while there has been much debate about the exact nature of the quantum state , the most persuasive interpretation is that the state represents the knowledge or belief of an observer about a given quantum system . using the state ,the observer can assign probabilities to the outcomes of any possible measurement of the system . from this point of view , a pure state represents a state of maximal knowledge ( or minimal ignorance ) about a system , and a mixed state represents a description with incomplete knowledge .whether the state also reflects the structure of an underlying physical reality is a subject of heated debate ; but at a practical level , we make state assignments based on our available information .if the state assignment reflects the information of an observer , it follows reasonably that two observers with different information will make different state assignments .this can arise naturally in a number of ways ; for example , one observer may have access to certain measurement outcomes that another observer does not , or may have additional information about how the system was prepared . in question was examined , and a necessary and sufficient condition was found which must be satisfied by any two or more state assignments which reflect differing information about the same physical system , provided that the information underlying all of these assignments is accurate and reliable .classically , the state of knowledge about a system is described by a probability distribution .new information about the system can be acquired by performing a measurement , and updating the state assignment based on a measurement result is done using the bayes rule .classically , two probability distributions can be compatible so long as they are not actually contradictory .furthermore , it is possible to pool the information of different observers using only their individual states of knowledge ( probability distributions ) , _ provided _ that the information of different observers is independent . in this case, other information ( such as how the knowledge was obtained ) is not required .a reasonable question is whether similar conditions hold in quantum mechanics .can we take the state assignment to reflect all information about a quantum system ?if so , under what conditions ?what assumptions do we have to make about how the information underlying the state assignments was acquired ?  in this paper , we look at different ways that a state can be derived from underlying information , and show that in general the state assignment does not completely summarize all the information used to derive it .in particular , we consider two significantly different ways of constructing the state . in the first case ,the information is derived from measurement .each party obtains his or her information by making a measurement . in the second case ,each observer is given _ classical _ information about how the system was prepared , for example from the third party who prepared it .the method of pooling information becomes clear once we know how the information was obtained , but different types of information can result in very different ways of acquiring a joint state assignment .however , this does not rule out the possibility that within a very sharply defined context , it may be possible and useful to both define reasonable measures of compatibility , and give rules for for pooling state assignments.  there have been various studies concerning compatibility of state assignments and pooling of information .quite recently , the authors in ref . used the conditional states formalism to show that earlier results can be recast in this more general formalism .our results differs from theirs in that the main focus of this paper is an attempt to _ quantify _ the degree of compatibility in the assigned quantum states .we also note that the problem of pooling information , so that a unique joint state assignment can be derived , is very sensitive to the type of information used to make the state assignment in the first place , and illustrate this with a couple of representative examples .  in sec .[ sec2 ] , we briefly describe the necessary and sufficient condition for two different states to be compatible .we discuss the possible existence of a compatibility measure ( or measures ) that quantifies how much the two states are compatible . in sec .[ sec3 ] , we show two significantly different ways of obtaining the state assignment , and argue that the state does not encapsulate the whole information in general . in sec .[ smr ] , we explore a particular compatibility measure , and show that this measure is actually a distance measure between two states , while in sec .[ sec5 ] we show how if we assume that the state assignments were derived from a particular type of measurement , there is a simple rule for forming a joint state assignment . in sec .[ sdp_sec ] we extend this measure to compatibility of states for observers , and show that it is the solution of a semidefinite program ; we then define similar compatibility measures for two more restrictive notions of compatibility , post - peierls ( pp ) and equal support ( es ) compatibility .we conclude the paper in sec .[ sec7 ] .it seems necessary to make a bit more concrete what we mean when we talk about different observers have different _ information _ about a given system .what , exactly , do we mean by information in this context ?  for clarity , we will draw somewhat arbitrary distinctions between _ information _ , _ knowledge _ , and _belief_. when does it make sense even to talk about compatibility and state pooling ?arguably , two observers will only have a basis for comparison between their state assignments if they start from a fairly substantial base of shared knowledge about the system .for example , the two observers might both know that the give system was prepared by a given type of experimental apparatus , or by performing a particular generalized measurement .this would represent their shared knowledge base about the system .given that shared knowledge , the two observers might still have different _ information _ about the system .for instance , they might each have partial , but different , information about the settings of the device that prepared the system ; or they might each have partial , but different , information about the outcome of the generalized measurement .when we refer to _ information _ in the discussion that follows , we will mostly be using it in this rather restrictive sense that is , some amount of numerical data about the system , or more generally , a probability distribution over such data .if the two observers do not have a shared knowledge base then it becomes harder to compare their state assignments , or for such comparisons to even make sense .if two observers think that the system was prepared in radically different ways , then it is hard to see how they could share their knowledge , even if their state assignments were almost identical .this is where knowledge shades over into belief .a number of authors have pointed out that two observers in possession of the same facts might nevertheless arrive at quite different state assignments if they have different _ prior distributions _ or _ priors_. ( see , for example , . )  in most discussions of compatibility including that in this paper we are implicitly assuming that the observers with compatible state assignments begin with identical ( or very similar ) priors about the system , and a strong base of shared knowledge .their state assignments differ due to the acquisition of different data about the system .while there is certainly no necessity that different observers should start from the same prior quite the contrary in practice , it is quite common in science for observers to come to consensus about particular experimental systems , generally by sharing large amounts of data from repeated trials or preparations .it would be interesting to study the idea of compatibility for observers with different priors , and in particular the process whereby consensus can be reached , but that is beyond the scope of the current paper .let and be two different assignments of density matrix to the same system . if we assume that these assignments were made by two observers with different information , what restriction does this place on the state assignments ?we call two states that satisfy such a restriction _ compatible_. a necessary and sufficient criterion for and to be compatible is that _ the intersection of their supports _ is nonempty : to show that this condition is necessary , we assume that the information used to derive state descriptions and is _ accurate and reliable_. the two observers should then be able to combine their information to produce a joint state description . since their information is accurate and reliable , any measurement result which is ruled out ( i.e. , assigned zero probability ) by either party should also be ruled out in the joint state . this then implies  ( _ a ) , & ( _ b ) ( _ ab ) + & \{(_a),(_b)}(_ab ) + & ( _ ab)(_a)(_b ) .  sufficiency is proven by constructing an explicit situation in which and could arise as state estimates by two observers making different measurements .we introduce purifications of and with two ancillas , and then combine them into a single state .the given state then results from alice and bob each measuring his or her own ancilla and getting a particular outcome .see for details .the compatibility criterion in eq .