You are an expert at summarizing long articles. Proceed to summarize the following text: 
classical spin models have called the attention of statistical - mechanics and magnetism researchers throughout many years  @xcite . many techniques have been used in their study , both analytical and numerical , in such a way that a reasonable knowledge of their equilibrium thermodynamics has been achieved , and specially , of their critical properties . among those models , one could mention the classical spin models , which present the xy ( @xmath17 ) and heisenberg ( @xmath18 ) as particular cases . an interesting formulation of a @xmath19-vector classical model comes when one adds a kinetic term to its hamiltonian , i.e. , the spin variables may be interpreted as classical rotators ( see , e.g. , refs . @xcite ) , so that the terminology `` inertial model '' is currently used . this additional term does not pose difficulties in the calculation of equilibrium properties within a canonical - ensemble approach , but it turns possible to derive equations of motion for each rotator , which can be integrated by means of a molecular - dynamics procedure . in this way , the dynamical behaviour of these models can be investigated numerically without the need of introducing any particular type of probabilistic transition rules for changing the microscopic states . the inertial ferromagnetic xy model was introduced in ref . @xcite , within a fully - coupled framework ( i.e. , infinite - range interactions ) , a limit where the mean - field approach becomes exact . mostly referred to as `` hamiltonian mean field '' ( hmf ) model , it became paradigmatic in the study of the dynamical behaviour of classical many - body hamiltonian systems , and it has given rise to a large amount of works  @xcite . one of the most interesting features in the hmf model concerns the appearance of metastable states , for some particular initial conditions , whose lifetime grows by increasing the total number of rotators , that could be interpreted as a breakdown of ergodicity in the thermodynamic limit . a generalization of the hmf model was proposed in ref . @xcite , by introducing a distance  @xmath1 between rotators at sites  @xmath2 and  @xmath3 of a given lattice . moreover , the two - body interaction was assumed to decay with  @xmath1 , like a power - law , @xmath4 ( @xmath5 ) . the index  @xmath6 controls the range of the interactions , in such a way that one recovers both the hmf and nearest - neighbour - interaction models in the particular limits @xmath7 and @xmath8 , respectively . in between these two limits , one finds an important change of behaviour in the thermodynamic quantities , yielding two physically distinct regimes , namely , the long- ( @xmath20 ) and short - range ( @xmath21 ) interaction regimes  @xcite . in the latter , one has the usual extensive and intensive quantities , whereas in the former , one may find also nonextensive thermodynamic quantities  @xcite . this generalization is usually referred to as  @xmath6-xy or  @xmath6-hmf model , and it also has been studied by several groups  @xcite . the dynamics of the fully - coupled inertial version of the ferromagnetic heisenberg model has been much less investigated in the literature  @xcite . metastable , or quasi - stationary states ( qsss ) , also occur for this system , similar to those that appear in the hmf model ; their lifetime diverge by increasing the total number of rotators  @xmath0 , which implies that the order in which the thermodynamic limit ( @xmath22 ) and the infinite - time limit ( @xmath23 ) are considered , becomes important . more specifically , if we let @xmath22 first , the system remains trapped in these qsss , never reaching the final boltzmann - gibbs ( bg ) equilibrium state , most probably being the qss itself the final state in such a case . in these metastable states , thermodynamical quantities , like temperature and magnetization , do not coincide with the canonical - ensemble predictions . herein , we will modify the fully - coupled inertial ferromagnetic heisenberg model , by introducing a distance  @xmath1 between rotators at sites @xmath2 and  @xmath3 of a given lattice . analogously to the @xmath6-hmf model  @xcite , the rotator - rotator interaction will be taken to decay with the distance like a power - law , @xmath4 ( @xmath5 ) . the interaction part of this hamiltonian has already been studied analytically in ref . @xcite , within a canonical - ensemble approach to the equilibrium state of the corresponding model , where it was shown that if the two - body interaction is appropriately scaled , a universal thermodynamical behaviour is achieved for @xmath24 , e.g. , relations involving temperature  @xmath25 , magnetization  @xmath26 , and internal energy  @xmath9 become @xmath6-independent in this interval . apart from this study of the equilibrium state , an investigation of the dynamical behaviour of the above - mentioned heisenberg model , and particularly , how the qsss may be affected by the exponent  @xmath6 , has never been addressed in the literature , to our knowledge . in the present work we study this model on a ring , i.e , @xmath27 with periodic boundary conditions , using molecular - dynamics simulations . we perform a detailed analysis of the qsss behaviour by varying the energy , number of particles , and range of the interaction , i.e. , the exponent  @xmath6 . in the next section we define the model , the appropriate scaling for the interactions , the canonical - ensemble solution , equations of motion , and initial conditions to be used in the molecular - dynamics procedure . in section  [ sec : results ] we present the results of our simulations , showing an agreement with some analytical results known in the literature for the equilibrium state , in some particular limits ; most importantly , we show the existence of qsss for energies  @xmath9 below criticality ( @xmath10 ) . such qsss , which were verified herein for initial conditions corresponding to zero magnetization , appear for values of  @xmath6 in the range @xmath12 , being characterized by durations  @xmath11 that increase for increasing values of  @xmath0 , like those found numerically and