studies of the properties and response of matter under extreme conditions rely increasingly on lasers to induce dynamic loading by ablation . compared with the canonical technique of impact - induced shocks ,laser ablation is more convenient for studying phenomena on shorter time scales ( nanoseconds rather than microseconds ) , for investigating the properties of single crystals and the basic microstructural processes , and for developing new diagnostics .however , laser ablation has potential disadvantages including uncertainties in loading history , spatial variations in irradiance from speckles or the overall intensity envelope , and the possibility of heating the sample under investigation by hot electrons or bremsstrahlung x - rays .a thorough understanding of the processes of ablative loading is necessary to assure that the response of the sample can be distinguished from phenomena associated with the ablation process .  in many experiments ,dynamic loading can be induced by ablation of the sample material itself .previous studies have demonstrated that spatial variations and preheat were negligible , for ablatively - generated shocks from to 1tpa in elements and compounds , and for ramp compression . in other experiments , it is highly desirable to use the laser energy to ablate a well - controlled layer of a different material , in contact with the sample .advantages of a separate ablator include the ability to restrict multi - electron - volt ablation temperatures to elements of low atomic number irrespective of the sample material , reducing the risk of generating hard x - rays ; smoothing of small - scale spatial variations by beam dispersion , radiation transport , and mechanical equilibration within the ablation plasma ; and the ability to use a single ablator of known plasma properties for experiments on less well - understood sample materials .a current interest in dynamic loading experiments is the use of x - ray diffraction to study the sample material in the compressed state . with ramp compression, the material of highest compression is closest to the ablation surface , so it is desirable to pass the x - rays in and out through the ablator and the ablation plume .it is thus particularly desirable to restrict the ablator to elements of low atomic number , dissimilar to the sample .when the sample material is ablated directly , heat conduction from the plasma and re - condensation of ablated material at the end of the laser pulse can lead to material with a locally different microstructure , an added complication when interpreting the microstructure of recovered samples .a dissimilar ablator acts as a palliative in this case . by appropriate choice of material ,an ablator may also increase the efficiency of conversion of laser energy to pressure in the sample .an ablator layer can however complicate dynamic loading experiments .the time at which the load is first applied to the sample depends on the ablator thickness and the laser irradiance .the mechanical impedance mismatch between the ablator and the sample causes wave interactions , leading to multiple waves in the sample which must be discriminated from the response of the sample . if the composition and properties of the ablator material are not adequately known , it introduces additional uncertainty in the experiments .plastics are attractive ablator materials as they are stable to handle and readily available with compositions of low atomic number , such as polystyrene ( c) and polyparaxylylene - n ( parylene - n ) ( c) , and polyethene ( c) . deposition and coating techniques exist which can produce layers of uniform and well - characterized thickness .polystyrene and parylene - n ablators have been used as a coating for glass spheres in inertial confinement fusion experiments .previous studies have investigated the properties of the ablation plasma including two dimensional effects in planar targets , and have investigated the generation and mitigation of preheat in the sample .significant discrepancies have been identified between pressures inferred from experiments and predicted in radiation hydrodynamics simulations .parylene layers have been used in material dynamics experiments at lower irradiances , again with some discrepancy between simulation and experiment .  herewe present studies of the behavior of plastic ablators under conditions relevant to shock physics and material dynamics experiments , including sensitivities to uncertainties in the composition or equation of state of the ablator , and wave interactions caused by the impedance mismatch with the sample .the effect of wave interactions is presented in a form convenient for the design of laser - driven shock experiments .the response of a plastic layer to intense laser irradiance , and the effect of the impedance mismatch between the plastic and a denser sample material behind , were investigated using numerical simulations .laser - matter interaction is non - linear and time - dependent , so spatially - resolved radiation hydrodynamics is needed to integrate the coupled radiation transport and continuum dynamics equations forward in time . in the regime of interest , laser ablation and dynamic loading can be simulated accurately by assuming three temperature hydrodynamics ( ions , electrons , and radiation ) , including thermal conduction and radiation diffusion , and calculating the absorption of the laser energy in the expanding plasma cloud through the electrical conductivity . thermal conduction and radiation diffusionare necessary only near the ablated surface of the plastic : the bulk of the ablator and the sample behind are affected only by the hydrodynamics .material dynamics experiments are designed to apply a simple loading history to the sample : ideally close to one - dimensional in space , i.e. with a time - dependence that is independent of position over a region of the sample surface .experiments are usually designed so that edge effects , two- or three - dimensional dependencies in the loading history , do not affect some portion of the sample for long enough for the one - dimensional response to be measured .we therefore use one - dimensional radiation hydrodynamics , and estimate the size of the region affected by the edges .radiation hydrodynamics is the simultaneous solution of the continuum dynamics , heat conduction , and radiation transport equations . using the lagrangian frame of reference , in which time derivatives are expressed along characteristics moving locally with the material , the equations of continuum dynamics are where is time , the mass density , the particle velocity , the pressure , and the specific internal energy .these equations are closed by a mechanical equation of state ( eos ) .each equation may also have a source term ; here , heat conduction and radiation transport provide source terms in the velocity and energy equations .heat conduction and radiation transport involve temperatures for the ions , electrons , and radiation field . in the regime of interest , laser and thermal radiation couples principally to the electrons , which equilibrate with the ions according to a compression- and temperature - dependent rate .a thermal eos relates to ; in practice we used an eos of the form , using as one of the state parameters and calculating and when needed in the hydrodynamics .the equation of heat conduction is ,\ ] ] where is the thermal conductivity .radiation transport was calculated for an equilibrium radiation distribution ( represented by a single radiation temperature field ) the equation of single - temperature radiation diffusion is where , and are frequency - averaged absorption and scattering coefficients , is the integrated power in the planck distribution at , the radiation flux  is the stefan - boltzmann constant , and is the speed of light . the absorption coefficient was approximated as the rosseland mean opacity , and the emission coefficient as the planck mean opacity .the transport and deposition of incident laser energy was calculated as the attenuation of a free - propagating flux , using the electrical conductivity to calculate the attenuation , with energy deposited as heating of the electrons .  in our context ,radiation hydrodynamics is an initial value problem , where the material fields and are specified over a region at some time , time - dependent boundary conditions are specified for the continuum or and radiation flux , and the continuum and radiation fields are integrated for .simulations were performed using the hyades radiation hydrocode , version 01.05.11 .this hydrocode used a one - dimensional ( 1d ) lagrangian finite - difference discretization of the material and leapfrog time integration , with shock waves stabilized using artificial viscosity .the radiation flux limiter was set to 0.03 of the free stream value - a common choice for simulations of this type .the ablation properties of plastic coatings were investigated by radiation hydrodynamics simulations as described above , to predict the relationship between irradiance and pressure , ablation rates , and the sensitivity to composition of the plastic .the properties of the plastic were represented through models for its pressure - volume - energy equation of state ( eos ) , its conductivities , and its opacity .wide - ranging eos were taken from the sesame database .opacities for radiation diffusion were also taken from sesame .conductivities for laser deposition and heat conduction were calculated using the thomas - fermi ionization model ; this was found previously to be accurate for direct drive shock simulations on samples of a wide range of atomic numbers .a significant difficulty with parylene - n is the only eos model available was constructed by density scaling from an eos for parylene - c , which was fitted to shock hugoniot data at 300 - 600gpa with an uncertainty %  . in parylene - c, one h is substituted by cl , and density scaling will not necessarily capture the different chemical behavior . to investigate the uncertainty in pressures predicted in ablative loading experiments , the sensitivity to eos and opacity was investigated by performing simulations for parylene - c and parylene - d , in which an additional h atom is substituted by cl , and also for polystyrene , which has the same composition as parylene - n though a different molecular structure .the molecular structure should become unimportant once the plastic has been converted to plasma .the models for parylene - d ( sesame tables 7770 and 17770 ) were developed earlier and have been used more commonly in radiation hydrodynamics simulations , so these models were used for the studies of general ablation behavior described next .  to allow a fine spatial resolution of the surface to be ablated , without requiring the same resolution in the bulk of the plastic where it remains solid ,the material was discretized with geometrically - expanding cells .the expansion factor between successive cells was around 1 - 5% : small enough to avoid inaccuracy in the hydrodynamic equations .the smallest cell , at the surface where ablation starts , was chosen to have an initial size in the few nanometer range . numerical convergence with respect to spatial resolutionwas tested by performing simulations with 200 or 300 zones , with an expansion ratio of 2.5% or 2.0% respectively , giving zone widths of around 9.0 or 2.6 nm at the ablation surface .the ablation pressure was around 5% higher in the finer - mesh simulations , indicating an adequate level of mesh convergence .the finer mesh was used for all subsequent simulations .  for the ablation rate and pressure study ,the plastic was chosen to be 50 m thick , and the laser pulse was 10ns long with constant irradiance . to avoid numerical oscillations ,the irradiance was taken to rise from and fall to zero over 0.1ns , which is typical of the laser systems used for the material dynamics experiments of interest here .the critical surface of the plasma was predicted to remain within a few micrometers of the initial position of the ablator , so the plasma plume should be essentially 1d .release and cooling at late times would be faster than predicted using 1d simulations , because of lateral expansion .  