( [ compatibility ] ) is robust against sufficiently small distortions of the states .for example , given and , the two states remain compatible until .unfortunately , it is in some ways not very informative .the criterion is an all - or - nothing property ; it says only that two state assignments _ could _result from observers with differing information , but gives no clue how likely it is that they did .this makes it natural to seek a measure of compatibility that would indicate this likelihood .the measure would be zero for incompatible states , and go up to one for identical states .we discuss the possibility of such measures in secs .[ smr ] and [ sdp_sec ] .we note that other ( more restrictive ) notions of compatibility also exist : see ref . for an extensive discussion of this subject and a variety of different compatibility criteria .we briefly look at two of these other criteria in sec .[ sdp_sec ] .we think of the state assignment as being based on information possessed by the observer ; but information can come in many different forms . in this paper , we consider two different types of information in particular : classical information about the state preparation , in which the state assignment is made using the _ maximum entropy estimate _ , and information derived from measurements .   if an observer acquires classical information about the way a system was prepared , he or she can make a state assignment based on the _ maximum entropy principle _ : choosing the state with the highest entropy consistent with the given information .this takes a particularly simple form if the information is in the form of an expectation value .the observer can construct a state assignment by maximizing the von neumann entropy under the following two constraints :  [ constrain ] f ( ) & \ { } = o + g ( ) & \ { } = 1 .this is a constrained maximization problem , and can be solved with lagrange multipliers : the estimated state is where we can solve for and from ( [ constrain ] ) .this assignment is obviously based on the observer s knowledge ; further information can alter the assignment .if an additional piece of information is given , the state changes accordingly : if alice and bob wish to form a joint state assignment , they must share all the information they have about the state preparation and perform a maximum entropy assignment with this shared information .it is easy to see that as long as all the information provided to alice and bob is consistent that is , that there exists a state with all those expectation values their state assignments will be compatible . moreover , the entropy of the joint state must be less than or equal to the entropies of the individual states : and .consider now a different situation .suppose that alice and bob share two halves of an entangled state , which we write in the schmidt decomposition : with .alice has the reduced state which can accurately predict the probabilities for any measurement she makes .suppose bob makes a measurement in the basis , and gets outcome .then alice s state should immediately become .she can not , however , update her state until she learns the outcome of bob s measurement . until that point ,her original state assignment is the best she can make .   as mentioned in sec .1 , the state itself does not always include all the information an observer has about a system . for information acquired from measurements ,this was shown by jacobs , who demonstrated that if and are compatible state assignments , which are assumed to be obtained from the results of measurements by alice and bob , then it is possible to construct an initial state and choice of measurement such that the joint state assignment can be _ any _ density matrix at all , so long as .thus , in general more information is needed than and in order to construct ; and do not encapsulate all of alice and bob s information about the system .here we give another example showing that the same is true if alice and bob make their state assignments using classical information and the maximum entropy principle .suppose alice assigns the state of a qubit : where .this same state can be obtained from classical information in many ways . hereare two examples :  & a+b = a^2+b^2 ,  = c + & b+c = b^2+c^2 ,  = a.  suppose that bob makes his state assignment based on the classical information that .we can readily see that and are compatible .( indeed , any two qubit states are compatible unless they are two distinct pure state assignments . )now suppose that alice and bob share their information to obtain a joint state assignment .the joint state in cases 1 and 2 will be different :  &  _ ab = ( + ( a+b(b - d)/a)+d+c)/2 + &  _ ab = ( + a+d+(c+b(b - d)/c))/2 .this shows that knowledge of the state assignments and alone is not sufficient to know how to pool information in the case where the information is classical , as well .the state assignment does not encapsulate all information about the system .the compatibility criterion of is all - or - nothing : either two states are compatible , or they are not .this includes extreme examples like and . for any these statesare compatible , even though they are practically orthogonal for very small .this leads to the natural question : can we define _ measures _ of compatibility ?intuitively , while compatibility indicates that two state assignments _ could _ represent different information about the same system , such a measure would represent a probability that they actually _ do_. we should therefore expect the measure to go from 0 for incompatible states to 1 for identical states , with some kind of smooth behavior in between .  from our earlier discussion ,some caveats are clearly needed .since the states do not represent all information about the system , it is impossible to _ truly _ measure the compatibility based on the state assignments alone .for example , it is impossible to arrive at compatible state assignments using classical information which is contradictory ; e.g. , the two pieces of classical information and lead to compatible state assignments for a qubit , but are obviously inconsistent with each other .( this inconsistency comes from the assumptions of the maximum entropy procedure , where these expected values are taken to be guaranteed properties of the state . ) therefore , in defining a compatibility measure , one must implicitly assume that the state assignments represent information which was acquired in a consistent manner for example , by alice and bob performing measurements on the same system .one such measure was proposed by poulin and blume - kohout , in close analogy with the measure of distance between classical probability distributions .the correct interpretation of this measure is not exactly clear in the quantum case , but it does have the right sort of qualitative behavior , and reduces to a known measure in the case of essentially classical state ( i.e. , when and commute ) .in other cases , however , it is somewhat difficult to compute , since it requires taking a minimum over all pure state decompositions of density matrices .  herewe examine a different compatibility measure , which was originally suggested by kitaev .the original idea is to find a positive matrix that `` fits '' into both the state matrix and to the greatest degree possible .let and be compatible states . consider all positive matrices such that  [ mr ] _ a-&0 , + _ b-&0 .the compatibility measure is defined as   we have found no simple formula for this measure , in general .( in sec . [ sdp_sec ]we show that this value can be found as the solution of a semidefinite program . )however , it is possible to upper bound with the _ trace distance _ between and  ) : where is the trace distance : to see this inequality , define .we can rewrite eqs .( [ mr ] ) and ( [ kitaevmeasure ] ) in terms of :  [ mr1 ] & 0 , + _ b-_a+ & 0 , + _ a - & 0 ,  and this just rewrites the original definition of the measure in terms of the new operator .however , we can put an upper bound on this quantity by relaxing the third requirement above that . in this case, we can find an exact expression for ( [ bound ] ) . from , we know that can be expressed as where and are positive operators with orthogonal support .this implies that . we can satisfy ( [ mr1 ] ) and minimize by choosing then   it is easy to see that this measure has the right qualitative behavior for a measure of compatibility : it goes from for incompatible states ( where , in general , the above bound will be nonzero unless the states are actually orthogonal ) to for identical states ( at which point it will agree with the above bound ) .we will now show a couple of examples of this compatibility measure for states of qubits , in which the information which produces the state assignment is either classical information about the preparation , or the results of measurements . for our first example , suppose that charlie prepares the pure state where and .define an hermitian operator .charlie gives the expectation values to alice , and to bob .alice and bob s state estimates are , respectively , where and .the measure is then  k(_a,_b ) & = 1- + & = 1-(/2 ) .this expression obviously depends both on charlie s choice of state and his choice of observables to pass on to alice and bob .if we average this over all possible pure states , then we get a simple result which depends only on the choice of observable :  k_a , b&=_=11-(/2 ) + & = 1 - ( 1/3)(/2 ) .we plot this in fig .[ fig : me ] .the states and become identical when , and therefore are perfectly compatible .   