document: the study of nonequilibrium phase transitions is a topic of growing interest due to its application to a variety of complex systems@xmath3 : contact process , domain growth , catalysis , phase separation and transport phenomena . 
   although there is no general theory to account for nonequilibrium model systems , in recent years some progress has been achieved in understanding the stationary states of these systems employing approximate analytical methods and simulations . 
   some rigorous mathematical questions concerning the phase transitions of these complex interacting particle systems can be appreciated in the books of liggett@xmath4 and konno@xmath5 .    in this paper 
   we focus our attention on the phase transitions observed in the surface reaction model proposed by ziff , gulari and barshad@xmath6 ( zgb ) , which describes some kinetic aspects of the oxidation of @xmath7 over a catalytic surface . 
   in particular , here we consider a modified version of the zgb model , where we incorporate a random distribution of inert sites on the catalytic surface . 
   the original zgb model is an irreversible lattice model for surface reactions based on the langmuir - hinshelwood mechanism , where the reactants must be adsorbed before reacting . 
   the steps used to describe the zgb model ( a lattice markov process ) are the following : molecules of @xmath0 and @xmath1 from a gaseous phase can be adsorbed onto the sites of a regular square lattice of identical sites . 
   these molecules arrive at the surface according to their partial pressures in the gas mixture , that is , the probability of a @xmath0 molecule arriving is @xmath8 and @xmath9 for the @xmath1 molecule . 
   the @xmath0 molecule requires only a single vacant site to be adsorbed , while the @xmath1 is adsorbed if it finds a nearest - neighbor pair of empty sites . upon adsorption , the @xmath1 molecule dissociates and the two free @xmath10 atoms can react independently . if , after an adsorption step , a nearest - neighbor @xmath11 pair appears on the lattice , they immediately react , forming a @xmath2 molecule that goes to the gas phase , leaving two empty sites on the lattice . therefore , in this adsorption controlled limit , only a single parameter ( @xmath8 ) is sufficient to describe the dynamics of the model . 
   the simulations performed by ziff and co - workers have shown that the system exhibits two phase transitions between active and poisoned states : for @xmath12 , an o - poisoned state is formed , while for @xmath13 the lattice is poisoned by @xmath0 . for @xmath14 a reactive steady - state is found , in which a nonzero number of vacant sites is present in the lattice . at @xmath15 
   the transition is continuous , whereas at @xmath16 the transition is of the first - order type . using a mean field theory , dickman@xmath17 qualitatively reproduced the phase diagram of the zgb model and showed that , at the level of site approximation , only the first - order transition appears . 
   however , employing the pair approximation , both continuous and first - order transitions are obtained . 
   we are interested on the effects of inert sites on the phase transitions of the zgb model . 
   we have investigated in detail the dependence of the phase transitions on the concentration of inert sites . 
   this problem presents some experimental interest in the automobile industry , where lead particles are deposited over the catalyst during the exhaust of the gases after combustion . 
   this affects the efficiency of the catalytic surface due to the pinning of these lead particles on the surface , forbidding the adsorption of @xmath0 and @xmath1 molecules at the lead positions and reducing the reaction paths . 
   hovi and co - workers@xmath18 , have studied by computer simulations the effect of preadsorbed poison and promoters on the irreversible zgb model . 
   they calculated the coverage of species as a function of the concentration of inert sites for a wide range of values , finding the interesting result that the first - order transition changes to a continuous one at a critical value of the concentration . 
   corts and valencia@xmath19 have also reported some results concerning random impurities distributed over the catalyst , in which they observed the change of the first - order transition into a continuous one as one increases the concentration of impurities . 
   albano@xmath20 simulated the zgb model on incipient percolation clusters ( ipc s ) with a fractal dimension of 1.90 . 
   he showed that both transitions , at @xmath15 and @xmath16 are continuous , and that for an infinite lattice , in which @xmath8 is larger than 0.408 , the reactions stop at finite times because the ipc s are poisoned by pure @xmath0 . 
   casties et al.@xmath21 also performed a monte carlo simulation of the @xmath0 oxidation on probabilistic fractals . 