analytically in previous works of the similar model of xy rotators  @xcite . finally , in section  [ sec : conclusions ] we present our main conclusions . first , let us define the model in terms of  @xmath0 localized interacting classical rotators , on a @xmath28-dimensional hypercubic lattice , through the hamiltonian @xmath29 where @xmath5 and @xmath30 represents a vector with  @xmath18 components , assigned to the rotator at site  @xmath2 ( similar to a classical heisenberg spin variable ) , allowed to change its direction continuously inside a 3-dimensional sphere of unity radius , leading to the constraint @xmath31  . moreover , @xmath32 depicts the corresponding angular momentum ( we are considering moments of inertia equal to unit , so that angular momenta and angular velocities are equivalent quantities ) . the coupling constants may be identified as @xmath33 , where  @xmath1 measures the ( dimensionless ) distance between rotators  @xmath2 and  @xmath3 , defined as the minimal one , given that periodic boundary conditions will be considered . one should notice that the interaction term defined in eq . ( [ eq : hamiltonian_heisenberg ] ) is long - ranged for @xmath20 , in the sense that if  @xmath34 the partition function does not admit a well - defined thermodynamic limit  @xcite , e.g. , the internal energy per particle diverges in the limit  @xmath35 , so that the system is said to be nonextensive  @xcite . consequently , in order to calculate the thermodynamic limit adequately whenever  @xmath20 , the quantity  @xmath36 has to be defined appropriately to ensure a total energy proportional to the system size  @xmath0 . for  @xmath7 this is attained with  @xmath37 , the so - called kac s prescription  @xcite . for a general @xmath38 , the extensivity of the hamiltonian in  ( [ eq : hamiltonian_heisenberg ] ) is achieved through the following choice for  @xmath36 , @xmath39 this may be seen intuitively if we note that for  @xmath0 large , @xmath40 , if @xmath41 ( hence ,  @xmath42 for @xmath7 ) , @xmath43 , if @xmath44 , and @xmath45 , if @xmath46 ; therefore , this proposal yields the interaction term in the hamiltonian  ( [ eq : hamiltonian_heisenberg ] ) proportional to the system size  @xmath0 , for all @xmath47 . the need of the scaling in eq . ( [ eq : ntil_definicao ] ) , for systems characterized by long - range interactions , began to be realized within a generalized ferrofluid model  @xcite ( see also  @xcite ) . this proposal was applied successfully to a controllable - range interaction ( @xmath6-dependent ) curie - weiss model , where it was shown numerically that by considering such a scaling , the magnetization per particle follows a thermodynamical equation of state ,  @xmath48 , that becomes independent of  @xmath6 in the nonextensive regime  @xcite . moreover , applications of this scaling for lennard - jones - like systems were carried in ref . in addition to the aforementioned works , the correctness of the conjectural scaling of eq . ( [ eq : ntil_definicao ] ) has been verified in several other systems , suggesting that it should be valid for a wide variety of systems with long - range interactions  @xcite .    a systematic analysis of eq . has been particularly undertaken for the @xmath6-xy model  @xcite , whose hamiltonian is identical to eq .  , but @xmath17 , i.e. , @xmath49 is treated as a two - dimensional classical rotator , allowed to change its direction continuously inside a circle of unit radius . for this system , it was shown both analytically  @xcite and numerically  @xcite that , for @xmath24 , the prescription  ( [ eq : ntil_definicao ] ) associates to the @xmath6-xy model the same thermodynamical behaviour as the one previously known for the case @xmath7  @xcite . furthermore , the work of ref . @xcite has extended such a result to a general @xmath19-dimensional  @xmath49 ( @xmath50 ) unit vector . although the model of eq . ( [ eq : hamiltonian_heisenberg ] ) has been defined on a @xmath28-dimensional hypercubic lattice , from now on we will restrict ourselves to a ring , i.e. , a one - dimensional chain with periodic boundary conditions . based on its successful use for the above - mentioned systems , we will consider the scaling of eq . ( [ eq : ntil_definicao ] ) for the present problem as well .      within a canonical - ensemble solution , the kinetic term  @xmath51 of the hamiltonian in eq . ( [ eq : hamiltonian_heisenberg ] ) does not bring difficulties in the evaluation of average values . hence , the nontrivial part for the calculation of equilibrium properties of the hamiltonian  ( [ eq : hamiltonian_heisenberg ] ) comes from the interaction term  @xmath52 . this term may become troublesome to deal analytically for dimensions @xmath53 , and to our knowledge , it has been investigated analytically only for @xmath27 , i.e. , the linear chain , in some particular regimes of @xmath6 , namely , @xmath54 and @xmath8 . the model on an open linear chain with nearest - neighbour interactions ( limit @xmath55 ) was solved exactly in ref . @xcite , whose solution is equivalent to the one considering periodic boundary conditions , in the thermodynamic limit  @xcite ( see also refs . @xcite ) . on the other hand , the fully - coupled limit ( @xmath7 ) corresponds to the case where the mean - field approach becomes exact , so that the model may be solved through a relatively easy calculation ( see , e.g. , ref . @xcite ) . as mentioned above , using the scaling of eq . ( [ eq : ntil_definicao ] ) the solution for @xmath7 applies to all @xmath24  @xcite . therefore , considering the model of hamiltonian  ( [ eq : hamiltonian_heisenberg ] ) defined on a ring , i.e. , a one - dimensional chain with periodic boundary conditions , the solutions described above may be summarized , yielding for the internal energy and magnetization per particle respectively ( we work with units such that @xmath56 ) , @xmath57 } & ; \qquad \textrm{if}\hspace{0.3 cm } 0 \leq \alpha < 1~ , \\ \\ t \,+\ , \displaystyle{\frac{1}{2}}{\left[1-l{\left(1/2t\right ) } \right ] }     & ; \qquad \textrm{if}\hspace{0.3 cm } \alpha \to \infty~ ,   \end{array}\r.