with the development of new architectures and massively parallel codes , molecular dynamics ( md ) simulations have been applied to systems of increasing sizes , up to billions of atoms , and times , up to a few nanoseconds  .but md simulations still can not reach the time and length scales at which some complex phenomena , such as reactive waves in molecular systems , occur .a variety of mesoscopic methods have been designed to stretch these scales by several orders of magnitude , with decreasing predictive power compared to md .they generally consider fewer degrees of freedom and allow for larger timesteps since they do not need to track the interatomic vibrations and use softer potentials .dissipative particle dynamics ( dpd ) has been introduced as a coarse - grained mesoscopic model  which represents groups of atoms , typically forming one or several molecules , by a single particle .dpd particles interact through some potential energy ( usually , a soft potential ) while dissipative and stochastic forces are added to take into account the missing degrees of freedom .the magnitude of the fluctuation and dissipation forces are related through a fluctuation - dissipation relation , ensuring the sampling of the canonical ensemble  .dpd can however only be used when the temperature in the system is fixed , which prevents its application to nonequilibrium situations such as shock waves .dpd with conserved energy ( dpde )  has been introduced to cover such situations .the coarse - grained internal degrees of freedom in dpd are represented in this model by a single variable , called internal energy , which exchanges energy with the external degrees of freedom through a dissipation and fluctuation mechanism .this ensures the conservation of the total energy in the system and allows for its use in nonequilibrium situations such as the simulation of shock waves and reactive waves  .  however , dpde has some shortcomings on its own .while it is possible to choose the friction parameters without perturbing the equilibrium properties , as long as an appropriate fluctuation - dissipation relation is satisfied , it is still unclear whether it is possible to retrieve various dynamical properties such as the equilibration time between the internal and external degrees of freedom , and transport coefficients of the fluid ( _ e.g. _ self - diffusion , thermal conductivity , shear viscosity , )  .moreover , the fluctuations in dpde do not scale with the level of coarse - graining chosen to model the system and it is also questionable if one can satisfactorily represent several non - bonded particles by a single coarse - grained particle  .recent works such as  aim at providing firm theoretical grounds for the replacement of a single molecule by a thermal blob obeying dpde - like equations .such methods thus remains quite atomistic in their validity domains and do not allow to freely choose the level of resolution to be used .on the other hand , smoothed particle hydrodynamics ( sph )  is a particular lagrangian method to solve the navier - stokes equations .it has become a popular method in astrophysics  and in meshless simulations of fluid dynamics  .it is easy to implement since particles are used as interpolation nodes and no mesh is involved .however it does not account for thermal fluctuations which can be of importance for phenomena such as hydrodynamic instabilities  or for small fluid particles .  smoothed dissipative particle dynamics ( sdpd )  has been introduced to overcome the shortcomings of the previously mentioned methods .it blends the sph discretization of the navier - stokes equations with the thermal fluctuations of mesoscopic models , and thus allows to deal with hydrodynamics at the nanoscale .sdpd has in particular been used to study colloids  or polymer suspensions  .it has been shown that sdpd yields a consistent diffusion coefficient for a colloid in a sdpd bath  at any resolution .one of the main improvement of sdpd over dpde is the ability to select the desired level of resolution in the model by playing with a parameter fixing the number of molecules one mesoparticle stands for .this paves the way for multiscale simulations based on a concurrent coupling between models at different coarse - grained level , such as md and sdpd  , sdpd and navier - stokes  or even sdpd at various resolutions  .our aim in this work is to justify the use of sdpd in a multiscale setting by showing that sdpd can reproduce the thermodynamic properties of a microscopic system at any resolution .the method relies on fitting an equation of state obtained by md simulations to convey the physical information needed in sdpd .we will handle both equilibrium properties and dynamic properties in a nonequilibrium situation .we also present a new numerical scheme which preserve the invariants of the system for the integration of the full sdpd dynamics involving the evolutions of the positions , momenta and internal energies .this article is organized as follows .we first present in section [ sec : equations ] a reformulation of the original sdpd equations in terms of the internal energy rather than entropy to increase the structural similarity between sdpd and dpde . in section [ sec : schemes ], we introduce a numerical scheme inspired by the ones used for dpde and study the energy conservation it provides . section [ sec : results ] is devoted to the numerical study of the size consistency .we show that we are able to reproduce the microscopic equilibrium properties such as pressure or temperature for a wide range of resolutions in section [ sec : pg ] and [ sec : lj ] .we also study the effect of the sdpd resolution in the simulation of shock waves in section [ sec : shocks ] .some conclusions are gathered in section [ sec : conclusion ] , while more technical arguments are postponed to the appendix .at the hydrodynamic scale , the dynamics of the fluid is governed by the navier - stokes equations , which read in their lagrangian form when the heat conduction is neglected ( for time and position in a domain ) : the material derivative used in the lagrangian description is defined as the unknowns are the density of the fluid , its velocity , its internal energy and the stress tensor : where is the pressure of the fluid , the shear viscosity and the bulk viscosity .  in the following ,we first present the sph discretization of the navier - stokes equations in section [ sec : sph ] before introducing the original sdpd equations proposed by espaol and revenga  in section [ sec : original - sdpd ] .we then formulate these equations in terms of internal energy variables in section [ sec : nrj - sdpd ] by considering a fluctuation / dissipation mechanism structurally very similar to the one appearing in dpde  .this reformulation in particular allows us to more easily construct stable and accurate numerical schemes which preserve the invariants of the system , such as the energy ( see section [ sec : schemes ] ) .it also provides a first step towards a more straightforward coupling with dpde although this issue is not discussed in this work . in section [ sec : thermo ] , we study the thermodynamic properties of the reformulated sdpd equations .   smoothed particle hydrodynamics  is a lagrangian discretization of the navier - stokes equations ( [ eq : navier - stokes ] ) on a finite number of fluid particles which play the role of interpolation nodes .these fluid particles are associated with a portion of fluid of mass .they are located at positions and have a momentum .the internal degrees of freedom are represented by an entropy .   in the sph discretization ,the field variables are approximated as a weighted average of their values at the particle positions .the weighting function is often referred to as a smoothing kernel function .the kernel function is generally chosen non negative , normalized as , regular and with compact support  .we denote by the smoothing length of the kernel so that if . in the sequel , we use the notation .a variety of smoothing kernels have been introduced for the sph discretization and compared in the litterature  .a precise analysis of the kernel properties is however made difficult by the superposition of the two approximations made in the sph discretization : the kernel approximation which consists in replacing the function by a smoothed version of it and the particle approximation where the integral in equation  is replaced by a summation over the particles . the kernel approximation can be studied analytically and its accuracy improved by the choice of higher order kernels .the particle approximation depends by nature on the particle configuration and can lead to undesired behaviors and instabilities such as particle clustering  .numerical simulations are thus required to evaluate the quality of the kernel . in this work , since we do not focus on an exhaustive study of the influence of the kernel on the thermodynamic properties , we considered only two kernels : the lucy function proposed early on  : and , alternatively , a cubic spline  , whose expression reads     \displaystyle \frac{16}{\pi h^3 } \left(1-\frac{r}{h}\right)^3 & \displaystyle \text{ if } \frac{h}{2 } \leq r \leq h,\\[1em ]    0 & \displaystyle  \text { if } r \geq h.   \end{array }  \right.