the work of brauchart , hardin and saff  as a recipe for approximating the equilibrium measure .the second method uses equation . is the dimension of the discretized problem arising from  . [cols="^,^,^,^,^",options="header " , ]   a plot of for , , and . for each manifold we ve overlaid the best fit of the form .,height=340 ]conjectures [ conj : simple ] and [ conj : ks - ext ] are expressed in terms of an integral over the equilibrium measure and a coefficient derived from a sum over a hexagonal lattice .the formulation of these conjectures does not make any assumption about the location or structure of the defects .this would imply that , if stable configurations differ from the minimal configuration only in the structure and location of defects , then conjectures [ conj : simple ] and [ conj : ks - ext ] should approximate the average stable energy as well .this is our second test of the conjectures . in the top of figure[ fig : variation_s1 ] we see that the difference between the average energy of stable configurations and the lowest observed energy is bounded by three ten - thousandths of the conjectured term . in the bottom of figure [ fig : variation_s1 ]we see that this difference between the average and minimal energies is substantially larger when compared to the empirically obtained linear term ( ) for the minimal energy .indeed for our data at the average and minimal energy differ by 30% of the linear term .the conclusion is that the first and second terms given by the transfinite diameter and the conjectured term will predict energies of stable and minimal configurations well , but the empirically obtained linear third term reflects properties of the minimal configuration that are absent in the stable configurations .we assume that these properties are the location and structure of the defects .the top plot shows the difference between the average energy of stable configurations and the minimal observed energy divided by the conjectured term . in both plots the axis is broken to effectively display the data point at .the bottom plot shows the same energy difference divided by the empirically obtained linear third term .we have rescaled the right section of the lower plot and included a single line , plotted in both scales as reference .the error bars in this plot , and following plots of this type , are the standard error of the mean of the energy of the stable configurations.,height=340 ]   the problem of minimizing the energy is equivalent to the problem of maximizing the product of pairwise distances of points , and has received considerable attention from the mathematics community .the seventh of smale s eighteen problems for the twenty first century  is to develop an algorithm that will generate rapidly a configuration , , that satisfies for some constant that does not depend on .one challenge in solving this problem is estimating to at least .rakhmanov , saff and zhou made progress in this direction by bounding the linear term ( * ? ? ?* theorems 3.1 and 3.2 ) by defining as and showing in the same paper , those authors conjecture that   we fit to our minimal energies and find a best fit for , and .the value of we obtain is in reasonable agreement with the value of obtained empirically by brauchart , hardin and saff  , and in stronger agreement with the value of given in conjecture 4  .this is the observed minimal logarithmic energy minus a five term asymptotic expansion for the minimal energy .we see evidence of a term that decreases with .,height=340 ]  we fit over a range of because the data with which we have to work has behavior for that is not captured in equation .we plot the difference of the observed lowest energy and the five term asymptotic expansion in figure [ fig : s0 ] .it is worth noting that , for , the magnitude of this five term residual is less than while the value of is about million .the top plot shows the difference of the average and lowest observed energies divided by the linear term in an asymptotic expansion .the bottom plots shows the same difference divided by the logarithmic term.,height=340 ]  in figure [ fig : variation_s0 ] we compare the difference between the average and minimal observed energies with the terms in the asymptotic expansion . for the data available ,this energy difference is bounded by about one percent of the empirically obtained linear term , as is shown in the top plot .that is , the difference between the average energy of the stable configurations and the minimal observed energy is growing roughly as .it is worth comparing this with figure 2 of  where the energy of constructively generated spiral point configurations differs from an estimate of the minimal energy by roughly .  the qualitative interpretation that the data in the upper plot in figure [ fig : variation_s0 ] are bounded while the data in the lower plot are growing implies that the first three terms in the asymptotic expansion describe the energy of stable configurations as well as the energy of minimal configurations , while the logarithmic term in the asymptotic expansion will reflect properties of the minimal configurations that are absent in most stable configurations .this implies that solving smale s seventh problem will require some understanding of the defects .the top plot shows the residual after the three term expansion given by conjecture 5 from  .the bottom plot shows the residual with three additional terms ., height=340 ]  the riesz kernel is not locally integrable on a -manifold and the potential theoretic arguments can not provide a first order term .initial results for the leading order term on the sphere are given by kuijlaars and saff ( * ? ? ?