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recently there has been much experimental and theoretical interest in the properties of granular media . in such systems the thermal energy available is not sufficient to allow the rearrangement of a single particle and hence the system is effectively at zero temperature in the thermal sense . the fact that the problem is not trivial lies in the fact that such systems have an exponentially large number of such metastable states , which may be also called blocked or jammed configurations . edwards associated an entropy to these configurations @xmath3 where @xmath4 is the total number of metastable states of the system @xcite . it is reasonable to assume that in complex systems such as granular media @xmath5 is extensive meaning that @xmath6 where @xmath7 is the entropy per particle , alternatively one may work with an entropy per unit of volume which is clearly a more natural choice in granular media . because the system has an extensive number of blocked configurations , if it is prepared from a random initial state it will lower its energy via only energy lowering rearrangements until it becomes stuck in a metastable state . normally this first encountered blocked state will not be that of lowest energy ( or most dense packing ) . in order to change the state of the system an external perturbation such as tapping or shearing is required . in between perturbations the system relaxes into new configurations . a natural and practically very important question concerning this sort of dynamics is : what are the properties of the steady state regime obtained via such mechanical perturbation schemes ? recently it has been shown that spin glasses and ferromagnets on random graphs have an extensive entropy of metastable states and one may calculate this entropy at fixed values of the energy @xcite . therefore , though they are quite different physically to granular materials , these systems have an extensive entropy of metastable states as do granular media . the motivation of this paper is to see if one can understand certain steady state properties of mechanically perturbed systems in terms of their organization of metastable states . the possibility of using spin glasses as a paradigm for granular material was first introduced in @xcite . let us recall an example of an experiment on a system of hard spheres reported in @xcite . a system of dry hard soda glass spheres is placed in a glass tube . the system is tapped by using a piston to move the tube vertically through a sine cycle . the tapping parameter @xmath8 is defined to be the ratio of the maximal acceleration due to the piston in the cycle to @xmath9 the acceleration due to gravity . after an initial irreversible curve , obtained by increasing the tapping rate slowly , the system arrives on a reversible curve where the density is a monotonic function of @xmath8 , the highest packing densities being obtained at lowest tapping rate . numerical simulations on granular media @xcite reveal similar behavior ( though the irreversible part of the experimental curve corresponding to a loosely packed _ fluffy _ metastable state was not seen ) . it was also observed that at small tapping the relaxation to the final density is extremely slow and is well fitted by an inverse logarithmic decay of the form @xmath10 where @xmath11 ( the final density ) , @xmath12 , @xmath13 ( the characteristic relaxation time ) and @xmath14 are fitting parameters . it should however be remarked that the behavior of granular systems is strongly dependent on the tapping mechanism and that horizontal shearing @xcite leads to behavior qualitatively different to vertical tapping .    in this paper we extend and elaborate a preliminary report of the results of @xcite . the philosophy of the paper is to examine spin glasses as paradigms for granular media . here the quantity corresponding to the density is the energy of the system . we allow the system to evolve under a random sequential zero temperature single spin flip dynamics where only moves which reduce the energy are allowed . when the system is blocked we tap it with strength @xmath15 $ ] , that is to say each spin is flipped with a probability @xmath0 , the updating at this point being parallel . the system is then evolved by the zero temperature dynamics until it becomes once again stuck , the tapping is then repeated . physically this corresponds to assuming that in granular media the relaxation time to a new metastable state is much shorter than the time between taps . a similar , though not identical , tapping dynamics has also been introduced independently in the context of three spin ferromagnetic interactions on thin hypergraphs @xcite , also in the goal of studying the dynamics of granular media . we find that a stationary regime is reached after a sufficiently large number of taps , characterized by a steady state energy @xmath1 ( analogous to the stationary density  the same analogy as used in @xcite ) . the initial dynamics from the random initial configuration into the first metastable