   they observed a change in the character of the transition at @xmath16 from first order on regular lattices to second order on percolation clusters ( for @xmath22 larger than @xmath23 , which is the percolation threshold on the square lattice ) .    in this work 
   we have performed mean - field ( site and pair approximations ) calculations and monte carlo simulations for different values of the concentration of inert sites . 
   the model studied here is a variant of the original zgb model , where inert sites are randomly distributed over the lattice . 
   our approach is close related to that presented by vigil and willmore@xmath24 to study the effects of spatial correlations on the oscillatory behavior of a modified zgb model , where defects are continually added and desorbed from the surface . in their studies , they considered the mean - field site and pair approximations , as well as monte carlo simulations . in the present work we have determined the phase diagram for different concentrations , and the spinodal and transition lines as a function of the concentration of inert sites . 
   we have constructed hysteresis curves to find the critical concentration at which the first - order transition changes into a continuous one . 
   this paper is organized as follows : in sec . 
   ii we present the results obtained within the site approximation ; in sec . 
   iii we introduce the pair approximation equations and show the results obtained using this scheme ; sec . 
   iv presents the results of simulations , and finally , in sec . 
   v we present our conclusions . 
   we take a square lattice as our catalytic surface . 
   a fraction @xmath25 of the sites is randomly distributed over the lattice representing the pinned inert sites . 
   the remaining sites of the lattice can be vacant , or occupied by either @xmath10 atoms or @xmath0 molecules . 
   the zgb model is described by the following steps : @xmath26 where the labels @xmath27 and @xmath28 denote the gaseous phase and an adsorbed reactant on the surface , respectively , and @xmath29 indicates a vacant site . steps ( 1 ) and ( 2 ) indicate the adsorption of the species , whereas the third step is the proper reaction , between distinct species located at adjacent sites of the lattice . in the site approximation the time evolution equations of the concentrations are given by @xmath30 where @xmath31 , @xmath32 and @xmath33 represent , respectively , the coverages of @xmath10 , @xmath0 and blank sites in the lattice . @xmath8 gives the arrival probability of a @xmath0 molecule . 
   in addition , there is the following constraint among the concentrations @xmath34    the steady - state solutions of the above system of equations are given by @xmath35 , that corresponds to a poisoned surface , and @xmath36 inserting eq . 
   ( 7 ) into eq . 
   ( 4 ) we obtain an expression for the steady - state values of the concentration @xmath32 : @xmath37    we exhibit in fig . 
   1 a typical diagram for the coverages of @xmath0 , @xmath10 and vacant sites obtained for @xmath38 . 
   this diagram was obtained by integrating the equations of motion for the @xmath32 and @xmath31 concentrations , starting from an initial condition in which the number of empty sites is @xmath39 . 
   the site approximation does not give any continuous transition for all values of the concentration of inert sites . 
   this was already pointed out by dickman@xmath17 for the zgb model without inert sites . 
   we observe in fig . 
   1 , that the limit of stability of the reactive phase is @xmath40 , which corresponds to the spinodal point . 
   therefore , a reactive steady - state is found for all values of @xmath41 . for values of @xmath42 , 
   the system becomes poisoned , with a large amount of @xmath0 and a small concentration of @xmath10 atoms . 
   the presence of @xmath10 atoms in the region @xmath42 is due to the inert sites that can block some oxygen , and also to the simplicity of the site approximation , which does not forbid the formation of @xmath43 nearest - neighbor pairs in the lattice . 
   the tolerance of these @xmath43 pairs also explains the absence of the continuous phase transition , which is observed in the simulations . 
   2 is a plot of the solutions @xmath32 of eq.(8 ) versus the parameter @xmath44 for different values of concentration @xmath25 of inert sites . 
   we obtain two solutions , which we call @xmath45 and @xmath46 , that join together at the spinodal point . 
   for instance , for @xmath47 the value we find is @xmath48 , which furnishes the value @xmath49 . 
   we also note in fig . 