\ ] ]    @xmath58    where @xmath59 denotes the modulus of the magnetization vector , @xmath60 in these equations , @xmath61 represents the langevin function , @xmath62 and @xmath63 is the modified bessel function of the first kind . the analytical results presented above are displayed as solid lines in fig . [ fig : curvas_t_e_m ] , where one finds a continuous phase transition separating the ferromagnetic and paramagnetic states for values of  @xmath6 in the interval @xmath64 ; for  @xmath8 , there is no phase transition at a finite temperature , so that only the disordered phase with  @xmath65 exists . in the first case , at sufficiently high temperatures ( or equivalently , high enough energies ) , the directions of the  @xmath66 become randomly distributed , corresponding to the paramagnetic disordered phase , where one has the order parameter  @xmath65 . at the fundamental state all spins are parallel , corresponding to the ferromagnetically fully ordered case with  @xmath67 . hence , for all cases @xmath64 , in between these two regimes , a continuous phase transition occurs at a critical temperature @xmath68 , with an associated critical energy @xmath69 . herein we work with cartesian components for the spin variables and angular momenta , written respectively as @xmath70 and @xmath71 , yielding @xmath72 variables for each rotator . hence , the set of @xmath73 equations to be solved numerically can be cast in the following form  @xcite ,    @xmath74}~ ,      \\ \nonumber \\ \label{eq : equacoes_movimentos } \dot{{\mathbf{s}}}_i & = & { \mathbf{l}}_i \times { \mathbf{s}}_i~,\end{aligned}\ ] ]    for @xmath75 . one notices that , in the particular case @xmath7 , the expression for  @xmath76 used in previous works  @xcite is recovered , namely , @xmath77 [ where @xmath78 represents the magnetization per particle of eq . ( [ eq : magnetizacao_definicao ] ) ] . it is important to remind that eqs .   and  , written in the cartesian representation , are not canonical equations of motion , since @xmath79 and @xmath80 ( @xmath81 ) do not represent canonically conjugate pairs . alternatively , one could also work along the line of refs . @xcite , by using spherical coordinates in order to write the spin variable as @xmath82 and the squared angular momentum in the corresponding lagrangian as @xmath83 . in this representation , one can derive equations of motion through the usual hamiltonian formalism , where each rotator is characterized by two angles , @xmath84 and @xmath85 , and their canonically conjugate momenta , @xmath86 and @xmath87 . although the use of spherical coordinates , instead of the cartesian ones of eqs . ( [ eq : equacoes_movimentol ] ) and ( [ eq : equacoes_movimentos ] ) , seems to be a natural way to handle the problem , the denominator @xmath88 that appears in  @xmath89 becomes hard to be tackled numerically . indeed , as @xmath90 approaches zero , one needs to decrease the time step used in the integrations of the equations of motion , in such a way that the computational time may increase substantially . this difficulty has been discussed in the literature by several authors  @xcite , where numerical techniques for circumventing it were proposed . based on the above - mentioned reasons , herein we shall deal with the equations of motion expressed in terms of cartesian components , as in eqs . ( [ eq : equacoes_movimentol ] ) and ( [ eq : equacoes_movimentos ] ) . it is straightforward to verify that , in addition to the total energy , the total angular momentum , @xmath91 , as well as the norm of each spin , @xmath92}^{1/2}$ ] , are also integrals of motion . indeed , evaluating the time derivative of  @xmath93 and taking into account that  @xmath94 , we get @xmath95 similarly for @xmath96 , @xmath97 it should be stressed that the equation of motion  ( [ eq : equacoes_movimentol ] ) corresponds precisely to the euler equation for a linear rigid body of unit length ( and unit inertial moments ) . in such system the angular momentum  @xmath98 is always perpendicular to  @xmath49 ( the  molecular " axis ) , yielding the constraint @xmath99 , to be incorporated in the initial state . once this additional constraint is imposed , it will hold throughout the whole time evolution , since the product @xmath100 is also a constant of motion , as we can verify by using eqs . ( [ eq : equacoes_movimentol ] ) and ( [ eq : equacoes_movimentos ] ) , @xmath101      all our molecular - dynamical simulations were carried for a single copy of the system defined by eq . ( [ eq : hamiltonian_heisenberg ] ) , considering fixed values of the total number of rotators  @xmath0 and energy per particle  @xmath9 . since we are applying periodic boundary conditions , the model of eq . ( [ eq : hamiltonian_heisenberg ] ) becomes defined on a ring , with  @xmath1 corresponding to the minimal dimensionless distance between rotators  @xmath2 and  @xmath3 , taking the values @xmath102 , for each rotator  @xmath2 . to integrate the  @xmath73 equations of motion [ eqs ( [ eq : equacoes_movimentol ] ) and ( [ eq : equacoes_movimentos ] ) ] we have used a standard fourth - order runge - kutta scheme , with an integration step chosen in such a way to yield conservation of the energy per particle within a relative fluctuation smaller than @xmath103 ; this was achieved with a time step typically and larger values of  @xmath0 , by using a fast - fourier - transform algorithm . to implement this technique we have used the fftw library http://www.fftw.org . ] of  @xmath104 . the initial conditions used were such that each angular momentum component  @xmath80 ( @xmath81 ) was drawn at random from a uniform distribution , and then rescaled to yield zero total angular momentum for the whole system . in what concerns the variables @xmath79 , the initial conditions corresponded to those of zero magnetization , achieved numerically as  @xmath105 ( typically @xmath106 ) . for this , the components @xmath107 and @xmath108 were also drawn at random from a uniform distribution within the interval  @xmath109}$ ] , whereas the components @xmath110 were set in such way to preserve the constraint  @xmath111 @xmath112 , i.e. , through @xmath113)$ ] in these works , at the beginning of the simulations . however , it was verified numerically that @xmath114 was also a constant