cost estimation is a major issue for operators , especially when their commercial expansion leads them to deploy networks on new territories often using new technologies .today s world of mobility generates an amount of challenging problems but does not eliminate the need to consider fixed connections that either constitute whole networks or are part of the infrastructure of mobile ones .high speed optical connections are nowadays required wherever possible in order to satisfy customers .the problem for an operator lies in the words `` wherever possible '' that describe the technical feasibility as well as the financial impact of potential solutions ; obviously the last one should be minimal .we consider here the design from scratch of a fixed network deployment at urban scale under cost and efficiency constraints .the description of a realistic telecommunication network deployment involves a huge number of parameters not all equally well documented , spanning fields from geography , regional development , sociology , technology , etc one does not a priori know which of them will play a crucial role for a given analysis such as cost estimation in our case .for example , the physical links of fixed networks ( optical fibres or copper wires ) are located along the streets and therefore one can assume that it is compulsory to include the underlying road system in the formulation of the problem .but what is its real impact on the sought result ? andwhat precise characteristics of the street system is it important to consider ?exhaustive reconstructions of deployment solutions can not answer this for computational reasons ( precise input data are often missing and computing time to find optimal solutions is prohibitive ) and due to the difficulty to extract the impact of each parameter from a set of solutions .once the architecture and deployment rules are known , optimization methods compute optimal locations for nodes and optimal ways for paths in order to minimize the cost of a network given the area of deployment .however , for planning purposes on a territory , other methods must be used to determine globally what would be the potentially best architectures and technical choices prior to the above mentioned detailed spatial localization of equipment .one class of such methods uses graph theory and complex networks theory to compute an estimation of the number of nodes and of the trenching length needed for a given urban area and a simple architecture .another class of such methods are macroscopic models based on * _ stochastic geometry_*. stochastic geometry is the mathematical field of spatial probability and has a wide range of applications . in particular, it allows an instantaneous analysis of modern telecommunication networks , which performances strongly depend on the spatial arrangement of their components , while focusing on global results rather than a detailed description of the network .this approach makes it possible to take into account and to understand the statistical complexity of the system by extracting its structuring parameters and explicitly showing how they impact the statistics of geometry dependent quantities . here, one is not interested in the precise location of network equipment , but on statistics on quantities that are meaningful for the operator such as distances from customers to equipment , or costs , that can decide on the economic interest and/or technical feasibility of a network design on a particular area .the idea of macroscopic models for fixed network is to replace actual locations of objects by random processes that take into account the main principles of their positioning while introducing a variability that represents the effect of numerous and unavoidable local constraints that compel in reality to slightly modify the planned locations .random processes deal with geometrical objects of any shapes .their definition explicitly describes some rules and associated parameters to randomly generate realizations of sets of these objects .so called `` functionals '' ( for example the distances between pairs of points ) can then be computed on each realization of the set .the aim of stochastic geometry is to derive closed form expressions for the output of the functional , depending on the type and parameters of the processes . in this way, statistics on the geometrical properties of sets of objects can be obtained instantaneously and independently of their number of elements .this paper focuses on cost estimation of network at urban scale using the techniques of stochastic geometry .it is organized as follows : section [ sec : methods ] recalls the context of existing models developed in orange labs and explains the underlying mathematical objects involved ; section [ sec : implementation ] describes how those objects are organized and integrated in a user - friendly tool to build a faithful macroscopic model of the implementation of a realistic network and its costs ; section [ sec : results ] provides and discusses results on real cities and section [ sec : conclusion ] opens some perspectives .in a simple cost function for a hierarchical core network which nodes are located in the plane is considered and optimized to deduce the characteristics of the middle layer . to address access networks that describe connections to the customers, the location of networks equipment must be estimated considering the existing road system since the network nodes are located on or close to the streets . to take advantage of the power of stochastic modelling , all the parts of the model including the street system have to be randomized .subsections below briefly present the work done in the past years concerning the randomization , respectively of the street system , the typical cell , the links and the capacity trees .discussions and details can be found in the references .we call * _ node _ * a geographical site containing various pieces of equipment , designed to perform specific functions in the network . nodes of similar functions are referred to by their type such as pb ( branching point ) or pm ( shared access point ). this section will be devoted to the model of the elementary two levels network .its architecture is the simplest one , involving low level nodes ( ll ) , high level nodes ( hl ) and their formal * _ link _ * ( l ) ( fig .[ fig : fig2net ] ) .ll and hl must be of different types .engineering rules ensure that any node ll is connected to a unique node hl and specify how the nodes and the physical path of the link are spatially located .a * _ 2net _ * is the formal representation of a number of hl nodes and their links l deployed on the territory to be served .note that the characteristics of the links ( number , length ... ) depend on both hl and ll , but their description does not include any node characteristics .we shall explain below how a 2net can be modelled by a unique spatial random object , the * _ typical cell _ * and its content .any hierarchical architecture can be represented as an assembly of 2nets . for example a three levels purely hierarchical architecture from ll to hl is built form ( hl , ll_2 , l_2 ) and ( hl_1 , ll , l_1 ) by identification of ll_2 and hl_1 . the generic aspect of the 2net notion allows us to conduct generic computations on all network characteristics whatever the architecture ( see section [ subsec : network_design ] ) .the morphology of an urban street system reflects the history of the city . ignoring dead - ends ,the streets segments of a planar map constitute the edges of a planar tessellation .the cells of the tessellation are generally not uniform : their area , number of edges , etc can vary from one cell to the other . to cope with this spatial variability , and ignoring the finite area of the map, it is viewed as a realization of a random stationary tessellation that is fully determined by three independent scalar geometrical quantities averaged per unit area .we chose to represent the morphology of the map by the four - vector composed of quantities than can be easily estimated : the mean number of crossings , of edges , of cells and the mean length of streets .the mean number of edges is included for checking purpose and should equal the mean number of crossings plus cells .simple homogeneous random models as poisson vorono , poisson line and poisson delaunay tessellation ( pvt , plt and pdt ) are good candidates to capture the morphology of a street system .each one is fully described by a unique scalar intensity parameter .their corresponding theoretical vector can be expressed as a function of , for example : , a * _ fitting procedure _ * applied to the city map selects the best random model chosen in a set of available models and estimates its parameter by minimizing a distance between and .realizations of the selected model look quite different from the real data , but averaged quantities are close .for example : 1855 crossings , 2937 edges , 1083 cells and 439 km streets measured in an area 53.9 km of the french city of pau , correspond to a pvt , =18.2 km and theoretical averages : 1961 crossings , 2941 edges , 980 cells and 460 km streets . to cope with the frequent non stationarity of cities , a * _ segmentation _ * procedure automatically partitions the city in as many homogeneous parts as desired and fits the best model for each part .simple poisson models are sufficient for our final purpose , but better fits can be obtained considering iterated , stit or poisson gabriel tessellations .   