\ ] ] the field variables are then approximated as where denotes the value of the field on the particle  .the approximation of the gradient is obtained by deriving equation , which yields in order to have more explicit expressions , we introduce the function such that . for the lucy function ( [ eq : sdpd - lucy - w ] ) , while , for the cubic spline ( [ eq : sdpd - cubic - w ] ) ,     \displaystyle \frac{48}{\pi h^5 } \frac1{r } \left(1-{\frac{r}{h}}\right)^2 & \displaystyle \text{ if } \frac{h}{2 } \leq r \leq h,\\[1em ]    0 & \displaystyle \text { if } r \geq h.   \end{array }   \right.\ ] ] the gradient approximation can then be rewritten as   in order to simplify the notation , we define the following quantities for two particles and : we can associate a density and volume to each particle as the corresponding approximations of the density gradient evaluated at the particle points read     -m \sum\limits_{j=1}^n f_{ij}{\boldsymbol{r}}_{ij } & \text { if } j = i .\end{array }  \right.\ ] ]   as in the navier - stokes equations , an equation of state is required to close the set of equations provided by the sph discretization .this equation of state relates the internal energy associated with particle with its density ( as defined by ) and its entropy through an internal energy function the equation of state can be computed by microscopic simulations or by an analytic expression modeling the material behavior ( see section [ sec : results ] for some examples ) .it is then possible to define , in accordance with the equation of state , a temperature pressure and heat capacity at constant volume (s,\rho).\ ] ] we assign to each particle the corresponding temperature , pressure and heat capacity as to simplify the notation , we omit in sections [ sec : eom - sph ] and [ sec : original - sdpd ] the dependence of , and on the variables and .the sph discretization can be split into two elementary dynamics , the first one being a conservative dynamics derived from the pressure gradient in the stress tensor  and the last one a dissipative dynamics stemming from the viscous terms in .  the elementary force between particles and arising from the discretization of the pressure gradient in the navier - stokes momentum equation reads the corresponding total force is actually conservative since it can be rewritten in a hamiltonian form as where this allows us to write the conservative part of the dynamics in hamiltonian form as this dynamics preserves by construction the total momentum and the total energy .in order to give the expression of the viscous part of the dynamics , we define the relative velocity for a pair of particles and as the viscous terms in the momentum continuity equation result in an elementary pairwise dissipative force where the friction coefficients are defined from the fluid viscosities and appearing in the stress tensor ( [ eq : stress - tensor ] ) as the pairwise dissipative elementary dynamics can be written as the third equation on the entropy is such that the total energy is preserved .in addition , galilean invariance is ensured by the dynamics ( [ eq : sdpd - diss - espanol ] ) since .   smoothed dissipative particle dynamics  is a top - down mesoscopic method relying on the sph discretization of the navier - stokes equations with the addition of thermal fluctuations which are modeled by a stochastic force .sdpd is a set of stochastic differential equations for the variables used in the original sdpd equations : the positions , the momenta and the entropies for .the fluctuation terms read where the fluctuation amplitudes are given by these coefficients are determined in  through the generic framework  . here , for , are matrices of independent standard brownian motions such that .we denote by the symmetric traceless part of .the random fluctuation term is balanced by an extra dissipative force with   the dissipative and stochastic forces produce a variation of the entropy which is accounted for by a dynamics on the entropy in order to ensure the conservation of the energy .the elementary fluctuation / dissipation dynamics reads where .the dynamics ( [ eq : sdpd - fluct - espanol ] ) replaces the dissipative dynamics ( [ eq : sdpd - diss - espanol ] ) of the sph discretization .it can be shown that it preserves the energy and the total momentum .finally , the complete set of equations of motion for the original sdpd  is obtained by concatenating the conservative dynamics ( [ eq : sdpd - cons ] ) and the fluctuation / dissipation dynamics ( [ eq : sdpd - fluct - espanol ] ) as {{\rm d}}t \\   & + \left(a_{ij}{{\rm d}}{\boldsymbol{\overline{w}}}_{ij}+\frac13b_{ij}{\rm tr}({{\rm d}}{\boldsymbol{w}}_{ij}){{\boldsymbol{\mathrm{id}}}}\right){\boldsymbol{e}}_{ij},\\   t_i{{\rm d}}s_i = & \sum_{j\neq i } \frac12\mathscr{d}_{ij}\left(a_{ij}{\boldsymbol{v}}_{ij}^2+\left(\frac{a_{ij}}3+b_{ij}\right)({\boldsymbol{v}}_{ij}^t{\boldsymbol{e}}_{ij})^2\right){{\rm d}}t\\   & -\frac{8k_{\rm b}}{m}\frac{t_it_j}{t_i+t_j}\left(\frac{10}3a_{ij}+b_{ij}\right){{\rm d}}t\\   & -\frac12 { \boldsymbol{v}}_{ij}^t\left(a_{ij}{{\rm d}}{\boldsymbol{\overline{w}}}_{ij}+\frac13b_{ij}{\rm tr}({{\rm d}}{\boldsymbol{w}}_{ij}){{\boldsymbol{\mathrm{id}}}}\right){\boldsymbol{e}}_{ij}.  \end{aligned }  \right.\ ] ] the dynamics  preserve the total momentum and the total energy since the elementary dynamics ( [ eq : sdpd - cons ] ) and ( [ eq : sdpd - fluct - espanol ] ) conserve these invariants .the generic framework  ensures that measures of the form are invariant for the dynamics .we propose in this section a reformulation of the original sdpd equations ( [ eq : sdpd - espanol ] ) in terms of positions , momenta and energies .the corresponding phase space is denoted by .we also propose a somewhat simpler expression for the fluctuation term in analogy with the stochastic term used in dpde  .the new expressions for the fluctuation / dissipation forces are chosen to ensure the same invariant measure as in the original sdpd equations .moreover , the same friction forces appear in both formulations .the interest of this reformulation is twofold : first , it allows for a better control of the energy conservation in the integration scheme , second , it is a first step towards a more straightforward coupling with dpde .  as in section[ sec : sph ] , we need an input equation of state in order to close the equations . in the energy formulation , the equation of state links the entropy with the internal energy and the density of the particles as the pressure , temperature and heat capacity are then determined accordingly as (\varepsilon,\rho),\\   \mathcal{p}(\varepsilon,\rho ) & = -\frac{\rho^2}{m}\left[\frac{\partial_{\rho}\mathcal{s}}{\partial_{\varepsilon}\mathcal{s}}\right](\varepsilon,\rho),\\   \mathcal{c}(\varepsilon,\rho ) & = -\left[\frac{(\partial_{\varepsilon } \mathcal{s})^2}{\partial_{\varepsilon}^2\mathcal{s}}\right](\varepsilon,\rho ) ,  \end{aligned}\ ] ] and assigned to each particle as as in the previous sections , we now omit the dependence of , and on the variables and .we keep the conservative part of the dynamics ( [ eq : sdpd - cons ] ) . in order to reformulate it in terms of internal energies ,we compute the associated variation in the energy , expressed in the new set of independent variables as since a change of variables should not change the energy , defined by  and are related by the dynamics ( [ eq : sdpd - cons ] ) being isentropic , the induced energy variation is simply given by from the definition ( [ eq : sdpd - rho - v ] ) of the density and volume of a particle , the infinitesimal volume variation reads with given by therefore , the conservative part of the dynamics can be rewritten as let us emphasize that the indices and do not play a symmetrical role in the evolutions of the internal energies , although the conservative forces are symmetric .this asymmetry is a consequence of the fact that the energy variation ( [ eq : energy - variation ] ) for a given particle only involves the volume variation of this particle .  in the spirit of dpde, we choose a pairwise fluctuation and dissipation term for of the following form {{\rm d}}t -\frac12 { \boldsymbol{v}}_{ij}^t{\boldsymbol{\sigma}}_{ij}{{\rm d}}{\boldsymbol{b}}_{ij},\\   { { \rm d}}\varepsilon_j & = \frac12\left[{\boldsymbol{v}}_{ij}^t{\boldsymbol{\gamma}}_{ij}{\boldsymbol{v}}_{ij } - \frac{{\textrm{tr}}({\boldsymbol{\sigma}}_{ij}{\boldsymbol{\sigma}}_{ij}^t)}{m}\right]{{\rm d}}t -\frac12 { \boldsymbol{v}}_{ij}^t{\boldsymbol{\sigma}}_{ij}{{\rm d}}{\boldsymbol{b}}_{ij } ,  \end{aligned }  \right.\ ] ] where is a -dimensional vector of standard brownian motions , and are symmetric matrices . in the dynamics, the equations acting on the momenta preserve the total momentum in the system .furthermore , as in dpde , the equations for the energy variables are determined to ensure the conservation of the total energy . as , it calculus yields the resulting equations in .  for the friction and fluctuation coefficients , we consider matrices of the form with the projection matrices and given by   using such a decomposition for the friction and fluctuation matrices , the dynamics ( [ eq : sdpd - simple - fluct ] ) is obtained by superposing the following dynamics for : {{\rm d}}t\\   & -\frac12 \sigma_{ij}^{\theta}{\boldsymbol{v}}_{ij}^t{{\boldsymbol{p}}}_{ij}^{\theta}{{\rm d}}{\boldsymbol{b}}_{ij},\\   { { \rm d}}\varepsilon_j = & \ , \frac12\left[\gamma_{ij}^{\theta}{\boldsymbol{v}}_{ij}^t{{\boldsymbol{p}}}^{\theta}{\boldsymbol{v}}_{ij } - \frac{(\sigma_{ij}^{\theta})^2}{m}{\textrm{tr}}({{\boldsymbol{p}}}_{ij}^{\theta})\right]{{\rm d}}t\\   & -\frac12 \sigma_{ij}^{\theta}{\boldsymbol{v}}_{ij}^t{{\boldsymbol{p}}}_{ij}^{\theta}{{\rm d}}{\boldsymbol{b}}_{ij}.  \end{aligned }  \right.\ ] ] the choice with the coefficients and defined in section [ sec : eom - sph ] ,ensures that measures of the form are left invariant by the elementary dynamics  ( see appendix [ sec : app - minv ] for the proof ) . note that is just obtained from the measure defined in  by the change of variables . as our derivation shows , other choicesare possible for the coefficients and ( see  in section [ sec : minv - fd ] ) .this may be of interest since it is possible to choose a constant fluctuation magnitude while the friction coefficient still depends on the configuration of the system through the positions and the internal energies and as +  \frac{(\sigma_{ij}^{\theta})^2}{k_{\rm b } } \frac{t_i+t_j}{t_it_j } \right).\ ] ] such a choice would further increase the similarity with dpde and simplify the numerical discretization .however , in this work , we stick to the choice ( [ eq : sdpd - gamma - sigma ] ) which yields the same friction terms as in the original sdpd equations .  as a result , the sdpd equations of motion reformulated in the position , momentum and internal energy variables read {{\rm d}}t\\   & - \frac12 { \boldsymbol{v}}_{ij}^t{\boldsymbol{\sigma}}_{ij}{{\rm d}}{\boldsymbol{b}}_{ij } ,  \end{aligned }  \right.\ ] ] with and given by  and ( [ eq : sdpd - gamma - sigma ] ) .the dynamics  preserves the total momentum and the total energy since all the elementary sub - dynamics ensure these conservations ( see appendix [ sec : app - cons ] for the proof ) .let us also emphasize that the reformulated dynamics involves only a -dimensional brownian motion for each pair instead of the -dimensional brownian motion appearing in the original dynamics .we present in this section expressions for the estimators of thermodynamic quantities like temperature and pressure .following the same ideas as for dpde  , we rely on a thermodynamic equivalence with an appropriate canonical measure to make the computations tractable .although the dynamics ( [ eq : sdpd - energy ] ) leaves any measure of the form  invariant ( as shown in appendix [ sec : app - minv ] ) , there are no mathematical results about its ergodicity since the fluctuation may be degenerate . even for dpd , ergodicity is known to hold only for simple one - dimensional systems  . since the total energy and the total momentum are conserved , we assume the ergodicity of the dynamics  with respect to the measure with the initial total energy , the initial momentum and a normalization constant . under this assumption , the average of some observable can be estimated as where is the solution at time of .we assume in the following that .this can be achieved by adopting the center of mass reference frame .  to justify the expressions of thermodynamic estimators of temperature , it is convenient to introduce the canonical measure +\frac{s_i(\varepsilon_i,{\boldsymbol{q}})}{k_{\rm b}}\right)}{t_i(\varepsilon_i,{\boldsymbol{q}})}\,{{\rm d}}{\boldsymbol{q}}\,{{\rm d}}{\boldsymbol{p}}\,{{\rm d}}\varepsilon ,  \end{aligned}\ ] ] where is chosen such that and a normalization constant . in the thermodynamic limit , and expected to be equivalent in the same way that the microcanonical and canonical measures are equivalent for systems described only in terms of and .  under the canonical measure ( [ eq : sdpd - minveq ] ) , the thermodynamic temperature can be estimated from the kinetic energy as which motivates the use of the kinetic temperature as an estimator of . under some assumptions on the equation of state , which hold for instance for the ideal gas equation of state ( see equation  below ) , namely {}-\infty,\\   & \quad \mathcal{s}(\rho,\varepsilon ) - k_{\rm b}\beta\varepsilon\xrightarrow[\varepsilon\to+\infty]{}-\infty ,  \end{aligned}\ ] ]the internal temperature also provides an estimator of the thermodynamic temperature since the temperature can therefore be estimated from the average of the internal temperature in the system as let us stress that the internal temperature estimator for sdpd relies on an arithmetic average in contrast to dpde where an harmonic mean should be used  .the thermodynamic pressure in the system is defined as the derivative of the free energy with respect to the total volume of the system . with the previous ergodicity assumption, the pressure can be estimated as where is the kinetic pressure and is the virial pressure a detailed proof of the equalities ( [ eq : estim - tint ] ) and ( [ eq : estim - pressure ] ) can be read in appendix [ sec : app - tp ] .one of the important feature of sdpd is that it is possible to prescribe a size for the particles , which enables a multiscale approach  . in the following ,we study the behavior of sdpd when the mass of the particles varies ( see section [ sec : results ] ) . in this perspective, the mass of the fluid particles is given by , where is the mass of one microscopic particle ( _ i.e. _ a molecule ) .it is in fact convenient to define a system of reduced units for each size : where is the mass unit , the length unit , the energy unit and the average density of the fluid .with such a set of reduced units , the time unit is   the smoothing length defining the cut - off radius in  and  also needs to be adapted to the size of the sdpd particles so that the approximations ( [ eq : sph - approx ] ) continue to make sense . in order to keep the average number of neighbors roughly constant in the smoothing sum, should be rescaled as in this work , we have taken , which correspond to a typical number of 60 - 70 neighbors , a commonly accepted number  .to our knowledge , there are very few works providing numerical schemes for the integration of the full energy - conserving sdpd such as gatsonis _ et al . _ who mix a velocity - verlet scheme for the update of positions and momenta and a runge - kutta scheme for the entropy updates . in such works ,no specific attention is devoted to the preservation of the invariants such as the energy .let us also mention that most of the work published for sdpd resort to a simplified version of the dynamics where internal temperatures are kept fixed at .the equations of motion are then integrated using a verlet scheme or more specific splitting schemes designed for sdpd  .these schemes are very much inspired by integration schemes designed for dpd such as the shardlow splitting scheme  .when considering the dynamics ( [ eq : sdpd - espanol ] ) or ( [ eq : sdpd - energy ] ) , the preservation of the invariants , especially the energy , requires some care in the design of the numerical scheme .other desirable properties include stability , accuracy and parallelizability .exhibiting a scheme satisfying all these constraints is not an easy task .there is , to our knowledge , no numerical scheme for sdpd able to meet these requirements . though the development of such an integration scheme is not the purpose of this work , we suggest in section [ sec : splitting ] a convenient scheme inspired by works done for dpde . the increased similarity of the reformulated dynamics ( [ eq : sdpd - energy ] ) with dpde makes it indeed possible to resort to similar integration schemes in both methods .we analyze the properties of the scheme in terms of energy conservation in section [ sec : vnum ] .a more detailed comparison with existing schemes ( such as  ) and adaptions of other dpde schemes to the sdpd setting is currently in progress .we propose a numerical scheme to integrate the stochastic dynamics ( [ eq : sdpd - energy ] ) obtained from the superposition of the elementary dynamics ( [ eq : sdpd - nrj - cons ] ) and ( [ eq : sdpd - simple - fluct ] ) .we denote by the time step .since the sdpd and dpde equations have a similar structure , we follow the ideas introduced for the discretization of the dpd equations  and their adaptation to the dpde model  .the corresponding schemes are splitting schemes , which are a popular method to integrate differential equations ( as proposed in  ) and stochastic differential equations .the scheme presented in the following is based on a trotter splitting of the dynamics .first the conservative dynamics  is integrated with a velocity - verlet scheme ( see  below ) during a time , then the fluctuation / dissipation part  is approximately evolved during a time by successive pairwise updates ( see  below ) .we first consider the conservative dynamics  formulated in terms of entropies as .since it is of hamiltonian nature , it is natural to solve this part of the dynamics with a symplectic scheme such as the well - known velocity verlet scheme  which is able to conserve the energy in average in the long term  : since the entropy of each particle is preserved by the integration of this reversible dynamics , the internal energy after this step can be computed by inverting the equation of state as with the updated density .if no analytic form is available for , a numerical inversion may be required ( see section [ sec : lj ] for more details ) .  