of motion , in such a way that eq . ( [ eq : ddtlisi ] ) remained true @xmath112 . ] this procedure leads to unnormalized spins , which are then normalized by dividing each component by @xmath116  ; finally , all @xmath80 s were rescaled again to obtain precisely the desired total energy  @xmath9 . consistently with eqs . ( [ eq : solucao_u ] ) and ( [ eq : solucao_m ] ) , initial configurations with zero magnetization like the one just described above , only applies to a total energy @xmath117 . although we are dealing with three cartesian components for the angular momenta ( @xmath118 and @xmath119 ) , due to the constraint  @xmath111 only two of these components are independent . let us then define the instantaneous kinetic temperature as given by @xmath120 . since we are working with a single copy of the system defined by eq . ( [ eq : hamiltonian_heisenberg ] ) , the time average at the equilibrium state should be identified with the thermodynamic temperature , @xmath121 , and in a similar way , one has for the equilibrium magnetization , @xmath122 . as it will be shown later on , such time averages should be computed after a sufficiently long time , where both  @xmath123 and @xmath124 fluctuate around the values predicted by boltzmann - gibbs ( bg ) statistical mechanics . in this regime , where ergodicity is expected to hold , time averages and ensemble averages should coincide .    in figure [ fig : curvas_t_e_m ] we present results from analytical calculations ( full lines ) and data from numerical simulations in the long - time limit ( symbols ) , for the controllable - range inertial heisenberg model of eq . ( [ eq : hamiltonian_heisenberg ] ) . in figure [ fig : curvas_t_e_m](a ) we represent the kinetic temperature versus the internal energy  @xmath9 , whereas in fig .  [ fig : curvas_t_e_m](b ) we do the same for the magnetization . as mentioned above , at equilibrium , an average over the quantity  @xmath123 is expected to coincide with the temperature  @xmath25 ( obtained from the equipartition theorem ) , so that for the analytical results presented , the vertical axis of fig . [ fig : curvas_t_e_m](a ) corresponds precisely to the equilibrium temperature  @xmath25 , whereas for the numerical ones , it represents the quantity computed from a time average . all data shown in fig .  [ fig : curvas_t_e_m ] for both temperature and magnetization coincide with the analytical results , showing that the equilibrium state considered numerically should be the one predicted by bg theory . the canonical - ensemble solution of the case @xmath7 ( see , e.g. , ref . @xcite ) is represented by the full line in fig . [ fig : curvas_t_e_m](a ) ( usually known as the caloric curve ) , presenting a discontinuity in its slope at the critical point @xmath125 , where the magnetization becomes zero . according to ref . @xcite , the equilibrium analytical solution of the case @xmath7 should apply to any @xmath126 , so that the corresponding full lines in figs .  [ fig : curvas_t_e_m](a ) and  ( b ) hold for any  @xmath6 in this range . such a universal behaviour was verified numerically for the @xmath6-xy model  @xcite , and it is now established herein for the heisenberg model , as displayed in fig . [ fig : curvas_t_e_m ] for typical values of @xmath126 . for completeness , in fig . [ fig : curvas_t_e_m](a ) we also present the solution corresponding to the quasi - stationary state  ( qss ) hat appears for @xmath10 , characterized by zero magnetization , leading to @xmath127 , and presenting a lower kinetic temperature than the corresponding equilibrium solution . it was essentially within the energy range @xmath128 that qsss have been investigated within the present numerical investigation . in the limit @xmath55 our model recovers the nearest - neighbour - interaction inertial heisenberg model on a linear chain , for which the interaction part was solved exactly in refs . @xcite , and shown to present zero magnetization for all @xmath129 ; consequently , its caloric curve exhibits the smooth behaviour indicated in fig . [ fig : curvas_t_e_m](a ) . in this case , our simulations for  @xmath130 are in full agreement with the results of the analytical solutions .        in figure [ fig : t_vs_tempo_varios_u ] we exhibit the time evolution of  @xmath131 ( conveniently rescaled in each case by the corresponding equilibrium temperature  @xmath132 ) for typical values of energies , @xmath133 , and @xmath134 , and @xmath135 . simulations were carried for a single copy of @xmath136 rotators , considering the above - defined initial conditions , for sufficiently long times ( up to times slightly larger than @xmath137 in some cases ) . for all energies investigated , in the range @xmath138 , we have verified the existence of qsss and after these , a crossover is observed to a state whose temperature and magnetization coincide with those obtained analytically within bg statistical mechanics . for a given value of @xmath6 one has that : ( i ) the lower energies have produced qsss with smaller durations  @xmath11 , and one observes that  @xmath11 increases substantially as one approaches the critical energy ; ( ii ) the gaps separating  @xmath131 of these qsss and their corresponding values of equilibrium temperatures  @xmath132 are larger for smaller energies , dropping to zero as @xmath139 . on the other hand , for a fixed total energy , higher values of  @xmath6 yield smaller  @xmath11 , and it will be shown later on that these durations tend to zero as @xmath16 . therefore , the qsss disappear in both limits @xmath139 and @xmath16 . the existence of qsss for energies below , but sufficiently close to  @xmath140 , is well known to occur in the hmf and @xmath6-xy ( with @xmath141 ) models , and it has been found analytically and numerically by many authors  @xcite . moreover , the most interesting characteristic of such states concerns an increase in their lifetime with the total number of rotators  @xmath0 ; this property will also be verified for the present model .        