in the simulations , the surface of the plastic was heated to plasma conditions rapidly after the start of the laser pulse .the plasma then absorbed the incident laser energy , forming a nearly steady state plume close to the plastic .the plasma pressure and reaction to plasma flow induced an elevated pressure in the remaining condensed - phase plastic , driving a shock wave .as was found previously , a laser pulse of constant irradiance did not induce a constant driving pressure in the condensed region .instead , there was an initial spike of pressure tens of percent higher , decaying rapidly to a more constant sustained pressure .this sustained pressure typically decreased by a few percent during the laser pulse .the eos used exhibited simple , concave behavior , so the decaying pressure in the spike and the plateau was able to catch up and erode the higher pressure states ahead , reducing the shock pressure monotonically as it propagated through the material .after the end of the pulse , the ablation pressure fell rapidly , and a rarefaction wave propagated into the shocked material , releasing the pressure .for the thickness and pulse duration chosen , and over the irradiance range considered , the shock reached the rear surface of the plastic before the rarefaction caught up , so a second rarefaction wave from the rear surface also propagated into the material .[ fig : ch1f ] )   applied for 10ns .the undisturbed material extends from 0 to 50 m , and the laser drive impinges from the left . ]  temperatures in the ablation plume were predicted to be of order 10 - 100ev .the mass density and time scales were low enough that conductive heating of the condensed - phase ablator was not significant .[ fig : ch1 ft ] )   applied for 10ns .the undisturbed material extends from 0 to 50 m , and the laser drive impinges from the left . ]the relationship between irradiance and ablation pressure was investigated by performing simulations with the same 10ns laser pulse duration , and varying the irradiance between 0.1 and 100pw / m . ; 1pw / m=100gw / cm . ]the sustained pressure following the initial peak was represented well by a power law in irradiance : ^{0.833}\ ] ] the fitting uncertainty in the parameters is 1.5% , and the uncertainty in the irradiance - pressure values is around 5% , because of temporal variations in the pressure .this result is consistent with previous studies on elemental metals , though the exponent here is larger , and the irradiance - pressure relation is better reproduced by the power law fit despite covering a wider range of irradiance .the exponent is consistent with experimental and calculational results obtained for polystyrene , and the prefactor is similar to the value implied by these results .the exponent is also consistent with more recent simulation results for parylene .  in this regime of irradiance ,laser and thermal radiation penetrates the material much more slowly than the shock wave passes through , so there is a region of compressed material heated only by the passage of the shock between the ablation surface and the undisturbed material ahead of the shock .it is important to know the rate at which material is ablated , to ensure that the ablator is thick enough for a laser pulse of given irradiance and duration .the ablation rate was extracted from the simulations by observing the time of first outward motion of points in the material . for convenience in choosing a thickness of ablator ,a lagrangian ablation rate was calculated , i.e. with respect to the original position of the material .the ablation - supported shock compresses material before it is ablated , so the ablation rate , with respect to the instantaneously compressed material , is lower than the lagrangian rate . for laser irradiances from 0.1 and 10pw / m ,the ablation rates were fitted well by a straight line , indicating that a constant irradiance produces a constant ablation rate . at an irradiance of 100pw / m ,the ablation rate decreased significantly over the 10ns of the laser pulse .ablation was predicted to continue at a similar rate for around 20% of the duration of the laser pulse after its end , because of retained heat in the ablation plume . after this time, the ablation rate decreased rapidly ( fig .[ fig : ablratecmp ] ) .the individual ablation rates were reproduced accurately by the relation ^{0.651},\ ] ] valid for between 0.1 and 100pw / m . above 10pw/ m , this relation ceases to be valid over the full 10ns investigated : at 100pw / m it is valid for the first 1ns .the fitting uncertainty in the parameters is 0.5% .note that 1km / s=1 / ns . this result is similar to the results reported for the ablation of polystyrene , though our simulations predict a larger prefactor . in the previous results ,the depth of material ablated was inferred indirectly from measurements of the ablation plasma .there is significant uncertainty in the eos of matter , particularly in the warm dense matter regime of the ablation plume .the uncertainty is correspondingly greater for a material for which no specific eos has been developed , as discussed above .it was found previously that pressures induced in the condensed region were fairly insensitive to the details of the plasma eos .the sensitivity was investigated here by performing equivalent sets of simulations using the eos and opacity for parylene - c from the sesame library : ( tables 7771 and 17620 ) , and also polystyrene ( tables 7590 and 17593 ) .the two parylene models gave the same ablation pressure to within 1 - 2% , and the shock speeds were almost identical .the polystyrene model gave a smaller initial pressure spike and a sustained pressure that was up to several percent lower , but with less of a decrease over the pulse duration and hence a slower decay as it propagated through the plastic .the shock speed in polystyrene was significantly different : slower at low pressures and faster at high pressures .these variations are all small compared with typical uncertainties in mean irradiance in laser ablation experiments .[ fig : pablcmp ] . )the sensitivity to ablator material was investigated further by performing equivalent simulations for polyethene ( tables 7171 and 17171 ) .the ablation pressures were 5 - 10% lower than for parylene - d , but within 5%of polystyrene .the pressure profile was generally similar to that of polystyrene .[ fig : pablcmp ] . )    for x - ray diffraction experiments , the timing of the x - ray pulse with respect to the shock - loading of the sample is particularly important .the effect of composition and eos was significantly more sensitive than the ablation pressure ( fig .[ fig : hugpus ] ) .experimentally , it has been found that some laser light may be transmitted through the plastic coating before the opaque plasma sheath forms , inducing earlier ablation of the sample material behind by a small amount of the laser energy .this early transmission was not predicted in the radiation hydrodynamics simulations , which ( as is common ) did not include an adequate model of the line opacity and breakdown of the plastic .a palliative is to deposit a thin , metal flashing on the surface of the plastic to prevent early transmission . a typical flashing is 100 nm of al , thoughother metals have been used .the al layer potentially has an effect on the ablation pressure , apart from the palliative effect .this effect was predicted by performing simulations in which the first 100 nm of the plastic was replaced by al . at 0.1pw/ m , the presence of the al increased the sustained ablation pressure by 10 - 15% and made it more constant , but introduced slight wave reverberations in the compressed ablator ( fig .[ fig : alch1cmp ] ) .the difference was smaller at 1pw / m , and negligible at 10pw / m and above .   applied for 10ns .the undisturbed material extends from 0 to 50 m , and the laser drive impinges from the left . ]  there is also a potential effect from x - ray preheating of the sample , induced by k - shell radiation from the ablator or flashing .x - ray preheating is very sensitive to intensity variations in the laser beam , as x - ray yield is highly non - linear in laser irradiance .the fluence of preheating x - rays is therefore very sensitive to the beam quality of the specific laser system , including any phase plate used to reduce large - scale intensity variations .the degree of x - ray heating also depends on the opacity of the sample material . experimental evidence and theoretical predictions for x - ray preheating in such experimentsis mixed : simulations have indicated that preheating may be significant in our regime of interest , but whereas experiments have shown little or no preheating . while preheating is a potential concern when the initial temperature of the sample is important ,it has a much smaller effect on wave interactions caused by the presence of the ablator , so we do not consider it further in the present work .  at the end of the laser pulse ,the pressure at the ablation surface drops rapidly as the ablation plume expands and cools .until two - dimensional expansion takes effect , the pressure gradient in the ablation plume is inversely proportional to the pulse length , so the rate at which the pressure falls is inversely proportional to the pulse length .the ablation pressure can be made constant in time by adjusting the temporal shape of the laser pulse .many , though not all , large laser systems allow the power history to be controlled , so a desired temporal shape can be delivered to some finite level of precision . on lasers with large numbers of beams but a simple shape from each ( such as a constant power ) ,a smooth temporal shape can be approximated by altering the energy and relative timing of the beams .the principle of pulse shaping for constant ablation pressure was demonstrated for parylene - n ablators by performing simulations in which the laser irradiance was adjusted .these simulations were performed for a laser pulse where the time - dependent irradiance was represented in tabular form , with table entries at the start and end of the pulse of equal value for a constant irradiance .the procedure followed was first to adjust the irradiance at the end of the pulse , giving a linear ramp starting at the initial irradiance , and then to add extra points between the first and last and change their values to remove shorter time scale variations in pressure .the sustained ablation pressure was made constant by ramping the irradiance up in time , and the initial pressure spike was removed by reducing the irradiance early in time and following a more gentle approach to the main part of the pulse .this procedure was demonstrated previously for direct ablation of be .calculationally , it was possible to make the induced pressure constant to any desired precision by adjusting the laser irradiance history ( fig .[ fig : pulseshape ] ) . for irradiances from 0.1 to 10pw / m ,the ramp was 10 - 30% of the mean irradiance .  . ]when the ablator is in contact with a sample , the impedance mismatch between the ablator and the sample lead to wave interactions which change the pressure history in the ablator and may lead to multiple waves in the sample . in most cases of interest , the sample has a higher shock impedance than the plastic ablator , so the ablation - induced shock reflects a shock from the interface between the ablator and the sample .  in this discussion of wave interactions ,we consider a simplified situation where the laser irradiance history has been adjusted to give a constant drive pressure in the ablator , which will induce a constant initial shock pressure in the sample .early in the laser pulse , an ablation - driven shock is induced , driving a transmitted shock in the sample and a reflected shock in the ablator . if the laser pulse is long enough , when the reflected shock reaches the ablation surface , double - shocked ablator material is then ablated , resulting in an ablation pressure that is slightly higher than before , but generally lower than the double - shocked pressure .the resulting ablation surface release wave propagates through the ablator and is transmitted into the sample , reflecting a further weak release wave back through the ablator . when the transmitted shock reaches the free surface of the sample , a release wave propagates backward into the sample . when the laser pulse ends , a strong release wave propagates through any remaining ablator and forward into the sample . where the strong release waves interact ,the sample material is subjected to tension , and spall may occur .