to deal with the fluctuation / dissipation part, we use a modification of the shardlow scheme  which relies on a splitting of the fluctuation / dissipation into pairwise elementary dynamics , each pair being handled successively . if we consider the internal energies to be constant ,the equations on the momenta become an ornstein - uhlenbeck process that can be solved analytically . as suggested by marsh  , and later used in  ,it is possible to ensure the conservation of the energy by a redistribution of the kinetic energy variation induced by the dissipative and stochastic forces in the internal energies .we define the kinetic energy of the pair of particles and as we also introduce , for   for a random gaussian vector , we define the updated momenta as    { \boldsymbol{p}}_j  \end{pmatrix}\\  & \quad +  \sum\limits_{\theta\in\{\parallel,\perp\ } } { { \boldsymbol{p}}}_{ij}^{\theta}\left[\frac{m}2  ( \alpha_{ij}^{\theta}-1 ) { \boldsymbol{v}}_{ij } + \zeta_{ij}^{\theta } { \boldsymbol{g}}_{ij } \right ]  \begin{pmatrix }   { \boldsymbol{1}}\\[.5em ]   { \boldsymbol{-1 } }  \end{pmatrix } , \end{aligned}\ ] ] and the induced kinetic energy variation when changing to as we would like to stress that and include cut - off functions in their expressions , which limits the range of the fluctuation / dissipation interactions and allows an evaluation of the corresponding forces scaling linearly with the system size . the integration of the elementary fluctuation / dissipation dynamics  then proceeds as follows .we denote by and the momentum and internal energy at the moment just before the dynamics for the pair is integrated : and have thus been obtained by performing a step of the verlet scheme  and by integrating the previous pairs .we generate a vector of 3 standard gaussian variables .the integration of the pair consists in replacing by     \displaystyle \overline{\varepsilon}_i^{n , ij } + \frac12\delta_{{\boldsymbol{\pi}}}e_{\rm kin}({\boldsymbol{\overline{p}}}_i^{n , ij},{\boldsymbol{\overline{p}}}_j^{n , ij},\overline{\varepsilon}_i^{n , ij},\overline{\varepsilon}_j^{n , ij},{\boldsymbol{g}}_{ij}^n)\\[.5em ]    \displaystyle \overline{\varepsilon}_j^{n , ij } + \frac12\delta_{{\boldsymbol{\pi}}}e_{\rm kin}({\boldsymbol{\overline{p}}}_i^{n , ij},{\boldsymbol{\overline{p}}}_j^{n , ij},\overline{\varepsilon}_i^{n , ij},\overline{\varepsilon}_j^{n , ij},{\boldsymbol{g}}_{ij}^n )   \end{pmatrix}.\ ] ] as a consequence of the energy redistribution , the total energy is exactly conserved by these elementary updates .the final scheme for the integration of the reformulated dynamics consists in the superposition of the verlet scheme  and of the shardlow - like scheme  for all pairs .this scheme is essentially sequential as it requires considering each pair of particles one after another .the availability of parallel integration schemes is obviously crucial to apply the method to large physical systems .it is still a challenge to design an accurate and efficient parallel algorithm for this kind of stochastic dynamics .some recent results were obtained for dpde with a parallelization of shardlow - like algorithms  or with the splitting with energy reinjection integration ( ser ) which allows for an easy parallelization at the cost of a larger energy drift  .the ser scheme has been introduced for dpde but the similarities between the dpde and sdpd equations make it possible to adapt ser for the dynamics .  the two parts of the scheme separately ensure a good energy conservation : the velocity - verlet scheme  preserves energy in average while the algorithm  preserves it exactly .however , the overall scheme obtained by superposing  and  can lead to energy drifts as observed in dpde  ( see section [ sec : vnum ] ) .a standard numerical analysis shows that the scheme is of weak order 1 so that the error on average properties is of order  .   to validate our numerical scheme , we consider the ideal gas , whose equation of state is given by where is the size of the sdpd particles .we recall that this equation of state satisfies the conditions .the equation of state is formulated in the reduced units introduced in section [ sec : scaling - sdpd ] .we first check the energy conservation for different particle sizes .the scheme presented in section [ sec : splitting ] displays a linear drift in energy with respect to the simulation time ( see the inset in figure [ fig : sdpd - energy - drift ] ) .the results have been obtained by averaging over realizations of the dynamics for a total time for each time step and size .we choose a unitary mass  kg .the system is initialized as follows : the particles are initially located on a simple cubic lattice at density  kg.m .their velocities are distributed according to a normal distribution with variance corresponding to a temperature  k. the internal energies are chosen so that with the density evaluated from the initial distribution of the positions . unless otherwise specified , all the numerical results presented in this work are obtained by resorting to this initialization method .also , in all the simulations of this work , the bulk viscosity is neglected while the shear viscosity is chosen as  pa.s .  to analyze the error in the energy conservation , we compute , for each timestep and size , the drift rate defined as the slope of the energy drift .we evaluate by fitting a linear function on the energy with a least - square minimization .the drift rate is represented in figure [ fig : sdpd - energy - drift ] as a function of for , and .   as a function of the time step for the integration of sdpd with the scheme presented in section[ sec : splitting ] .the inset represents the average time - dependent energy drift for . ]assuming that the drift can be written as we find , by performing a least - square fit in a log - log scale , that is independent of , with .the prefactor varies with the mass of the particle . herewe estimate , and .this suggests a way to choose the timestep to obtain a given drift rate : we perform a preliminary run with some timestep and measure the drift rate in this simulation .the power law , independent of , then allows us to estimate the timestep needed to keep the energy drift below a given threshold .the results of figure [ fig : sdpd - energy - drift ] also show that , for larger particle sizes , we can increase the timestep in reduced units for a given drift rate , which allows us to integrate over longer times when using a coarse resolution .in this section , we study how the properties predicted by sdpd are influenced by the choice of the resolution , namely the size of the particles .we first study the size consistency of sdpd at equilibrium for both the ideal gas equation of state  ( see section [ sec : pg ] ) and an equation of state optimized for a lennard - jones fluid  ( see section [ sec : lj ] ) .we then consider nonequilibrium situations with the simulation of shock waves in section [ sec : shocks ] .we first study a sdpd system with the ideal gas equation of state  at different masses with  kg .we run simulations of a system of particles initialized according to the method described in section [ sec : vnum ] at density  kg.m and temperature  k. the number of iterations is fixed to .the timestep is chosen for each such that the drift in relative energy is less than after iterations , which gives for in reduced units .  under the invariant measure ,the momenta are distributed according to a normal distribution with mean and variance ( so that the variance of the velocities scales as ) , which is well recovered in our simulations .moreover , thanks to the analytic form of the equation of state , we can determine the theoretical distribution of the internal energies . by integrating out the positions and momenta in the invariant measure , the marginal law for the internal energy is given by where is the heat capacity in the equation of state  and is the gamma function .we check that we recover this distribution in our simulations for , and in figure [ fig : pg - sdpd - eint ] .   of the particles .the simulation results ( histograms ) are compared to analytic distribution ( solid line ) . ]we now study the evolution of the average pressure and temperature with respect to the particle size for the lucy kernel  and the cubic kernel .the pressure is estimated according to .there are two temperature estimators : the kinetic temperature  and the internal temperature .we compare the simulation results to the values given by the equation of state .small biases ( around ) are observed for small particle sizes but , as predicted by equations  and , the kinetic and internal temperatures are in excellent agreement with the equation of state as soon as .there is no theoretical result predicting a perfect agreement of the pressure obtained by sdpd with the equation of state .however , the results obtained from sdpd simulations with both the lucy kernel and the cubic kernel match the pressure from the equation of state with a maximum of difference for sizes varying from to . the thermodynamic limit for pressureis reached for sizes .let us however mention that we observe some metastability issues at large masses for the lucy kernel due to particle clumping .this leads us to prefer the cubic kernel in the following computations .+   we discuss in this section the size consistency of properties estimated for a more realistic fluid of lennard - jones type . in a fully atomistic model, particles interactions can be modeled by a pairwise potential of lennard - jones type : .