the qsss shown in fig . [ fig : t_vs_tempo_varios_u ] appear throughout the whole range @xmath12 , as exhibited in fig . [ fig : t_vs_tempo_varios_alphas ] for several values of  @xmath6 . each case corresponds to molecular - dynamics simulations of a single realization of @xmath136 heisenberg rotators defined by eq . ( [ eq : hamiltonian_heisenberg ] ) , with a total energy @xmath142 . one sees that the lifetime of these qsss decrease for increasing values of  @xmath6 in the range @xmath12 and that all qsss occur at a kinetic temperature corresponding to zero magnetization ( or slightly larger than zero ) , i.e. , @xmath143 . after these qsss , the system approaches the bg equilibrium temperature ( @xmath144 ) , associated to its energy through the caloric curve of fig . [ fig : curvas_t_e_m ] . at @xmath145 one notices a crossover from the existence of these metastable states to a smooth gradual increase in the kinetic temperature , such that for @xmath146 there is no clear evidence of qsss . finally , for a sufficiently high value of  @xmath6 ( e.g. , @xmath130 ) , the long - time behaviour of the kinetic temperature  @xmath131 gradually approaches the bg equilibrium temperature of the corresponding  @xmath55 model , i.e. , the nearest - neighbour - interaction model ( @xmath147 ) . our numerical results however do not exclude a possible finite - size dependence of these short - lived qsss for  @xmath148 , as already found in the  @xmath6-xy case  @xcite . however , no dependence on @xmath0 is expected for @xmath149 , since the interaction potential becomes effectively short range in this limit . an important question concerns the range of energies over which the above - mentioned qsss exist . in figure [ fig : curva_t_u_qss ] we present part of the caloric curve for @xmath12 , corresponding to energies @xmath150 ( full line ) . numerical simulations for a single copy of @xmath136 rotators , in the cases @xmath134 , and @xmath135 , show that the qsss characterized by @xmath65 ( symbols along the dashed line ) exist within the range @xmath151  , in agreement with the results of fig . [ fig : t_vs_tempo_varios_u ] . however , as one goes further below the critical point , roughly for @xmath152 , qsss are still found numerically , although characterized by finite values of magnetization , in spite of the initial conditions of zero magnetization considered . such states are understood due to the prevalence of the ferromagnetic couplings for sufficiently low values of  @xmath9 , and they may be observed more clearly in the case @xmath153 , being sometimes difficult to be identified for higher values of  @xmath6 ( e.g. , @xmath154 ) . it should be noticed that within a small energy range ( typically @xmath155 ) , the qsss yield a negative microcanonical specific heat . although this represents a well - known feature for the hmf model ( see discussion in ref . @xcite ) , it is the first time that such a result is verified for a system of heisenberg rotators , to our knowledge . the numerical data of fig . [ fig : curva_t_u_qss ] suggests that a negative specific heat should occur for @xmath126 . the most interesting feature of the qsss presented in figs . [ fig : t_vs_tempo_varios_u ] and  [ fig : t_vs_tempo_varios_alphas ] concerns an increase in  @xmath11 for increasing  @xmath0 , as shown in fig . [ fig : t_vs_tempo_3_alphas ] , where we analyse the qsss for an energy  @xmath142 in the cases @xmath134 , and  @xmath135 , by varying the system sizes . it should be mentioned that , in order to identify qsss clearly within the present approach , we had to simulate systems with sufficiently high values of  @xmath0 ( essentially , @xmath156 or higher ) . hence , in fig .  [ fig : t_vs_tempo_3_alphas ] such a behaviour is presented for typical values of  @xmath0 in the range from @xmath156 up to @xmath157 , for the smaller values of  @xmath6 ( i.e. , @xmath153 and  @xmath158 ) ; however , in the case @xmath159 the observation of qsss becomes more difficult , in such a way that one needs to simulate larger values of  @xmath0 ( e.g. , @xmath160 or higher ) for this purpose . the growth of  @xmath11 for increasing  @xmath0 represents a well - known feature in models of rotators , like the hmf and @xmath6-xy ( with @xmath141 ) models  @xcite , as well as the inertial heisenberg model  @xcite , yielding significant physical consequences , some of them directly related to the applicability of bg statistical mechanics . more specifically , the order in which the thermodynamic limit ( @xmath22 ) and the infinite - time limit ( @xmath23 ) are considered becomes mostly relevant in the present case , in such a way that if the thermodynamic limit is carried first , one should remain in a  qss forever . this may be viewed as a breakdown of ergodicity , with the system being trapped in  a  qss , never reaching the  bg equilibrium state ; consequently , time averages and ensemble averages do not coincide , violating a basic assumption of bg statistical mechanics . the qss represents the final state in this case ; whether it should be interpreted as a truly equilibrium thermodynamical one represents an important question beyond the scope of the present work . herein we have defined the duration  @xmath11 of a given qss ( e.g. , those exhibited in fig . [ fig : t_vs_tempo_3_alphas ] ) , as the time at which  @xmath131 presents its halfway between the values at the qss and its corresponding  @xmath132 . the fact that we are computing this time interval starting from @xmath161 , including a short transient regime before reaching the qss , does not affect our final results , since this small transient becomes negligible in the final computation of  @xmath11 , for sufficiently large values of  @xmath0 . according to this , the growth of  @xmath11 with the total number of rotators  @xmath0 , for a given energy ( @xmath142 ) and three typical values of  @xmath6 ( @xmath134 , and @xmath135 ) , is exhibited in fig . [ fig : behavior_qss_n ] . in the three cases investigated , these behaviours are fitted by power laws , @xmath13 ( as shown in the respective inset , where the same data is presented in a log - log representation ) , with the exponent  @xmath14 decreasing as  @xmath6 increases , e.g. , @xmath162 , @xmath163 , and @xmath164 . in the first case , our estimate agrees with the one of ref . @xcite ( see also  @xcite ) , where @xmath165 was computed . curiously , such an