versus angle ,scaledwidth=50.0% ]  in our second example , the information comes from measurement .assume that the initial state is unknown to alice and bob , so that they assume a completely mixed state .that is , alice performs projective measurement with outcome and , a_0= ,  a_1= , where and . for is a measurement ; for it is an measurement .bob also performs a projective measurement of : b_0= ,  b_1= .however , as in the paper of jacobs , we assume that alice and bob do nt know the order of their measurements , and that they each know only the outcomes of their own measurements .their state estimates are thus  _ a^i=_j=0^ 1,p_i=_j=0 ^ 1p_ij + _ b^j=_i=0 ^ 1,p_j=_i=0 ^ 1p_ij +  where .let , and =\sum_{i , j}p_{ij}k_{i , j} ] .being positive semidefinite implies that is hermitian , so by setting we obtain an satisfying the constraints of eq .( [ k state pp ] ) such that >0 $ ] .this implies that and the set is also pp compatible .thus , we have the following result :   .   a third compatibility measure for quantum states , stronger than both bfm and pp compatibility ,is that of _ equal support _ ( es ) .a set of states is es compatible if and only if the states have the same support .again , we can define a measure of es compatibility by an sdp :  the es compatibility of a set , , can be defined as :   it can be shown ( appendix [ es sdp appendix ] ) that the dual sdp is : , \\\textrm{subject to : } & \sum_{i=1}^{d}\alpha_i\geq1 ; \\ & \sum_{j=1}^{k}\rho_j \sum_{i=1}^{k } m_i \geq \begin{pmatrix } \alpha_1 & & \\ & \ddots & \\ & & \alpha_d \end{pmatrix};\\ & \alpha_i\in\mathbb{r } , \alpha_i\geq0 ,  m_i\geq 0,\  \forall i. \label{k state es dual } \end{split}\end{aligned}\ ] ] again , the primal and dual sdps will give the same value , and will return 0 if the states in are es incompatible as they do not have equal support .caves _ et al ._ show that if a set of states is es compatible , then it is also bfm compatible .equivalently , if the states are bfm incompatible , then they are also es incompatible. this can be rederived by comparing the sdps in eqs .( [ k state bfm dual ] ) and ( [ k state es dual ] ) .if is bfm incompatible , then there exists a set that satisfies the constraints of eq .( [ k state bfm dual ] ) such that : =0.\ ] ] by picking the set to be the eigenvalues of , will be a feasible solution to the sdp in eq .( [ k state es dual ] ) that returns .therefore , is also es incompatible .the fact that a feasible solution to the bfm dual sdp can be used as the basis for a feasible solution to the es dual gives :   .the problem of state compatibility is rather subtle , because state assignments do not perfectly reflect the information used to create them . different measures of compatibility ( and different methods of pooling information ) may make sense for different ways that alice and bob may have acquired their information .it is certainly impossible to define a measure of compatibility without some assumption about what type of information was used to produce the state assignments . by contrast , the qualitative criterion of only requires that the states _ might _ describe the same system .looking at different methods of acquiring information , however , remains an interesting question . within a particular assumption about information gathering, it may make sense to define measures of compatibility and methods of pooling states , and these might be useful in practice .we defined one such measure which is the solution of a semidefinite program and one such pooling method , which have an interpretation based on the assumption that the observers state assignments derive from information acquired by measurements of a particular form .there is still considerable room for development in this area .moreover , as discussed in sec .[ sdp_sec ] , there are other quite different approaches to the problem of state compatibility , based on the existence of measurements that discriminate between pairs of states ; these approaches lead to an entire hierarchy of compatibility conditions .we have shown that for two of these criteria ( pp and es compatibility ) , we can define measures of compatibility based on sdps , just as for bfm compatibility. it would be interesting to determine whether compatibility measures for the other definitions given in ( weak and weak ) also can be formulated as sdps .the status of these other idea of compatibility leaves much to be explored , and may be the key to understanding compatibility and the ability ( or inability ) to achieve consensus between observers with different priors .we would like to thank alexei kitaev , michael nielsen , and rdiger schack , for very useful conversations , and robin blume - kohout , carl caves , jennifer dodd , jerry finkelstein , chris fuchs , kurt jacobs , david mermin , david poulin , and howard wiseman for their interactions and feedback .this work was supported in part by nsf career grant no .mh is currently supported by an arc future fellowship under grant ft140100574 .99  t. brun , j. finkelstein , and n. mermin , phys .a 65 , 032315 ( 2002 ) .k. jacobs , quantum inf process 1 , 73 ( 2002 ) . c. caves , c. fuchs , and r. schack , phys .a 66 , 062111 ( 2002 ) .d. poulin and r. blume - kohout , phys .a 67 , 010101 ( 2003 ) .f. herbut , j. phys . a : math .37 , 5243 ( 2004 ) .k. jacobs , phys .a 72 , 044101 ( 2005 ) .r. w. spekkens and h. m. wiseman , phys .a 75 , 042104 ( 2007 ) .m. s. leifer and r. w. spekkens , arxiv:1110.1085 ( 2011 ) . m. s. leifer and r. w. spekkens , arxiv:1107.5849 , ( 2011 ) .e. t. jaynes , phys .rev * 106 * , 620 ( 1957 ) ; e. t. jaynes , phys . rev . * 108 * , 171 ( 1957 ) .a. kitaev , private communication .m. a. nielsen , private communication .m. a. nielsen and i. l. chuang _ quantum computation and quantum information _( cambridge university press , cambridge , 2000 ) . l. vandenberghe and s. boyd , siam review 38 , 49 ( 1996 ). j. watrous , lecture notes , cs 766/qic 820 theory of quantum information , university of waterloo .see lecture 7 , semidefinite programming , and lecture 8 , semidefinite programs for fidelity and optimal measurements ( fall , 2011 ) .j. lfberg , in computer aided control systems design , 2004 ieee international symposium on ( ieee , 2004 ) pp .. j. f. sturm , optimization methods and software 11 , 625 ( 1999 ) .s. bandyopadhyay , r. jain , j. oppenheim , and c. perry , phys .rev . a 89 , 022336 ( 2014 ) .here we give the form of a semidefinite program and the relation between the primal and dual problems as formulated in .an sdp is formed of three elements , . and are hermitian matrices and is a hermicity preserving superoperator .  using these , we define the primal problem to be : ,\\ \textrm{subject to : } & \quad \phi(x)\leq b ; \quad x\geq0 .\end{split}\end{aligned}\ ] ] the related dual is then given by : ,\\ \textrm{subject to } : & \quad \phi^{*}(y)\geq a ; \quad y\geq0 .\end{split}\end{aligned}\ ] ] here is the dual map to , given by : = .[ phi * equation ]   here we show that the sdp given in eq .( [ k state es dual ] ) is the dual to that defined in eq .( [ k state es ] ) .first we rewrite eq .( [ k state es ] ) so that it has the same structure as eq .( [ prime ] ) .this leads to :  .\\ \textrm{subject to : } & \quad\lambda-\lambda_i \leq 0 , \quad\forall i ; \\ & \quad  \begin{pmatrix } \lambda_1 & & \\ & \ddots & \\ & & \lambda_d \end{pmatrix } \sum_{j=1}^{k } \rho_j \leq \rho_i , \quad\forall i;\\ & \quad\lambda\geq0 ; \quad \lambda_i\geq0 , \  \forall i. \end{split } \label{es sdp rewrite}\end{aligned}\ ] ]   * is a by matrix : * is a by matrix where the first entries on the diagonal are 0 , and the remaining matrix is block diagonal with the blocks given by : * , the variable matrix , is a by matrix : * is a by matrix whose first entries on the diagonal we label by , and the remaining block diagonal with the elements we denote by : * the map is given by :   using eq .( [ phi * equation ] ) , we see that must satisfy : \\ = & \tr\left [ \begin{pmatrix } \lambda & & & \\ & \lambda_1 & & \\ & & \ddots & \\ & & & \lambda_d \end{pmatrix}\phi^*\left ( \begin{pmatrix } \alpha_1 & & & & & \\ & \ddots & & & & \\ & & \alpha_d & & & \\ & & & m_1 & & \\ & & & & \ddots & \\ & & & & & m_k \end{pmatrix } \right ) \right ] , \end{split}\ ] ] and hence produces a by matrix :

Abstract: we say that two ( or more ) state assignments for one and the same quantum system are _ compatible _ if they could represent the assignments of observers with differing information about the system . a criterion for compatibility was proposed in [ _ phys .  rev .  a 65 , 032315 ( 2002 ) _ ] ; however , this leaves unanswered the question of whether there are _ degrees _ of compatibility which could be represented by some quantitative measure , and whether there is a straightforward procedure whereby the observers can pool their information to arrive at a unique joint state assignment . we argue that such measures are only sensible given some assumption about what kind of information was used in making the state assignments in the first place , and that in general state assignments do not represent all of the information possessed by the observers . however , we examine one particular measure , and show that it has a straightforward interpretation , assuming that the information was acquired from a particular type of measurement , and that in this case there is a natural rule for pooling information . we extend this measure to compatibility of states for observers , and show that the value is the solution to a semidefinite program . similar compatibility measures can be defined for alternative notions of state compatibility , including _ post - peierls _ ( pp ) and _ equal support _ ( es ) compatibility .