( figs [ fig : ablwavext ] and [ fig : ablwaveupp ] . )the impedance mismatch between most metals and plastics such as parylene and polyethene means that a given ablation pressure in the plastic induces a considerably higher pressure in a metal sample .the pressure enhancement was estimated for the plastic eos used above and published eos for al and cu , by finding the intersection between the secondary shock hugoniot for a given initial shock state in the plastic and the principal hugoniot in the metal in pressure - particle speed space .the eos of al was as above ; that for cu was sesame table 3330 .shock hugoniots and their intersections were found numerically . repeating this procedure for a range of ablation shock pressures , relations were found between the pressure in the plastic and the pressure in the metal ( fig .[ fig : impedmismatch ] ) .radiation hydrodynamics simulations were performed to predict the integrated effect of ablative loading via a plastic ablator on the loading history experienced by the sample .as for the ablator - only simulations , a geometrically expanding spatial discretization was used .the sample was chosen to be 25 m thick .simulations were performed for ablators 10 and 20 m thick , with al flashing .the laser pulse was taken to be 3ns long , with constant irradiance , which are specifications often used for material dynamics experiments at the janus laser . with a laser pulse of constant irradiance, the pressure was predicted to decrease with time during the pulse , though the compression varied less .simulations were performed for irradiances of 5 , 10 , 20 , and 50pw / m .the peak pressure in the sample was in agreement with the impedance mismatch calculations , and decayed as the shock propagated through the sample . with a thicker ablator ,the pressure in the sample was more constant .( figs [ fig : sampleprofcmpal ] to [ fig : ablthickcmp ] . )m of parylene - n ablator , flashed with 100 nm of al , and driven with a laser pulse of 10pw / m for 3ns . before the start of the laser drive ,the ablator extended from 0 to 20 m , and the sample from 20 to 45 m .the profile at 1ns shows the shock in the ablator . at 3ns ,the shock has been transmitted into the sample , almost doubling the pressure but exaggerating its deviation from flatness . at 4ns, the shock has almost reached the free surface of the sample , its peak has decayed and become less flat , and the ablation surface release is evident as a tilted step between 40 and 50gpa on release . by 5ns, the shock has reached the free surface of the sample and started to release . ]m of parylene - n ablator , flashed with 100 nm of al , and driven with a laser pulse of 10pw / m for 3ns . before the start of the laser drive ,the ablator extended from 0 to 20 m , and the sample from 20 to 45 m .the profile at 1ns shows the shock in the ablator . at 3ns ,the shock has been transmitted into the sample , almost trebling the pressure but exaggerating its deviation from flatness . at 5ns, the shock has almost reached the free surface of the sample , and the ablation surface release has caught up with the shock and made it triangular . ]m ( solid ) and 20 m ( dashed ) ablators , with a 3ns drive pulse of 5 , 10 , and 20pw / m . ]  for comparison , simulations of ablative loading of uncoated samples were performed . for a given irradiance ,the pressure induced in the sample was significantly lower when ablated directly , though the duration of the period of high pressure was greater .ablation of plastic was predicted to generate a slightly lower pressure than ablation of the metal sample , but the large impedance mismatch more than compensated for the reduced ablation pressure . with an ablator , the pressure fell more slowly at the end of the laser pulse . with an ablator ,the pressure varied less during the initial shock , with a constant laser irradiance .the inclusion of the low density ablator increased the coupling of laser energy to the sample .( figs [ fig : pablcmpal ] and [ fig : pablcmpcu ] . )  .the pressure history was calculated at the ablator - sample interface when the ablator was present , and at a lagrangian position 1 m inside the sample when there was no ablator . ]  .the pressure history was calculated at the ablator - sample interface when the ablator was present , and at a lagrangian position 1 m inside the sample when there was no ablator . ]  one way to think of the function of the ablator is that it converts laser energy into kinetic and potential energy in the moving , compressed ablator , which is then partly transferred to the sample . for a given irradiance , laser pulse length , and sample material, there is an optimum thickness for any ablator to maximize the duration of the peak shock pressure applied to the sample .the optimum thickness is such that the laser pulse ends when the reflected shock reaches the ablation surface .when the ablation rate is a constant in time , this relationship can usefully be expressed as an effective speed : microns of ablator per nanoseconds of laser pulse ; this is a useful quantity when designing ablation experiments .the optimum relationship between thickness and pulse length depends on the sample material too , because the impedance mismatch affects the pressure of the reflected shock and therefore its transit time .it should be emphasized that this ` ablator drive speed ' does not represent any individual physical wave speed : it is a composite of the initial ablator shock speed , the speed of the reflected shock through the pre - compressed ablator , and the ablation rate . if the laser pulse is the optimum length for the ablator thickness , or longer , then the duration of the higher pressure drive experienced by the sample at the interface with the ablator can be related to the duration of the laser pulse , as a load duration factor .if the laser pulse is longer than the optimum for a given ablator thickness , the principal pressure pulse applied to the sample does not become longer , but the sample pressure releases to the ablation pressure , and this lower pressure is sustained for longer .there is a minimum laser pulse duration for the peak pressure to reach the sample .if the laser pulse is longer than this , but shorter than the ablator thickness optimum , the duration of the pressure pulse at the sample varies proportionately from zero to the maximum .the initial mass density of the ablator is .the ablation - driven shock travels at a speed and compresses the ablator to a mass density , ablating material at a rate with respect to the uncompressed ablator ( the quantity calculated above ) or of the compressed ablator . the shock reflected from the sampleback through the ablator travels at a speed and compresses the ablator further to a mass density . in this state , the speed of sound is : this is the speed of the head of the ablation surface rarefaction . for an ablator of initial , uncompressed thickness , the transit time of the initial ablation shock is : this is the time that loading starts in the sample after the start of the laser pulse .the transit time of the reflected shock is the optimum laser pulse length is , so the ablator drive speed is the residual thickness of ablator when the reflected shock reaches the ablation surface is \frac{\rho_a}{\rho_r},\ ] ] so the transit time of the ablation surface release is the time for which the initial pressure is applied to the sample is thus so the load duration factor is   if the sound speed in the ablation shocked state is , traveling through the compressed ablator , the minimum pulse length for the peak pressure to reach the sample can be found by calculating the point in the ablator at which the head of a release wave at the end of the laser pulse would catch up with the ablation shock .the minimum time is proportional to the sample thickness , so an effective ` catch - up speed ' can be defined for a given ablation pressure : where , , and are all in the principal hugoniot state for the ablator at the ablation pressure . for a given laser pulse duration , the maximum depth in the ablatorbefore the ablation shock starts to decay is then .this calculation can also be used to calculate the maximum sample thickness for a supported shock in experiments where the sample itself is ablated . for an ablator of given thickness , the minimum laser pulse duration . for laser pulse durations between this and ,the duration of the peak pressure pulse applied to the sample is   as a planar loading or unloading wave propagates through a sample , release waves from the edge will usually erode the shocked region .the head of the release wave initiated by any lateral variation in the material or loading conditions is an expanding circle with respect to the shocked material ( fig .[ fig : latrel ] ) : a cylinder or torus in two - dimensional plane or axisymmetric geometry respectively .the lateral variation in a projectile impact experiment is the edge of the projectile on the impact surface ; for laser ablation it is the edge of the focal spot though unlike a projectile impact , the pressure does not release to zero because the ablation of laterally - released material may induce a pressure almost as high as that from the shock - compressed ablator .the speed of sound in the shocked region is greater than the shock speed , so the release wave erodes the shock .the hydrodynamic flow is self - similar , so the erosion of the shock by the release wave can be characterized by the angle at which the planar region of the shock is eroded . is often assumed to be , but this is not generally the case . as with the catch - up speed , these angles can be calculated given the sound speed on the principal hugoniot , by considering the propagation of a disturbance from the edge of the planar shock region , traveling at , across material moving with the particle speed parallel with the shock , and the resulting speed of the disturbance as it moves across successive positions of the shock itself : an equivalent analysis has been used previously to measure the sound speed on the hugoniot . for a circular laser spot of diameter ,the diameter of the region of the sample initially subjected to the maximum shock pressure is where is the thickness of the ablator .subsequent release depends also on the sound and particle speed in the double - shocked ablator material .the same analysis , using the hugoniot state in the sample , can be used to predict the lateral erosion of the shock in the sample itself .the edge release angle is important when designing experiments to study phenomena at the leading edge of the shock . for experiments on release from the shocked state , including tensile damage ,an important design parameter is the scale rate at which lateral release propagates radially through the shocked material , .may be greater or less than one .a steady shock is always subsonic with respect to the flow behind .however , with respect to the shocked material , the shock moves at , which does not exclude . ]the ablator drive speed , load duration factor , catch - up speed , edge angle , and lateral release ratio were calculated using a generalized shock dynamics algorithm to determine the principal and secondary hugoniot states , using the same eos as above ( figs [ fig : drivespeeds ] to [ fig : latsonicity ] ) . as was found previously for direct ablation of metals ,the catch - up speed exhibited a minimum value around the bulk modulus of the ablated material : at the minimum , the ablator thickness should be smallest for a given laser pulse duration .the ratio was greater than 3 for all relevant pressures , and rose past 10 for ablation pressures above 40gpa : lateral release in the ablator may be a concern in experiments with thick ablators , though the pressure may not drop rapidly once release starts as it is sustained by the ablation .radiation hydrodynamics simulations were performed of the integrated system , with consistent results for the duration of the peak shock state in the sample . it should be noted that these quantities depend on material properties in states significantly off the principal shock hugoniot of the ablator .the hydrodynamic relations above are valid so long as radiation and charged particle transport are not significant within the compressed ablator or the sample .the radiation hydrodynamics simulations indicated that radiation transport is unlikely to be significant in this regime ; they did not include electron transport . for ablatively driven experiments on a range of materials at a range of shock states , a wide range of off - hugoniot states may be involved .these properties are often not known at all accurately from experimental measurements .it would be more efficient to use experimental measurements to validate and adjust wide - ranging theoretical models , rather than attempting to