estimate has also been found numerically for qsss of the hmf model , with similar initial conditions ( zero magnetization )  @xcite , being supported through analytical approaches in ref . one should mention that in the case @xmath159 the qsss are more difficult to be observed , appearing only for sufficiently high values of  @xmath0 , such that data points for smaller  @xmath0 are not exhibited . in spite of such difficulties , our simulations suggest that one should have @xmath15 in the limit @xmath16 . although relevant , it is not possible to propose a particular scaling behaviour describing the decrease of @xmath14 with respect to @xmath6 , on the basis of the present numerical simulations . such a scaling deserves further computational efforts , from which one could obtain a larger number of data points for an appropriate plot of @xmath14 versus @xmath6 ( see , e.g. , ref . @xcite for the xy case ) . the behaviours of the duration  @xmath11 of the present qsss with respect to the total energy  @xmath9 , as well as its decrease with the parameter  @xmath6 are presented in fig . [ fig : behavior_qss_ua ] . in figure [ fig : behavior_qss_ua](a ) we exhibit data of simulations of @xmath136 rotators , for three values of  @xmath6 ( @xmath134 , and @xmath135 ) , varying the energy in the range @xmath166 . these results show that  @xmath11 grows exponentially for increasing  @xmath9 in this range , as can be seen more clearly through the log - linear representation in the inset . although  @xmath11 may become very large as @xmath167 , from the plots of fig . [ fig : t_vs_tempo_varios_u ] one sees that the gap between the kinetic temperature of each qss and the equilibrium state , represented by  @xmath132 , decreases as  @xmath9 approaches  @xmath140 . hence , in this case , the qsss disappear by means of the difference between these two quantities , which goes to zero as @xmath167 , so that for @xmath168 one finds no qsss . the dependence of  @xmath11 on the interaction - range parameter  @xmath6 ( @xmath169 ) is presented in fig . [ fig : behavior_qss_ua](b ) , where we exhibit data of simulations for the energy  @xmath142 and three values for the total number of rotators ( @xmath170 , and @xmath171 ) . in the log - linear representation , one sees that these results do strongly depend on  @xmath0 , although they are all suggestive that @xmath172 as @xmath16 ; more particularly , the linear - linear representation of the inset indicates this tendency more clearly in the case @xmath136 . we have carried molecular - dynamics simulations on a one - dimensional hamiltonian system , composed by  @xmath0 classical localized heisenberg rotators on a ring . we have considered the two - body interaction characterized by a distance  @xmath1 between rotators at sites  @xmath2 and  @xmath3 , decaying with the distance  @xmath1 as a power law , @xmath4 ( @xmath5 ) . in this way , one may control the range of the interactions by changing the parameter  @xmath6 , and specially , two well - known cases may be recovered , namely , the fully - coupled ( i.e. , mean - field limit ) and nearest - neighbour - interaction models , in the limits  @xmath7 and @xmath8 , respectively . for the first time in the literature , we have numerically established the validity of the correct scaling for spin dimensionality @xmath173 . considering this scaling , one obtains the same thermodynamical behaviour for any  @xmath6 in the range @xmath174 , as analytically predicted in ref . @xcite . we have investigated the dynamics of the model by following the time evolution of a single copy of rotators , for energies  @xmath9 below its critical value ( @xmath10 ) , considering initial conditions corresponding to zero magnetization . analogously to what happens for the similar model of xy rotators , we have verified the existence of quasi - stationary states ( qsss ) for values of  @xmath6 in the range @xmath12 . but , in contrast to the xy rotators , the allowed number of initial conditions of the present model is substantially larger . the particular types of initial conditions that will lead the system to robust qsss was discussed , representing a very relevant matter for further analytical investigations . for a given energy  @xmath9 , our numerical analysis indicated that the durations  @xmath11 grow by increasing  @xmath0 , following a power law , @xmath13 , where the exponent @xmath14 gets reduced for increasing values of  @xmath6 in the range @xmath12 ; particularly , our results suggest that @xmath15 as @xmath16 . the relevance of establishing the growth of @xmath11 with  @xmath0 accurately , achieved herein by considering sufficiently large values of  @xmath0 , may be assessed if one takes into account the fact that in spite of the large amount of effort dedicated to the @xmath6-xy model , this matter is still controversial for this system ( see , e.g. , refs . the particular scaling behaviour , describing the decrease of @xmath14 with respect to @xmath6 , requires further computational efforts and is left for future investigations . moreover , for the initial conditions and energy range considered , we have shown that the duration of the qsss grows exponentially with the energy , @xmath175 , for  @xmath0 and  @xmath6 fixed . our simulations have indicated that the gaps separating the kinetic temperature of these qsss and their corresponding equilibrium temperatures  @xmath132 become larger for smaller energies , dropping to zero as @xmath139 . in this way , the qsss disappear by means of the difference between these two quantities , which goes to zero as  @xmath167 , so that for @xmath168 one finds no qsss . we have found qsss over a wide range of energies , typically for @xmath176 ( @xmath177 ) , although these states presented zero magnetization throughout a smaller interval , @xmath151 . it should be mentioned that we have chosen to explore more deeply the qsss associated with a given value of energy ( @xmath142 ) , where such states could be observed easily through simulations of a single copy of rotators . however , the results presented herein for this particular energy should be valid for any similar qss ( characterized by zero magnetization ) in the range @xmath151 . by going