in this paper , we present a general method of constructing the reduced description for dissipative systems of reaction kinetics .our approach is based on the method of invariant manifold which was developed in end of 1980th - beginning of 1990th by .its realization for a generic dissipative systems was discussed by .this method was applied to a set of problems of classical kinetic theory based on the boltzmann kinetic equation .the method of invariant manifold was successfully applied to a derivation of reduced description for kinetic equations of polymeric solutions .it was also been tested on systems of chemical kinetics .the goal of nonequilibrium statistical physics is the understanding of how a system with many degrees of freedom acquires a description with a few degrees of freedom .this should lead to reliable methods of extracting the macroscopic description from a detailed microscopic description .meanwhile this general problem is still far from the final solution , it is reasonable to study simplified models , where , on the one hand , a detailed description is accessible to numerics , on the other hand , analytical methods designed to the solution of problems in real systems can be tested .  in this paperwe address the well known class of finite - dimensional systems known from the theory of reaction kinetics .these are equations governing a complex relaxation in perfectly stirred closed chemically active mixtures .dissipative properties of such systems are characterized with a global convex lyapunov function ( thermodynamic potential ) which implements the second law of thermodynamics : as the time tends to infinity , the system reaches the unique equilibrium state while in the course of the transition the lyapunov function decreases monotonically .  while the limiting behavior of the dissipative systems just described is certainly very simple , there are still interesting questions to be asked about .one of these questions is closely related to the above general problem of nonequilibrium statistical physics .indeed , evidence of numerical integration of such systems often demonstrates that the relaxation has a certain geometrical structure in the phase space .namely , typical individual trajectories tend to manifolds of lower dimension , and further proceed to the equilibrium essentially along these manifolds .thus , such systems demonstrate a dimensional reduction , and therefore establish a more macroscopic description after some time since the beginning of the relaxation .there are two intuitive ideas behind our approach , and we shall now discuss them informally .objects to be considered below are manifolds ( surfaces ) in the phase space of the reaction kinetic system ( the phase space is usually a convex polytope in a finite - dimensional real space ) .the ` ideal ' picture of the reduced description we have in mind is as follows : a typical phase trajectory , , where is the time , and is an element of the phase space , consists of two pronounced segments .the first segment connects the beginning of the trajectory , , with a certain point , , on the manifold ( rigorously speaking , we should think of not on but in a small neighborhood of but this is inessential for the ideal picture ) .the second segment belongs to , and connects the point with the equilibrium , .thus , the manifolds appearing in our ideal picture are `` patterns '' formed by the segments of individual trajectories , and the goal of the reduced description is to `` filter out '' this manifold .there are two important features behind this ideal picture .the first feature is the _ invariance _ of the manifold : once the individual trajectory has started on , it does not leaves anymore .the second feature is the _ projecting _ : the phase points outside will be projected onto .furthermore , the dissipativity of the system provides an _ additional _ information about this ideal picture : regardless of what happens on the manifold , the function was decreasing along each individual trajectory before it reached .this ideal picture is the guide to extract slow invariant manifolds .the paper is organized as follows . in the section [ outline ] , we review the reaction kinetics ( section [ kinetics ] ) , and discuss the main methods of model reduction in chemical kinetics ( section [ reduction_review ] ) . in particular , we present two general versions of extending partially equilibrium manifolds to a single relaxation time model in the whole phase space , and develop a thermodynamically consistent version of the intrinsic low - dimensional manifold ( ildm ) approach . in the section [ mimg ] we introduce the method of invariant manifold in the way appropriate to this class of nonequilibrium systems . in the sections [ thermo ] and [ dc ] we give some details on the two relatively independent parts of the method , the thermodynamic projector , and the iterations for solving the invariance equation .we also introduce a general symmetric linearization procedure for the invariance equation , and discuss its relevance to the picture of decomposition of motions . in the section [ mim ] , these two procedures are combined into an unique algorithm . in the section [ ex ] , we demonstrate an illustrative example of computations for a model catalytic reaction . in the section [ parameterization ]we demonstrate how the thermodynamic projector is constructed without the a priori parameterization of the manifold .this result is essentially used in the section [ grid ] where we introduce a computationally effective grid - based method to construct invariant manifolds . in the section[ open ] we describe an extension of the method of invariant manifold to open systems .finally , results are discussed in the section [ conclusion ] .we begin with an outline of the reaction kinetics ( for details see e. g. the book of ) .let us consider a closed system with chemical species , participating in a complex reaction .the complex reaction is represented by the following stoichiometric mechanism : where the index enumerates the reaction steps , and where integers , and , are stoichiometric coefficients . for each reaction step , we introduce vectors and with components and .notation stands for the vector with integer components ( the stoichiometric vector ) .we adopt an abbreviated notation for the standard scalar product of the -component vectors :   the system is described by the -component concentration vector , where the component represents the concentration of the specie .conservation laws impose linear constraints on admissible vectors ( balances ) : where are fixed and linearly independent vectors , and are given scalars .let us denote as the set of vectors which satisfy the conservation laws ( [ conser ] ) : the phase space of the system is the intersection of the cone of -dimensional vectors with nonnegative components , with the set , and . in the sequel , we term a vector the state of the system .in addition , we assume that each of the conservation laws is supported by each elementary reaction step , that is for each pair of vectors and .reaction kinetic equations describe variations of the states in time . given the stoichiometric mechanism ( [ stoi ] ) ,the reaction kinetic equations read : where dot denotes the time derivative , and is the reaction rate function of the step . in particular, the mass action law suggests the polynomial form of the reaction rates : where and are the constants of the direct and of the inverse reactions rates of the reaction step .the phase space is positive - invariant of the system ( [ reaction ] ) : if , then for all the times .  in the sequel, we assume that the kinetic equation ( [ reaction ] ) describes evolution towards the unique equilibrium state , , in the interior of the phase space .furthermore , we assume that there exists a strictly convex function which decreases monotonically in time due to eq.([reaction ] ) : here is the vector of partial derivatives , and the convexity assumes that the matrices are positive definite for all .in addition , we assume that the matrices ( [ matrix ] ) are invertible if is taken in the interior of the phase space .the function is the lyapunov function of the system ( [ reaction ] ) , and is the point of global minimum of the function in the phase space .otherwise stated , the manifold of equilibrium states is the solution to the variational problem , for each fixed value of the conserved quantities , the solution is unique . in many cases , however , it is convenient to consider the whole equilibrium manifold , keeping the conserved quantities as parameters .for example , for perfect systems in a constant volume under a constant temperature , the lyapunov function reads : .\ ] ]  it is important to stress that in eq .( [ gfun ] ) is an _arbitrary _ equilibrium of the system , under arbitrary values of the balances . in order to compute ,it is unnecessary to calculate the specific equilibrium which corresponds to the initial state .moreover , for ideal systems , function is constructed from the thermodynamic data of individual species , and , as the result of this construction , it turns out that it has the form of eq .( [ gfun ] ) .let us mention here the classical formula for the free energy : ,\ ] ] where is the volume of the system , is the temperature , is the quantum volume of one mole of the specie , is the avogadro number , is the mass of the molecule of , , and is the free energy of the internal degrees of freedom per mole of .  finally , we recall an important generalization of the mass action law ( [ mal ] ) , known as the marcelin - de donder kinetic function .this generalization was developed by based on ideas of the thermodynamic theory of affinity .we use the kinetic function suggested by . within this approach , the functions are constructed as follows : for a given strictly convex function , and for a given stoichiometric mechanism ( [ stoi ] ) , we define the gain ( ) and the loss ( ) rates of the step , ,\quad w_s^{-}=\varphi_s^-\exp[\langle \nabla g,\mbox{\boldmath}_s\rangle],\ ] ] where are kinetic factors. the marcelin - de donder kinetic function reads : , and the right hand side of the kinetic equation ( [ reaction ] ) becomes , \!