measure the properties of the ablator so generally .experiments should include the principal hugoniot of the ablator , and also off - hugoniot states induced by , for example , reflected shocks from high - density samples .these data could be acquired using laser loading experiments with velocimetry measurements .transparent samples would allow the velocity history to be measured at the interface .whether transparent or opaque , the samples should be thick enough to prevent release waves from their free surface from perturbing the states inside the ablator , which would complicate the experiments .measurements would include the amplitude and duration of the shock in the sample .the sensitivity of ablation pressure in ch - based plastic ablators to uncertainties in equation of state or composition was predicted to be 5 - 10% , which is small compared with typical uncertainties in laser irradiance . a laser pulse of constant irradiancewas predicted to induce an ablation pressure that decreased by several percent per nanosecond , but was significantly more constant than the pressure history induced by direct ablation of metal samples .a simple relationship between laser irradiance and shock pressure was deduced from the simulations , valid for irradiances from 0.1 - 100pw / m .al flashing , used to prevent early - time shine - through , was predicted to make at most a few percent difference in loading history once dielectric breakdown had occurred , increasing the pressure and flattening the pressure history at lower irradiances .it was demonstrated that the irradiance history can be adjusted to produce a more constant pressure history .the impedance mismatch between plastic ablators and metal samples induces a stronger transmitted shock in the sample , by a factor of around 1.8 for al and 2.5 for cu .the shock reflected into the ablator interacts with the ablation surface to produce a stepped pressure history in the sample , the pressure reducing to closer to the ablation pressure .this composite loading history should be taken into account when designing and interpreting material dynamics experiments .the shock pressure is applied to the sample for a significantly shorter time than the laser pulse duration , because the successive waves in the ablator are faster than the initial ablation - driven shock .a compact method was found for representing the optimum ablator thickness for a given laser pulse duration ( or vice versa ) , as an effective speed .a similar method was found to represent the relationship between the laser pulse duration and the duration of the initial , peak pressure pulse applied to the sample .these relations are helpful in the design and interpretation of ablatively driven shock experiments .we would like to acknowledge the contributions of jeff colvin for comments and advice on radiation hydrodynamics simulations and the manuscript , david young for providing the equation of state and comments for parylene - n , and jon larson for advice on hyades simulations , and flavien lambert for comments on the manuscript .this work was performed in support of laboratory - directed research and development project 06-si-004 ( principal investigator : hector lorenzana ) , under the auspices of the u.s .department of energy under contracts w-7405-eng-48 and de - ac52 - 07na27344 .

Abstract: radiation hydrodynamics simulations were used to study the effect of plastic ablators in laser - driven shock experiments . the sensitivity to composition and equation of state was found to be 5 - 10% in ablation pressure . as was found for metals , a laser pulse of constant irradiance gave a pressure history which decreased by several percent per nanosecond . the pressure history could be made more constant by adjusting the irradiance history . the impedance mismatch with the sample gave an increase in the pressure transmitted into the sample , for a reduction of several tens of percent in the duration of the peak load applied to the sample , and structured the release history by adding a release step to a pressure close to the ablation pressure . algebraic relations were found between the laser pulse duration , the ablator thickness , and the duration of the peak pressure applied to the sample , involving quantities calculated from the equations of state of the ablator and sample using shock dynamics .


in communication networks and computing systems , resource allocation ( in some contexts also referred to as resource scheduling , pooling , or sharing ) is a phase , in a network protocol or system management stack , when a group of individual users or clients have to receive a portion of the resource in order to operate a service .resource allocation becomes a challenging problem when the available resource is limited and not enough to fully satisfy users demand .in such situations , resource allocation algorithms need to ensure a form of fairness .such situations emerge in a variety of contexts , such as wireless access  , competitive routing  , transport control  .the common methodology adopted in the literature is to , on the one hand , determine allocation rules such that they satisfy desirable properties  , and , on the other hand , analyse the fairness of a given allocation through indices , the most commonly used being the jain s index  . allocation rules and indices of fairness are commonly justified by some fairness criteria .for instance , among two equivalent users demanding the same amount of resource , it makes sense not to discriminate and to give to each of them the same portion of the resource . in some cases, it can be desirable to guarantee at least a minimum amount of the resource so that the maximum number of users can be served .  in the networking literature ,the resource allocation problem is historically solved as a single - decision maker problem in which users are possibly not aware of the other users demands and of the total amount of available resource .it follows that the most natural and intuitive way to quantify the user satisfaction is through the proportion of the demand that is satisfied by an allocation .large literature exists indeed in the networking area on proportional resource allocations for many practical situations , from wireless networks to transport connection management  .  in this paper , we are particularly interested instead in cooperative networking contexts such that users can be aware of other users demands and the available amount . as such , rational users shall compute their satisfaction also based on the presence of other users .in fact , such networking contexts with demand and resource availability awareness are making surface in wired and wireless network environments with an increasing level of programmability , i.e. , using software - defined radio and network platforms that expose novel ( northbound ) interfaces to users to disseminate information and pilot network resource allocations .our main idea is defining a new notion of user satisfaction for such interactive resource allocation situations with demand and resource awareness .[ es ]  let us briefly clarify our motivation with the following allocation example .a user asks a quantity of resource that is bigger than the resource itself ( as b in fig .classical fairness indices  ,  ,  tend to qualify the user satisfaction as maximum when obtains exactly what it asks . in the case where asks more than the available amount, it can not reach the maximum satisfaction due to the fact that its demand exceeds the available resource . instead , in demand and resource awareness conditions , it would be more reasonable that its satisfaction is maximum when it obtains all the available resource .furthermore , if all the other users together ask a quantity of good inferior to the resource , a minimum portion of it , equal to the difference between the resource and the sum of the demands of all the others , is guaranteed to . under a dual reasoning, it also appears more acceptable that the minimum satisfaction of a user is reached when it receives the minimum portion of the available resource , instead of when it receives zero .if users are in complete information context the classical approach can lead to not reasonable outcomes .  in this perspective , in order to better describe the user satisfaction as a function of the available resource , and to capture the interactions due to the networking context ( e.g. , networked users may be aware of respective demands , may ally in the formulation of their demands , etc ) , we propose to model the resource allocation problem as a coalitional game .accordingly , we define a new satisfaction rate for users , able to adapt to various configurations of the demands . furthermore , we define a new resource allocation rule , called the mood value , based on the idea that the most fair allocation is the one that equalizes the satisfaction of each player .indeed , regardless of the level of satisfaction , each player is not discriminated if its satisfaction is the same than the one of all the others .we also provide an interpretation of this approach positioning it with respect to classical traffic theory  .the paper is organized as follows .section [ background ] presents the state of the art on the topic . in section [ satisf ] a new satisfaction rateis proposed . in section [ moodvalue ] the mood value and a new fairness index are described . in section [ traffictheory ]we provide an interpretation of the mood value with a traffic theory methodology .section [ results ] presents some numerical examples .finally , section [ concl ] concludes the paper .a resource allocation problem can be characterized by a pair , in which is the vector of demands ( claims ) from users ( claimants ) and is the resource ( estate ) that should be shared between them .the set of users is .the resource allocation is a challenging problem when is not enough to satisfy all the demands ( ) .an allocation is a solution vector that satisfies three basic properties :  * _ non - negativity _ :each user should receive at least zero . * _ demands boundedness _ :each user can not receive more than its demand . * _efficiency _ : the sum of all allocations should be .  an allocation rule is a function that associates a unique allocation vector to each .many resource allocation rules are proposed in the literature and each of them is characterized by a set of properties that justify the use of the given rule in order to find a solution of the allocation problem  . in computer networks , the most well - known rules are : the proportional rule and the weighted proportional rule  , the max - min fair allocation ( mmf )  ,  , and the -fair allocation  .each of these allocation rules , result of an optimization problem and/or an iterative algorithm , follows a fairness criterion .the _ weighted proportional allocation rule _ is based on the idea that a logarithmic utility function captures well the individual evaluation of the worth of the resource  .one way to compute it is via the maximization of subject to demand boundness and efficiency constraints .when is equal to the resulting allocation is called simply proportional and when is equal to we obtain the allocation that actually produces allocations proportional to the demands ; hence in the following , we refer to the latter rule as proportional instead of the previous ( not weighted ) one .the idea behind the _ max - min fairness ( mmf ) allocation _ is to maximize firstly the minimum allocation ; secondly , the second lowest allocation , and so on  .this solution coincides with the only feasible allocation such that , if the allocation of some users is increased , the allocation of some other users with smaller or equal amount is decreased .  more generally , it is possible to obtain a family of allocation rules maximizing a parametric utility function . the _ -fair utility _ function is defined as  . if the solution of the optimization problem coincides with the weighted proportional allocation with equal to , if with the minimum delay potential allocation , that is the allocation obtained minimizing the total potential delay  , and if with the max - min fair allocation .recently game theory has been applied to communication systems in order to model network interactions .for example , in a cooperative game model is proposed to select a fair allocation of the transmission rate in multiple access channels and in the authors studied , using coalitional game theory , the cooperation between rational users in wireless networks .  