further below the critical point , roughly for @xmath152 , qsss were still found numerically , but in contrast to the above - mentioned ones , they are characterized by finite values of magnetization , in spite of the initial conditions of zero magnetization considered . within a small energy range ( typically @xmath155 ) , the numerical data of the kinetic temperature of these qsss versus  @xmath9 yielded a negative microcanonical specific heat . although this represents a well - known feature for the hmf model , it is the first time that such a result has been verified for a system of heisenberg rotators , to our knowledge . the particular case @xmath153 of the present analysis ( carried herein for the cartesian components of angular momenta and spin variables ) , coincides precisely with the one of ref . @xcite , which have considered angular variables and their corresponding canonically - conjugated angular momenta . herein , we have extended these results for power - law decaying interactions among spin variables , by investigating situations characterized by @xmath178 , showing that interesting qsss appear for any  @xmath6 in the range @xmath12 . due to the wide diversity of possibilities for initial conditions in both spins and angular momenta in the present model , investigations for qsss under initial conditions different from the ones considered herein would be highly desirable . 99 t.  nakamura , _ statistical theory of hindered rotation in molecular crystals _ , http://dx.doi.org/10.1143/jpsj.7.264[j . * 7 * , 264269 ] http://dx.doi.org/10.1143/jpsj.7.264[(1952 ) ] ; m.  e.  fisher , _ magnetism in one - dimensional systems  the heisenberg model for infinite spin _ , http://dx.doi.org/10.1119/1.1970340[am . j.  phys . * 32 * , 343346 ( 1964 ) . ] g.  s.  joyce , _ classical heisenberg model _ , http://dx.doi.org/10.1103/physrev.155.478[phys . rev .  * 155 * , 478491 ( 1967 ) ] . h.  e.  stanley , _ exact solution for a linear chain of isotropically interacting classical spins of arbitrary dimensionality _ , http://dx.doi.org/10.1103/physrev.179.570[phys . rev .  * 179 * , 570577 ( 1969 ) ] . m.  antoni and s.  ruffo , _ clustering and relaxation in hamiltonian long - range dynamics _ , http://dx.doi.org/10.1103/physreve.52.2361[phys . e ] http://dx.doi.org/10.1103/physreve.52.2361[*52 * , 23612374 ( 1995 ) ] .    c.  anteneodo and c.  tsallis , _ breakdown of exponential sensitivity to initial conditions : role of the range of interactions _ , http://dx.doi.org/10.1103/physrevlett.80.5313[phys . lett .  * 80 * , 53135316 ( 1998 ) ] . f.  d.  nobre and c.  tsallis , _ classical infinite - range - interaction heisenberg ferromagnetic model : metastability and sensitivity to initial conditions _ , http://dx.doi.org/10.1103/physreve.68.036115[phys . e * 68 * , 036115 ( 2003 ) ] . y.  y.  yamaguchi , j.  barr , f.  bouchet , t.  dauxois , and s.  ruffo , _ stability criteria of the vlasov equation and quasi - stationary states of the hmf model _ , http://dx.doi.org/10.1016/j.physa.2004.01.041[physica a * 337 * , 3666 ( 2004 ) ] . a.  pluchino , v.  latora , and a.  rapisarda , _ glassy dynamics in the hmf model _ , http://dx.doi.org/10.1016/j.physa.2004.04.006[physica a * 340 * , 187195 ] http://dx.doi.org/10.1016/j.physa.2004.04.006[(2004 ) ] ; _ effective spin  glass hamiltonian for the anomalous dynamics of the hmf model _ , http://dx.doi.org/10.1016/j.physa.2006.03.067[physica a ] http://dx.doi.org/10.1016/j.physa.2006.03.067[*370 * , 573584 ( 2006 ) ] ; _ metastable states , anomalous distributions and correlations in the hmf model _ , http://dx.doi.org/10.1016/j.physd.2004.01.029[physica d * 193 * , 315328 ( 2004 ) ] . v.  latora , a.  rapisarda , and s.  ruffo , _ lyapunov instability and finite size effects in a system with long - range forces _ , http://dx.doi.org/10.1103/physrevlett.80.692[phys . lett .  * 80 * , 692695 ( 1998 ) ] ; _ chaos and statistical mechanics in the hamiltonian mean field model _ , http://dx.doi.org/10.1016/s0167-2789(98)00217-6[physica d * 131 * , 3854 ( 1999 ) ] . moyano and c.  anteneodo , _ diffusive anomalies in a long - range hamiltonian system _ , http://dx.doi.org/10.1103/physreve.74.021118[phys .  rev . ] http://dx.doi.org/10.1103/physreve.74.021118[e * 74 * , 021118 ( 2006 ) ] . a.  pluchino , a.  rapisarda , and c.  tsallis , _ nonergodicity and central - limit behavior for long - range hamiltonians _ , http://dx.doi.org/10.1209/0295-5075/80/26002[europhys . 80 * , 26002 ( 2007 ) ] ; _ a closer look at the indications of @xmath179-generalized central limit theorem behavior in quasi - stationary states of the hmf model _ , http://dx.doi.org/10.1016/j.physa.2008.01.112[physica a * 387 * , 31213128 ( 2008 ) ] . chavanis and a.  campa , _ inhomogeneous tsallis distributions in the hmf model _ , http://dx.doi.org/10.1140/epjb/e2010-00243-x[eur . j. ] http://dx.doi.org/10.1140/epjb/e2010-00243-x[b * 76 * , 581611 ( 2010 ) ] ; a.  campa and p .- h . chavanis , _ caloric curves fitted by polytropic distributions in the hmf model _ , http://dx.doi.org/10.1140/epjb/e2013-30947-0[eur . j.  b * 86 * , 170 ( 2013 ) ] . w.  ettoumi and m .- c . firpo , action diffusion and lifetimes of quasistationary states in the hamiltonian mean - field model . http://dx.doi.org/10.1103/physreve.87.030102[phys . e * 87 * , 030102 ( 2013 ) ] . a.  campa , t.  dauxois , and s.  ruffo , _ statistical mechanics and dynamics of solvable models with long - range interactions _ , http://dx.doi.org/10.1016/j.physrep.2009.07.001[phys .  rep .  * 480 * , 57159 ( 2009 ) ] .    t.  m.  rocha filho , m.  a.  amato , a.  e.  santana , a.  figueiredo and j.  r.  steiner , _ dynamics and physical interpretation of quasistationary states in systems with long - range interactions _ , http://dx.doi.org/10.1103/physreve.89.032116[phys .  