the above defined 2net will be deployed in an area covered by a street system fitted by a random process . in order to anchor this 2net to the territory , one has to specify ( i ) how to locate the nodes with respect to the streets and ( ii ) how to associate a unique hl to any ll , thus defining the hl serving zone with respect to lls ( fig .[ fig : fig2net ] ) .the process for nodes is doubly stochastic ( poisson - cox process ) since they are assumed to be randomly positioned following a linear poisson process on the random process for streets .processes for hls and lls are assumed independent and are described by intensity parameters and under homogeneity assumption ( see appendix ) .any ll located in the geographical area called the serving zone of the hl will be connected to it .the serving zone of a hl is modelled as its vorono cell ( set of points of the plane closest to this hl than to any other ) constructed with the euclidean metric .any ll is thus almost surely connected to a single hl .note that the set of all serving zones is a poisson - cox - vorono - tessellation ( pcvt ) of the plane , different from a pvt since the location of hls is restricted to the streets .we define the * _ typical cell _ * as the random object which characteristics obey the statistics of the cells of pcvt .fast algorithms for simulation of these cells are described in .they are based on realizations of the street system supporting the hl and stopping criteria from neighbouring hls locations .links and capacity trees discussed below are mathematically defined relatively to those typical cells .more complicated models using non euclidean metric for the typical cells and gibbs models for the location of nodes , could be thought more realistic .but the simpler assumptions above proved to be quite effective and sufficient for our macroscopic approach .the links between a hl and all the lls included in its serving zone are assumed to follow the shortest paths along the streets .their number as well as their lengths are random . explicit formulas relating the average to the parameters of the streets and nodes random models were proposed in .this is not enough for practical purpose on topics such as eligibility ( % of customers closer than a given length threshold to the highest node , ensuring a good quality for communication ) that require to know the probability distribution of the length .closed parametric forms for the probability distribution of ( see appendix ) are derived in .the distribution does not depend on and may be understood as the distribution of lengths viewed from a typical ll , or as the distribution of all the lengths of links l in the network .the formulas include limit values and are valid in the whole range of parameters. they can be plugged in a software , avoiding lengthy simulations , and adapt to all possible cases .this abstract construction has been successfully tested in by comparing simulation results to histograms for lengths of links obtained from real networks .the parametric formulas provide a quantitative and explicit appreciation of the structuring features of our problem that are the morphology of the street system ( represented by the random model ) and the nature of the physical path of the link ( in straight line or along the streets ) .however costs can not be directly inferred from the distance distributions and one has to consider the multipoint to point tree rooted in hl to all the lls of its serving zone .this tree has a meaning only if the physical paths of the links follow an underlying street system ; then more than one link can share a common street segment . in optical networks ,optical fibres are bundled in cables .the cabling policy selects which size of cable(s ) will accommodate the * _ capacity _ * of each street segment , i.e. the number of fibres running along it .since the cost of cables depends on their size ( for example a 12-fibre cable is quite cheaper than two 6-fibre ones ) it is a major issue to study * _ capacity trees _ * rooted in hls . cabling itselfis considered as network engineering and will be addressed in section [ subsec : cable ] . in the framework of stochastic modelling , random capacity trees on street systems are extremely difficult to handle .tentative definitions for structuring characteristics , such as typical sub trees , were investigated in together with estimators for the distribution function of their length . since an hlis located almost surely on a street segment and not on a crossing , a capacity tree is composed of two half trees that are statistically dependent because they are constrained to live in the same cell . a parametric copula description for their lengths as well as asymptotic results for infinitely sparse or dense networks is studied in .sparse random trees with a fixed number of two lls are fully explored in .since we do not have for now generic parametric formulas for the joint probability distributions of the various elements that will enter our cost function ( number of cables merging , number of kilometres of cables for each cable capacity ) , we have to simulate the capacity trees .the input parameters are , and the capacity incoming from each ll , assuming that all lls bring in a common deterministic capacity .future work could introduce some variability in this capacity .a capacity tree is obtained from a realization of the typical cell and of the lls ( fig .[ fig : typicaltree ] ) .the root of the tree is the hl .the final leaves are either an edge of the cell or a `` midpoint '' at which two different paths of equal length to the root hl exist .note that a path to the hl may need to borrow streets outside the cell .a street element of constant capacity defines a `` section '' of the tree .starting from a final leaf , the capacity changes ( i ) by adding each time we pass a ll ( ii ) by adding the capacities of incoming branches at a street crossing .the vertices of the tree are the street crossings .the set of sections of non zero capacity is the portion of the road system covered by the network .the sections are indexed and each realization of the tree is stored in a text file containing information on vertices ( planar location , index of parent vertex ) , contour of the typical cell , distance of lls to hl , etc this allows a full reconstruction of the tree and offers the possibility of statistical analysis of all possible single or joined characteristics . for our purpose , only the length of covered streets and the joint distribution ( capacity , length of sections ) will be used .the limit cases mentioned above are not yet included in the tool that restricts the simulation to a given range of parameters , avoiding cases of too dense or too sparse networks .the sections above introduce typical random objects related to the description of an elementary two levels network .to model the deployment of this network in a realistic setting , one has to precise the geographical area to be covered ( whole city or part of it ) and the expected number of serving zones ( or equivalently of hl nodes ) to consider .added to the model for roads , this characterizes the parameters for the typical cell of averaged area and the intensity ( see appendix ) . in real networks ,the serving zones in are not identical and present a variability that is assumed to be correctly represented by the statistics of this typical cell .rigorously , the covering of by typical cells should include some constraints on the sum of their areas and border effects should also be considered . for the sought applications ,these corrections can be neglected and the very simple approach of combining the characteristics of typical cells of identical parameters is sufficient .  to summarize , realistic access networks have their lowest level directly connected to the customers .they can be built by combining elementary 2nets simply assuming for each of them that the serving zones are vorono areas in which the physical links follow shortest paths on the streets or in the plane .the above mentioned results mainly concern technical and geometrical aspects of the network ( nodes , links and capacity trees ) .  