moreover , it is possible to analyze the allocation problem as a transferable utility ( tu ) game , which is defined as a pair , where denotes the set of _ players _ and is the _characteristic function _ ,( by convention , ) .bankruptcy games  , in particular , deal with situations where the number of claimed resource exceeds that available .a _ bankruptcy game _ is a tu - game in which the value of the coalition is given by where represents the estate to be divided and is a vector of claims satisfying the condition  .the bankruptcy game is _ superadditive _ , that is : it is also _ supermodular _ ( or , equivalently , _ convex _ ) , that is :   a classical set - value solution for a tu - game is the _ core _ , which is is defined as the set of _ allocation vectors _ for which no coalition has an incentive to leave the grand coalition , i.e. :   a _ one - point solution _ ( or simply a _ solution _ ) for a class of coalitional games is a function that assigns a payoff vector to every coalitional game in the class . a well - known solution for tu - games is the _ shapley value _ of a game , defined as the weighted mean of the players marginal contributions over all possible coalitions and computed as follows :    with where denotes the cardinality of .another well studied solution for tu - games is the nucleolus , based on the idea of minimizing the maximum discontent . given a tu - game and an allocation ,let be the _ excess _ of coalition over the allocation , and let be the _ lexicographic _ order on .given an imputation , is the vector that arranges in decreasing order the excess of the non - empty coalitions over the imputation .the _ nucleolus _ is defined as the _ imputation _ ( i.e. , and for each ) such that for all imputations of the game .given a bankruptcy game , many other solutions can be proposed . as already introduced in the previous section , the _ proportional allocation _ assigns to player an allocation equal to .for example , it is worth mentioning the _ constrained equal loss _ ( cel ) allocation that divides equally the difference between the sum of the demands and , under the constraint that no player receives a negative amount .the evaluation of the fairness of the allocations , used as an important system performance metric especially in networking , can be useful to discriminate among allocation rules and to evaluate the level of justice in the repartition of the resources .jain  introduces a formula aimed at providing a quantitative measure of the fairness of a resource sharing allocation .given an allocation problem and an allocation , the _ jain s fairness index _is: ^ 2 % } {    \bigg/\biggl [ n\sum\limits_{i=1}^n \bigl(\frac{x_i}{c_i}\bigr)^2\biggr ] } % } \ ] ]  the jain s index is bounded between and  .the maximum fairness is measured when all the users obtain the same fraction of demand and the minimum fairness is measured when it exists only one user that receives all the resource .the jain s index has the following good properties :  * _ population size independence _ : applicable to any user set , finite or infinite . *_ scale and metric independence _ : not affected by the scale .* _ boundedness _ : can be expressed as a percentage . *_ continuity _ : able to capture any change in the allocation .the index considers the proportion of demand and it gives the maximum fairness to the allocation for which all the users receive the same proportion of the demand , regardless of the type of allocation problem , it suggests to allocate the resources in a proportional way even when this allocation is not the most suitable to solve the problem .another well - know index of fairness is the atkinson s index ; contrary to the jain s index , it measures the degree of inequality of a given allocation , taking value equal to 0 when the system is fair in the mmf sense , and 1 when it is totally unfair .let be the situation of fig .[ es ] with and .the discussed allocation rules provide values in table [ tab1 ] along with the jain s index and 1-atkinson s index in order to have a measure of fairness .c|c|c|c|c|c  ' '' ''  user demands & prop .& mmf & shapley & nucleolus & cel +  ' '' ''  a : 3 & 1.67&3&1.5&1&0 +  ' '' ''  b : 13&7.22&5&7.5&8&10 +  ' '' ''c : 2&1.11&2&1&1&0 + jain s index & 1 & 0.882 & 0.995&0.946&0.333 + atkinson s index & 0.844 & 0.965 & 0.821 & 0.777 & 0.333 +  [ tab1 ]  the axiomatic theory of fairness proposed in  shows that it exists an unique family of fairness measures , which includes the jain s and the atkinson s indices , satisfying a set of reasonable axioms . in the rest of the paper, we consider only the jain s index because it is the one classically used in networking applications .mmf - driven inequality indices find their most appropriate use in socio - economical contexts , because they are linked to the concept of welfare of an income distribution .furthermore the jain s index is based on the idea of summarizing the information about the users satisfaction , which is close to our methodology of redefining users satisfaction under demand and resource awareness , as discussed in the following section .in this section , we propose a game - theoretic approach to evaluate the satisfaction of a user for an allocation .a crucial issue in resource allocation is to jointly :  * find the best solution in terms of a certain goal ; * evaluate its fairness by referring to a fairness index .  with this purpose ,it is important to evaluate the individual satisfaction rates and to summarize the information given by each of them with a global fairness index .  a natural way to quantifythe satisfaction of a user , as proposed by jain , is through the proportion of the demand that is satisfied by an allocation .  given the user with demand and an allocation , the _ demand fraction satisfaction ( dfs ) rate _ of is :   this rate takes value between 0 and 1 since it represents the percentage of the demand that is satisfied .unavoidably , this way to quantify the user satisfaction makes the weighted proportional allocation the fairest one since it allocates proportionally to the demand .there are , however , situations in which the common sense does not suggest to allocate in a proportional way ; e.g. , if there is a big gap between the demands , in order to protect the weaker users and guarantee them a minimum portion of the estate , the mmf allocation can be preferable . furthermore , as mentioned in the introduction , the presence of other users should rationally be considered not to distort the satisfaction of each user , in case of awareness about other users demand and the available demands .  for these reasons , we aim at defining an alternative satisfaction rate such that it satisfies the following two properties we name demand relativeness and relative null satisfaction :  * _ demand relativeness _ :a user is fully satisfied when it receives its maximal right , based on the available resource ; * _ relative null satisfaction _ : a user has null satisfaction when it receives exactly its minimal right , based on other users demands and the available resource .the minimal right for a player is the difference between the available amount and the sum of the demands of the other users ( i.e. , taking a worst case assumption that the others get the totality of their demand ) , and the maximal right is equal to the maximum available resource , i.e. , if , or it is equal to e otherwise . remembering the definition of the characteristic function of a bankruptcy game we have that :  * the _ minimal right _ for player is * the _ maximal right _ for player is   thus we introduce the player satisfaction ( ps ) rate , which satisfies the above two properties by considering the value of the bankruptcy game associated to the allocation problem .  given a bankruptcy game such that and an allocation , the _ player satisfaction ( ps ) rate _ for is : where : , . if the player satisfaction rate is , . ] for with the interval  ] .if a solution belongs to a core it holds : and .thus and are the minimum and the maximum value that an allocation in the core can take . if then , if then .[ relaz ] it is possible to summarize the bankruptcy regimes of the rate in four possible cases as in table [ casi ] .  c|c|r||c|r  ' '' ''  & & + & _ps _ & _ case _ & _ ps _ & _ case _ +  ' '' ''   & & gm & & gg +  ' '' ''  & & mm & & mg +  let us treat each possible cases of table [ casi ] :  * _ case gm : , _ + using the definition of bankruptcy game , it holds : + .it follows . *case gg : , _ + using the definition of bankruptcy game , it holds : + .it follows * _ case mm : , _ + as in case mg , .it follows . *_ case mg : , _ + as in case gg , .it follows .the ps rate differentiates 4 possible cases we name gm , gg , mm , mg .if a player asks less than we call it _ moderate player ( m ) _ while if it asks more than it is a _ greedy player ( g)_. in similar way , if the sum of the demand of a group of players exceeds , that means , the group is a _ group of greedy players ( g ) _ otherwise if we have a _ group of moderate players ( m)_.  proposition [ relaz ] highlights that , not only there is a relation between the dfs rate and the ps rate , the satisfaction of a user should be modified when it is considered as a player inside a cooperative game . in particular , we can notice that for case gm the ps rate coincides with the dfs one , i.e. , ; for case gg , the user satisfaction measured with the ps rate is higher than when it is measured with the dfs rate , i.e. , ; in the mg case , we have instead that .we can also notice that the denominator of the ps rate is always different from zero . in casesgm and gg this is obviously true , in case mm the denominator is zero when but in this case we set and in case mg the denominator is zero when that is impossible .furthermore , from proposition [ relaz ] it follows that if an allocation , i.e. a solution of an allocation problem that satisfies efficiency , non - negativity and demand boundedness , is an imputation , then  ] . and the mood value is given by :   let  .it follows : due to the efficiency property it holds : thus . since is the mood value iff  :  and remains proved .  from wecan notice that the mood depends only on the game setting , thus , given a bankruptcy game , we can know a priori the value of the mood that produces a fair allocation . knowing , oncan easily calculate the mood value .the formula shows that each user receives the minimum possible allocation plus a portion of the quantity .the nearer to 1 is the mood , the greater is the happiness of each user , and the closer to the maximum the allocation is .in fact , when is equal to 1 , the player receives exactly , that is the maximum portion of resource that it can get , being inside a bankruptcy game .the mood value owns some interesting properties .it is an allocation thus it satisfies non - negativity , demand boundedness and efficiency property ; it is stable , that means it belongs to the core of the game ( prop .[ propcore ] ) and it guarantees more than minimal right to each player ( . furthermore it satisfies the following property : if and then .this implies the equal treatments of equals (  ) and equal treatment of greedy claimants ( given a bankruptcy game , let g be the set of greedy players , i.e. such that : if then  ) .[ propcore ] the mood value belongs to the core of .we should prove that , .+ if the condition holds due to the fact that , .now consider the case .suppose that .for the efficiency property it holds , implying , which yields a contradiction with the fact that , according to the mood value solution , each user receives at most its demand .   differently from the other allocation solutions inspired by game theory , in order to calculate this new allocation , only the value of coalitions , i.e. , the ones formed by the single players and the ones containing players , is needed .the time complexity of mood value computation is dominated by the complexity of computing that is . in dynamic situations ,i.e. when the value of each of the coalitions has to be updated at each slot of time , the complexity is therefore , but it can be reduced to where pre - computation is possible .this makes the mood value the best allocation rule in terms of time complexity together with the proportional allocation : the shapley value has a time complexity of , while iterative algorithms for the computation of mmf and cel allocations have a time complexity ; the nucleolus computation is a np - hard problem .  