state is examined analytically for the one dimensional @xmath2 spin glass or ferromagnet ( the two are equivalent by a gauge transformation ) . we call this the initial fall and the average energy of the first metastable state visited @xmath16 is computed . we then develop a mean field theory for the dynamics under falling then tapping , interestingly this theory appears to be exact in the case of the one dimensional system and one may calculate @xmath1 within this scheme , the results being in excellent agreement with the numerical simulations . numerically we examine the tapping of spin glasses and ferromagnets of higher connectivity . for the spin glass we find that @xmath1 is , as in the experiments , a decreasing function of @xmath0 . for small @xmath0 we define the exponent @xmath17 by @xmath18 , with @xmath19 constant . in the one dimensional case we show analytically that @xmath20 , hence @xmath21 , whereas for spin glasses on thin graphs for connectivity superior to two we find that @xmath22 . however for @xmath23 we find that the time to reach the steady state is extremely long and not accessible numerically . in this slow dynamical regime we find a slow relaxation of the time dependent energy , reminiscent of that observed in experiments on granular media @xcite and hence compatible with eq . ( [ eqlog ] ) . in the case of the ferromagnet we find numerically that there exists a critical value @xmath24 of @xmath0 such that for @xmath25 , @xmath26 where @xmath27 is the energy of the ground state and the inequality is strict , and that for @xmath28 @xmath29 . hence in the ferromagnetic system there is a first order phase transition under tapping dynamics ( in contrast to the usual thermodynamic ferromagnetic transition in these systems which is second order @xcite ) . there have of course been many models studied to understand the compaction process in granular media @xcite , which reproduce many of the experimental features . here the spin glass is clearly far from a realistic realization of a granular media , however the fact that it has extensive entropy of blocked states and the obviously natural form of the tapping dynamics implemented makes it a natural testing ground for ideas about dynamics and possible thermodynamics of systems such as granular media . moreover , it has been argued in @xcite that the slow compaction regime is well explained if we assume that particles can rearrange themselves in such a way as to create a particle size void , which is quickly filled by a new grain . this mechanism involves crossing of energy barriers and leads to a logarithmic compaction before the asymptotic steady state regime @xcite . we expect that the local rearrangements which occur during the tapping dynamics on spin glasses random graphs will be lead to behavior analogous to the slow glassy dynamics of systems as granular media . of course one would ultimately like to obtain a theoretical understanding of the asymptotic , steady state regime of lightly tapped granular media . edwards has proposed @xcite that a light tapping dynamics on granular type systems leads to a steady state whose properties are determined by a flat measure over the blocked or metastable states satisfying the macroscopic constraints involved ( _ e.g. _ fixed internal energy and compactivity ) . this idea has recently attracted much interest and has been examined in the context of various models @xcite . in this paper we shall concentrate simply on the asymptotic energy of the final tapped state , the study of the dynamics leading to this final regime is deferred for further investigation @xcite . we shall see that the calculation of the edwards entropy as a function of energy gives us a possible explanation of the first order ferromagnetic transition . the models we shall consider are spin systems on random thin graphs . a random thin graph is a collection of @xmath30 points , each point being linked to exactly @xmath31 of its neighbors , @xmath31 therefore being the connectivity of the graph . the distribution of metastable states in these systems has been recently considered in @xcite . the spin glass / ferromagnet model we shall consider has the hamiltonian @xmath32 where the @xmath33 are ising spins , @xmath34 is equal to one if the sites @xmath35 and @xmath36 are connected and zero otherwise . the fact that the local connectivity is fixed as @xmath31 imposes the local constraints @xmath37 , for all sites @xmath35 . in the spin glass case the @xmath38 are taken from a binary distribution where @xmath39 with probability half and @xmath40 with probability half . in the ferromagnetic case @xmath41 . here we define a metastable state as a spin configuration where any single spin flip does not increase the energy of the system . mathematically the total number of these metastable states is expressed as : @xmath42 it should be pointed out here that the definition of metastable states is of course dependent on the dynamics of the system , in contrast with micro - states in classical statistical mechanics . whether or not , in certain cases , the information about the dynamics encoded in the calculation of the entropy of metastable states is enough to allow one to predict the properties of the steady state regime is an open question .    