the theoretical framework being complete , in order to achieve our goal to build an user - friendly tool for cost estimation , one still has to :  * enable the user to select a deployment territory , to design an architecture and to specify engineering rules , * specify how demography and road data should be selected and processed to be proposed as input for the tool , * enable the user to enter precise formulas allowing to compute the costs , * compare the results to data obtained in an operational context and explain the observed discrepancies .two tools were developed with the aim to allow a user to benchmark different architectures and several engineering parameters for each architecture , in order to compare them and to determine the optimal ( with respect to costs and eligibility ) architecture and engineering rules to deploy in a given city .the * _ osmminer _ * prototype ( fig .[ fig : osmminer ] ) is used first to acquire the territory data , choose the best model to fit them and compute the parameters of the model ( section [ subsec : territory ] ) .then , proceeding with the * _ ntstool _ * prototype , the user selects a territory and designs or selects an architecture ( fig .[ fig : ntstool ] ) ; the tool then implements the methodology described above to compute the distance and attenuation distributions and to estimate the deployment costs .ntstool is designed to be generic for urban areas and can potentially address a variety of situations .sections [ subsec : network_design ] to [ subsec : costs ] below focus on optical access networks .the road system of the territory is extracted from the open street map ( osm ) database since it has a worldwide coverage with a very detailed description of the various types of roads .its rich typology allows to select the types of streets that will support the network wires while suppressing the useless or duplicated ones ( eg . cycle ways or paths ) .moreover , osm comes with several free applications and is free of charge .osm also provides information about buildings , but they are not always complete nor accurate .the administrative boundary of a given municipality generally differs from the `` build - up '' area limit inside which the streets are more or less continuously lined with buildings .the `` build - up '' area sometimes includes part of the neighbouring municipalities and contains most of the population of the city .it is thus the area of interest from the point of view of network deployment .a java code has been developed to automatically determine this limit using information on buildings .since there are regions where no information on the buildings is available , the limit can also be deduced from the road system information only .the method is to sum up the number of buildings or streets in each square of a regular grid of adjustable size and to first eliminate the squares with a sum less than a threshold given by the user , and second the isolated squares ( those with no more than one neighbour ) .a concave hull of the set of remaining squares is determined using and an algorithm directly inspired by .the limit ( fig .[ fig : osmminer ] ) is given in the form of a standard osm `` .poly '' file which can then be tailored if necessary by free tools like josm .  finally , to recover a planar graph for the town , i.e. the streets inside the build - up area ,the longitude - latitude coordinates given by osm are projected on the tangent plane at the center of the territory ( which is small enough to neglect distortions ) . following the methodology described in section [ subsec : street ] ,dead - ends are removed in order to get a tessellation , the town is segmented in as many parts as desired , and the best model for each part is determined with its intensity parameter ( for now , one can chose among three tessellation models : pvt , pdt and plt ) .the results are stored in a `` * * _ voirie _ * * '' file constituted by as many lines as there are parts of the town , plus one line representing the town as a whole ( table [ tab : voirie ] ) ..example of a `` voirie '' file for the french city of nantes .part1 is the dense core of the city . [cols="^,^,^,^",options="header " , ]the theoretical results developed in previous works were implemented in two user - friendly prototypes for cost estimation of fixed access network deployment on any urban area with very short computing time .the idea is to let the user try several architectures and scenarios in order to compare them and determine the best one according to its criteria . on a specific use case ( deploying optical fibre ) the costs obtained are very close to those computed by an orange optimization tool based on an entirely different approach .the nts methodology based on statistical modelling proved to be quite efficient even on the small areas considered in the comparison .the tools are modular and will be enriched by non - isotropic models for streets ( manhattan ) and extended to rural areas .the range of applications of osmminer / ntstool prototypes is quite large since they make it possible to consider any architecture and scenario on any urban or rural area .we are currently considering using their results on the fibre networks linking mobile base stations to antennas , as well as on electrical networks .another possible application is to evaluate the distance to drive in order to maintain all the mobile sites in a country , depending on the number of maintenance centres : this could allow us to better challenge the costs declared by the sub - contractors .a two levels network with a number of high nodes hl and of low nodes ll is to be deployed on a territory of area ( km ) .let us assume that the fitting process identified a pvt model of intensity ( km ) for the streets and recall that the corresponding morphological vector is where ( km ) is the average length of streets par unit area .the averaged area of a hl serving zone is then since everything is assumed homogeneous .it contains a mean length of streets ( km ) and is assimilated to the area of the typical `` poisson - cox - vorono '' cell centred on hl .this typical cell also contains by construction a unique hl thus giving ( km ) for the linear intensity of the random process that models hl locations on the streets .a key parameter is the dimensionless ratio a high ( resp .low ) value of describes a typical cell with a dense ( resp .sparse ) street system .  using scaling arguments , palm theory and statistical analysis of extensive simulation data, it is shown that for the probability distribution of length ( km ) writes : where are known polynomial functions and is the truncated weibull probability distribution with .explicit formulas exist for the cumulative function , quantile and average of distribution .theoretical considerations provide formulas for .similar expressions hold for the others plt and pdt models , using the associated expression for .  *acknowledgments * the authors would like to thank m jean - philippe lanquetin and m patrick boss ( orange engineers ) , as well as m yu hao , ms wenting bao and ms lena schmidt ( orange interns ) who participated in the development of the two prototypes .        mller j. ( 1989 ) .random tessellations in r ._ adv appl probab _( 21):37 - 73 .gloaguen c. , fleischer f. , schmidt h. , schmidt v. ( 2006 ) .fitting of stochastic telecommunication network models via distance measures and monte - carlo tests ._ telecommun syst _ 31(4):353 - 377 .        gloaguen c. , coup p. , maier r. , schmidt v. ( 2002 ) .stochastic modelling of urban access networks .networks , 10th international telecommunication network strategy and planning symposium .munich , germany .gloaguen c. , schmidt h. , thiedmann r. , lanquetin j.p . ,schmidt v. ( 2007 ) .comparison of networks trees in deterministic and random settings using different connection rules .limassol , cyprus .arcep ( 2015 ) . recueil de spcifications fonctionnelles et techniques sur les rseaux en fibre optique jusqu ' l ' abonn en dehors des zones trs denses . version 3 .accessed november 28th , 2016 .

Abstract: using theoretical results presented in former papers , two prototypes were developed , aiming at estimating within minutes the cost of a fibre network deployment on a given territory . the first one helps defining the limit of an urban territory and computes mathematical parameters representing its street system . the second one gives global information on a fixed network deployment on this territory , namely the probability distributions of distances and attenuation from a node of the network to the final customer , and an evaluation of the deployment cost , once given an architecture and engineering rules . this allows a final user to easily design various network architectures in the tool , to compare different deployment scenarios , and to optimize the budget and the efficiency of the network in a few minutes . the results were compared on two real french urban territories ( in tours and rouen ) to those given by an optimization tool currently used by orange .