rev . e ] http://dx.doi.org/10.1103/physreve.89.032116[*89 * 032116 ( 2014 ) ] ; t.  m.  rocha filho , a.  e.  santana , m.  a.  amato and a.  figueiredo , _ scaling of the dynamics of homogeneous states of one - dimensional long - range interacting systems _ , http://dx.doi.org/10.1103/physreve.90.032133[phys . e * 90 * ] http://dx.doi.org/10.1103/physreve.90.032133[032133 ( 2014 ) ] . p.  jund , s.  g.  kim , and c.  tsallis , _ crossover from extensive to nonextensive behavior driven by long - range interactions _ , http://dx.doi.org/10.1103/physrevb.52.50[phys . b * 52 * , 5053 ( 1995 ) ] . a.  campa , a.  giansanti , d.  moroni , and c.  tsallis , _ classical spin systems with long - range interactions : universal reduction of mixing _ , http://dx.doi.org/10.1016/s0375-9601(01)00440-6[phys . a * 286 * , 251256 ( 2001 ) ] .      c.  tsallis , _ introduction to nonextensive statistical mechanics  approaching a complex world _ http://dx.doi.org/10.1007/978-0-387-85359-8[(springer , ] http://dx.doi.org/10.1007/978-0-387-85359-8[new york , 2009 ) ] . l.  j.  l.  cirto , v.  r.  v.  assis , and c.  tsallis , _ influence of the interaction range on the thermostatistics of a classical many - body system _ , http://dx.doi.org/10.1016/j.physa.2013.09.002[physica a * 393 * , 286296 ( 2014 ) ] . firpo and s.  ruffo , _ chaos suppression in the large size limit for long - range systems _ , http://dx.doi.org/10.1088/0305-4470/34/37/102[j . a ] http://dx.doi.org/10.1088/0305-4470/34/37/102[*34 * , l511 ( 2001 ) ] ; c.  anteneodo and r.  o.  vallejos , _ scaling laws for the largest lyapunov exponent in long - range systems : a random matrix approach _ , http://dx.doi.org/10.1103/physreve.65.016210[phys .  rev . e * 65 * , 016210 ( 2001 ) ] . f.  d.  nobre and c.  tsallis , _ metastable states of the classical inertial infinite - range - interaction heisenberg ferromagnet : role of initial conditions _ , http://dx.doi.org/10.1016/j.physa.2004.06.036[physica a * 344 * , 587594 ( 2004 ) ] . a.  campa , a.  giansanti , and d.  moroni , _ canonical solution of classical magnetic models with long - range couplings _ , http://dx.doi.org/10.1088/0305-4470/36/25/301[j . a  math .  gen .  * 36 * , 6897 ( 2003 ) ] . a.  giansanti , d.  moroni , and a.  campa , _ universal behavior in the static and dynamic properties of the @xmath6-xy model _ , http://dx.doi.org/10.1016/s0960-0779(01)00022-4[chaos soliton . * 13 * , 407416 ( 2002 ) ] . s.  a.  cannas and f.  a.  tamarit , _ long - range interactions and nonextensivity in ferromagnetic spin models _ , http://dx.doi.org/10.1103/physrevb.54.r12661[phys . b * 54 * , r12661r12664 ( 1996 ) ] ; l.  c.  sampaio , m.  p.  de  albuquerque , and f.  s.  de  menezes , _ nonextensivity and tsallis statistics in magnetic systems _ , http://dx.doi.org/10.1103/physrevb.55.5611[phys . b * 55 * , 56115614 ] http://dx.doi.org/10.1103/physrevb.55.5611[(1997 ) ] . j.  r.  grigera , _ extensive and non - extensive thermodynamics . a molecular dynamic test _ , http://dx.doi.org/10.1016/0375-9601(96)00276-9[phys . a ] http://dx.doi.org/10.1016/0375-9601(96)00276-9[*217 * , 4751 ( 1996 ) ] ; s.  curilef and c.  tsallis , _ critical temperature and nonextensivity in long - range interacting lennard jones - like fluids _ , http://dx.doi.org/10.1016/s0375-9601(99)00727-6[phys . a * 264 * , 270275 ( 1999 ) ] . j.  d.  parsons , _ linear chain of classical spins with arbitrary isotropic nearest - neighbor interaction _ , http://dx.doi.org/10.1103/physrevb.16.2311[phys . ] http://dx.doi.org/10.1103/physrevb.16.2311[rev . b * 16 * , 23112312 ( 1977 ) ] . j.  barojas , d.  levesque , and b.  quentrec , _ simulation of diatomic homonuclear liquids _ , http://dx.doi.org/10.1103/physreva.7.1092[phys . a ] http://dx.doi.org/10.1103/physreva.7.1092[*7 * , 10921105 ( 1973 ) ] . d.  j.  evans , _ on the representation of orientation space _ , http://dx.doi.org/10.1080/00268977700101751[mol . phys .  * 34 * , 317325 ( 1977 ) ] ; d.  j.  evans and s.  murad , _ singularity free algorithm for molecular dynamics simulation of rigid polyato _ , http://dx.doi.org/10.1080/00268977700101761[mol . phys . ] http://dx.doi.org/10.1080/00268977700101761[*34 * , 327331 ( 1977 ) ] . m.  p.  allen and d.  j.  tildesley , _ computer simulation of liquids _ ( oxford university press , oxford , 1987 ) . a.  turchi , d.  fanelli and x.  leoncini , _ existence of quasi - stationary states at the long range threshold _ , http://dx.doi.org/10.1016/j.cnsns.2011.03.013[commun . nonlinear sci . simulat . * 16 * 47184724 ( 2011 ) ] ; w.  ettoumi and m .- c . firpo , _ stochastic treatment of finite-@xmath0 effects in mean - field systems and its application to the lifetimes of coherent structures _ , http://dx.doi.org/10.1103/physreve.84.030103[phys . e * 84 * 030103 ( 2011 ) ] . s.  gupta and d.  mukamel , _ quasistationarity in a model of classical spins with long - range interactions _ , http://dx.doi.org/10.1088/1742-5468/2011/03/p03015[j . : theor .  exp . p03015 ( 2011 ) ] . r.  bachelard and m.  kastner , _ universal threshold for the dynamical behavior of lattice systems with long - range interactions _ , http://dx.doi.org/10.1103/physrevlett.110.170603[phys . 170603 * 110 * ( 2013 ) ] .
a numerical analysis of a one - dimensional hamiltonian system , composed by  @xmath0 classical localized heisenberg rotators on a ring , is presented . 
 a distance  @xmath1 between rotators at sites  @xmath2 and  @xmath3 is introduced , such that the corresponding two - body interaction decays with  @xmath1 as a power - law , @xmath4 ( @xmath5 ) . 
 the index  @xmath6 controls the range of the interactions , in such a way that one recovers both the fully - coupled ( i.e. , mean - field limit ) and nearest - neighbour - interaction models in the particular limits @xmath7 and @xmath8 , respectively . 
 the dynamics of the model is investigated for energies  @xmath9 below its critical value ( @xmath10 ) , with initial conditions corresponding to zero magnetization . 
 the presence of quasi - stationary states ( qsss ) , whose durations @xmath11 increase for increasing values of  @xmath0 , is verified for values of  @xmath6 in the range @xmath12 , like the ones found for the similar model of xy rotators . 
 moreover , for a given energy  @xmath9 , our numerical analysis indicates that @xmath13 , where the exponent @xmath14 decreases for increasing  @xmath6 in the range @xmath12 , and particularly , our results suggest that @xmath15 as @xmath16 . 
 the growth of @xmath11 with  @xmath0 could be interpreted as a breakdown of ergodicity , which is shown herein to occur for any value of  @xmath6 in this interval .