\ ] ] we use the standard parameters for argon (  m ,  j ,  kg ) . for sdpd, we use the equation of state for lennard - jones fluids presented in  .it is based on microscopic simulations carried out with md in the nvt ensemble .the helmholtz free energy is fitted as a function of density and temperature on a modified form of the benedict - webb - rubin equation of state  , with 33 parameters .the useful quantities for sdpd , like the internal energy , the entropy , the pressure and the heat capacity , are then deduced from the free energy as since we use the internal energy as our primary variable , we perform a newton inversion algorithm to find the temperature corresponding to a given internal energy and density whenever we need to compute the associated pressure or temperature .denoting the temperature at iteration by , we initialize the algorithm with , with the thermodynamic temperature , and iterate until the relative residual decreases below a threshold . usually only a few iterations are required for convergence .the functional form of the equation of state  diverges for small temperatures .since a few particles can , at few occasions , reach this small temperature domain due to fluctuations , we continuously extend the equation of state for by choosing the heat capacity to be independent of temperature , _i.e. _ : we then write the energy and entropy in the regime as this enhances the stability of our simulations where we use  k.  we run simulations for systems of particles initialized at temperature  k and density  kg.m as in section [ sec : vnum ] .we use the cubic kernel and the reduced units defined in section [ sec : scaling - sdpd ] .the number of iterations is fixed to .the timestep is chosen for each such that the drift in energy is less than after iterations , which gives for in reduced units .we plot the distributions of normalized internal energies , densities and pressures for several masses ( see figure [ fig : ljc - dist ] ) .for each distribution , we find the appropriate scaling of their expected values and variance with respect to the particle size by fitting as a second - order polynomial in and as a power law in ( see table [ tab : rescaling - dist ] ) .+  +  the distributions are then rescaled as where is the distribution function for some quantity .the rescaled distributions are represented as an inset in figure [ fig : ljc - dist ] .we notice that , for , the rescaled distributions collapse to a single distribution independent of the particle size . as increases ,the mesoparticles stand for a larger collection of molecules , so that some effective averaging process takes place .standard results from probability theory suggest that the variances scale as and that the distribution should tend to a normal distribution .our numerical results are in excellent agreement with this prediction , the distributions becoming more symmetric for larger sizes with a variance inversely proportional to .we check the consistency of the sdpd simulations with the reference equation of state by plotting the average pressure , internal and kinetic temperatures with respect to the mass of the fluid particles in figure [ fig : lj - consistency ] .+  +  for large masses , the thermal fluctuations included in the sdpd equations are no longer significant and the estimated pressure converges to some limiting pressure which is lower than the pressure predicted by the equation of state .however , we also note a discrepancy in the estimation of the density as we observe an average mean density of  kg.m for instead of the expected value  kg.m .as the size of the particles decreases , the agreement between the pressure obtained from sdpd simulations and the pressure predicted by the equation of state remains within of difference even for or , _i.e. _ for masses of the order of only a few multiples of .we turn in this section to the study of the consistency of sdpd in nonequilibrium situations such as shock waves .we consider a system of particles initialized on a simple cubic lattice .we use periodic boundary condition in the - and -directions whereas two walls are located at each end of the system in the -direction . each wallis composed of 3 layers of `` virtual '' sdpd particles arranged in a cubic lattice , using the same ideas as bian _et al . _ . the positions and momenta of the virtual particlesare not updated with the dynamics but are kept fixed within the walls .these virtual particles enable us to evaluate the density of the sdpd particles in the neighborhood of the walls , as well as the conservative forces  acting on the actual particles . in order to ensure that walls are not permeable , these particles induce a repulsive force deriving from a truncated lennard - jones potential : \mathds{1}_{r \leq 2^{\frac16}\sigma_{\rm rlj}}.\ ] ] we set and in the reduced units .we use the lennard - jones equation of state  in these simulations .the system is initialized at a temperature  k and a density  kg.m . to produce a sustained shock wave in the system, the bottom wall is set in motion at a velocity  m.s in the -direction and continues moving at velocity throughout the simulation .we run the simulation with particle sizes varying from to .  in order to avoid the effects due to the presence of the walls, we only consider the information arising from particles located at a distance larger than from a wall .making use of the stationarity of shock waves in the reference frame of the shock front , it is possible to average profiles over time .we split the simulation box into a number of slices regularly distributed along the -axis and compute average quantities in the slices .we determine the position of the shock front at every step as the point where the mean particles velocity along the -axis is the closest to the velocity .the various profiles are then averaged by setting the position of the shock front as the reference frame ( ) .table [ tab : shock-500-ljv ] summarizes the main physical properties estimated with the simulations : the velocity of the shock front along with the thermodynamic properties in the shocked state ( the density , the pressure and the internal temperature ) .they are compared to the corresponding values obtained via direct md simulation and to the values predicted by the rankine - hugoniot relations applied to the equation of state  .assuming that the evolution can be described in an effective manner by a one - dimensional euler system ( in particular , viscosity effects can be neglected ) , the rankine - hugoniot conditions allow to predict the thermodynamic properties in the shocked state , knowing the initial thermodynamic state , the velocity of the shock wave and the velocity of the particles in the shocked region .these conditions are obtained from the conservation laws for mass , momentum and energy . the density , pressure and internal energy per unit mass in the shocked state are respectively predicted to be where , and are the density , pressure and internal energy per unit mass in the initial unshocked region .we find that sdpd gives similar results for all the resolutions which are considered .these results are consistent with the predictions obtained from the rankine - hugoniot relations. they also agree with md within a -difference margin for pressure , density and temperature .while no particular bias is observed for temperature , the density observed in sdpd is slightly higher and the pressure slightly lower than the md results .the shock velocity is also a bit underestimated and seems to decay for larger particles .since we consider a bulk material , the viscosity has no effect on the average properties in the shocked state .as such , we obtain quasi identical results when  pa.s ( as presented in table [ tab : shock-500-ljv ] ) or  pa.s  .[tab : shock-500-ljv]average observables in the shocked state .the wall velocity is fixed to  m.s and the viscosity parameter for sdpd is set to  pa.s . [ cols="^,^,^,^,^",options="header " , ]   following  , we compute the one - dimensional navier - stokes shock wave profile . in this 1d stationary setting , the navier - stokes conservation equations  simplify to simple differential equations : where is the distance to the shock front ( located at ) .the unknowns are the density , the velocity , and the internal energy per unit mass , while the pressure is given by the equation of state .we choose the unshocked material to be at and the shocked fluid at .this determines the velocity in the unshocked state to be and the velocity in the shocked state .equations  are integrated with a finite differences scheme on a domain 12 & 12#1212_12%12[1][0] in link:\doibase 10.1145/1362622.1362700 [ _ _ ] ( ) pp .link:\doibase  10.1098/rsta.2009.0218 [ * * ,  ( ) ] http://stacks.iop.org/0295-5075/19/i=3/a=001 [ * * , ( ) ] http://stacks.iop.org/0295-5075/30/i=4/a=001 [ * * ,  ( ) ] http://stacks.iop.org/0295-5075/40/i=2/a=141 [ * * ,  ( ) ] http://stacks.iop.org/0295-5075/40/i=6/a=631 [ * * ,  ( ) ] http://stacks.iop.org/0295-5075/76/i=5/a=849 [ * * ,  ( ) ] http://stacks.iop.org/0295-5075/78/i=6/a=68001 [ * * , ( ) ] http://stacks.iop.org/0295-5075/96/i=6/a=68007 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.94.014301 [ * * ,  ( ) ] \doibase http://dx.doi.org/10.1063/1.4891308 [ * * , ( ) ] link:\doibase  10.1021/jz500756s [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/144/6/10.1063/1.4941332 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.98.267801 [ * * , ( ) ] link:\doibase 10.1039/c5sm03038b [ * * ,  ( ) ] http://adsabs.harvard.edu/abs/1977aj.....82.1013l [ * * ,  ( ) ] http://adsabs.harvard.edu/abs/1977mnras.181..375g [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999184710345 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999197957764 [ * * ,  ( ) ] link:\doibase 