the fact that , in our definition of metastable states , we include the marginal case ( where the energy change is zero ) implies that here @xmath43 , the heaviside step function , is taken such that @xmath44 . in the context of granular media , where friction plays an important role , this is a natural choice as one certainly needs a non zero force in order to make a grain move . with this definition , the total number of metastable states of internal energy @xmath45 per spin is formally given by @xmath46 the corresponding edwards entropy per spin , at fixed energy @xmath45 per spin , is then given by @xmath47 we remark that by a gauge transformation the one dimensional ferromagnet and @xmath2 spin glass are equivalent and place ourselves for transparency in the context of the ferromagnet . let us remark that the zero temperature glauber dynamics of the one dimensional ferromagnet can be explicitly solved @xcite , here diffusion of domain walls occurs and the dynamics does not get blocked . in the glauber case one may close the dynamical equations , however here such a closure scheme does not seem possible . the zero temperature kawasaki dynamics ( conserving the total magnetization ) of the one dimensional ising model , where the system can freeze , has been solved in @xcite .    to solve the dynamics of the one dimensional ferromagnet we consider the dynamics from the point of view of the bonds . we define a fault of length @xmath48 to be a sequence of @xmath48 neighboring adjacent domain walls . the zero temperature dynamics takes place within these faults via the flipping of one of the @xmath49 spins contained between the @xmath48 domain walls . we define @xmath50 to be the indicator function that starting from bond @xmath35 there are exactly @xmath48 consecutive domain walls ( there being no domain wall on bond @xmath51 and no domain wall on bond @xmath52 but all the intervening bonds have a domain wall ) . in the initial configuration we take the probability that a given spin is different to its left neighbor ( that is to say the probability of a domain being between two spins ) to be @xmath53 . hence if @xmath54 we have an initially ferromagnetic configuration , if @xmath55 it is an antiferromagnetic configuration , the case @xmath56 corresponds to a completely random configuration of maximal entropy . the total energy of the initial configuration is then given by @xmath57 as the energy is given by the ground state energy @xmath58 plus two times the number of domain walls ( excitations ) . defining @xmath59 to be the probability that one has @xmath48 consecutive domain walls starting from a given site ( hence @xmath60 where @xmath61 indicates the average over the initial conditions ) , with the initial conditions introduced above one finds @xmath62 ( the length of the defaults has a geometric distribution ) . the initial energy per site of a configuration generated in this manner is therefore ( using the translational invariance of the system when @xmath63 ) given by @xmath64    for the distribution of initial configurations considered here we find therefore that @xmath65    we define by @xmath66 the average number of isolated domain walls left by a fault of @xmath48 consecutive domain walls after the zero temperature dynamics described above has finished . the final energy of the system per site @xmath16 is therefore @xmath67    by the definition of the spin dynamics , domain walls disappear by pairs of two neighboring domain walls . it is clear that @xmath68 , @xmath69 and we set @xmath70 . within such a fault the dynamics proceeds by flipping one of the @xmath49 spins between the domain walls . by recurrence , after a random flip we obtain @xmath71 we solve equation ( [ eq : rec ] ) by introducing the generating functional @xmath72 the resulting equation for @xmath73 is @xmath74 solving this with the appropriate boundary conditions one obtains    @xmath75    and substituting this into eq . ( [ eqef ] ) yields @xmath76 thus giving the result @xmath77    this yields a value of @xmath16 for the completely random initial configuration where , @xmath78 , of @xmath79 . in fact the value of @xmath16 is maximal for the case @xmath78 . for the totally antiferromagnetic initial condition , where @xmath80 , here we find @xmath81 . clearly when @xmath82 the system is already in its ground state and we find @xmath83 as we should . we note that these values ( and those for all @xmath53 ) have been checked with and are in perfect agreement with our numerical simulations . this calculation with the one dimensional ferromagnet demonstrates two important points :    * the final value of the energy @xmath16 depends strongly on the initial configuration . * the system does not fall into a state of energy corresponding to the maximum of @xmath84 . in @xcite it was shown that @xmath85 where @xmath86 is a concave function peaked at @xmath87 . hence even if the total number of metastable states is dominated ( in the statistical sense ) by those of energy @xmath88 , generic initial conditions always seem to lead to an energy lower than this @xcite . in @xcite the value of @xmath16 for a variety of zero temperature dynamics ( sequential , greedy and reluctant ) in the fully connected sherrington kirkpatrick ( sk ) spin glass model @xcite was studied , similar behavior was found . when the one dimensional system is tapped we find results in line with those described later for the spin glasses at higher connectivity . the curve of @xmath1 , the asymptotic stationary value of the energy at a given @xmath0 , is shown in fig.