   2 that , at the spinodal point , the concentration of @xmath32 molecules remains the same irrespective of the value we choose for @xmath25 . 
   this special value is @xmath50 . 
   then , the net effect of adding @xmath25 is to shift the curves horizontally . in this site approximation , solutions are possible only for values of @xmath51 . 
   this happens because above this value the solution would correspond to the non - physical value @xmath52 . 
   so , the meaning of the two solutions in fig . 
   2 is the following : the branch @xmath46 represents the stable steady - state solutions whereas the @xmath45 branch gives the unstable solutions . 
   these solutions were obtained after numerical integration of the equations of motion for @xmath32 and @xmath31 , starting from the state described by @xmath39 . for the initial condition @xmath53 and @xmath32 larger than @xmath45 the system evolves to the poisoned state . the initial condition @xmath53 and @xmath32 less than @xmath45 drives the system to the lower stable reactive solution @xmath46 .    fig . 
   2 also shows that , as we approach the spinodal point for any value of @xmath25 , the region of stability becomes narrower . 
   then , we expect that for some value of @xmath41 a first - order transition occurs , that is , the concentration @xmath32 must increase abruptally from a small value ( reactive phase ) to a large value ( poisoned phase ) . 
   unfortunately , we can not use here the usual thermodynamic considerations based on the minimization of a suitable thermodynamic potential . in order to find this first - order transition 
   we adopt the same kinetic criteria employed by dickman@xmath17 , which was borrowed from the work of ziff et al@xmath6 . 
   the phase transition was determined by choosing an initial state where half of the lattice was empty and the other half was completely filled with @xmath0 . in this work 
   we choose as our initial state , to solve the equations of motion for @xmath32 and @xmath31 , the values @xmath54 . 
   it is clear that this choice is not the same as that considered by ziff et al . , because we can not discriminate which sites are empty or not . 
   the phase boundary is defined at the special value @xmath16 where the solution of the equations of motion changes from the reactive to the poisoned state as we vary the value of @xmath8 for the same initial condition , as established above . 
   for @xmath47 we obtain the same value found by dickman . we exhibit in fig . 
   3 the results obtained for the first - order transition and the spinodal points for @xmath38 . 
   the spinodal was obtained from the initial condition @xmath39 , and the first - order transition from the condition @xmath54 . 
   for this particular value of @xmath38 , we have @xmath55 and @xmath56 . 
   we have considered all values of the concentration of inert sites , and fig . 
   4 shows the values of @xmath57 ( dashed line ) and @xmath16 ( full line ) as a function of the concentration of inert sites . at the particular value @xmath58 
   the two lines merge . for values of @xmath59 
   the transition still remains of the first - order type , although the number of vacant sites that stay in the active state is very small . 
   for instance , for @xmath60 , at the transition point ( @xmath61 ) , the number of vacant sites changes from @xmath62 in the active state , to @xmath63 in the poisoned state . throughout our analysis 
   we considered a given state to be active if the number of vacant sites is larger than @xmath64 . 
   we also exhibit in fig . 
   5 the number of vacant sites @xmath33 at the active state as a function of the number of inert sites @xmath25 , at the transition and spinodal points . 
   we observe that for all values of @xmath65 , the number of vacant sites at the spinodal point is always larger than that at the transition point . 
   let us consider the application of the pair approximation procedure to this zgb model that includes inert sites . here 
   we introduce the pair probability @xmath66 of a random nearest neighbor pair of sites being occupied by species @xmath67 and @xmath68 . 
   we have the following types of species : @xmath29 , @xmath69 , @xmath70 , and @xmath10 , which represent , respectively , vacant , inert , carbon monoxide , and oxygen . as in the previous treatments 
   @xmath71 we need to consider only the changes that occur at a particular central pair in the lattice . in the table below we display the allowed and forbbiden ( indicated by @xmath72 ) nearest - neighbor pairs in the present model . 
   @xmath73    the next table also exhibits all the possible transitions among pairs . 
   we obtain @xmath74 independent transitions , labelled by numbers in the range @xmath75 . in the table transitions indicated by @xmath72 
   are prohibited . 