the catalysis of chemical reactions by surfaces plays a crucial role in a vast range of physical , chemical and biological systems . in many cases ,the surfaces are of macroscopic dimensions and the reactants appear in large quantities . under these conditions, the law of large numbers applies and fluctuations in the surface concentrations of reactants and their reaction rates become negligible . as a result , these reactions can be analyzed using rate equation models that incorporate the mean - field approximation and ignore fluctuations .consider the case of a surface , which is partitioned into microscopic domains , that are disconnected , namely reactants can not diffuse between them .the population of reactive atoms and molecules in each domain becomes small and fluctuations become important . as a result ,rate equations fail and the simulation of these reactions requires stochastic methods such as direct integration of the master equation or monte carlo ( mc ) simulations . while mc simulations require the accumulation of statistical data over long times , the master equation provides the probability distribution from which the reaction rates can be obtained directly . in certain cases , the master equation can be solved using a generating function .the set of coupled ordinary differential equations is then transformed into a single partial differential equation for the generating function .this equation can solved numerically and in a few cases can also be solved analytically .the master equation can also be approximated using the fokker - planck equation .this is a partial differential equation in which the population sizes of the reactive species are represented by continuous variables .the master equation and related methods described above are useful for simple reaction networks , which involve few reactive species . however , as the number of reactive species increases , the number of equations in the master equation quickly proliferates .this makes the master equation infeasible for complex networks including a large number of reactive species .the recently introduced multiplane method provides a dramatic reduction in the number of equations . in this method , the reaction networks are described by graphs , where each node represents a species and each edge represents a reaction . typically , these networks are sparse , namely most species react only with few other species . in the multiplane methodone breaks the network into a set of fully connected subnetworks ( cliques ) .lower - dimensional master equations are constructed for the marginal probability distributions associated with the cliques , with suitable couplings between them .this enables the simulation of large networks much beyond the feasibility limit of the master equation .however , it turns out that the construction of the multiplane equations for complex networks is difficult .  in this paperwe present a method for stochastic simulations of surface reaction networks , which is based on moment equations .this method exhibits crucial advantages over the multiplane method .the number of equations is further reduced to one equation for each reactive species ( node ) and one equation for each reaction ( edge ) .thus , for typical sparse networks the complexity of the stochastic simulation becomes comparable to that of the rate equations .unlike the master equation ( and the multiplane method ) there is no need to adjust the cutoffs , namely the same set of equations is used under all physical conditions .moreover , for any given network the moment equations can be easily constructed using a diagrammatic approach .this enables to automate the construction of the set of equations - a feature which is essential in the case of complex networks .furthermore , the moment equations are linear in terms of the moments .thus , the stability and convergence properties can be easily controlled and the steady state solution can be obtained by standard algebraic methods .  below we consider some examples of surface reaction systems in which fluctuations play an important role . in surface catalysis systems ,the surface is often partitioned into facets on nanometric dimensions , with little diffusion between the facets . under these conditionsthe number of reactive atoms and molecules residing on a facet is small and fluctuations are strong . .another important example of chemical reaction networks that require stochastic analysis appears in the field of interstellar chemistry .some chemical reactions in interstellar clouds take place on the surfaces of dust grains .these include molecular hydrogen formation as well as reaction networks that form ice mantles and certain organic molecules .unlike gas phase reactions in cold clouds that mainly produce unsaturated molecules , surface processes are dominated by hydrogen - addition reactions that result in saturated , hydrogen - rich molecules , such as water ( h ) , ammonia ( nh ) , methane ( ch ) , formaldehyde ( h ) and methanol ( ch ) . in particular , recent experiments have shown that methanol can not be efficiently produced by gas phase reactions .on the other hand , there are indications that it can be efficiently produced on ice - coated grains .therefore , the ability to perform efficient simulations of chemical reactions on interstellar grains is of great importance .  due to the submicron size of the grains and the low gas density , the populations of reactive species per grain are small and strongly fluctuate .therefore , rate equations are not suitable and stochastic methods such as the direct integration of the master equation or mc simulations are required . as discussed above , these methods apply in the case of small networks but become infeasible for large networks .  herewe demonstrate the moment - equation method for a set of astrophysically relevant networks of increasing complexity , culminating in the network that describes the production of methanol .it is shown that in spite of the small number of equations and the ease of their construction , the accuracy of the results is not compromised .the results of the moment equations are in excellent agreement with the master equation and coincide with the rate equations in the limit of large grains .  the paper is organized as follows . in sec .ii we consider the production rate of molecular hydrogen using the rate equation , master equation and moment equation methods . in sec .iii we consider the simulation of complex reaction networks using the moment equations .the construction of the set of moment equations using diagrammatic methods is presented in sec .iv , followed by a summary in sec . v.consider the reaction h that takes place on the surfaces of interstellar dust grains .hydrogen atoms from the gas phase collide and stick to the surface of a dust grain .they diffuse between adsorption sites on the surface and when two of them encounter each other they recombine and form an h molecule , which desorbs into the gas phase .  for simplicity , we assume that the grains are spherical and denote their diameter by .the cross - section of a grain is and its surface area is .the density of adsorption sites on the surface of a grain is denoted by ( sites ) .thus , the number of adsorption sites on the grain is .each grain is exposed to a flux ( s ) of h atoms , where ( ) is the density of h atoms in the gas phase and ( cm s ) is their average velocity , which is determined by the gas temperature .it is also useful to define the flux in units of monolayers ( ml ) per second , namely ( ml s ) . clearly , this flux is independent of the grain size .the desorption rate of h atoms from the grain is given by , where is the attempt rate ( standardly taken to be s ) , is the activation energy barrier for desorption of an h atom , and ( k ) is the surface temperature .the hopping rate of adsorbed atoms between adjacent sites on the surface is , where is the activation energy barrier for hopping of the atoms .molecular hydrogen formation on astrophysically relevant surfaces of silicates , carbon and ice has been studied by laboratory experiments during the past decade . from the results of these experiments one can extract , for each surface , the energy barriers defined above and the density of adsorption sites . in the simulations presented below , the density of adsorption sites was ( sites ) , the activation energies were mev for diffusion and mev for desorption .these parameters are rounded values of the experimental results for hydrogen recombination on silicates .the grain temperature was taken as . herewe assume that diffusion occurs only by thermal hopping , in agreement with the experimental results .for small grains , it is convenient to replace the hopping rate ( hops s ) by the sweeping rate , which is approximately the inverse of the time it takes for an h atom to visit nearly all the adsorption sites on the grain surface ( for a more precise evaluation of the sweeping rate see refs .the rate equation for this reaction takes the form    where is the average population size of h atoms residing on the grain . the first term on the right hand side of eq .