- \varphi_s^-\exp[\langle{\mbox{\boldmath}}g,{\mbox{\boldmath}}_s\rangle]\}.\ ] ] for the marcelin - de donder reaction rate ( [ mdd ] ) , the dissipation inequality ( [ htheorem ] ) reads : \left\{\varphi_s^+ e^{\langle{\mbox{\boldmath}}g,{\mbox{\boldmath}}_s\rangle}- \varphi_s^-e^{\langle{\mbox{\boldmath}}g,{\mbox{\boldmath}}_s\rangle}\right\}\le 0.\ ] ] the kinetic factors should satisfy certain conditions in order to make valid the dissipation inequality ( [ hmdd ] ) .a well known sufficient condition is the detail balance : other sufficient conditions are discussed in detail elsewhere . for the function of the form ( [ gfun ] ), the marcelin - de donder equation casts into the more familiar mass action law form ( [ mal ] ) .   what does it mean , `` to reduce the description of a chemical system '' ? this means the following :  1 .  to shorten the list of species .this , in turn , can be achieved in two ways : + \(i ) to eliminate inessential components from the list ; + \(ii ) to lump some of the species into integrated components . 2 .  to shorten the list of reactions .this also can be done in several ways : + \(i ) to eliminate inessential reactions , those which do not significantly influence the reaction process ; + \(ii ) to assume that some of the reactions `` have been already completed '' , and that the equilibrium has been reached along their paths ( this leads to dimensional reduction because the rate constants of the `` completed '' reactions are not used thereafter , what one needs are equilibrium constants only ) .3 .  to decompose the motions into fast and slow , into independent ( almost - independent ) and slaved etc . as the result of such a decomposition, the system admits a study `` in parts '' .after that , results of this study are combined into a joint picture .there are several approaches which fall into this category : the famous method of the quasi - steady state ( qss ) , pioneered by bodenstein and semenov and explored in considerable detail by many authors , in particular , by , and many others ; the quasi - equilibrium approximation ; methods of sensitivity analysis ; methods based on the derivation of the so - called intrinsic low - dimensional manifolds ( ildm , as suggested by ) .our method of invariant manifold ( mim , ) also belongs to this kind of methods .why to reduce description in the times of supercomputers ?first , in order to gain understanding . in the process of reducing the description one is often able to extract the essential , and the mechanisms of the processes under study become more transparent .second , if one is given the detailed description of the system , then one should be able also to solve the initial - value problem for this system .but what should one do in the case where the the system is representing just a point in a three - dimensional flow ?the problem of reduction becomes particularly important for modeling the spatially distributed physical and chemical processes .third , without reducing the kinetic model , it is impossible to construct this model .this statement seems paradoxal only at the first glance : how come , the model is first simplified , and is constructed only after the simplification is done ?however , in practice , the typical for a mathematician statement of the problem , ( let the system of differential equations be _ given _, then ... ) is rather rarely applicable in the chemical engineering science for detailed kinetics .some reactions are known precisely , some other - only hypothetically .some intermediate species are well studied , some others - not , it is not known much about them .situation is even worse with the reaction rates .quite on the contrary , the thermodynamic data ( energies , enthalpies , entropies , chemical potentials etc ) for sufficiently rarefied systems are quite reliable .final identification of the model is always done on the basis of comparison with the experiment and with a help of fitting . for this purpose , it is extremely important to reduce the dimension of the system , and to reduce the number of tunable parameters .the normal logics of modeling for the purpose of chemical engineering science is the following : exceedingly detailed but coarse with respect to parameters system reduction fitting reduced model with specified parameters ( cycles are allowed in this scheme , with returns from fitting to more detailed models etc ) . a more radical viewpoint is also possible : in the chemical engineering science , detailed kinetics is impossible , useless , and it does not exist .for a recently published discussion on this topic see ; .  alas , with a mathematical statement of the problem related to reduction , we all have to begin with the usual : let the system of differential equations be given ... .enormous difficulties related to the question of how well the original system is modeling the real kinetics remain out of focus of these studies .our present work is devoted to studying reductions in a given system of kinetic equations to invariant manifolds of slow motions .we begin with a brief discussion of existing approaches ._ quasi - equilibrium with respect to reactions _ is constructed as follows : from the list of reactions ( [ stoi ] ) , one selects those which are assumed to equilibrate first .let they be indexed with the numbers .the quasi - equilibrium manifold is defined by the system of equations , this system of equations looks particularly elegant when written in terms of conjugated ( dual ) variables , : in terms of conjugated variables , the quasi - equilibrium manifold forms a linear subspace .this subspace , , is the orthogonal completement to the linear envelope of vectors , .  _quasi - equilibrium with respect to species _ is constructed practically in the same way but without selecting the subset of reactions . for a given set of species , , one assumes that they evolve fast to equilibrium , and remain there .formally , this means that in the -dimensional subspace of the space of concentrations with the coordinates , one constructs the subspace which is defined by the balance equations , . in terms of the conjugated variables , the quasi - equilibrium manifold , ,is defined by equations , the same quasi - equilibrium manifold can be also defined with the help of fictitious reactions : let be a basis in . then eq . ( [ qe1 ] )may be rewritten as follows :   _ illustration : _ quasi - equilibrium with respect to reactions in hydrogen oxidation : let us assume equilibrium with respect to dissociation reactions , , and , , in some subdomain of reaction conditions .this gives : quasi - equilibrium with respect to species : for the same reaction , let us assume equilibrium over , , , and , in a subgomain of reaction conditions .subspace is defined by balance constraints : subspace is two - dimensional .its basis , in the coordinates , , , and reads : corresponding eq .( [ qe2 ] ) is :   _ general construction of the quasi - equilibrium manifold _ : in the space of concentration , one defines a subspace which satisfies the balance constraints : the orthogonal complement of in the space with coordinates defines then the quasi - equilibrium manifold . for the actual computations ,one requires the inversion from to .duality structure is well studied by many authors .  _quasi - equilibrium projector ._ it is not sufficient to just derive the manifold , it is also required to define a _projector _ which would transform the vector field defined on the space of concentrations to a vector field on the manifold .quasi - equilibrium manifold consists of points which minimize on the affine spaces of the form .these affine planes are hypothetic planes of fast motions ( is decreasing in the course of the fast motions ) . therefore , the quasi - equilibrium projector maps the whole space of concentrations on parallel to .the vector field is also projected onto the tangent space of parallel to .thus , the quasi - equilibrium approximation implies the decomposition of motions into the fast - parallel to , and the slow - along the quasi - equilibrium manifold . in order to construct the quasi - equilibrium approximation , knowledge of reaction rate constants of `` fast '' reactionsis not required ( stoichiometric vectors of all these fast reaction are in , , thus , knowledge of suffices ) , one only needs some confidence in that they all are sufficiently fast .the quasi - equilibrium manifold itself is constructed based on the knowledge of and of .dynamics on the quasi - equilibrium manifold is defined as the quasi - equilibrium projection of the `` slow component '' of kinetic equations ( [ reaction ] ) .the assumption behind the quasi - equilibrium is the hypothesis of the decomposition of motions into fast and slow .the quasi - equilibrium approximation itself describes slow motions .however , sometimes it becomes necessary to restore to the whole system , and to take into account the fast motions as well . with this , it is desirable to keep intact one of the important advantages of the quasi - equilibrium approximation - its independence of the rate constants of fast reactions .for this purpose , the detailed fast kinetics is replaced by a model equation ( _ single relaxation time approximation _ ) .  _ quasi - equilibrium models _ ( qem ) are constructed as follows : for each concentration vector , consider the affine manifold , .its intersection with the quasi - equilibrium manifold consists of one point .this point delivers the minimum to on .let us denote this point as .the equation of the quasi - equilibrium model reads : +\sum_{{\rm slow}}{\mbox{\boldmath}}_{s}w_s({\mbox{\boldmath}}^*_l({\mbox{\boldmath}})),\ ] ] where is the relaxation time of the fast subsystem .rates of slow reactions are computed in the points ( the second term in the right hand side of eq .