in terms of spatial complexity , the mood value , proportional , mmf and cel allocations can be considered as equivalent and in the order of . instead, the shapley value and the nucleolus computations have a spatial complexity of .   considering the observed good properties that make the jain s index a strong fairness index , we propose its modification replacing the dfs rate of the jain s index with the ps rate .the resulting new fairness index we propose takes value 1 when all the users have the same satisfaction , i.e. , when the allocation is the mood value .given a problem and an allocation , the _ players fairness index _ is : ^ 2 \bigg/ n\sum\limits_{i=1}^n \bigl(ps_i\bigr)^2 } \ ] ]  the players fairness index takes value in  ] and that is always not negative .the maximum fairness is measured when all the users have the same ps rate , i.e. :   ^ 2 = \bigl ( n ps_i \bigr)^2 \rightarrow n\sum\limits_{i=1}^n \bigl(ps_i\bigr)^2=nn \bigl(ps_i\bigr)^2. ]  for core allocations , takes value in the same interval of making possible a comparison between the two indices . furthermore, this index maintains all the good properties of the jain s index : the population size independence , the scale and metric independence , the boundedness and the continuity .in the already cited seminal works about the definition of proportional and weighted proportional allocations in network communications , network optimization models are defined using as goal the maximization of an utility function .a typical application is the bandwidth sharing between elastic applications  , i.e. , protocols able to adapt the transmission rate upon detection of packet loss . in this contextwe show how it is possible to revisit the mood value as a value resulting of the sum of the minimum allocation and the result of a weighted proportional allocation formulation where the weights are not the original demands , but new demands re - scaled accordingly to the maximum possible allocation knowing the available resource , and the minimum allocation under the awareness of other user s demands and the available resource .more precisely , the mood value can be computed as the result of the following 4-step algorithm .: we assign to each user the minimal right . :we set the new value of the estate + and the new demands .+  + : we solve the following optimization problem     + : the mood value coincides with the sum of the minimal right and the allocation given by step 3 : .  we should prove that the result of the optimization problem is .the lagrangian of the problem is where the vector and are the lagrangian multipliers ( or shadow prices ) , is the vector of the demands  ] ._ step 2 : _ ,  ] _ step 4 : _ $ ] .the algorithm shows that the mood value firstly assign the minimal right ( step 1 ) and secondly , considering the new allocation problem resulting after the first assignment ( step 2 ) , it allocates in a proportional way the resources ( step 3 ) .the proportion of resource to allocated is the mood .we provides two ways to compute the mood value : ( [ xm ] ) and the 4-step algorithm of section [ traffictheory ] .it is clear that the computation of the mood value throught the formula ( [ xm ] ) is less complex than the one using the 4 steps algorithm .we tested the mood value and the new fairness index in a few significant configurations comparing them with the classical allocations and the jain s index .we considered two demands distributions : ( i ) a uniform distribution , and ( ii ) a weibull distribution .the former can be considered as a baseline , while the latter a maybe more realistic one . to create the weibull distribution , taking inspiration from cellular ( ofdma ) resource allocation studies we emulated an indoor scenario of femtocells using the winner ii channel model  : generating in a uniform way 10000 users around the cell station between 3 and 100 m , we associate resource blocks ( rbs ) to each of them with a transmit power between 1 and 100 db ; fig . [ scenario ]is the resulting rb histogram distribution , which is well fit by a weibull distribution for with scale parameter and shape parameter .the range for demand generation is between 0 and 6000 units .it is worth noting that the weibull distribution is quite close to the pareto distribution ( both are exponential ones ) , its and discrete variations ( e.g. , zipf s one ) , for example used in in - network content caching resource allocation  .[ scenario 2 ]  [ scenario ]  [ scenari ]  [ fig_simrand ]  we run different instances with a ratio of ( available resource ) ranging from to of the global demand .we first simulate 300 bankruptcy games with 3 and 5 users .[ scenari ] show the users configuration as a function of the available resource . with 3 users ( fig .[ scenari]a , c ) , for low value of almost all are greedy players ( gg case ) due to the fact that the resource is small ; increasing the number of moderate players ( gm ) increases but also some users in configuration mg appear .in fact , increasing some greedy players become moderate while the others remain greedy ones ; some of them are greedy inside a group of greedy users ( gg ) , while some others greedy inside a group of moderate ones ( mg ) .when the available resource is higher than half of the global demand , greedy players gg disappear and the number of moderate players increases . in particular , users mm appear and they become the majority when the resource is large . with 5 users ( fig . [scenari]b , d ) , we find a similar trend than with 3 users in the number of moderate players that increases when increasing .however , mg users are few ; in fact , it holds that it can exist at most one mg user in a game and , due to the higher number of users in the system , it is very unlikely that there exists only a player mg in the system such that the sum of the demands of the other players exceeds .thus , with a number of users higher than 5 , one can practically reduce the number of user cases from 4 to 3 .for this reason , in order to capture all the possible scenarios , we choose a low number of user for the first round of simulations .[ fig_simrand ] , [ fig_sim1 ] and [ fig_sim2 ] show the results of the first simulations .we consider the six allocations discussed before : proportional , shapley , nucleolus , mood value , mmf and cel .we calculate the jain s fairness index and the players fairness index and we plot , for each ratio of and each index , the mean value in between the first and third quantile lines .the jain s index is depicted with the red color and square points while the players fairness index with the blue one and round points .due to space limit , we do not show the case of 3 users with uniform demand distribution because similar to the 3-user one with the weibull distribution .[ fig_sim1 ]  [ fig_sim2 ]  in the 3-user scenario ( fig .[ fig_sim1 ] ) it is possible to notice differences between the result obtained by the classical jain s index and our new players fairness ( pf ) index .it is worth recalling that the jain s index has value 1 when the allocation is proportional while the pf index is 1 when the allocation is the mood value .we can notice that the pf index considers the mmf allocation as a fair one when the available resource is small ( high congestion ) , i.e. , when there are many greedy users .in fact the mmf allocation and the mood value , are close : in such cases , both have the property of treating equally the greedy claimant , giving them the same portion of resource , independently of their demands .instead , when increases , the mmf one is not fair any longer because it satisfies more the two users with less claim while it gives the minimal right to the one with bigger claim ; in fact , in such cases the mood value becomes closer to the proportional allocation , to the shapley value and to the nucleolus .the similarity between the proportional allocation and the mood value is due to the fact that the correct way to measure the satisfaction of moderate players is through the dfs rate and increasing the number of moderate players increases .it follows that the allocation equalizing the dfs rates , i.e. , the proportional one , is close to the one equalizing the ps rates of user , i.e. , the mood value .  with 5 users ( fig .[ fig_simrand ] and fig .[ fig_sim2 ] ) , we can notice that , due to the fact that the group mg is little , when reach the of the global demand the proportional is equal to the mood value and it shows a pf index equal to 1 .furthermore the mmf allocation show again a high ( pf ) fairness when the resource is little ( ) , i.e. , when there are many greedy players .  [ rateslow ]  with a second round of simulations we want to see how the demand fraction ( dfs ) and player ( ps ) satisfaction rates are distributed with an increasing number of users .results are shown in figure [ rateslow ] as box - plots ( minimum , quantiles and maximum , without outliers ) , for two regimes ( high congestion of a 5% /demand ratio , and low congestion of a 95% ratio ) .we can notice that the value of the satisfaction is low when the resource is small ( fig .. [ rateslow]a , b , c ) while it is higher when the congestion is low ( fig .[ rateslow]d , e , f ) . with 4 users , in terms of user satisfaction rate ,the mood value is close to the mmf allocation when the congestion is high , and to the proportional allocation when the congestion is low . as the number of players grows , the absolute difference between allocation in terms of distribution of the satisfaction decrease .in both the congestion situations ( equal to 5% and 95% of the demand ) the shapley value is the closest allocation to the mood value in terms of ps rate .  summarizing ,the simulations show that the mood value is able to nicely weight the nature ( greedy or moderate ) of users and of user groups .in particular it is close to the mmf allocation when the resource is scarce and to the proportional allocation when the resource is close to the global demand .furthermore , it is worth noticing that with respect to classical game - theoretical allocation rules ( shapley value , nucleolus ) , the results show that the mood value shows a similar good behavior in terms of fairness , with the key advantage of having a much lower computation time complexity .we proposed a game - theoretical approach to analyze and solve resource allocation problems , going beyond classical approaches that do not explore the setting where users can be aware of other users demand and the available resource .in particular , we proposed a new way of quantifying the user satisfaction and a new fairness index as enhancement of the jain s index , describing and comparing their mathematical properties in detail . accordingly to these new concepts , we propose a new resource allocation rule that meets the goal of providing the fairest resource allocation , we called the mood value , which we position with respect to game theory metrics as well the common theory of fair allocation in networks .  finally , we test our ideas via numerical simulations of representative demand distributions , showing at which extent the mood value can approach and differ from max - min - fairness , weighted proportional , constrained - equal loss , shapley value and nucleolus allocations .99 w. saad et al .`` coalitional games for distributed collaborative spectrum sensing in cognitive radio networks . 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Abstract: in networking and computing , resource allocation is typically addressed using classical sharing protocols as , for instance , the proportional division rule , the max - min fair allocation , or other solutions inspired by cooperative game theory . in this paper , we argue that , describing the resource allocation problem as a cooperative game , such classical resource allocation approaches , as well as associated notions of fairness , show important limitations . we identify in the individual satisfaction rate the key aspect of the challenge of defining a new notion of fairness and , consequently , a resource allocation algorithm more appropriate for the cooperative context . we generalize the concept of user satisfaction considering the set of admissible solutions for bankruptcy games . we adapt the jain s fairness index to include the new user satisfaction rate . accordingly , we propose a new allocation rule we call mood value . for each user it equalizes our novel game - theoretic definition of user satisfaction with respect to a distribution of the resource . we test the mood value and the new fairness index through extensive simulations showing how they better support the fairness analysis .