10.1073/pnas.0702871104 [ * * ,  ( ) ] http://link.aps.org/doi/10.1103/physreve.67.026705 [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/130/3/10.1063/1.3050100 [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/pof2/24/1/10.1063/1.3676244 [ * * ,  ( ) ] http://link.aps.org/doi/10.1103/physreve.77.066703 [ * * , ( ) ] http://scitation.aip.org/content/aip/journal/jcp/130/2/10.1063/1.3058437 [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/142/4/10.1063/1.4905720 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s1877050913005851 [ * * ,  ( ) ] ,  http://scitation.aip.org/content/aip/journal/jcp/138/23/10.1063/1.4810754 [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/144/8/10.1063/1.4942499 [ * * ,  ( ) ] _ _ ( , ) http://www.sciencedirect.com/science/article/pii/s0377042702008695 [ * * ,  ( ) ] \doibase http://dx.doi.org/10.1006/jcph.1996.0128 [ * * ,  ( ) ] \doibase http://dx.doi.org/10.1016/j.jcp.2010.12.011 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999100964398 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999185710108 [ * * ,  ( ) ] link:\doibase 10.1111/j.1365 - 2966.2010.16577.x [ * * , ( ) ] http://journals.aps.org/pre/abstract/10.1103/physreve.56.6620 [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/144/2/10.1063/1.4937797 [ * * ,  ( ) ] http://www.worldscientific.com/doi/abs/10.1142/s0219493706001670 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999113006037 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/s0021999110001592 [ * * ,  ( ) ] http://dx.doi.org/10.1137/s1064827501392879 [ * * ,  ( ) ] http://dx.doi.org/10.1090/s0002-9939-1959-0108732-6 [ * * ( ) ] http://epubs.siam.org/doi/abs/10.1137/0705041 [ * * ,  ( ) ] http://link.aps.org/doi/10.1103/physrev.159.98 [ * * ,  ( ) ] _ _ , , vol .( , ) _ _ ,  ph.d . thesis ,  ( ) http://www.sciencedirect.com/science/article/pii/s001046551400109x [ * * ,  ( ) ] http://scitation.aip.org/content/aip/journal/jcp/135/20/10.1063/1.3660209 [ * * ,  ( ) ] link:\doibase 10.1093/imanum / dru056 [ * * ,  ( ) ] _ _ ,  ph.d . thesis ,  ( ) http://dx.doi.org/10.1080/00268979300100411 [ * * ,  ( ) ] link:\doibase 10.1017/s0962492916000039 [ * * , ( ) ] http://scitation.aip.org/content/aip/journal/jpcrd/2/4/10.1063/1.3253132 [ * * ,  ( ) ] link:\doibase 10.1103/physreve.47.r24 [ * * , ( ) ] http://scitation.aip.org/content/aip/journal/jcp/133/11/10.1063/1.3486088 [ * * ,  ( ) ] http://www.sciencedirect.com/science/article/pii/0021999183900360 [ * * ,  ( ) ]

Abstract: smoothed dissipative particle dynamics ( sdpd ) is a mesoscopic method which allows to select the level of resolution at which a fluid is simulated . in this work , we study the consistency of the resulting thermodynamic properties as a function of the size of the mesoparticles , both at equilibrium and out of equilibrium . we also propose a reformulation of the sdpd equations in terms of energy variables . this increases the similarities with dissipative particle dynamics with energy conservation and opens the way for a coupling between the two methods . finally , we present a numerical scheme for sdpd that ensures the conservation of the invariants of the dynamics . numerical simulations illustrate this approach .


the largest part of astronomical research directly relies on data taken at various kinds of telescopes of variable size - from those having mirrors of 8 or 10 meters in diameter to backyard amateur telescopes .however , serious comlications arise when one attempts to gain information on the internal structure of any kind of celestial bodies .the reason for these complications is simple : the interiors of * any * kind of objects are unobservable .for example , nobody has ever gone ( and nobody ever will ) into the center of jupiter to measure the temperature there .surfaces of many kinds of objects are observable . however , the `` state '' of a surface of any given object is a consequence of at least two factors : processes occuring in the interior of the object but also of any possible external influences .the aim of this lecture is to discuss to some extent one particular `` subfield '' within the field of modelling of the internal structure of celestial objects - the internal structure of the asteroids .apart from the introduction , this contribution has three more parts . in the following section a reviewwill be given of the present knowledge on the internal structure of the asteroids .in particular , a _ vocabulary _ of the definitions of the most important notions used in the literature on this topic will be presented .knowledge on the interiors of the asteroids will be discussed there from the point of view of astronomy .the third section is devoted to a discussion of possible applications of solid state physics to studies of the interiors of the asteroids .the fourth and final section contains the conclusions .the first asteroid,1 ceres , has been discovered on the first night of the century - january , by father giuseppe piazzi from the astronomical observatory in palermo .the number of discoveries slowly rose throughout the century while only visual techniques were used in searches . around wolf in wienna introduced photography in asteroid observation , which accelerated the pace of new discoveries .the total number of objects listed in the _ ast - dys _ data base kept at the department of mathematics of the university of pisa ( italy ) is at the time of this writing ( beginning of april 2006 ) around * a quarter of a milion * objects . to be precise ,on april  there were 129436 numbered and 112511 unnumbered asteroids .studies of asteroids are at present performed for two strong motives .fundamentally , it assumed in the research community that the asteroids represent nearly unchanged primordial material from the earliest phases of the existence of our planetary system . studying themgives the opportunity to get knowledge about the origin and chemical composition of the protoplanetary disk , its changes with heliocentric distance , various mixing processes ...  there exists also a `` practical '' motivation .namely , orbit calculations have shown that there exists a small group of objects called `` potentially hazardous asteroids '' ( pha ) .there are at present known pha objects . in their motionthey can come closer than a.u . to the earth or even less ( which amounts to nearly earth - moon distances ) .it is not impossible than one of these objects collides with the earth , provoking various dammage and destruction .the amount of destruction resulting from such a colision would be a function of the kinematical parameters of the motion of the object , its * chemical composition * and the geographical coordinates of the impact point .we have now encountered the first purely observational problem of modern astronomy related to asteroid interiors .high resolution observations show that most asteroids have irregular shapes , as if they were outcomes of a catastrophic collision .the importance of collisions in asteroid studies is demonstrated by the existence of families - groupings of asteroids with nearly identical orbits .for a recent review of asteroid family indentifications see , for example,(bendjoya and zappala,2002 ) .how does one determine the mean mass density of an asteroid without assuming the sphericity of its shape ?masses can be determined from asteroid , planetary and spacecraft perturbations .the volume can be determined from direct immaging from ground based observatories and/or spacecraft , and the bulk density can then be calculated in a simple way .this procedure has been applied to the biggest and best observed asteroids , and some of the results are presented in the following table ( hilton 2002 ) .ccccc name & density & + & [ kg m & + 1 ceres & + 2 pallas & + 4 vesta & + 45 eugenia & + 433 eros & +  taking into account the error bars , it follows that the mass density of asteroid eugenia may be * lower * than the density of water .another interesting example of a low density asteroid is patroclus .its density has recently been determined as kg m ( marchis et.al,2006),which is again lower than the density of water .these two asteroids are examples of a class of asteroids called `` rubble piles '' - they consist of rocks loosely bound together , with gaps between them .it is usually assumed that `` rubble pile '' asteroids are a result of catastrophic collisions and subsequent reaccumulation of the debris . for a recent review of the subject of catastrophic disruption of the asteroidssee for example ( michel , benz and richardson,2004 ) .a line of study aiming at drawing some conclusions on the interior of the asteroids is the analysis of their shapes .this demands extremely high resolution data , avaliable at just a handful of facilities throughout the world .an interesting example concerns ceres . using data avaliable in the middle of the eighties , and a particular theoretical framework ,the present author has concluded that this asteroid is undifferentiated ( elebonovi,1987 ) .however , a recent analysis of hst data shows that there are strong indications that ceres is differentiated ( thomas , parker , mcfadden et al.,2005 ) .another asteroid with differentiated interior is vesta ( for example keil,2002 ) .  