( [ eq : rate_1specie ] ) describes the incoming flux of h atoms .the second term accounts for the desorption of h atoms from the surface , which is proportional to the number of adsorbed atoms .the third term accounts for the depletion in the population of adsorbed h atoms due to the recombination process .the production rate of h molecules on a single grain , is given by    for simplicity we assume that these molecules desorb from the surface upon formation .  the rate equation ( [ eq : rate_1specie ] ) can be solved either analytically or by numerical integration . the steady state solution can be obtained by setting the left hand side of eq .( [ eq : rate_1specie ] ) to zero . under steady state conditions ,the average population size of h atoms on a grain is   \label{eq : rate_ss}\ ] ]  and the formation rate of molecules per grain is   ^ 2 ,  \label{eq : eta_ss}\ ] ]  where . note that is independent of the grain size . under these conditions, one can define the recombination efficiency    which is the fraction of adsorbed atoms that come out in molecular form .note that .one can identify two limits . in the limit where the desorption process is dominant and the recombination efficiency is low . in this case , the recombination rate per grain is and the recombination efficiency is .since depends quadratically on the flux , this regime is referred to as second order kinetics . in the opposite limit , where , the recombination process is dominant and the efficiency is .this is the regime of first order kinetics , in which depends linearly on  .  under given flux and surface temperature , for grains that are large enough to hold many h atoms , eq .( [ eq : rate_1specie ] ) provides a good description of the recombination process .however , in the limit of small grains and low flux , may be reduced to order unity or less . under these conditions , eq .( [ eq : rate_1specie ] ) becomes unsuitable , because it ignores the discrete nature of the population of adsorbed atoms and its fluctuations .to account correctly for the recombination rate on small grains , simulations using the master equation are required .the dynamical variables of the master equation are the probabilities of having a population of hydrogen atoms on the grain . in the case of hydrogen recombination, the master equation takes the form   + w  \left [ ( n + 1 ) p  ( n + 1 ) - n  p ( n ) \right ]  \nonumber \\ & + & a  [ ( n + 2 ) ( n + 1 ) p ( n + 2 )  - n ( n -1 ) p ( n ) ] , \label{eq : master_1specie}\end{aligned}\ ] ]  where . the first term on the right hand side of eq . ( [ eq : master_1specie ] ) describes the effect of the incoming flux .the probability increases when an h atom is adsorbed by a grain that already has adsorbed h atoms , and decreases when it is adsorbed on a grain with atoms .the second term accounts for the desorption process .the third term describes the recombination process .the recombination rate is proportional to the number of pairs of h atoms on the grain , namely .therefore , the h production rate can be expressed in terms of the moments of as    in numerical simulations the master equation must be truncated in order to keep the number of equations finite . a convenient way to achieve this is to assign an upper cutoff on the population size .the number of equations is thus .the truncated master equation is valid if the probability to have a larger population than the assigned cutoff is vanishingly small .therefore , the upper cutoff should be chosen according to the parameters of the simulation .as noted above , the population size of the adsorbed h atoms is given by the first moment of , while the reaction rate can be expressed in terms of the difference between the second moment and the first moment .therefore , a closed set of equations for the time derivatives of these first and second moments could directly provide all the information that we need about the population size and reaction rates .such equations can be obtained from the master equation using the identity    where is an integer . inserting according to eq .( [ eq : master_1specie ] ) , one obtains the moment equations    this is a set of coupled differential equations , which are linear in the moments . although we have written the equations only for the first two moments , the right hand side includes the third moment for which we have no equation .in order to close the set of equations one must express the third moment in terms of the first two moments .different such expressions have been proposed .for example , in the context of birth - death processes the relation was used .this choice makes the moment equations nonlinear , with possible effects on their stability .another common choice is to assume that the third central moment is zero ( which is exact for symmetric distributions ) and use this relation in order to express the third moment in terms of the first and second moments .  herewe set up the closure condition by imposing a highly restrictive cutoff on the master equation .the cutoff is set at .this is the minimal cutoff that still enables the recombination process to take place . under this cutoff, one obtains the following relation between the first three moments     using this result , one can bring the moment equations [ eq .( [ eq : momentsh ] ) ] into a closed form :    numerical integration of these equations provides the first two moments , from which the population size and reaction rate are obtained .the steady state solution of the moment equations can be obtained by setting the derivatives on the left hand side of eq .( [ eq : moment_2 ] ) to zero .one obtains    using eq .( [ eq : master_production ] ) we find that the recombination rate is   .\label{eq : r_moment_ss}\ ] ]  in this system one can identify two characteristic spatial scales , namely and  .the limit of small grains is obtained when is smaller than both and . in this limitthe populations size of atoms on the grain is small , the rate equation fails while the moment equations are accurate .the limit of large grains is obtained when is larger than both and . in this limitthe population size of atoms on the grain is large , the rate equation applies and the reaction rate satisfies .since and , the limit of large grains can be expressed by    we recall that the moment equations were derived by imposing a strict cutoff on the master equation .one may thus expect that the equations will apply only in the case of small grains , where this cutoff is suitable .however , it turns out that the moment equations are valid much beyond this limit .below we show that the reaction rate obtained from the moment equations is accurate for both small and large grains .  in fig .[ fig1 ] we present the population size of h atoms and the reaction rate , vs. grain diameter , obtained by the moment equations ( circles ) .the flux of h atoms was ( atoms s ) .the parameters used in fig .[ fig1 ] satisfy the conditions for second - order kinetics , namely .the results are in perfect agreement with the master equation ( solid lines ) for the entire range of grain sizes . for small grains , the rate equation ( dashed lines ) over - estimates the recombination rate , but coincides with the master equation and the moment equations for large grains . in fig .[ fig2 ] we present the population size of h atoms and the reaction rate , vs. grain diameter , under the conditions of first - order kinetics , where .these conditions were obtained by increasing the flux to ( atoms s ) .the moment equations still provide accurate results for the reaction rate on grains of all sizes .however , for large grains the population size of adsorbed h atoms , obtained from the moment equations , deviates from the master equation results .  