([fig1 ] ) from @xmath89 systems of size @xmath90 spins . the time taken to reach a stationary value for @xmath1 were rapid for larger @xmath0 but for small @xmath0 there is a very slow relaxation to the final asymptotic state which is of the form @xmath91 , where @xmath92 is the number of taps . this is easily understood as at very slight tapping order @xmath0 effects dominate at early times , this means that : ( i ) isolated pairs of domain walls within large domains are immediately destroyed once tapping is stopped . ( ii ) flipping a spin either side of a domain wall creates domain wall diffusion and with this annihilation by coalescence of domain walls . hence the dynamics at small @xmath0 and early times is qualitatively the same as that for the low temperature ising model coarsening @xcite .    in order to go beyond our first calculation of @xmath16 and solve the tapping dynamics we consider a mean field theory for the dynamics of a system of connectivity @xmath31 . we shall see that at @xmath93 this theory gives the analytic result ( [ ef ] ) and reproduces to within numerical errors the numerical tapping results . once again we concentrate on the dynamics on the bonds . for a given site define @xmath94 to be the difference between the number of unsatisfied and satisfied bonds . hence @xmath94 is the local field on the spin at this site and @xmath95 . if @xmath96 then the spin can flip bringing about the change @xmath97 . in addition we denote by @xmath98 the probability that the site of interest has local field @xmath94 after a total of @xmath99 attempted random sequential spin flips under the zero temperature falling dynamics ( that is to say the dynamics in between taps ) . we define @xmath100 and @xmath101 the probabilities that a given spin can flip conditional on the fact that the bond with a given neighboring site is not satisfied or satisfied respectively . formally we have    @xmath102    we may turn around this conditional probability using bayes theorem to obtain    @xmath103    given that a site has local field @xmath94 , it must have @xmath104 unsatisfied bonds and @xmath105 satisfied bonds . therefore we find that @xmath106 and @xmath107 putting these results into eq . ( [ eqfpm ] ) then gives @xmath108    if we are interested in the spin at site @xmath35 the possibilities between time @xmath99 and @xmath109 are    * the spin at site @xmath35 is chosen and @xmath110 , then the spin at site @xmath35 will flip and @xmath94 goes to @xmath111 . * the spin at site @xmath35 is chosen and @xmath112 , then the spin at site @xmath35 can not flip and @xmath94 does not change . * a neighbor of site @xmath35 with positive local field is chosen and so flips . in this case , @xmath94 goes to @xmath113 or to @xmath114 depending whether or not the bond with site @xmath35 was satisfied or not satisfied . * a neighbor of site @xmath35 with negative or zero local field is chosen and so does not flip . in this case , @xmath94 does not change * one chooses neither the spin at site @xmath35 nor any of its neighbors , and so @xmath94 stays @xmath94 .    assuming that the distribution at every site is given by @xmath98 and assuming independence between the values of @xmath94 from site to site ( the mean field approximation ) we obtain    @xmath115    in this equation we have to define @xmath116 , the choice compatible with the conservation of probability is @xmath117 . taking the limit @xmath118 we may introduce the continuous time @xmath119 and obtain @xmath120 the average energy per site at time @xmath13 is then given by @xmath121 and one can show that the above mean field equation ( [ eqn ] ) respects the exact identity for the evolution of the average energy per spin @xmath122 the case where @xmath93 ( the one dimensional case ) is accessible to analytic solution and we proceed by defining    @xmath123    one finds that    @xmath124    and the full mean field evolution equations become @xmath125    if we look for a stationary solution of ( [ eqn1 ] ) and ( [ eqn2 ] ) , we find @xmath126 , which expresses the fact that when @xmath13 is infinite , the system is in a metastable state . to solve ( [ eqn1 ] ) and ( [ eqn2 ] ) , we introduce @xmath127 so that @xmath128 . then @xmath127 obeys the equation : @xmath129 with the initial condition @xmath130 . then ( [ eqn2 ] ) becomes :    @xmath131    and @xmath132 and @xmath133 are given