the causal dynamical triangulations ( cdt ) approach to quantum gravity has emerged as a breath of fresh air in the effort to quantize gravity . in cdt , developed by ambjrn , jurkiewicz and loll , we have an attempt at quantizing gravity that relies solely on ideas and techniques that have been known to physicists for quite some time path integrals , causality , simplicial manifolds , finite - size scaling effects , etc .the formulation of cdt , while being extremely straightforward , has yielded some important results , such as the _ emergence _ of classical spacetime and dimensional reduction of spacetimes at short distances .so far , however , all the results that have been reported for cdt have either come from the original group headed by ambjrn , jurkiewicz and loll or from groups that have used the code or data sets from the original group . in this paper , we present the first independent verification of the cdt algorithm , using code developed from scratch in a completely independent fashion .we have been able to reproduce the important results reported for cdt and set the stage for further results , which will be reported elsewhere .the outline of this paper is as follows . in section [ sec : cdt ]we present a brief overview of the causal dynamical triangulation approach .readers interested in greater detail should refer to and .in section [ sec : num - impl ] we describe our cdt implementation , at a fairly high level . finally , in section [ sec : results ] , we present the results from our simulation .cdt is based on the gravitational path integral ^{is_{eh}[g]}\ ] ] where =\frac{1}{16\pi g}\int d^nx\sqrt{-g}(r-2\lambda)\ ] ] is the einstein - hilbert action . here and are the spatial metrics on the initial and final slice and represents all possible ( spacetime ) metrics that satisfy the initial and final conditions . by all possible metrics , we mean metrics that are not related to each other by diffeomorphisms .to define the action in the path integral , cdt uses regge calculus as its starting point .the basic idea of regge calculus is to approximate a smooth manifold ( for example spacetime ) with a piecewise linear manifold , with curvature restricted to subspaces of co - dimension two .the geometrical objects used to approximate -dimensional manifolds are -simplices , which consist of -dimensional faces , -dimensional hinges or `` bones '' ( on which the curvature of the manifold resides ) , and so on down to 0-dimensional simplices ( points ) .a -simplex has points .  for manifolds without a boundary , a discrete version of the einstein - hilbert actioncan be formulated using  to get where is the volume of a hinge , is the volume of a top level d - simplex , and is the deficit angle at .cdt assumes globally hyperbolic spacetimes .the existence of cauchy hypersurfaces enables us to use -simplices that span adjacent spatial slices as the building blocks for our -dimensional spacetimes .a time - like -simplex , where , has points on the lower time slice and points the adjacent higher time slice . for a space - like simplex ,all points are on the same spatial slice .no space - like -simplices are present in -dimensional spacetimes . but both space - like and time - like types of lower dimensional simplices are present in the spacetime .the simplicial building blocks in cdt have an inherent causal structure built into them . to preserve this causal structure ,cdt does not allow topology changes of the spatial slices .the lengths of the space - like and time - like links ( 1-simplices ) are defined as where is the lattice spacing and is the asymmetry parameter . in 2 + 1 dimensions, we have three types of 3-simplices , denoted by , and . in 3 + 1 dimensions, we have four types of 4-simplices , denoted by .the volume of each type of simplex and the dihedral angles subtended between the faces of these simplices are given in .using these , we can formulate the lorentzian version of the regge action in as where is the dihedral angle subtended at the hinge .equation ( [ eqn : cdt - analogue - regge - action ] ) is a reformulation of ( [ eqn : regge - action ] ) with space - like and time - like simplices distinguished .  in cdt ,wick rotation is defined as  this leads to a pure imaginary value for the action .the gravitational path integral ( [ eqn : grav - path - integral ] ) in cdt is replaced by a sum over distinct triangulations ^{is } \rightarrow \sum_t\frac{1}{c(t)}e^{is(t)}\ ] ] where is the order of the automorphism group of the triangulation .the wick rotated action ) and ( [ eqn:2p1-kappa0-kappa3 ] ) are derived for . for the action will have terms involving and , where and are the number of ( 3,1 ) and ( 2,2 ) 3-simplices . however , in 2 + 1 dimensions , there are only two independent bulk degrees of freedom , so the action still takes the form ( [ eqn:2p1-wick - rotated - action ] ) , although with ( [ eqn:2p1-kappa0-kappa3 ] ) altered .see for details . ] for a triangulation in 2 + 1 dimensions is given by where and are the number of 0-simplices and 3-simplices , respectively .equation ( [ eqn:2p1-wick - rotated - action ] ) can be derived from ( [ eqn : cdt - analogue - regge - action ] ) by substituting the appropriate expressions for the simplex volumes and dihedral angles , using topological identities for ( 2 + 1)-dimensional lorentzian triangulations and then performing a wick rotation ( [ eqn : wick - rot ] ) .the topological identities eliminate all but two of the bulk variables .  substituting the wick rotated action in ( [ eqn : sum - over - triangulations ] )gives us the partition function :   a similar procedure in 3 + 1 dimensions leads to the wick rotated action and the corresponding partition function where and is a complicated function that depends on , with corresponding to . is the number of plus simplices , and is the number of plus simplices .  to generate the distinct triangulations required for evaluating the partition function and the expectation value of any observable , the cdt approach uses a set of ergodic moves .an ergodic move applied to a triangulation yields another distinct triangulation , one that is not related to the first one by a diffeomorphism .ergodicity ensures that we can go from one triangulation to any other triangulation by repeated application of moves .ergodic moves also preserve the topology of the manifold , as well as the simplicial manifold property .the moves are described in detail in .in our numerical implementation , each -simplex is identified by a unique i d and is stored as a list ( containing sublists ) of points  .... ( ty lo hi ( p0 p1 . . .pd ) ( n0 n1 . .nd ) ) ....  here ` ty ` identifies the type of the simplex ` ty = 1,2, ... ,d ` . `lo ` and ` hi ` are the lower and higher time slices spanned by the simplex .the ` pis ` are the points that make up the simplex , and the ` nis ` are the ids of the neighboring simplices . the simplex ` nk ` is connected on the ` ( p0 ... pd ) - ( pk ) ` face .the points are used only as unique labels to identify the vertices of a simplex .while we distinguish the points on the lower time slice from the points on the higher time slice , the ordering of the points in any given slice is immaterial .  with this choice for the simplex data structure ,it is fairly straightforward to implement the ergodic moves described in detail in .it is essential that the moves maintain the simplical property of the manifold .this requires that two simplices should be connected across a single face and that a -simplex should be shared between exactly two -simplices .  to generate the distinct triangulations using the ergodic moves, we apply the metropolis algorithm .the metropolis algorithm uses a markov chain that , upon reaching equilibrium , satisfies the following detailed balance condition where is the probability of being in a particular triangulated geometry and is the transition probability of going from triangulation to triangulation . the probability of being in a particular triangulation is proportional to the boltzmann weight . the detailed balance condition in this casecan be written as : the simplest choice for the transition probability that satisfies this condition is thus a move that increases the action may be accepted or rejected , while moves that decrease the action or leave the action unchanged are always accepted .we initialize the simulation by specifying the number of time slices and the total volume of the spacetime .the initialization process starts by triangulating each spatial slice as a -sphere .the and the slices are identified to enforce the periodic boundary conditions in time .this is done for technical convenience , rather than due to any physical argument .next , the triangulations on adjacent slices are connected .we then apply the volume increasing subset of the ergodic moves until the total volume reaches the desired target volume .once the simulation has been initialized , we have to run simulation for about 100,000 sweeps to thermalize the spacetimes .a sweep is defined as attempted moves , where is number of -simplices in the spacetime .thermalization ensures that the system is sufficiently randomized , and has moved away from the highly structured initial conditions .once thermalization is completed , we start saving spacetimes every 100 sweeps , to build our ensembles .in 2 + 1 dimensions , the phase structure of the cdt model can be explored by tracking the dependence of an order parameter on coupling constants .the order parameter selected is the ratio of the number of 3-simplices to the total number of 3-simplices in the spacetime : the dependence of on the coupling constant is shown in figure [ fig : op_vs_kzero ] .   