* theorem 3 ) .these results were generalized to a class of sets that include manifolds by hardin and saff ( * ? ? ?* theorem 2.4 ) . combining these results with conjecture 5 from , one has an asymptotic expansion of the form the conjectured value for is   we fit the available data to and find that .however , the difference between the observed minimal energies and the best fit , shown in the top of figure [ fig : s2 ] , has considerable structure .one hypothesis is that the form of the expression used for the fit is not correct . making the arbitrary decision to include the same sequence of terms found in the expansion for the logarithmic energy , we fit to our data , and when we fit the above , we found and .the residuals associated with the best fit of this augmented asymptotic expansion is shown in the lower plot of figure [ fig : s2 ] .these two plots are the ratios of the difference between the average energy of stable configurations and the minimal observed energy to two terms in the asymptotic expansion for the energy .the top plot shows this difference compared to the empirically obtained and the bottom plot compares this difference with the empirically obtained term.,height=340 ]  figure [ fig : variation_s2 ] shows the growth of the difference between the average energy of stable configurations with the minimal observed energy divided by the term in the top plot and an empirically obtained term in the bottom plot .if one accepts that the data in the top plot is bounded , and the data in the bottom plot is growing , then one would conclude that the first two terms in the asymptotic expansion for the energy describe the energy of stable configurations as well as the minimal energy to about three parts in one thousand , while the next term , possibly an term , would reflect properties of the minimal configurations absent in most stable configurations .the difference between the observed minimal energy and the two term expansion for the energy.,height=340 ]  the riesz kernel , like , is not locally integrable on -manifolds .early progress toward the leading order term for the asymptotic expansion of minimal -point energy on the sphere ( * ? ? ?* theorem 2 ) shows that , if the leading order term exists for any , the leading order term has the form .kuijlaars and saff further conjecture that where is again the hexagonal lattice and is the associated zeta function the sum of the reciprocals of the non - zero distances in raised to the argument .the existence of the limit in , and hence the first order term , was established for a broad class of sets by hardin and saff  and strengthened by borodachov , hardin and saff  , although the value of the limit has still not been proven .the natural assumption of a local hexagonal structure is implicit in the conjecture as is the hexagonal lattice .we compute this leading term , via the factorization presented  to get a value of .the second order term is conjectured ( * ? ? ?* conjecture 3 ) to be where is given as the analytic extension , in , of to the case .following ( * ? ? ?* equation 10 ) we compute the coefficient as .fitting the expression with fixed at the value given in , to our data for gives a value of .the addition of terms of the form does not substantially change the value for obtained through such a fitting procedure . if we fit expression  to the data and let vary we obtain and .the difference between the observed lowest energy and the fit , shown in figure [ fig : s3 ] shows considerable structure , suggesting that either the form to which we fit is not correct , or that the energies with which are working are not minimal .we plot the difference between the average and minimal energies in figure [ fig : variation_s3 ] .the upper plot suggests that this difference is small compared to the leading order term .the lower plot compares this difference to the conjectured second order term .this difference is about percent of the conjectured second order term at .however , the difference between the empirically obtained coefficient for the second order term and the conjectured coefficient is percent of the conjectured second order term .if our measurement of the second order coefficient differs from the conjectured value because our lowest observed energies are not the minimal energies , then the minimal energies differ from the lowest observed energies by several times the difference between the average and minimal energies .   