by    @xmath134    the probability to have a positive value for the local fields then goes to zero at infinite @xmath13 as expected and the limit of @xmath135 is @xmath136 . if we consider the geometric initial conditions used in the previous exact calculation of @xmath16 , the induced initial conditions are : @xmath137 , @xmath138 and @xmath139 . in this case we obtain @xmath140 reproducing the exact result ( [ ef ] ) . tapping the system with tapping probability @xmath0 , starting from the values @xmath141 , we obtain the new _ tapped _ values @xmath142 . defining @xmath143 , the relations between the old and _ tapped _ probabilities are :    @xmath144    then , after another zero temperature evolution of the system , it reaches a new local energy probability distribution with @xmath145 and :    @xmath146    at this stage of computation one should remark that this recursive equation contains one of the main features of our numerical simulations , that is _ reversibility_. indeed , the process involved in ( [ eqn4o ] ) will reach an asymptotic value which is independent of the initial conditions and depends on @xmath0 . hence in the steady state regime under tapping , the probability @xmath147 ( the subscript _ s _ indicating steady state ) for sites to have zero local field is solution of the fixed - point equation :    @xmath148    this equation can be solved numerically and the result is shown in fig . ( [ fig1 ] ) in comparison with the numerical simulations which we see is excellent . the small @xmath0 behavior of @xmath1 from ( [ eqn4 ] ) is : @xmath149 , indicating that in this case @xmath21 . given the mean field nature of the above calculation we have used we do not expect this approximation to correctly describe the approach towards the steady state , by direct comparison with the numerical simulations we have verified that this is indeed the case . let us remark here that a defect of the mean field approximation scheme is that it can not distinguish between a spin glass and a ferromagnet , this is clearly not a problem for the one dimensional situation where the two are identical . the systems which we study are @xmath2 spin glasses or uniform ferromagnets on random graphs with fixed connectivity @xmath31 . let us first recall some analytical results of @xcite . it has been found that the mean number of metastable states increases exponentially with the number of sites in both cases . in addition in @xcite an annealed approximation to the edwards entropy per spin of metastable states at fixed energy @xmath45 was carried out : @xmath150 which may be exact for the ferromagnet as the calculated entropy is always positive . moreover , there is an energy threshold @xmath88 above which the results are the same for the @xmath2 spin glass and the ferromagnet ; below the ferromagnet has more metastable states and a non - zero magnetization . hence , as far as the energy density of metastable states is concerned , both ferromagnet and spin glass are the same above @xmath88  that is the effect of loop frustration is negligible . in this regime , one also suspects that the zero temperature dynamics are the same . in particular , numerical simulations with @xmath89 samples of @xmath151 sites for connectivities of @xmath152 , @xmath153 and @xmath154 have found the same @xmath16 for the spin glass and ferromagnet with very good accuracy ( the relative error is about @xmath155 ) . the results are show in table ( 1 ) . the result of tapping experiments on the systems with @xmath156 is displayed in fig.([fig2 ] ) . there is some critical tapping rate , @xmath24 , above which the curves of @xmath1 versus @xmath0 are the same for the spin glass and the ferromagnet . moreover , the ferromagnet is subject to a phase transition under tapping dynamics at @xmath24 such that for @xmath157 , the steady state reached is the ground state . finite size effects have been studied and revealed that the transition is first order ( in as far that @xmath158 ) , in contrast to the usual thermodynamic ferromagnetic transition in these systems which is second order @xcite . for the ferromagnet in the region close to @xmath24 one finds an excellent scaling of the energy as a function of @xmath30 , @xmath159 as shown in the inset of fig . ( [ fig2 ] ) . this scaling may be used to optimize the determination of @xmath24 . the first order nature of the transition may be seen explicitly by looking at the histogram over time ( in the steady state regime ) of the average energy per spin at @xmath160 ; one sees in fig . ( [ fig3 ] ) two separated peaks in the distribution and not a single peak which splits into two as one would expect for a second order transition . near @xmath24 , for systems of finite size , there is therefore coexistence of the two phases . the time dependence of the average energy per spin in the simulation leading to the histogram fig . ( [ fig3 ] ) is shown in fig.