studying the internal structure of small bodies opens the possibility of getting answers to several important scientific questions ( binzel , ahearn , asphaug et al.,2003 ) . briefly speaking ,some of these can be formulated as follows :  1 .the limiting size at which planetary processes determine the internal structure .planetary processes `` operate '' in the big planets , and allow for some degree of differentiation of their interiors . on the other hand ,in the small bodies , such as the asteroids , dominate material characteristics and deffects , which are completely irrelevant on planetary scale .the transition between the two regimes is estimated to occur in objects of the diameter of km ( binzel , ahearn , asphaug et al.,2003 ) . roughly speaking ,this is the diameter of ceres .one should note here that the possibility of differentiation is also composition dependent;a small asteroid made up of material with higher atomic mass will have more chances of having a differentiated interior .2 .  which objects are `` survivors '' from the formation stages of the planetary system ? such objects would provide a sort of `` windows '' to the shapes and internal structures created in accretion .it is almost certain the two such asteroids exist ; these are ceres and vesta . however ,identification of the smallest asteroids which survived since accretion is an open problem .3 .  what is the most probable size of rubble piles and monoliths ? a detailed answer to this question depends on the interplay between material strength and gravitation.it is virtually certain that all the smaller asteroids have been shattered by collisions .this problem involves a chain of issues related to material science so it is completely open to study form the viewpoint of solid state physics .  in this sectionwe have briefly reviewed the possibilities of `` pure astronomy '' in getting knowledge on the interiors of the asteroids .the remaining part of this contribution is devoted to the possibilites of solid state physis , both theoretical and experimental , in getting information on this subject .for a physicist , an asteroid is a solid body of known orbit , mass , volume , density , albedo , reflection spectrum .starting from the reflection spectrum it is possible , in principle , to derive the chemical composition of the surface .numerically , this means that ( in principle ) one can obtain the mean atomic mass of the surface layer , the ionic charge and the ionic mass of the surface of the object .observations show that collisions played an important role in the evolution of the planetary system , so at the beginning we have to introduce definitions of several notions .   * * stress * is the ratio of the force applied on a body to the cross section of the surface of a body normal to the direction of the force .* * strain * is the geometrical expression of the action of stress .  * * material strength * is the capacity of a material to withstand stress and strain .  * * porosity * is the ratio of the volume of all the pores on the object to the volume of the object .  * * tensile strength * is the force required to pull a material object to the point where it breaks up .  * * fragmentation * is the separation of a body into pieces under the action of stress .these definitions are in a sense `` borrowed '' from solid state physics .pertaining to asteroids proper , one frequently encounters the following notions :  * * monoliths * are asteroids consisting of one single block of material formed in the early stages of the solar system .  * * agregates * are asteroids made up of material of increased porosity compared to the monoliths .  * * shattered * asteroids are those whose interiors are dominated by joints and cracks .  * * rubble piles * are asteroids which were shattered ( by colisions ) and afterwards reassembled .standard solid state physics ( such as landau and lifchitz,1976 ) gives the following expression for the lattice energy per unit volume of a crystal equating this energy with the kinetic energy per unit volume exchnaged in a collision , one could get an estimate of the kinetic energy needed for complete destruction of an asteroid .all the symbols in eq.(1 ) have their standard meanings , and is the mean velocity of sound in the material .this quantity is related to the equation of state ( eos ) of the solid making up the asteroid by where and denote the pressure and the mass density of the material .as the aim is to relate asteroid studies as much as possible to solid state physics , the obvious question is how can one measure the pressure dependence of the value of .the instrument used in most high presure experiments in the last 50 years is the diamond anvil cell ( dac),presented on the following figure taken from ( jayaraman,1983 ) .the basic principles of functioning of a dac are discussed in the literature.a recent example is ( elebonovi,2006 ) .basically , a specimen a hydrostatic and a pressure sensor are inserted in a small hole drilled in a thin metal plate ( called _ the gasket _ ) which is inserted between the diamonds .the specimen is excited by an incoming laser beam , and its deexcitation spectrum is recorded at different values of external pressure . due to the small sizes of the diamonds ,the hole in the gasket is small , and so are the specimen and the pressure sensor . filling a dac ( that is , inserting the specimen , pressure sensor and some hydrostatic , in the hole in the gasket ) is extremely delicate .a dac can be used to experimentally determine the equation of state of any given material ( in asteroid studies the limitation is obviously to solids ) .the equation of state is usually expressed in the form of an expression linking the pressure and the density , so knowing it would give the possibility to determine the mean velocity of sound , and by eq.(1 ) the lattice energy per unit volume.equating this to the kinetic energy exchanged in a collision would give the kinetic energy needed for a complete destruction of an asteroid of given chemical composition . in principle , such a calculation can be performed , but its result would be experimentally useless.  a much more interesting calculation would be a determination of the energy needed to produce a fracture in a material of given chemical composition relevant for the asteroids.the basic aim of such a calculation would be the fracture stress - this is `` simply '' the stress needed to make a fracture in a given material .a small crack which for some reason occurs in a material will increase the stress in the region of the crack tip compared to the stress in the region far from the crack .a fracture in a material will occur when the stress at the tip of the crack becomes equal to the `` cohesive stress '' ( tilley,2004 ) .it can be shown that a fracture in a material occurs when the stress is equal to in this expression denotes young s modulus of the material, is the surface energy, is the radius of curvature of the crack tip, is the interatomic spacing at which stress is equal to zero and the length of the crack is .this equation is seemingly simple .however , each of the quantities which enter into it is in itself a result of a separate calculation of a different degree of complexity .the calculation of the young modulus starts from the expression for the lattice potential energy , which has the following form : where and are constants .the interatomic force is defined as and it can be shown ( tilley,2004 ) that the young modulus is given by the symbol denotes the separation between the atoms in the equilibrium configuration .  calculating the surface energy of any given material is a complicated task .studying the occurence of fractures in different types of asteroids , and connecting them to the collisions which produced them is , astronomically speaking , a very interesting subject .however , in advance of it , a series of calculations aiming at determining all the quantities in eq.(3 ) would have to be performed .this series of calculations is completely in the domain of solid state physics , and ( in a sense ) opens the door to an interplay of solid state physics and `` pure '' astronomy in asteroid studies .there exist different proofs that planets in our solar system have a certain number of layers in their interiors . for a recent review of the subject see , for example , stacey ( 2005 ) .the origin of these layers is basically ascribed to the influence of high pressure and/or temperature on the behaviour of materials in deep planetary interiors .related to the asteroids , the obvious question is are there physically reliable chances that differentiation occurs in their interiors ?quantum mechanics , laboratory and geophysical experiments show that high external pressure changes the electronic structure of atoms and molecules making up any material .observation ( in principle ) gives the mass , density and surface chemical composition of an asteroid .a simple calculation can give an estimate of the central pressure in the object .using the elementary definition of hydrostatic pressure , the central pressure in an asteroid of mass and radius can be estimated as on the other hand , quantum mechanics combined with statistical physics gives the possibility of estimating the pressure needed for changing the electronic structure of any given atom or molecule . in this way, one can theoretically estimate whether or not an asteroid made up of a material of mean atomic mass can be expected to be differentiated .the distribution of the pressure , density and temperature with depth can be obtained by any of the existing equations of state of a solid under high pressure .  