   @xmath76    then , we write the equations relating the probability of each element with the corresponding pair probabilities : @xmath77    the pair probabilities also satisfy the constraint @xmath78    next , we need to write the time evolution equations for the pair probabilities . examining the latter table we can construct the desired equations of evolution . 
   we explicitly write the equations of motion for the pair probabilities @xmath79 . 
   @xmath80 where @xmath81 to @xmath82 are the transition rates . 
   the factors of two arising in the equation of motion for @xmath83 are due to the fact that the pair probabilities @xmath84 and @xmath85 are equal by symmetry . for instance , from the pair @xmath86 we can obtain , with the same probability , the different configurations @xmath87 and @xmath88 . in general , the expressions for the transition rates are lengthy , and we present these transition rates in the appendix .    in this pair approximation 
   we can not obtain analytical solutions as we have done in the site approximation . 
   we solved the coupled set of eight nonlinear equations by the fourth - order runge - kutta method , searching for the stationary solutions . 
   we considered the two different initial conditions as in the case of the site approximation . 
   let us first consider the evolution from the initial state where @xmath89 , in which only the pairs @xmath86 , @xmath90 and @xmath91 are present in the lattice at @xmath92 . 
   figure 6 shows the diagram of the model for @xmath93 . for @xmath94 the lattice poisons with oxygen . in the range @xmath95 there is an active region , and for @xmath96 the lattice poisons with @xmath0 . 
   when @xmath97 , we found the same figures obtained by dickman in his pair approximation . 
   for instance , the site and pair approximations give the same value for the spinodal point @xmath57 . 
   however , when we consider some inert sites in the lattice , the spinodal point found in the site approximation is always smaller than that obtained within the pair approximation . for this particular value , @xmath93 , the site approximation yields @xmath98 , whereas @xmath56 is obtained by the pair approximation . 
   the value of @xmath8 at the continuous transition , which now arises in this pair approximation , decreases slightly with increasing values of the concentration of inert sites . 
   we also considered the solutions evolving from an initial condition where half of the free sites ( @xmath89 ) is filled with @xmath0 molecules and the other half left empty . in order to be close to the initial condition used in the simulation , we chose for the initial pair conditions @xmath99 , @xmath100 and @xmath101 , which mimics a division of the lattice into two parts : on one side of the lattice we would have inert sites and @xmath0 molecules and , on the other side , vacant and inert sites . 
   if @xmath97 , we found for the transition between the active and co - poisoned states the value @xmath102 , which agrees with the value found in the simulations . 
   7 displays the concentration of @xmath0 molecules at the transition point for which @xmath93 . in this pair approximation , 
   the values of @xmath57 and @xmath16 are very close . 
   we also show in fig . 
   8 the concentration of vacant sites as a function of the concentration of inert sites , at the transition point , and also at the spinodal point . 
   both curves join at @xmath103 , and for @xmath104 , we can not observe any active state . as in the site approximation , an active state 
   is defined only if @xmath105 . 
   then , the calculations performed within the pair approximation give results that are very similar to those obtained by the site approximation , concerning the spinodal and transition points . 
   in addition , it was observed that initial conditions do not affect the point in which the continuous phase transition occurs . in fig . 
   9 we exhibit the phase diagram for this zgb model with inert sites . the size of the reactive window decreases as we increase the concentration of inert sites . 
   we have plotted the transition line for the first - order transition and for the spinodal line , which gives the limit of stability of the reactive phase . 
   the line separating the active and o - poisoned phases is a continuous transition line . 
   we have performed monte carlo simulations in the zgb model with inert sites in order to check the results we have obtained in the site and pair approximations . 
   the simulations were carried for different values of the concentration of inert sites @xmath106 . for small values of @xmath106 
   , we considered square lattices of linear size @xmath107 , but for large values of @xmath106 we have used lattices of linear size up to @xmath108 . the first step in the simulation is to randomly distribute the selected fraction @xmath106 of inert sites in the lattice . 
   all simulations then started with a fraction of empty sites equal to @xmath89 . 
   the @xmath0 molecules arrive at the surface with a probability @xmath8 and the @xmath1 molecules with probability @xmath109 . 
   the rules for adsorption and reaction of the species are exactly the same as in the original zgb model@xmath6 . 
   since adsorption of oxygen requires two nearest neighbor empty sites , the effect of the inert sites is to favour the adsorption of @xmath0 relatively to that of @xmath1 molecules . in general , we have taken @xmath110 monte carlo steps ( mcs ) to attain the stationary states , and @xmath110 more to calculate the concentration averages at the stationary states . 
   one mcs is equal to @xmath111 trials of deposition of species , where @xmath112 is the linear size of the lattice . to speed up the simulations we worked with a suitable list of empty sites . 
   we exhibit in fig . 