([fig4 ] ) . one sees that the system tunnels between the two coexisting states . the typical time for this tunneling increases as the system size increases , indicating , in thermodynamic language , a _ free energy _ barrier between the two phases . as the system size is increased the occupation of the intermediate states of energy between the two phase of energy @xmath27 and @xmath161 ( between the two peaks in fig.([fig3 ] ) ) is suppressed . we have also measured @xmath1 by studying single systems of very large size ( @xmath162 ) , the results are shown in fig . ( [ fig5 ] ) . here again above @xmath24 the curve for the spin glass and the ferromagnet are completely indistinguishable and the ferromagnet reaches the ground state below @xmath24 . for such large sizes , one no longer sees a coexistence of two phases around @xmath24 as presumably the tunneling time has become much larger than the simulation time .    moreover , for @xmath162 , for @xmath163 the full temporal plots ( and not just the steady state values ) of @xmath164 and @xmath165 ( where @xmath92 is the number of taps ) are indistinguishable . for @xmath166 the two curves are identical up till a time @xmath167 , which depends on the initial configuration and the sequence of spins flipped during the tapping process , and diverge after @xmath167 , when the ferromagnetic system reaches quickly the ground state ( see fig.([fig6 ] ) ) . once the ferromagnetic system has broken the @xmath168 symmetry the easiest way to lower the energy is to flip the spins which are opposed to the global magnetization ( because they are more probable not to be in the direction of their local field ) until all the spins are @xmath169 or @xmath170 . identical behavior was found in the systems with @xmath171 and @xmath154 . the comparison of @xmath172 and @xmath173 for @xmath171 is shown in fig . ( [ fig5 ] ) . remark that if we compare the different values of @xmath24 when increasing @xmath31 , and considering only odd ( or even ) connectivities , we find that @xmath24 grows , and we expect that it goes to @xmath174 when @xmath31 is very large , as the metastable states are more and more magnetized when @xmath31 grows ( in the case of the fully connected ferromagnetic ising model , only the two ground states are metastable , so @xmath175 ) . the behavior of the spin glass systems is similar to that for the system with @xmath176 . the steady state energy @xmath172 is a monotonically decreasing and continuous function of @xmath0 . for small @xmath0 one finds that here @xmath177 giving @xmath22 in contrast with @xmath21 in the one dimensional case . _ a tentative explanation for the ferromagnetic transition _ : in @xcite it was also shown that for the ferromagnet the edwards entropy as a function of @xmath45 is concave for @xmath178 and convex for @xmath179 . the value of @xmath161 obtained from the tapping experiments are very close to those obtained for @xmath88 in @xcite , the energy at which @xmath180 becomes convex . the results are shown in table ( 1 ) . encouraged by this striking observation we will try to make a tentative link with a possible thermodynamics for such systems . if we imagine that the energy of the system is governed by a partition function inspired by the flat edwards measure over metastable states @xcite @xmath181 where @xmath182 is a lagrange multiplier corresponding to the inverse edwards temperature which depends solely on @xmath0 and not on @xmath45 and is a monotonically decreasing function of @xmath0 for @xmath183 $ ] . the monotonicity hypothesis is supported by the simulation results that @xmath1 decreases with decreasing @xmath0 . clearly the energy which dominates in the sum is that obeying @xmath184 however if this saddle gives a true maximum of the action one must have also that @xmath185 and hence the edwards entropy must be concave for the energy considered to be thermodynamically stable . hence for @xmath186 , this suggests that the only stable energy is the ground state .    if one would like to push the analogy of the thermodynamics of first order phase transitions to its limits and assume that @xmath182 is a continuous function of @xmath0 ( it is clear that if this is not the case one could trivially obtain a first order transition ) , one would also expect that the _ free energy _ of the two phases is equal at @xmath187 and hence @xmath188 where @xmath27 is the energy of the ground state of the ferromagnet given here by @xmath189 . it is also clear that @xmath190 hence one obtains @xmath191 however one also has that @xmath192 and hence one should obtain @xmath193 however the fact that the calculated annealed approximation for @xmath194 is convex for @xmath179 means that @xmath195 and hence the equality ( [ eq : fot ] ) is not respected . it is possible that the exact quenched calculation of @xmath194 would give a different value from the annealed calculation of @xcite , however one should not expect the result to be too different qualitatively to that of @xcite . in addition we remark that the annealed calculation @xmath196 gives an upper bound for the quenched value @xmath197 ( from jensen s inequality ) . also the value of @xmath88 found with this approximation seems to be greater than that found