( [ qemodel ] ) , whereas the rapid motion is taken into account by a simple relaxational term ( the first term in the right hand side of eq .( [ qemodel ] ) .the most famous model kinetic equation is the bgk equation in the theory of the boltzmann equation .the general theory of the quasi - equilibrium models , including proofs of their thermodynamic consistency , was constructed by .  _single relaxation time gradient models _ ( srtgm ) were considered by in the context of the lattice boltzmann method for hydrodynamics .these models are aimed at improving the obvious drawback of quasi - equilibrium models ( [ qemodel ] ) : in order to construct the qem , one needs to compute the function , this is a convex programming problem .it does not always has a closed - form solution .let is the orthonormal basis of .we denote as the matrix with the elements , where is the matrix of second derivatives of ( [ matrix ] ) .let be the inverse of .the single relaxation time gradient model has the form : the first term drives the system to the minimum of on , it does not require solving the problem ( [ qea ] ) , and its spectrum in the quasi - equilibrium is the same as in the quasi - equilibrium model ( [ qemodel ] ) . note that the slow component is evaluated in the `` current '' state .  the models ( [ qemodel ] ) and ( [ srtgm ] ) lift the quasi - equilibrium approximation to a kinetic equation by approximating the fast dynamics with a single `` reaction rate constant '' - relaxation time .the quasi - steady state approximation ( qss ) is a tool used in a huge amount of works .let us split the list of species in two groups : the basic and the intermediate ( radicals etc ) .concentration vectors are denoted accordingly , ( slow , basic species ) , and ( fast , intermediate species ) .the concentration vector is the direct sum , .the fast subsystem is eq .( [ reaction ] ) for the component at fixed values of .if it happens that this way defined fast subsystem relaxes to a stationary state , , then the assumption that is precisely the qss assumption .the slow subsystem is the part of the system ( [ reaction ] ) for , in the right hand side of which the component is replaced with .thus , , where bifurcations in the system ( [ fast ] ) under variation of as a parameter are confronted to kinetic critical phenomena .studies of more complicated dynamic phenomena in the fast subsystem ( [ fast ] ) require various techniques of averaging , stability analysis of the averaged quantities etc .various versions of the qss method are well possible , and are actually used widely , for example , the hierarchical qss method .there , one defines not a single fast subsystem but a hierarchy of them , .each subsystem is regarded as a slow system for all the foregoing subsystems , and it is regarded as a fast subsystem for the following members of the hierarchy . instead of one system of equations ( [ fast ] ) , a hierarchy of systems of lower - dimensional equations is considered , each of these subsystem is easier to study analytically .theory of singularly perturbed systems of ordinary differential equations is used to provide a mathematical background and further development of the qss approximation . in spite of a broad literature on this subject ,it remains , in general , unclear , what is the smallness parameter that separates the intermediate ( fast ) species from the basic ( slow ) .reaction rate constants can not be such a parameter ( unlike in the case of the quasi - equilibrium ) . indeed , intermediate species participate in the _ same _ reactions , as the basic species ( for example , , ) .it is therefore incorrect to state that evolve faster than . in the sense of reaction rate constants , is not faster .  for catalytic reactions , it is not difficult to figure out what is the smallness parameter that separates the intermediate species from the basic , and which allows to upgrade the qss assumption to a singular perturbation theory rigorously .this smallness parameter is the ratio of balances : intermediate species include the catalyst , and their total amount is simply significantly less than the amount of all the s .after renormalizing to the variables of one order of magnitude , the small parameter appears explicitly .  for usual radicals ,the origin of the smallness parameter is quite similar .there are much less radicals than the basic species ( otherwise , the qss assumption is inapplicable ) . in the case of radicals , however , the smallness parameter can not be extracted directly from balances ( [ conser ] ) .instead , one can come up with a thermodynamic estimate : function decreases in the course of reactions , whereupon we obtain the limiting estimate of concentrations of any specie : where is the initial composition .if the concentration of the radical r is small both initially and in the equilibrium , then it should remain small also along the path to the equilibrium .for example , in the case of ideal ( [ gfun ] ) under relevant conditions , for any , the following inequality is valid : \le g({\mbox{\boldmath}}(0)).\ ] ] inequality ( [ ineq_r ] ) provides the simplest ( but rather coarse ) thermodynamic estimate of in terms of and _ uniformly for . complete theory of thermodynamic estimates of dynamics has been developed by .one can also do computations without a priori estimations , if one accepts the qss assumption until the values stay sufficiently small .let us assume that an a priori estimate has been found , , for each . these estimate may depend on the initial conditions , thermodynamic data etc . with these estimates , we are able to renormalize the variables in the kinetic equations ( [ reaction ] ) in such a way that renormalized variables take their values from the unit segment  ] : .in the course of the optimization under certain restrictions one selects the optimal ( or appropriate ) reduced model . the sequential quadratic programming method and heuristic rules of sorting the reactions , substances etc were used . in the result ,for some stiff systems studied , one avoids typical pitfalls of the local sensitivity analysis . in simpler situationsthis method should give similar results as the local methods .one of the problems addressed by the sensitivity analysis is the selection of the important and discarding the unimportant reactions . suggested a simple principle to compare importance of different reactions according to their contribution to the entropy production ( or , which is the same , according to their contribution to ) .based on this principle , described domains of parameters in which the reaction of hydrogen oxidation , , proceeds due to different mechanisms . for each elementary reaction, he has derived the domain inside which the contribution of this reaction is essential ( nonnegligible ) . due to its simplicity , this entropy production principle is especially well suited for analysis of complex problems . in particular , recently , a version of the entropy production principle was used in the problem of selection of boundary conditions for grad s moment equations . for ideal systems ( [ gfun ] ) ,the contribution of the reaction to has a particularly simple form : for nonideal systems , the corresponding expressions ( [ hmdd ] ) are also not too complicated .in many cases , dynamics of the -dimensional system ( [ reaction ] ) leads to a manifold of a lower dimension .intuitively , a typical phase trajectory behaves as follows : given the initial state at , and after some period of time , the trajectory comes close to some low - dimensional manifold , and after that proceeds towards the equilibrium essentially along this manifold .the goal is to construct this manifold .the starting point of our approach is based on a formulation of the two main requirements :  ( i ) . _dynamic invariance _ : the manifold should be ( positively ) invariant under the dynamics of the originating system ( [ reaction ] ) : if , then for each ._ thermodynamic consistency of the reduced dynamics _ : let _ some _ ( not obligatory invariant ) manifold is considered as a manifold of reduced description .we should define a set of linear operators , , labeled by the states , which project the vectors , onto the tangent bundle of the manifold , thereby generating the induced vector field , , .this induced vector field on the tangent bundle of the manifold is identified with the reduced dynamics along the manifold .the thermodynamicity requirement for this induced vector field reads   in order to meet these requirements , the method of invariant manifold suggests two complementary procedures :  ( i ) . to treat the condition of dynamic invariance as an equation , and to solve it iteratively by a newton method .this procedure is geometric in its nature , and it does not use the time dependence and small parameters .  ( ii ) .given an approximate manifold of reduced description , to construct the projector satisfying the condition ( [ thermo ] ) in a way which does not depend on the vector field .we shall now outline both these procedures starting with the second .the solution consists , in the first place , in formulating the _ thermodynamic condition _ which should be met by the projectors : for each , let us consider the linear functional then the thermodynamic condition for the projectors reads : here is the null space of the projector , and is the hyperplane orthogonal to the vector .it has been shown that the condition ( [ therm1 ] ) is the necessary and sufficient condition to establish the thermodynamic induce vector field on the given manifold for all possible dissipative vector fields simultaneously .let us now turn to the requirement of invariance . by a definition ,the manifold is invariant with respect to the vector field if and only if the following equality is true : {\mbox{\boldmath}}({\mbox{\boldmath}})=0,\ { \rm for\ each\ } { \mbox{\boldmath}}\in{\mbox{\boldmath}}.