in conclusion, one can identify two kinetic regimes ( first and second order ) and two limits ( small and large grains ) . in the limit of small grains , the moment equations are valid , as expected , for both kinetic regimes ( table i ) . in the case of second order kinetics ,the moment equations are valid both for small grains and for large grains .they provide accurate results both for the average number of atoms on a grain and for for the reaction rate . in the case of large grains under conditions of first order kineticsthe situation is different .the moment equations still provide accurate results for the reaction rate but are not suitable for the evaluation of the population size on a grain . in the next section we present a more complete analysis of the moment equations and their validity .the moment equations were derived on the basis of a very strict cutoff , allowing at most two atoms to simultaneously reside on the surface of the grain .nevertheless , it turns out that the equations apply even under conditions in which the population size of adsorbed atoms is large . herewe examine the validity of the moment equations in the regimes of first and second order kinetics and in the limits of small and large grains .for sufficiently small grains the population of adsorbed atoms on a grain is small and is consistent with the strict cutoff imposed in the derivation of the moment equations .therefore in the limit of small grains the approximation of eq .( [ eq : third_moment ] ) is justified , and the moment equations are valid .  in the limit of large grainsthe population of adsorbed atoms on a grain is large and the rate equation becomes accurate .therefore , we will test the validity of the moment equations in this limit by comparison with the rate equations . in the limit of large grains the relations and are satisfied .we will first consider the case of second order kinetics , where .using the relations above , we find that eqs .( [ eq : moment_ss ] ) are reduced to and .thus the results of the moment equations for the population size and for the reaction rate coincide with the rate equations .  in the case of first order kinetics . in the limit of large grains, we obtain from eq .( [ eq : moment_ss ] ) that , which is consistent with the h production rate obtained from rate equations . on the other hand ,the expression for the first moment is reduced to , which is inconsistent with the rate equations , in which .these results are in line with the numerical results shown above , indicating that the moment equations are suitable for the evaluation of reaction rates on large grains both in first and second order kinetics . however , they provide the correct population size of atoms on large grains only in the case of second order kinetics and not in first order kinetics .  in fig .[ fig3](a ) we show the first moment , , as obtained from the moment equations ( circles ) vs. , for a very large grain of diameter mm , where .the results are compared with those of the rate equation ( dashed line ) , which under these conditions is accurate . for ( second order kinetics )the rate equation and the moment equations coincide . for ( first order kinetics )the moment - equation results for the average population size deviate from those of the rate equation .note that even when , where the moment equations apply , the population size of atoms per grain is very large , much beyond the cutoffs used in the construction of the equations . in fig .[ fig3](b ) we present the production rate of h molecules per grain vs. , as obtained from the moment equations ( circles ) .the results are in perfect agreement with those of the rate equations ( dashed line ) in the regime of first order kinetics as well as in the regime of second order kinetics .these results can be generalized to more complex networks .the analysis becomes more tedious , but the conclusions about the domain of validity of the moment equations remain the same . from a physical perspective , in first order kineticsthe reaction is dominant and desorption is suppressed , while in second order kinetics desorption is dominant and the reaction rate is low and diffusion limited .to generalize the analysis beyond the case of a single species , consider a simple chemical network that involves three reactive species : h and o atoms and oh molecules .for simplicity we denote the reactive species by h , o , oh , and the resulting non - reactive species by h , o , h .the reactions that take place in this network include h + o oh ( ) , h + h h ( ) , o + o o ( ) , and h + oh h ( ) . a graph of that networkis displayed in fig .[ fig4](a ) .now , the desorption rates of atomic and molecular species on the grain are given by  ] , where is the activation energy barrier for hopping of atoms ( or molecules ) .the sweeping rate will be .the master equation provides the time derivatives of the probabilities that a randomly chosen grain will have atoms or molecules of the reactive species .it takes the form    \nonumber \\  & & + \sum_{i=1}^3  w_i \left [ ( n_i+1 ) p( .. ,n_i+1 , .. ) - n_i p(n_1,n_2,n_3 ) \right ]  \nonumber \\  & & + \sum_{i=1}^2  a_i [ ( n_i+2)(n_i+1 ) p( .. ,n_i+2 , .. )  - n_i(n_i-1 ) p(n_1,n_2,n_3 ) ]  \nonumber\\  & & + ( a_1+a_2 ) [ ( n_1 + 1)(n_2 + 1 ) p(n_1 + 1,n_2 + 1,n_3 - 1 )  - n_1 n_2 p(n_1,n_2,n_3 ) ]  \nonumber \\  & & + ( a_1+a_3 ) [ ( n_1 + 1)(n_3 + 1 ) p(n_1 + 1,n_2,n_3 + 1 )  - n_1 n_3 p(n_1,n_2,n_3 ) ] .\label{eq : master }  \end{aligned}\ ] ]  the terms in the first sum describe the incoming flux , where ( atoms s ) is the flux _ per grain _ of the species .the second sum describes the effect of desorption .the third sum describes the effect of diffusion - mediated reactions between two identical atoms and the last two terms account for reactions between different species .the rate of each reaction is proportional to the number of pairs of atoms of the two species involved , and to the sum of their sweeping rates .the average population size of the species on the grain is , where , and , 2 or 3 .the production rate per grain ( molecules s ) of molecules produced by the reaction is given by , or by in case that .for complex networks with more than one species the truncation of the master equation demands setting upper cutoffs for all the reactive species , , , where is the number of reactive species .the number of coupled equations is then , which grows exponentially with the number of reactive species .this severely limits the applicability of the master equation to interstellar chemistry . to reduce the number of equations one tries to use the lowest possible cutoffs under the given conditions . in any case , to enable all reaction processes to take place , the cutoffs must satisfy for species that form homonuclear diatomic molecules ( h , o , etc . ) and for other species .  as in the case of hydrogen recombination ,the population sizes and reaction rates are completely determined by all the first moments and selected second moments of the distribution .therefore , a closed set of equations for the time derivatives of these first and second moments could provide the complete information on the population sizes and reaction rates . for the simple network considered here one needs equations for the time derivatives of the first moments , and of the second moments , , and .we obtain these equations from the master equation using the identity    where are integers . herewe show three of the resulting moment equations :    as expected , the right hand sides of these equations include third order moments for which we have no equations .if we add equations for their time derivatives , they will include fourth order moments , and again will not enable us to close the set of equations . in order to close the set of moment equationswe must express the third order moments in terms of the first and second order moments .we recall that in the case of a single reactive specie this was achieved by imposing an extremely restrictive cutoff on the master equation .generalizing it to our present system we choose the minimal cutoffs that do not terminate any of the reactions taking part in the network .these cutoffs allow at most two atoms or molecules to be adsorbed on the surface at any given time .furthermore , any two atoms or molecules that reside simultaneously on the surface must belong to species that react with each other . in our networkthese cutoffs allow only eight non - vanishing probabilities , namely , , , , , , , and .expressing the third moments , with these cutoffs , leads to the following results :    using these rules to modify eqs .( [ eq : momenth_h2o ] ) one obtains a closed set of the form    this set of equations is the minimal set required for the system .it includes one equation that accounts for the population size of each reactive specie and one equation that accounts for the reaction rate of each reaction . as in the hydrogen system, the equations were derived using extremely strict cutoffs , that are expected to apply only in the limit of small grains and low flux .nevertheless , the moment equations are valid well beyond the restrictive cutoffs under which they were derived . in fig .[ fig5](a ) we present the population sizes of h ( ) and o ( ) atoms on a grain vs. grain diameter , obtained from the moment equations . in fig .