numerically for @xmath161 by a very small amount ( @xmath198 for @xmath199 ) . if we accept the possibility that @xmath200 , a more probable hypothesis is the collapse of validity of edwards hypothesis in the region of energy where the @xmath168 symmetry is broken on metastable states , as has been found for the three dimensional random field ising model in @xcite . let us mention here that we measured the energy in the simulations over a few hundred time steps after the energy appeared to stop to decay . to be sure that the systems considered here were in a stationary regime ( and that the energy was not decaying extremely slowly as a function of time ) we measured the correlation function at different waiting times @xmath201 ( the number of taps after the initial preparation of the system ) , that is to say @xmath202 in the stationary regime this should be a function only of @xmath92 . in out of equilibrium systems the fact that the system is not in equilibrium shows up strongly as aging in the correlation function _ i.e. _ @xmath203 ( see @xcite and references within ) , even though the energy may be decaying so slowly that it appears to have reached its asymptotic equilibrium value . the time translational invariance of @xmath204 is thus quite a rigorous test of whether the steady state regime has been attained . for example for the case @xmath205 , with the waiting times @xmath206 and @xmath207 is shown in fig . ( [ fig7 ] ) , one sees , clearly that after the appropriate translation of the @xmath92 axis , the two functions collapse perfectly onto one another . one also sees that the decay of @xmath208 ( in the longtime regime we can now eliminate the @xmath201 dependence ) is exponential at large @xmath92 and also that @xmath208 decays to zero , indicating a form of ergodicity in the system . as pointed out in @xcite , this behavior of the correlation function seems a necessary condition for the validity of the scenario of edwards , that under tapping all metastable states satisfying the relevant macroscopic constraints ( fixed energy and compactivity ) are equiprobable in the stationary regime of gently tapped or perturbed system .    in order to test the accuracy of the mean field approximation at high energies ( where the system does not distinguish between the ferromagnet and the spin glass ) , we have compared the value of @xmath16 obtained in the numerical simulations with the result @xmath209 obtained by numerical integration of eq . ( [ eqn ] ) . the comparison is in table ( 1 ) and we see that the agreement is quite good .    finally we mention that we have also examined the reversibility of the tapping mechanism . if the system is tapped for a sufficiently long time , compatible with the relaxation times discussed above , the system is completely reversible . this reversibility was found in the experiments in @xcite once the system had left the initial fluffy state . granular media are a natural example of systems having an extensive entropy of metastable states . in such systems the role of thermal fluctuations are negligible and in order to evolve one must apply some external tapping mechanism . one would ultimately like to be able to formulate some sort of thermodynamics for such systems . the proposition of edwards @xcite for a thermodynamics of such systems is an important step in this direction and has had some success @xcite but it has been shown not to be generically true @xcite . a more general understanding of the asymptotic states of tapped systems has far reaching implications for computer science as the tapping mechanism studied here is similar to certain algorithms used in optimization problems . we have presented what appears to be an exact calculation of the steady state energy of a tapped one dimensional spin glass or ferromagnet . for this problem we have obtained the fixed point equations for the distribution of local fields under tapping . these equations also explain the reversibility observed in the numerical simulations . in a wide context of models we confirm the observations of @xcite , that if one reduces the _ strength _ of tapping , then the compaction process , corresponding here to the reduction of the energy of the system , becomes more efficient . the existence of a first order type phase transition for tapped ferromagnets on random thin graphs is of great interest , the possible explanation using the calculations of @xcite on the edwards entropy for this system indicates the possibility that one may eventually construct a more general theory for the thermodynamics and even phase transitions in tapped systems . one is tempted to speculate that generically the convexity of the edwards entropy below a certain energy threshold ( denoted here by @xmath88 ) leads to a _ collapse _ to the ground state energy @xmath27 , the metastable states in the intervening energy values being unable to support a stable thermodynamics . in terms of granular media this sort of transition would correspond to a transition between a random close packed state to a crystalline close packed state . it