\ ] ] in this expression is an _ arbitrary _projector on the tangent bundle of the manifold .it has been suggested to consider the condition ( [ invariance ] ) as an _ equation _ to be solved iteratively starting with some appropriate initial manifold .iterations for the invariance equation ( [ invariance ] ) are considered in the section [ dc ] .the next section presents construction of the thermodynamic projector using a specific parameterization of manifolds .in this section , denotes a generic manifold .first , it should be mentioned that _ any _ parameterization of generates a certain projector , and thereby a certain reduced dynamics .indeed , let us consider a set of independent functionals , and let us assume that they form a coordinate system on in such a way that , where is a vector function of the parameters . then the projector associated with this parameterization reads : where is the inverse to the matrix : this somewhat involved notation is intended to stress that the projector ( [ proj ] ) is dictated by the choice of the parameterization . subsequently , the induced vector field of the reduced dynamics is found by applying projectors ( [ proj ] ) on the vectors , thereby inducing the reduced dynamics in terms of the parameters as follows : depending on the choice of the parameterization , dynamic equations ( [ dyn ] ) are ( or are not ) consistent with the thermodynamic requirement ( [ thermo ] ) .the _ thermodynamic parameterization _ makes use of the condition ( [ therm1 ] ) in order to establish the thermodynamic projector . specializing to the case ( [ proj ] ) ,let us consider the linear functionals , then the condition ( [ therm1 ] ) takes the form : that is , the intersection of null spaces of the functionals ( [ derivative ] ) should belong to the null space of the differential of the lyapunov function , in each point of the manifold .  in practice , in order to construct the thermodynamic parameterization , we take the following set of functionals in each point of the manifold : it is required that vectors are linearly independent in each state .inclusion of the functionals ( [ func ] ) as a part of the system ( [ param_thermo ] ) and ( [ param_other ] ) implies the thermodynamic condition ( [ therm2 ] ) .also , any linear combination of the parameter set ( [ param_thermo ] ) , ( [ param_other ] ) will meet the thermodynamicity requirement .it is important to notice here that the thermodynamic condition is satisfied whatsoever the functionals are .this is very convenient for it gives an opportunity to take into account the conserved quantities correctly .the manifolds we are going to deal with should be consistent with the conservation laws ( [ conser ] ) .while the explicit characterization of the phase space is a problem on its own , in practice , it is customary to work in the space while keeping the constraints ( [ conser ] ) explicitly on each step of the construction .for this technical reason , it is convenient to consider manifolds of the dimension , where is the number of conservation laws , in the space rather than in the phase space .the thermodynamic parameterization is then concordant also with the conservation laws if of the linear functionals ( [ param_other ] ) are identified with the conservation laws . in the sequel , only projectors consistent with conservation laws are considered .very frequently , the manifold is represented as a -parametric family , where are coordinates on the manifold .the thermodynamic _ re - parameterization _ suggests a representation of the coordinates in terms of ( [ param_thermo ] ) , ( [ param_other ] ) .while the explicit construction of these functions may be a formidable task , we notice that the construction of the thermodynamic projector of the form ( [ proj ] ) and of the dynamic equations ( [ dyn ] ) is relatively easy because only the derivatives enter these expressions .this point was discussed in a detail by .finally , let us discuss how the thermodynamic projector is related to the decomposition of motions ._ assuming _ that the decomposition of motions near the manifold is true indeed , let us consider states which were initially close enough to the manifold . even without knowing the details about the evolution of the states towards , we know that the lyapunov function was decreasing in the course of this evolution .let us consider a set of states which contains all those vectors that have arrived ( in other words , have been projected ) into the point .then we observe that the state furnishes the minimum of the function on the set . if a state , and if it deviates small enough from the state so that the linear approximation is valid , then belongs to the affine hyperplane this hyperplane actually participates in the condition ( [ therm1 ] ) .the consideration was entitled ` thermodynamic ' because it describes the states as points of minimum of the function over the corresponding hyperplanes ( [ hyperplane ] ) .the thermodynamic projector is needed to induce the dynamics on a given manifold in such a way that the dissipation inequality ( [ thermo ] ) holds . coming back to the issue of constructing corrections, we should stress that the projector participating in the invariance condition ( [ invariance ] ) is arbitrary .it is convenient to make use of this point : when eq .( [ invariance ] ) is solved iteratively , the projector may be kept non thermodynamic unless the induced dynamics is explicitly needed .let us assume that we have chosen the initial manifold , , together with the associated projector , as the first approximation to the desired manifold of reduced description . though the choice of the initial approximation depends on the specific problem , it is often reasonable to consider quasi - equilibrium or quasi steady - state approximations . in most cases ,the manifold is not an invariant manifold .this means that does not satisfy the invariance condition ( [ invariance ] ) : {\mbox{\boldmath}}({\mbox{\boldmath}}_0)\ne0,\ { \rm for\ some\ } { \mbox{\boldmath}}_0\in{\mbox{\boldmath}}_0.\ ] ] therefore , we seek a correction . substituting and into the invariance equation ( [ invariance ] ) , and after the linearization in , we derive the following linear equation : \left[{\mbox{\boldmath}}({\mbox{\boldmath}}_0)+ { \mbox{\boldmath}}_{{\mbox{\boldmath}}_0}\delta{\mbox{\boldmath}}\right]=0,\ ] ] where is the matrix of first derivatives of the vector function , computed in the state .the system of linear algebraic equations ( [ method ] ) should be supplied with the additional condition .  in order to illustrate the nature of the eq .( [ method ] ) , let us consider the case of linear manifolds for linear systems . let a linear evolution equation is given in the finite - dimensional real space : , where is negatively definite symmetric matrix with a simple spectrum .let us further assume the quadratic lyapunov function , .the manifolds we consider are lines , , where is the unit vector , and is a scalar .the invariance equation for such manifolds reads : , and is simply a form of the eigenvalue problem for the operator .solutions to the latter equation are eigenvectors , corresponding to eigenvalues .assume that we have chosen a line , , defined by the unit vector , and that is not an eigenvector of .we seek another line , , where is another unit vector , , .the additional condition ( [ uni_con ] ) now reads : .then the eq .( [ method ] ) becomes [{\mbox{\boldmath}}_0+\delta{\mbox{\boldmath}}]=0 \gamma\gammahcxlcxcc\gamma\alpha\betahcxjclccpcjclccpc ] .the kinetic equation for the four component vector of concentrations , , has the form here are stoichiometric vectors , while functions are reaction rates : here are reaction rate constants .the system under consideration has two conservation laws , or , where and .the nonlinear system ( [ 1exequ ] ) is effectively two - dimensional , and we consider a one - dimensional reduced description .we have chosen the concentration of the specie as the variable of reduced description : , and , where .the initial manifold was taken as the quasi - equilibrium approximation , i.e. the vector function is the solution to the problem : the solution to the problem ( [ qe ] ) reads : the thermodynamic projector associated with the manifold ( [ 1ex1 ] ) reads : computing {\mbox{\boldmath}}({\mbox{\boldmath}}_0)jc ] .analytic results were compared with the results of the numerical integration .the following set of parameters was used : direct numerical integration of the system has demonstrated that the manifold in the plane attracts all individual trajectories .thus , the reduced description in this example should extract this manifold .