[ fig5](b ) we show the production rates of h ( circles ) , o ( triangles ) and h ( squares ) , obtained from the moment equations .the results are in excellent agreement with the master equation and in the limit of large grains they also coincide with the rate equations .the parameters used in the simulations are ( atoms s ) , and .the activation energies for diffusion and desorption were taken as , , , , and mev for h , o and oh respectively . the surface temperature of the grain was .for the species other than h there are no concrete experimental results , and the values reflect the tendency of heavier species to bind more strongly .the generalization to more complex networks is straightforward . as explained above, the cutoffs imposed on the master equation give rise to a minimal set of non - vanishing probabilities , that allow all the reactions to take place .these non - vanishing probabilities are for the each node , , and for each edge connecting nodes and . a loop connected to the node corresponds to the probability .all other probabilities vanish . in the realm of two - body reactionsadditional probabilities such as or are never required . under these restrictions, one can easily show that all the third moments that appear in the moment equations can be expressed in terms of second moments according to eq .( [ eq : barzel_rules ] ) .let us now examine how the moment equation method performs in a more complex network .recent laboratory experiments provide evidence that methanol in interstellar clouds is formed via reaction networks on dust grains .to simulate these networks , consider the case in which a flux of co molecules is added to the network shown in fig .[ fig4](a ) .this gives rise to the network shown in fig .[ fig4](b ) , which includes the following sequence of hydrogen addition reactions : h + co hco ( ) , h + hco h ( ) , h + h h ( ) and h + h ch ( ) .two other reactions that involve oxygen atoms also take place : o + co co ( ) and o + hco co + h ( ) .this network was studied before using the multiplane method , which required about a thousand equations compared to about a million equations in the master equation with similar cutoffs .the moment equations include one equation for each species ( node in the graph ) and one equation for each reaction ( edge in the graph ) , namely , the network shown in fig . [ fig4](b ) requires only 17 equations .we have performed extensive simulations of this network using the moment equations and found that they are in agreement with the master equation results . in fig .[ fig6](a ) we present the population sizes of h ( ) , o ( ) and co ( ) on a grain vs. grain diameter , as obtained from the moment equations . in fig .[ fig6](b ) we present the results of the moment equations for the formation rates of ch ( squares ) , h ( circles ) and o ( triangles ) and compare them to the master equation ( solid line ) and the rate equations ( dashed lines ) . as before ,the moment equations prove to be consistent with the master equation , and for large grains with the rate equations as well .in the simulations presented here the activation energies for diffusion and desorption of co , hco , h and h were taken as , , , , , and , mev , respectively .the flux of co molecules was taken as .  in complex reaction networksthe reduction in the number of equations becomes extremely significant .for each reactive species , ( denoted by a node in the graph ) , there is one equation for the time derivative of the first moment . for each reaction , between species and ( represented by an edge ) , there is one equation for the time derivative of the second moment .the number of equations is thus , where and are the numbers of nodes and edges , respectively .most reaction networks are sparse , namely the number of edges is of the same order of magnitude as the number of nodes . as a result ,the number of moment equations is roughly _ linearly _ dependent on the number of species .this is in contrast with the exponential growth of the master equation and the polynomial growth of the multiplane equations .in the previous section we described a procedure for the construction of the moment equations , that consists of three steps . in the first step the desired set of moment equationsis determined according to the network graph .this set includes one equation for the first moment associated with each species ( node ) and one equation for the second moment associated with each reaction ( edge or loop ) . in the second step the time derivative of each of these momentsis expressed as a suitable linear combination of first , second and third moments . in the third stepthe equations are brought into a closed form , by expressing all third moments in terms of first and second moments according to eq .( [ eq : barzel_rules ] ) .this procedure is suitable for simple networks but becomes difficult to execute as the network complexity increases .  herewe present a simple and highly efficient diagrammatic approach for the construction of the moment equations for any given network . consider the reaction networks shown in fig .each node in the graphs represents one of the reactive species , and each edge stands for a reaction between two species .a loop represents a reaction between two atoms / molecules of the same species .the list of moments that should be included in the moment equations is easily obtained from the graph : ( a ) for each each node , , there is an equation for the first moment ; ( b ) for each edge connecting the nodes and there is an equation for the second moment ; ( c ) for each loop attached to node there is an equation for the second moment .  to derive the moment equations from the graph of a given network ,we show how each element in the graph is expressed in the master equation and then translate it into a corresponding term in the moment equations .consider first the contribution of nodes .each node represents a species , that may be adsorbed on the surface or desorbed from it . in the master equation , these two processes are described by   \label{eq : node_master } \\ & + & w_i[(n_i+1)p(\dots , n_i+1 , \dots)- n_ip(\dots , n_i , \dots ) ] .\nonumber \end{aligned}\ ] ]  to construct the equation for we sum over all the terms of the master equation , according to eq . ([ eq : moment_abc ] ) .the terms above then make the following contribution to the moment equations    in case that the species reacts with itself , one needs to include an equation for . as before , a proper summation over the master equation is performed , but here the contribution of the node will be    if the species reacts with some other species , , it will be necessary to include an equation for .the contribution of the node to this equation is    note that in the equation for , one should also include the analogous term for the node .consider the contribution of edges to the moment equations .each edge represents a reaction between two different species . in the master equation , the term related to the edge between and is   . \nonumber\\\end{aligned}\ ] ]  the contributions of this term to the moment equations appear in all the equations for moments involving either species or species .for example , the contribution to the equation for is    after applying the rules of eq .( [ eq : barzel_rules ] ) , this term simplifies to .in general , each moment equation is obtained by a proper summation over terms of the master equation , using eq .( [ eq : moment_abc ] ) .each term in the master equation describes a certain process such as adsorption , desorption or reaction .consider the equations for and .when the summations are carried out , the only non - vanishing contributions are from processes that involve the species . similarly , consider the equation for . in this casethe only non - vanishing contributions are from processes that involve either the species or the species .this gives rise to a dramatic simplification in the construction of the moment equations . in the equation for any desired momentwe need to consider only those master equation terms related to the species that appear in that particular moment . in the diagrams , this means including only graph elements that are directly connected to the corresponding nodes .the complete set of translation rules from a given graph element to the corresponding terms in the moment equations is given in tables [ tab : moment_terms1 ] - [ tab : moment_terms3 ] .to demonstrate the construction of the moment equations we reconsider the reaction network shown in fig .