   10 the phase diagram of the model in the plane @xmath8 versus @xmath106 . 
   it is similar to that obtained within the pair approximation . 
   however , there is a fundamental difference between the transition line separating the active and co - poisoned phases in both approaches . in the pair approximation 
   the transition line is always of the first - order type , whereas in the simulations there is a critical concentration above which the transition becomes continuous . 
   we have done detailed simulations to find the critical concentration at which the transition becomes continuous . 
   we have found for the critical concentration of inert sites the value @xmath113 . 
   we arrived at this value by looking at the hysteresis loops in the curves of @xmath114 versus @xmath8 for different values of the concentration @xmath106 , as we can see in fig . 
   we proceed as follows : in fig . 
   11a we fixed the concentration of inert sites at the value @xmath115 and the curve with circles , which is the proper transition curve , was obtained from an initial state where @xmath89 , that is , with a lattice almost empty . 
   the curve with squares was determined from an initial state in which the lattice was almost covered by @xmath0 . 
   we have taken a fraction of only @xmath116 of randomly empty sites over the lattice at the starting time . 
   then , we clearly observe the hysteresis loop at the concentration @xmath106 , which implies that the transition is of first - order . on the other hand , fig . 
   11b , where the fraction of inert sites is @xmath117 , does not exhibit the hysteresis loop and the transition is clearly a continuous one . 
   the critical value of @xmath113 was obtained analysing the behavior of these curves in the range @xmath118 . as we have pointed out in the introduction , hovi et al.@xmath18 had already observed the change in the nature of this transition as a function of the concentration . 
   the phase boundary separating the active and the o - poisoned phases in fig . 
   10 is continuous for all values of @xmath106 . 
   we have checked this fact by observing that no hysteresis loop was found for any value of @xmath106 . 
   the width of the active phase decreases with increasing values of @xmath106 . for values of @xmath119 
   the lattice is poisoned ( absence of empty sites ) with different amounts of @xmath0 and @xmath10 species . due to finite size effects , this value is larger than the value 0.408 found by albano@xmath20 in the limit of very large ipc s .    we have also noted that the production rate of @xmath2 molecules attains its maximum value exactly at the first - order transition , for values of @xmath120 . if @xmath121 the maximum production rate of @xmath2 molecules is located inside of the reactive window . 
   this is seen in fig . 
   12 , where the circles indicate the points where the production rate of @xmath2 is maximum . in the site and 
   pair approximations this maximum occurs always at the phase boundary , irrespective of the value of @xmath106 . 
   13 shows the production rate @xmath122 of @xmath2 molecules as a function of @xmath106 . 
   as expected , the role of inert sites is also of blocking the reactions over the catalyst . 
   the maximum production rate occurs at a surface free of impurities . 
   we have studied the effects of a random distribution of inert sites on the phase diagram of the zgb model . 
   we determined the time evolution equations for the concentrations of the different species over the catalytic surface within an effective field theory , at the level of site and pair approximations , and also performed monte carlo simulations on the model . 
   we obtained the coverages of the species as function of the deposition rate of @xmath0 and of the concentration of inert sites . 
   in the site and pair approximations we found the transition line and the limit of stability of the reactive phase . in the site approximation , 
   the continuous transition between the o - poisoned and reactive states is absent for any values of the concentration of inert sites . 
   the width of the reactive window exhibits the same behavior , as a function of concentration of inert sites , in both pair approximation and monte carlo simulations . 