[ fig1 ] demonstrates the quasi equilibrium manifold ( [ 1ex1 ] ) and first two corrections found analytically .it is apparent that while the initial quasi - equilibrium approximation is in a poor agreement with the reduced description , the corrections _ rapidly _ improve the situation .this confirms our expectation of an advantage of using iteration methods in comparison to methods based on a small parameter expansions .formally , the method of invariant manifold does not require a global parameterization of the manifolds . however , in most of the cases , one makes use of a priori defined `` macroscopic '' variables .this is motivated by the choice of quasi - equilibrium initial approximations .let a manifold be defined in the phase space of the system , its tangent space in the point be . how to define the projector of the whole concentrations space onto without using any a priori parameterization of ?the basis of the answer to this question is the condition of thermodynamicity ( [ therm1 ] ) .let us denote as the concentration space , and consider the problem of the choice of the projector in the quadratic approximation to the thermodynamic potential :    where is the matrix of the second - order derivatives of ( [ matrix ] ) , , is the deviation of the concentration vector from the expansion point .let a linear subspace be given in the concentrations space ._ problem : _ for every , and for every , define a subspace such that : ( i ) is a complement of in : ( ii ) is the point of minimum of on : besides ( i ) and ( ii ) , we also impose the requirement of a _ maximal smoothness _ ( analyticity ) on as a function of and .requirement ( [ minq ] ) implies that is the quasi - equilibrium point for the given , while the problem in a whole is the _ inverse _ quasi - equilibrium problem : we construct such that will be the quasi - equilibrium manifold .then subspaces will actually be the kernels of the quasi - equilibrium projector .let be the orthonormalized with respect to scalar product basis of , vector be orthogonal to , , .condition ( [ minq ] ) implies that the vector is orthogonal to in the point .let us first consider the case .the requirement of analyticity of as the function of and implies , where is the orthogonal completement of with respect to scalar product .the constant solution , also satisfies ( [ minq ] ) .let us fix , and extend this latter solution to . with this, we obtain a basis , .here is the simplest construction of this basis : where is the first component in the expansion , .the rest of the basis elements , form the orthogonal completement of with respect to scalar product , is the line spanned by .  dependence ( [ ort1 ] ) on , and is singular : at , vector , and then is not the completement of in anymore . for ,dependence gives one of the solutions to the inverse quasi - equilibrium problem in the neighborhood of zero in .we are interested only in the limit ,   finally , let us define now the projector of the space onto .if , then is the orthogonal projector with respect to the scalar product :    if , then , according to eq .( [ lim1 ] ) , where is the orthonormal with respect to basis of , is orthogonal to , , , , .thus , for solving the invariance equation iteratively , one needs only projector ( [ result_projector ] ) , and one does not need a priori parameterization of anymore .grid - based approximations of manifolds are attractive from the computational perspective .since no a priori parameterization is required in the method of invariant manifold , in this section we develop its grid - based realization .let us consider a regular grid in , and its mapping into the concentrations space .it makes sense to consider only which map a finite part of the grid into the phase space .this part of the map is termed _essential_. extension of the map onto the rest of the nodes is done by a simple ( for example , linear ) extrapolation of the essential part ( in practice , one needs to extrapolate only onto the next neighbors of the essential nodes ) .let operators of grid differentiation where be defined for functions on the grid , where label grid coordinates . with this ,the tangent space to the image of the grid in the point is defined for each node of the grid : the grid is termed invariant if , for each essential node , for the essential nodes , we write down the invariance equation with the projector , : this equation is solved using the newton method as it was described above in the section [ mim ] .a good initial approximation is a linear map of the grid on the affine manifold corresponding to slow relaxation in the vicinity of the equilibrium .it is convenient to take this map isometric with respect to the metrics generated by the entropic scalar product in the equilibrium .if the vector field of the reduced model , , is defined on the nodes , then one can define the dynamics on the nodes . in order to do this ,we expand over : .the dynamics on the nodes is then defined by equations , .using interpolation , we can define the vector field within the essential cells of the grid ( those cells for which the all the nodes are essential ) .the system of equations thus obtained models the dynamics on the invariant manifold .the essence of this construction is that , by solving a set of uncomplicated linear equations arising from linearization of the invariance equations on the nodes one gets a reliable numerical scheme for constructing invariant manifolds .the use of the grid differentiation rather than a differentiable approximation to the manifold makes the scheme suited for parallel realizations .we stress it once again that such realizations are only possible if no a priori global parameterization of manifolds is required .further refinements of the scheme , taking into account the process of moving the inessential nodes into the phase space , and the opposite process of essential nodes leaving the phase space can be done based in the same way as for grid - based data analysis .one of the problems to be focused on when studying closed systems is to prepare extensions of the result for open or driven by flows systems .external flows are usually taken into account by addidinal terms in the kinetic equations ( [ reaction ] ) : _ zero - order approximation _ assumes that the flow does not change the invariant manifold .equations of the reduced dynamics , however , do change : instead of we substitute into eq .( [ dyn ] ) : zero - order approximation assumes that the fast dynamics in the closed system strongly couples the variables , so that flows can not influence this coupling ._ first - order approximation _ takes into account the shift of the invariant manifold by .equations for newton s iterations have the same form ( [ iteration ] ) but instead of the vector field they take into account the presence of the flow : ({\mbox{\boldmath}}+{\mbox{\boldmath}}'_{{\mbox{\boldmath}}}\delta{\mbox{\boldmath}})=0,\ { \mbox{\boldmath}}_{{\mbox{\boldmath}}}\delta{\mbox{\boldmath}}=0,\ ] ] where projector corresponds to the unperturbed manifold .the first - order approximation means that fluxes change the coupling between the variables ( concentrations ) .it is assumed that these new coupling is also set instantaneously ( neglect of inertia ) ._ various realizations of the first - order approximation in physical and chemical dynamics implement the viewpoint of an infinitely small chemical reactor driven by the flow . in other words ,this approximation is applicable in the lagrangian system of coordinates .transition to eulerian coordinates is possible but the relations between concentrations and the flow will change its form . in a contrast , the more simplistic zero - order approximation is equally applicable in both the coordinate system , if it is valid .in this paper , we have presented the method for constructing the invariant manifolds for reducing systems of chemical kinetics .our approach to computations of invariant manifolds of dissipative systems is close in spritit to the kolmogorov - arnold - moser theory of invariant tori of hamiltonian systems : we also base our consideration on the newton method instead of taylor series expansions , and systematically use duality structures .recently , a version of an approach based on the invarinace equations was rediscovered by .he was solving the invariance equation by a taylor series expansion .a counterpart of taylor series expansions for constructing the slow invariant manifolds in the classical kinetic theory is the famous chapman - enskog method .the question of how this compares to iteration methods was studied extensively for certain classes of grad moment equations .the thermodynamic parameterization and the selfadjoint linearization arise in a natural way in the problem of finding slowest invariant manifolds for closed systems .this also leads to various applications in different approaches to reducing the description , in particular , to a thermodynamically consistent version of the intrinsic low - dimensional manifold , and to model kinetic equations for lifting the reduced dynamics .use of the thermodynamic projector makes it unnecessary global parameterizations of manifolds , and thus leads to computationally promising grid - based realizations .arnold , v. i. ( 1963 ) .proof of a theorem of a. n. kolmogorov on the invariance of quasi - periodic motions under small perturbations of the hamiltonian .( english translation ) . _ russian mathematical surveys , _ * 18 * , 9 - 36 .bykov , v. i. , gorban , a. n. , & yablonskii , g. s. ( 1982 ) .description of nonisothermal reactions in terms of marcelin - de donder kinetics and its generalizations ._ react .lett . , _ * 20 * , 261 - 265 .chicone , c. , & swanson , r. ( 2000 ) .linearization via the lie derivative ._ electron .j.  diff .eqns . , _ monograph , 02 , http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu ( login : ftp )                      gorban , a. n. , zinovyev , a. yu . ( 2001 ) .visualization of data by method of elastic maps and its applications in genomics , economics and sociology .institut des hautes etudes scientifiques , preprint .ihes m/01/36 .online- version : http://www.ihes.fr/preprints/m01/resu/resu-m01-36.html .

Abstract: in this paper , we review the construction of low - dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold ( mim ) . mim is based on a formulation of the condition of invariance as an equation , and its solution by newton iterations . a review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low - dimensional manifolds . a grid - based version of mim is developed , and model extensions of low - dimensional dynamics are described . generalizations to open systems are suggested . the set of methods covered makes it possible to effectively reduce description in chemical kinetics .   kinetics , model reduction , invariant manifold , entropy , nonlinear dynamics , mathematical modeling