[ fig4](a ) .first , we identify the relevant moments associated with the nodes , edges and loops in the network graph . in this particular network ,the nodes call for , and .the edges call for and .finally , the two loops in the graph require the inclusion of and .altogether , there are seven equations .let us begin with the equation for .this species is represented by a node connected to two edges and one loop . using our diagrammatic scheme, this equation takes the form    this equation simply consists of all the graph elements related to the species .that diagrammatic sum can be translated , using table [ tab : moment_terms1 ] , giving rise to the equation    the same diagrammatic equation [ eq .( [ eq : diag1 ] ) ] is used when writing the equation for the time derivative of . however , the translation rules from graph elements to equation terms are different and are given in table [ tab : moment_terms2 ] .using these rules we obtain    next , we construct the moment equation for . this time we need to include some additional graph elements , namely , those related to , so the diagrammatic equation takes the form    this is an equation for a mixed second moment , that should be translated according to table [ tab : moment_terms3 ] .the resulting equation is    the last equation that we demonstrate explicitly is for the moment .this time the diagrammatic equation will include an additional graph element stating the fact that is produced by a reaction between the other two species in the network .the diagrammatic equation is    which , according to table [ tab : moment_terms1 ] translates to    this procedure enables an efficient and straightforward construction of the set of moment equations for any given graph , regardless of its complexity. the construction process can be easily automated into a computer program that receives the network structure as an input and provides the set of equations as an output .this opens the way to efficient stochastic modeling of complex reaction networks of any size .we have presented an efficient method for the stochastic simulation of complex reaction networks .the method is based on moment equations .it accounts correctly for the reaction rates not only for macroscopic populations , where rate equations apply , but also in the limit of low copy numbers where fluctuations dominate .the method exhibits several crucial advantages over existing techniques for stochastic simulations .the number of equations is reduced to the absolute minimum in a stochastic simulation , namely one equation for each reactive species and one equation for each reaction . forany given network the set of moment equations can be constructed easily and efficiently using a diagrammatic approach .the equations are linear , thus , for steady - state conditions they can be easily solved using algebraic methods .we have demonstrated the method for reactions taking place on dust - grain surfaces in interstellar clouds .we have shown that it applies very well even under the extreme interstellar conditions of low gas density and sub - micron grain sizes .the moment equations can be easily coupled to the rate equation models of interstellar gas - phase chemistry .therefore , the proposed method is expected to enable the incorporation of grain - surface chemistry into these models .we expect this method to be useful for the simulation of many other systems that exhibit a related mathematical structure .one such application is for reactions taking place on terrestrial surfaces such as metal catalysts in nanometric systems .another example is aerosol chemistry in stratospheric clouds , where complex reaction networks involving aerosol particles take place .the gas density is not nearly as low as in the interstellar clouds .however , the small size of the aerosol particles and the low density of some of the reactive species may require stochastic methods .another example is genetic networks in cells and bacteria .these networks describe the process of protein synthesis and its regulation .they include the interactions between genes , mrna s and proteins .since some of these components appear in low copy numbers , the simulation of these networks requires stochastic methods .graph element & equation term & reduced equation term + & & + & & + & & + & & + & ( ) & ( ) + & & + & & + & & + & & + & & + & & + & & + & & + & & + { } } \put(1,-1.875){\circle{1.5 } } \put(0,-2.875){\makebox(2,2){ } } \end{picture } } i(j)k ] & & + & & + & & +   molecules ( b ) on the surface of a grain vs. grain diameter , , under conditions of second - order kinetics .the number of adsorption sites , , is also shown .the results of the moment equations ( circles ) are in perfect agreement with the master equation ( solid line ) for all grain sizes . in the limit of large grainsthey also coincide with the rate equation ( dashed line ) ., width=480 ]   molecules ( b ) on the surface of a grain vs. grain diameter , , under conditions of first - order kinetics .the production rate obtained by the moment equations ( circles ) is in perfect agreement with the master equation ( solid line ) for all grain sizes .however , the population size deviates for large grains .the rate equation results are also shown ( dashed lines ) ., width=480 ]  , as obtained from the moment equations for a very large grain of diameter mm ( circles ) . under these conditionsthe rate equation is accurate . for ( second order kinetics ) ,the moment equations are in perfect agreement with the rate equation ( dashed line ) .this is in spite of the large population size . for ( first order kinetics ), the moment equations do not provide accurate results for the population size ; ( b ) the production rate of h molecules per grain vs. , as obtained from the moment equations ( circles ) .the results are in perfect agreement with those of the rate equation ( dashed line ) both for second order and for first order kinetics ( even when there is a deviation in the first moment ) . , width=432 ]    ) and o ( ) atoms on a grain vs. grain diameter , , obtained from the moment equations for the reaction network presented in fig .the results are in excellent agreement with the master equation ( solid line ) .the rate equation results are also shown ( dashed line ) .( b ) the production rates of h ( circles ) , h ( squares ) and o ( triangles ) per grain vs. , obtained by the moment equations .the results are in excellent agreement with the master equation ( solid line ) . in the limit of large grainsthey also coincide with the rate equations ( dashed line ) .note that the moment equations are valid both in the limit of small population sizes ( when is small ) , and even when the population is high ( large ) .the number of adsorption sites , , is also displayed ., width=480 ]  ) , o ( ) atoms and co molecules ( ) on a grain vs. grain diameter , , obtained from the moment equations for the reaction network presented in fig .the results are in excellent agreement with the master equation ( solid line ) .the rate equation results are also shown ( dashed line ) .( b ) the production rates of h ( circles ) , ch ( squares ) and o ( triangles ) molecules on a grain vs. , obtained from the moment equations .the results are in agreement with the master equation ( solid line ) . for small grains ,the rate equation results ( dashed line ) show significant deviations . , width=432 ]

Abstract: surfaces serve as highly efficient catalysts for a vast variety of chemical reactions . typically , such surface reactions involve billions of molecules which diffuse and react over macroscopic areas . therefore , stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations , which are based on the mean - field approximation . however , in case that the surface is partitioned into a large number of disconnected microscopic domains , the number of reactants in each domain becomes small and it strongly fluctuates . this is , in fact , the situation in the interstellar medium , where some crucial reactions take place on the surfaces of microscopic dust grains . in this case rate equations fail and the simulation of surface reactions requires stochastic methods such as the master equation . however , in the case of complex reaction networks , the master equation becomes infeasible because the number of equations proliferates exponentially . to solve this problem , we introduce a stochastic method based on moment equations . in this method the number of equations is dramatically reduced to just one equation for each reactive species and one equation for each reaction . moreover , the equations can be easily constructed using a diagrammatic approach . we demonstrate the method for a set of astrophysically relevant networks of increasing complexity . it is expected to be applicable in many other contexts in which problems that exhibit analogous structure appear , such as surface catalysis in nanoscale systems , aerosol chemistry in stratospheric clouds and genetic networks in cells .