   however , the transition between the reactive and co - poisoned phase is always of first - order in the site and pair approximations , whereas monte carlo simulations give a critical point where the transition changes nature . for values of the concentration of inert sites 
   less than the critical value , the transition is first - order and above this value , it changes to a continuous one . 
   the determination of this critical concentration was possible through the analysis of the hysteresis curves for different values of the concentration of inert sites . 
   the production rate of @xmath2 molecules is maximum at the first - order transition , in both site and pair approximations . 
   this is the case in the simulations , but the transition is of the first - order type . 
   when the concentration of inert sites is greater than the critical value , the maximum production rate of @xmath2 molecules moves towards the reactive window . 
   the overall effect of inert sites is to reduce the production of @xmath2 molecules . 
   we would like to thank ron dickman by his many valuable suggestions , and luis g. c. rego by the critical reading of the manuscript . 
   this work was supported by the brazilian agencies capes , cnpq and finep . 
   we present the transition rates in the pair approximation , which we used in section iii to solve the time evolution equations for the pair probabilities . 
   the transition rates are @xmath81 to @xmath82 , which are given by @xmath123                    @xmath132 + \frac 13\left ( \frac{p_{vo}}{p_v}\right ) ^2\left ( 1-\frac{p_{oo}}{p_o}\right )   \nonumber \\ & & + \left . \frac 
   23\frac{p_{vo}}{p_v}\left ( 1-\frac{p_{vo}}{p_v}\right ) \frac{% p_{oo}}{p_o}+\frac 14\left ( \frac{p_{vo}}{p_v}\right ) ^2\frac{p_{oo}}{p_o}% \right\ }   \nonumber\end{aligned}\ ] ]        @xmath135 + \frac{p_{vc}}{p_c}% \left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2\right . 
   \nonumber \\ & & + \left . \frac{p_{vc}}{p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 13% \left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] 
   \times \left . 
   \left [ \frac{p_{vc}% } { p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 23\left ( \frac{p_{vc}}{p_v}% \right ) ^2\right ] \right\ }    \nonumber\end{aligned}\ ] ]    @xmath136 \right\ }    \nonumber \\ 
   t_{14b } & = & 2(1-y_{co})p_{cd}\frac{p_{vc}}{p_c}\left\ { 2\frac{p_{vv}}{p_v}\left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) \left ( 1-\frac{p_{cd}}{p_d}\right ) + \frac   12\frac{p_{vc}}{p_v}\left ( 1-\frac{p_{cd}}{p_d}\right ) \right . 
   \right . 
   \nonumber \\ & & + \left . \frac 12\left ( 1-\frac{p_{vc}}{p_v}\right ) \frac{p_{cd}}{p_d}+% \frac 13\frac{p_{vc}}{p_v}\frac{p_{cd}}{p_d}\right ] + \frac{p_{vd}}{p_d}% \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2+\frac 22\frac{p_{vc}}{p_v}\frac{p_{vd}% } { p_d }   \nonumber \\ & & + \left . \left . 
   \frac 13\left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] \right\ }   \nonumber\end{aligned}\ ] ]    j. marro and r. dickman , _ nonequilibrium phase transitions in lattice models _ 
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summary: a random distribution of inert sites is introduced in the ziff - gulari - barshad model to study the phase transitions between active and poisoned states . 
   the adsorption of @xmath0 and @xmath1 molecules is not possible at the position of the inert sites . 
   this model is investigated in the site and pair approximations , as well as through monte carlo simulations . 
   we determine the mean coverages of the elements as a function of the dilution and show that the continuous transition between the active and o - poisoned state is slightly affected by moderate values of dilution in the pair approximation and in the simulations . on the other hand , from the analysis of the hysteresis curves , the transition between the active and co - poisoned states changes from first - order to continuous as one increases the concentration of inactive sites . 
   the observed transition in the site and pair approximations is always of first - order nature . 
   we also found the lines of transition and spinodal points as a function of the concentration of inert sites . 
   finally , the production rate of @xmath2 is calculated as a function of the dilution of sites . 
   pacs number(s ) : 05.70.ln , 05.70.fh , 82.65.jv , 82.20.mj