would be interesting to find other systems ( both theoretical and experimental ) showing the same collapse phenomena in order to test this idea . finally let us mention some of the open questions posed by this study we believe to be of interest for future investigation . clearly a general goal would be , in the spirit of edwards , to develop a thermodynamics to describe the stationary regime of tapped systems such as those studied here . the exact results presented on one dimensional systems here provide a completely analytic understanding of the tapping dynamics which one may be able to rederive from static considerations . indeed it has been shown @xcite in this simple context that several steady state observables may be predicted using edwards measure . the phase transition found in the case of the ferromagnetic systems studied here is extremely novel , an analytic understanding of this phenomena would be desirable , perhaps there exists a percolation type argument which would allow one to evaluate @xmath24 . also of interest is the decay of a system towards its final steady state energy . the slow logarithmic decay described by eq . ( [ eqlog ] ) has been used successfully to fit the experimental data of @xcite and the simulation data of @xcite . our preliminary study of finite connectivity spin glasses and the sk model @xcite indicates the presence of a slow dynamical regime for small values of the tapping parameter @xmath0 which is also compatible with a slow logarithmic decay , but the curves can also be well fitted by power law decays ( with the same number of fitting parameters ) . there exist however phenomenological arguments @xcite and exact calculations and simulations on toy models @xcite which support logarithmic decay .      tab . 1 . comparison of the numerical values of @xmath88 and @xmath161 for different values of the local connectivity c. the result for @xmath88 when @xmath199 is a truncation of the analytical value @xmath210 . 2 . numerical simulations of tapping experiments for the spin glass ( c ) and the ferromagnet ( ( a ) , ( b ) and ( d ) ) for @xmath199 for @xmath211 ( ( c ) and ( d ) ) , @xmath212 ( b ) and @xmath151 ( a ) . the inset shows the scaling @xmath213 for @xmath214 for @xmath215 , @xmath211 and @xmath212 . single run for a ferromagnet of local connectivity c=3 , for @xmath151 at @xmath216 . one sees that because the size is not too large , the energy switches between two values , one not far from that of the ground state @xmath27 and the other not far from @xmath88 . 5 . numerical simulations of tapping experiments for @xmath199 ( ( c ) : spin glass , ( d ) : ferromagnet ) and @xmath171 ( ( a ) : spin glass , ( b ) : ferromagnet ) . here , we have computed the asymptotic energy for only one sample of very big size @xmath162 . the results are quite the same as for @xmath217 and do not change if we average over several samples , which indicates that at these sizes , we are very near the thermodynamic limit and we can study only single runs to compute the energy . comparison of the energy versus time ( number of taps ) for the @xmath218 spin glass ( a ) and the ferromagnet ( b ) for @xmath162 spins at @xmath219 . we have displayed the magnetization for the ferromagnet ( c ) , whose absolute value increases with time , whereas that of the spin glass remains zero . correlation function for the @xmath2 spin - glass with local connectivity @xmath199 versus number of taps @xmath92 for two values of the waiting time @xmath201 : @xmath220 and @xmath221 . the tapping value is @xmath222 , the system contains @xmath211 spins and we have averaged over @xmath223 samples . with triangles is shown the right part of the curve , which corresponds to @xmath221 , shifted to the left by @xmath224 , it superimposes perfectly over the curve for @xmath220 demonstrating the time translation invariance of the correlation function .
a zero temperature dynamics of ising spin glasses and ferromagnets on random graphs of finite connectivity is considered , like granular media these systems have an extensive entropy of metastable states . 
 we consider the problem of what energy a randomly prepared spin system falls to before becoming stuck in a metastable state . 
 we then introduce a tapping mechanism , analogous to that of real experiments on granular media , this tapping , corresponding to flipping simultaneously any spin with probability @xmath0 , leads to stationary regime with a steady state energy @xmath1 . 
 we explicitly solve this problem for the one dimensional ferromagnet and @xmath2 spin glass and carry out extensive numerical simulations for spin systems of higher connectivity . 
 the link with the density of metastable states at fixed energy and the idea of edwards that one may construct a thermodynamics with a flat measure over metastable states is discussed . 
 in addition our simulations on the ferromagnetic systems reveal a novel first order transition , whereas the usual thermodynamic transition on these graphs is second order . 
 0.5 cm    pacs numbers : 05.20 , 75.10 nr , 81.05 rm .    2