versus   figure [ fig : op_vs_kzero ] shows a phase transition , at . in ,this phase transition was reported at a value of .the two results differ by a factor of two , for reasons we do not understand , but the qualitative results are identical . for , we observe that the spatial volumes of successive slices is not strongly correlated spacetime has effectively decoupled into disconnected spatial slices .this is shown in figure [ fig : decoupled_phase ] , where we have a taken a snapshot of a simulation in this decoupled phase .a snapshot of the simulation in the other phase , characterized by , clearly shows the emergence of a well defined , extended geometry phase .this is shown in figure [ fig : extended - phase ] .the simulation fluctuates around this extended geometry , a geometry that resembles a classical universe . this classical spacetime _ emerges _ from the simulation , and was not explicitly specified by us . in this phase , the volume of successive spatial slices is strongly correlated , and the path integral is dominated by metrics that are approximate solutions of the einstein equations .  a thick slice is defined as the portion of spacetime that is bounded by adjacent spatial slices .thick slices are labelled by half - integer values where , for example , a thick slice labelled by , is bounded by the spatial slices and .the volume of a thick slice is defined as the number of -simplices in that thick slice . in figures [ fig : decoupled_phase ] and [ fig : extended - phase ] time is along the horizontal axis .these figures were generated by considering the volumes of the thick slices for half - integer values ranging from to where is total number of time slices .we draw a circle with radius equal to the volume of the slice , and join circles corresponding to adjacent thick slices by linear interpolation .the axial symmetry in these figures is therefore a visualization artefact .figure [ fig:2p1-spatial - slice ] shows a typical spatial slice in 2 + 1 dimensions , embedded in a three dimensional space .    in 3 + 1 dimensions , while we are not aware of an order parameter analogous to the -dimensional case , we do observe three distinct phases , illustrated in figures [ fig:3p1-phase - a ] , [ fig:3p1-phase - b ] and [ fig:3p1-phase - c ] .all three figures are for spacetimes with and   ]  phase a is characterized by a splitting up of the spacetime in small pieces of irregular volume , with each piece not extending beyond a few time slices . in phase b ,all of the spacetime volume collapses into a single piece of minimal extension .both of these non - physical phases are characterized by a small value for the ratio .  ]phase c is the extended geometric phase , where there is a strong correlation between the volumes of successive spatial slices .  ]   one of the major contributions of the cdt approach has been the introduction of the spectral dimension as probe of the properties of spacetime , both on short and long scales .the diffusion equation for a -dimensional manifold with a smooth metric is given by  where is the diffusion time , is the laplace operator corresponding to and is the probability density of diffusion from to in a time .the diffusion processes we care about are ones that are initially peaked at a point  instead of probability density , we measure the return probability , which is defined as for a flat space with infinite volume , the solution of equation ( [ eqn : diffusion - equation ] ) is  where is the geodesic distance between and .the return probability for the flat space case is taking the logarithmic derivative of the equation ( [ eqn : flat - space - ret - prob ] ) , we get this definition of the spectral dimension can be extended to curved and/or finite volume spacetimes , with the addition of finite - size corrections , and allows one to define a generalized dimension on a wide variety of spaces that are not smooth manifolds   in cdt , the spectral dimension of the spacetime is measured by initiating a diffusion process from a randomly selected simplex .a single step in a random walk is generated by moving to randomly selected neighbor . by initiating a large number of walks with steps, we can calculate the probability of returning to the starting simplex .the return probability can be then used to compute the spectral dimension .the spectral dimension vs step size plot for a ( 2 + 1)-dimensional spacetime is shown in figure [ fig:2p1-spectral - dimension ] .,   we observe that the spectral dimension demonstrates a smooth transition from a value of about 2.4 at short scales up to a value of about 3.0 at larger values of , and then starts to fall for greater than about 400 .fitting the data to a function of the form we obtain fitting the data to , as suggested in , we obtain from this , we get and , which agrees well with and reported in .the smooth reduction of the dimension of the spacetime at shorter scales is one of the most interesting features of the cdt model .this is an indication of the highly non - classical behavior at small scales .the reduction in dimension for larger is due to the finite ( and rather small ) volume of the spacetimes being considered .the spectral dimension vs step size plot for a ( 3 + 1)-dimensional spacetime is shown in figure [ fig:3p1-spectral - dimension ] .  ,   the spectral dimension in this case ranges from a value of about 1.8 at short scales up to a value of about 4.0 at larger values of , and then starts to fall for greater than about 600 , with the functional form being given by : this is to be compared with the reported in .the spectral dimension of a spatial slice can be computed using a process identical to that used for determining the spectral dimension of the full spacetime .we select the slice with the maximum -volume and initiate diffusion processes from a randomly selected -simplex on this slice .the spectral dimension of the maximum volume slice for a ( 3 + 1)-dimensional spacetime is show in figure [ fig:3p1-sss ] . for short scales ,up to , the return probability differs considerably for even and odd step sizes , and the plot bifurcates .once the even and odd step size spectral dimensions converge , is approximately 1.5 , before a gradual reduction due to finite size scaling . in , a spatial slice spectral dimension of reported .this is value is computed by calculating the average return probability of all the spatial slices .the agreement of our results with value reported in indicates that each spatial slice has roughly the same spectral dimension .figure [ fig:2p1-sss ] shows a corresponding plot for ( 2 + 1)-dimensional spacetime . in this case , the even and odd plots converge at a much larger value of .the spectral dimension value then levels off at about 1.65     at the beginning of this section , we discussed the emergence of an extended phase , which we said was a solution to the classical einstein equations .however , the only evidence we have presented so far are figures [ fig : extended - phase ] and [ fig:3p1-phase - c ] , where the shape of the volume profile is indicative of such a solution .there is a rigorous , quantitative procedure , in which volume profiles of the spatial slices are fit to the euclidean de sitter solution , in 2 + 1 and 3 + 1 dimensions .this procedure is described in detail in .we have been able to verify these results in 2 + 1 and 3 + 1 dimensions ; our analysis will be presented in .the causal dynamical triangulations approach to quantum gravity has many promising features , and our computer simulation has been able to successfully reproduce these results in 2 + 1 and 3 + 1 dimensions .most of these results were first reported by ambjrn , jurkiewicz , and loll , and our simulation provides the first completely independent verification .we observe the emergence of a well defined extended geometric phase .this geometric phase shows a smoothly changing spectral dimension that ranges from a value close to 2.0 to a value larger than 3.0 for the ( 2 + 1)-dimensional spacetimes and from about 2.0 to 4.0 for the ( 3 + 1)-dimensional spacetimes .the spatial slices exhibit a non - classical behavior , as indicated by their spectral dimension plots .  our successful reproduction of many of the results reported by ambjrn , jurkiewicz , and loll , using a completely independent software implementation ,should improve confidence in the cdt model . with the exception of the value of the phase transition point in the ( 2 + 1)-dimensional model , our results are in good agreement with those reported in , and convince us of the overall validity of the cdt approach .having successful reproduced the basic results of the cdt model , we are now using our cdt implementation to explore some further interesting problems .these include investigating the quantum fluctuations in ( 2 + 1)-dimensional cdts and comparing the results with canonical quantization approaches , detailed analysis of the volume profiles in 2 + 1 and 3 + 1 dimensions and exploring horava - lifshitz gravity on the cdt lattice . in the near future , we plan to release our implementation into the public domain .i would like to thank professor steve carlip for his initial suggestion of cdt as a research topic , and for his guidance and patience for the duration of this project .i would like to thank the summer reu students , jun zhang , masha baryakhtar , david kamensky and christian anderson , for their valuable contributions to the project .i would also like to thank joshua cooperman , adam getchell , michael sachs and patrick zulkowski for helpful discussions regarding cdt and for their willingness to work with poorly documented code written in an unfamiliar programming language .

Abstract: the causal dynamical triangulation ( cdt ) approach to quantum gravity is a lattice approximation to the gravitational path integral . developed by ambjrn , jurkiewicz and loll , it has yielded some important results , notably the emergence of classical spacetime and short scale dimensional reduction . however , virtually all the results reported so far have been based on a single computer code . in this paper we present the first completely independent verification of the cdt algorithm , and report the successful reproduction of the emergence of classical spacetime and smooth reduction in the spectral dimension of the 2 + 1 and 3 + 1 dimensional spacetimes .