herewe present , in the top plot , the difference between the average of the energies of stable configurations and the minimal observed energy divided by the leading order ( ) term estimate .the bottom plot shows the difference between the highest and lowest observed energies divided by the difference between the observed and conjectured second order term.,height=340 ]we ve used numerically generated candidates for -energy minimizing configurations to assess conjectures for higher order terms in asymptotic expansions for the minimal -energy .in addition we ve developed a large library of stable configurations and compared the average of the energies of the stable configurations with the energies of the candidate minimal configurations to approximate a lower bound on the difference between the average and minimal energy .for we find that existing conjectures for the second order term on the sphere appear appear to hold when extended to the torus , and that the third term appears to be linear . for the sphere a straightforward fit suggests a value of as the coefficient of this linear term . for conjectured forms for the asymptotic expansion gave rise to an expression that agreed , for , with our observed minimal energies to one part in thirty million . using a fit for the linear termgives a value of , while the conjectured value is .for the conjectured form of the asymptotic expansion left considerable structure , suggesting that either the form of the fit was wrong or that the energies with which we had to work were not minimal .two fits , assuming different forms of the asymptotic expansion , gave values for the coefficient of the conjectured second order term of and .the conjectured value is for , the conjectured coefficient of the first order term is , while fitting our data gives .the second order term is conjectured to be .fitting our data suggests a coefficient of .   for the difference between the average and lowest observed energy was small compared to the term , and appeared to be growing compared to an empirically obtained linear term . for the case this difference appeared to be bounded when compared to the linear term , but growing when compared to the term .this suggests that an arbitrary sequence of stable configurations will not be a solution to smale s seventh problem . for difference was small compared to the term , but growing compared to .for this difference was small compared to the leading order term .because the stable configurations differ from minimal configurations in the location and structure of defects , we infer that the energy difference between stable states and minimal configurations is the energy scale at which defects play a role . and that theoretical models for the terms identified above will require an understanding of the role of defects .we want to compute where indicates integration with respect to area . for conveniencewe let we have we interpret as the limit as of the function . applying the poisson summation formulagives for some we compute as both and are rotationally symmetric , so we can choose and integrate in polar coordinates this change to polar coordinates leads to a convenient cancellation when  to get recognizing the right most integral as the laplace transform of the bessel function gives note also that and that which allows us to collect terms and write the quantity we would like to compute as the limit as of  the limit is well defined for each term .for the first term we have by monotone convergence .for the second term we have  by dominated convergence . by direct evaluation , the third and fourth terms are and we are left with we shall choose to be the hexagonal lattice , that is the lattice generated by the vectors and . in this case is generated by the vectors and .finally .31ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] * * ,  ( ) link:\doibase 10.1103/physrevb.79.224115 [ * * ( ) , 10.1103/physrevb.79.224115 ] link:\doibase 10.1145/235815.235821 [ * * ,  ( ) ] link:\doibase  10.1016/0021 - 9991(87)90140 - 9 [ * * , ( ) ] * * ,  ( ) * * ,  ( ) in  _ _ , , vol .( , ) pp .link:\doibase  10.1103/physrevb.53.r1740 [ * * ,  ( ) ] link:\doibase  10.1103/physrevlett.78.2681 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.79.1417 [ * * , ( ) ] link:\doibase  10.1103/physrevb.56.3640 [ * * ,  ( ) ] link:\doibase  10.1103/physrevb.74.212101 [ * * ( ) , 10.1103/physrevb.74.212101 ] `` , ''  ( ) `` , ''  ( ) link:\doibase  10.1103/physrevb.73.024115 [ * * ( ) , 10.1103/physrevb.73.024115 ] * * ,  ( ) _ _ ( , , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , , ) * * ,  ( ) * * ,  ( ) in link:\doibase 10.1090/conm/578/11483 [ _ _ ] , , vol . , , ( , , ) pp . _ _ , ed .( , , ) link:\doibase 10.1109/5.236187 [ * * ,  ( ) ] link:\doibase 10.1137/0914050 [ * * ,  ( ) ] in link:\doibase  10.1090/conm/481/09396 [ _ _ ] , , vol .( , , ) pp . `` ''  ( ) * * ,  ( ) ,  * * ,  ( )

Abstract: configurations of points on the two - sphere that are stable with respect to the riesz -energy have a structure that is largely hexagonal . these stable configurations differ from the configurations with the lowest reported -point -energy in the location and structure of defects within this hexagonal structure . these differences in energy between the stable and minimal configuration suggest that energy scale at which defects play a role . this work uses numerical experiments to report this difference as a function of , allowing us to infer the energy scale at which defects play a role . this work is presented in the context of established estimates for the minimal -point energy , and in particular we identify terms in these estimates that likely reflect defect structure .