high temperature can also change the electronic structure of atoms and molecules .physical reasoning shows that in collisions of asteroids transformation of a part of the kinetic energy of their relative motion into heat was certainly occuring . both asteroids participating in a collisionwould certainly heat up , the exact amount depending on their heat capacity . as a sort of `` experimental example '' it suffices to recall the recent impact of the deep impact probe into the nucleus of comet tempel 1.in this event the space probe disintegrated and actually evaporated , while the comet nucleus was heated and shaken .   in this lecturewe have so far discussed various aspects of gaining fundamental knowledge about the asteroids and , by analyzing them , on formation of the planetary system .there is also a very practical motive for the study of the asteroids , and this is the possibility that some of them impact into the earth and cause dammage of variable extent .of course , the possibility of collision with the earth exists also for comets or parts of asteroids .dammage occuring in such impacts could be of variable extent , ranging from localized consequences ( for example in the case of a meteorite falling on a house ) to a global catastrophe .such impacts have occured in history , and the best known and most recent example is the `` tunguska '' event from 1908 when a small asteroid exploded at a height of approximately kilometers ower the tunguska region of siberia .  thereexist now nearly 800 asteroids for which the probability of impacting into the earth is not identically zero .current data on this class of objects are avaliable at _ http://neo.jpl.nasa.gov_. studying their motion , predicting their close encounters with the earth and estimating the probability of impact in such passages is a problem in the domain of pure astronomy .hazards concerning the possibility of a neo asteroid impacting the earth have recently been discussed ( champan,2004 ) . in the remaining of this contribution we shall try to give some hints on what solid state physics and material science can contribute to the study of asteroid impacts into the earth .  for a material scientist , the collision of an asteroid with the earth can be simply described as the impact of a solid impactor into a bilayered target;the first layer of the target is a fluid ( the atmosphere ) and the second one is either fluid or solid ( that is the ocean or the continent ) .an asteroid approaches the atmosphere with certain velocity .the upper layers of the atmosphere , which it hits at first , are dense compared to the virtually empty interplanetary space .therefore , the asteroid heats up .the consequences of this heating depend on the heat capacity of the material of which the asteroid is composed , the speed and angle with which it hits the atmosphere and the density of the atmosphere .the heat capacity of a solid at low temperature is given by ( landau and lifchitz,1976 ) where is the mean velocity of sound in the interior of the asteroid , related to the equation of state of the material making up the asteroid and is the volume of the asteroid .  when it enters the atmosphere , the asteroid `` brakes '' .the resulting change of kinetic energy is spent on heating and forming cracks and fractures in the asteroid . equating the change of the kinetic energy of the asteroid to the sum of its heat capacity and the energy needed to form cracks and fractures in it ,gives the possibility to determine what will happen to the asteroid .that is , will it hit the surface unchanged , breakup into several pieces which will then hit the surface , or will it burn up in the atmosphere .if the incoming asteroid burns in the atmosphere , the only consequence on the surface will be the occurence of a shock wave , which will cause some localized dammage .the exact amount of dammage will depend on the speed , mass and chemical composition of the incoming object , and on the nature of the surface . a real `` experimental example '' of such an event occured in 1908 in the tunguska region of siberia .the shock wave which appeared destroyed the forest at a surface of more than km .no impact crater exists on the spot where the explosion occured .a small number of craters resulting from impact of asteroids in various geological periods has been identified on the earth ( kharif and pellinovsky,2005 ) .a more recent example of the impact of a comet into a planet was widely observed when comet shoemaker - levy impacted into jupier and left traces in its atmosphere .one can only immagine the consequences of such an impact into the earth .the situation is much more complicated in the case when the asteroid ( either intact or broken into pieces ) hits the surface of the earth .determining the characteristics of possible impactors is a complex long term task for astronomy but also solid state physics .astronomers can `` provide '' the optical spectrum of an object .however , transiting from the observed spectrum to the chemical composition and ( for example ) material strength is a problem for chemistry and solid state physics .there have been proposals to establish a data bank of physical and chemical data on possible impactors ( huebner and greenberg,2001 ) .if the impactor ( or possibly a `` chain of impactors '' ) hits the solid surface of the earth , it will certainly provoke heavy dammage on the spot where the impact occurs . an interesting question ( again for solid state physicists )is to determine the characteristics of a possible crater which will occur as a consequence of the impact .these calculations are extremely complex , because they have to take into account a multitude of factors : the velocity , impact angle , chemical composition , internal structure and porosity of the impactor , but also the chemical composition , porosity and other mechanical parameters of the soil .for a recent example of a theoretical study of crater formation taking into account the influence of porosity see ( wunnemann , collins and melosh 2006 ) .there have been attempts of experimental studies of impacts of small mass projectiles on large impacts .such experiments were often performed several decades ago , but for completely different motives ( for example , maurer and rinehart,1960 ) .  finally , if an asteroid imacts into any of the oceans , a tsunami will occur , the details depending on the multitude of parameters of the asteroid but also on the geographical coordinates of the impact .in this contribution we have discussed the problem of gaining knowledge on the internal structure of the asteroids .the main difficulty in this field is scarcity data on the masses , volumes and spectra of the asteroids .some known astronomical results were reviewed , and in particular a brief vocabulary of terms used in this field was presented .attempts were made throughout the text to indicate possibilities for people engaged in solid state and/or material science research to enter asteroid studies , because there exist numerous possibilities for their engagement .the author is grateful to professor osman demircan and the local organizing comitee for organizing a very interesting meeting in the beautiful setting of the ankara university summer campus .i am grateful to dr.zoran kneevi from the astronomical observatory in belgrade for helpful comments about a previous version of this manuscript .the preparation of this contribution was financed as a part of the research project 141007 of the ministry of science and protection of the environement of serbia .bendjoya , ph.and zappala , v .2002,in 2002 in asteroids iii , ed.by william.f.bottke jr.,alberto cellino , paolo paolicchi and richard p.binzel ( tucson : the university of arizona press ) p.613 .hilton , j.l .2002 in asteroids iii , ed.by william.f.bottke jr.,alberto cellino , paolo paolicchi and richard p.binzel ( tucson : the university of arizona press ) p.103 .marchis , f.,hestroffer , d.,descamps , p .2006 , nature,*439*,565 .michel , p.,benz , w.and richardson , d .2004,planet.space sci.,*52 * 1109 .elebonovi , v.1988,earth , moon and planets,*42*,297 .thomas , p.c.,parker , j.wm.,mcfadden , l.a.et al .2005,nature * 437*,224 .2002 in asteroids iii , ed.by william.f.bottke jr.,alberto cellino , paolo paolicchi and richard p.binzel ( tucson : the university of arizona press ) p.573 .binzel , r.p.,ahearn , m.,asphaug , e.et al .2003 , planetary and space sci.,*51 * 443 .landau , l.d.and lifchitz , e.m .1976,statistical physics,(nauka : moscow)p.217 .jayaraman , a .elebonovi , v .2006,in publ.obs.astron.belgrade * 80 * 131 and preprint astro - ph/0603067 at http://arxiv.org tilley , r. 2004,understanding solids : the science of materials,(j.wiley : chichester ) p.543 .stacey , f.d.2005 , rep.progr.phys.*68 * , 341 .champan , c.r .2004,earth and planet.sci.lett.,*222*,1 .kharif , c.and pellinovsky , e.,2005,c.r.physique,*6*,361 .huebner , w.f . andgreenber , j.m.,2001,adv.space.res.,*28*,1129 .maurer , w.c.and rinehart , j.s .1960,j.appl.phys.,*31 * 1247 .wunnemann , k.,collins , g.s . andmelosh , h.j . 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Abstract: knowledge of the internal structure of the asteroids is rather poorly developed , mainly due to lack of reliable data ( masses or densities and radii ) . this contribution has two aims : to review the basics of existing knowledge and explore the possibility of advancing the field by introducing and using in it some well known ideas from solid state physics .