aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Moment Closure - A Brief Review
Christian Kuehn ∗ September 30, 2015
Abstract
Moment closure methods appear in myriad scientific disciplines in the mod-elling of complex systems. The goal is to achieve a closed form of a large, usuallyeven infinite, set of coupled differential (or difference) equations. Each equationdescribes the evolution of one “moment”, a suitable coarse-grained quantity com-putable from the full state space. If the system is too large for analytical and/ornumerical methods, then one aims to reduce it by finding a moment closure re-lation expressing “higher-order moments” in terms of “lower-order moments”. Inthis brief review, we focus on highlighting how moment closure methods occur indifferent contexts. We also conjecture via a geometric explanation why it has beendifficult to rigorously justify many moment closure approximations although theywork very well in practice.
The idea of moment-based methods is most easily explained in the context of stochasticdynamical systems. Abstractly, such a system generates a time-indexed sequence ofrandom variables x = x ( t ) ∈ X , say for t ∈ [0 , + ∞ ) on a given state space X . Letus assume that the random variable x has a well-defined probability density function(PDF) p = p ( x, t ). Instead of trying to study the full PDF, it is a natural step to justfocus on certain moments m j = m j ( t ) such as the mean, the variance, and so on, where j ∈ J and J is an index set and M = { m j : j ∈ J } is a fixed finite-dimensional spaceof moments. In principle, we may consider any moment space M consisting of a choiceof coarse-grained variables approximating the full system, not just statistical moments.A typical moment-closure based study consists of four main steps:(S0) Moment Space:
Select the space M containing a hierarchy of moments m j .(S1) Moment Equations:
The next step is to derive evolution equations for the mo-ments m j . In the general case, such a system will be high-dimensional and fully coupled . ∗ Institute for Analysis and Scientific Computing, Vienna University of Technology, 1040 Vienna,Austria
Moment Closure:
The large, often even infinite-dimensional, system of momentequations has to be closed to make it tractable for analytical and numerical tech-niques. In the general case, the closed system will be nonlinear and it will only approximate the full system of all moments.(S3)
Justification & Verification:
One has to justify, why the expansion made instep (S1) and the approximation made in step (S2) are useful in the context ofthe problem considered. In particular, the choice of the m j and the approximationproperties of the closure have to be answered.Each of the steps (S0)-(S3) has its own difficulties. We shall not focus on (S0) asselecting what good ’moments’ or ’coarse-grained’ variables are creates its own set ofproblems. Instead, we consider some classical choices. (S1) is frequently a lengthy com-putation. Deriving relatively small moment systems tends to be a manageable task.For larger systems, computer algebra packages may help to carry out some of the cal-culations. Finding a good closure in (S2) is very difficult. Different approaches haveshown to be successful. The ideas frequently include heuristics, empirical/numerical ob-servations, physical first-principle considerations or a-priori assumptions. This partiallyexplains, why mathematically rigorous justifications in (S3) are relatively rare and usu-ally work for specific systems only. However, comparisons with numerical simulations ofparticle/agent-based models and comparisons with explicit special solutions have consis-tently shown that moment closure methods are an efficient tool. Here we shall also notconsider (S3) in detail and refer the reader to suitable case studies in the literature.Although moment closure ideas appear virtually across all quantitative scientific dis-ciplines, a unifying theory has not emerged yet. In this review, several lines of researchwill be highlighted. Frequently the focus of moment closure research is to optimize clo-sure methods with one particular application in mind. It is the hope that highlightingcommon principles will eventually lead to a better global understanding of the area.In Section 2 we introduce moment equations more formally. We show how to derivemoment equations via three fundamental approaches. In Section 3 the basic ideas formoment closure methods are outlined. The differences and similarities between differentclosure ideas are discussed. In Section 4 a survey of different applications is given. Asalready emphasized in the title of this review, we do not aim to be exhaustive here butrather try to indicate the common ideas across the enormous breadth of the area. Acknowledgements:
I would like to thank the Austrian Academy of Science ( ¨OAW)for support via an APART Fellowship and the EU/REA for support via a Marie-CurieIntegration Re-Integration Grant. Support by the Collaborative Research Center 910of the German Science Foundation (DFG) to attend the “International Conference onControl of Self-Organizing Nonlinear Systems” in 2014 is also gratefully acknowledged.Furthermore, I would like to thank Thomas Christen, Thilo Gross, Thomas House andan anonymous referee for very helpful feedback on various preprint versions of this work.
The derivation of moment equations will be explained in the context of three classicalexamples. Although the examples look quite different at first sight, we shall indicate2ow the procedures are related.
Consider a probability space (Ω , F , P ) and let W = W ( t ) ∈ R L be a vector of independentBrownian motions for t ∈ R . A system of stochastic differential equations (SDEs) drivenby W ( t ) for unknowns x = x ( t ) ∈ R N = X is given byd x = f ( x ) d t + F ( x ) d W (1)where f : R N → R N , F : R N → R N × L are assumed to be sufficiently smooth maps, andwe interpret the SDEs in the Itˆo sense [3, 72]. Alternatively, one may write (1) usingwhite noise, i.e., via the generalized derivative of Brownian motion, ξ := W ′ [3] as x ′ = f ( x ) + F ( x ) ξ, ′ = dd t . (2)For the equivalent Stratonovich formulation see [44]. Instead of studying (1)-(2) directly,one frequently focuses on certain moments of the distribution. For example, one maymake the choice to consider m j ( t ) := h x ( t ) j i = h x ( t ) j · · · x N ( t ) j N i , (3)where h·i denotes the expected (or mean) value and j ∈ J , j = ( j , . . . , j N ), j n ∈ N ,where J is a certain set of multi-indices so that M = { m j : j ∈ J } . Of course, itshould be noted that J can be potentially a very large set, e.g., for the cardinality of allmulti-indices up to order J we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( j ∈ N N : | j | = X n j n ≤ J )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) J + NJ (cid:19) = ( J + N )! J ! N ! . (4)However, the main steps to derive evolution equations for m j are similar for every fixedchoice of J, N . After defining m j = m j ( t ) (or any other “coarse-grained” variables),we may just differentiate m j . Consider as an example the case N = 1 = L , and J = { , , . . . , J } , where we write the multi-index simply as j = j ∈ N . Then averaging (2)yields m ′ = h x ′ i = h f ( x ) i + h F ( x ) ξ i , (5)which illustrates the problem that we may never hope to express the moment equationsexplicitly for any nonlinear SDE if f and/or F are not expressible as convergent powerseries, i.e., if they are not analytic. The term h F ( x ) ξ i is not necessarily equal to zero forgeneral nonlinearities F as R t F ( x ( s )) d W ( s ) is only a local martingale under relativelymild assumptions [72]. Suppose we simplify the situation drastically by assuming aquadratic polynomial f and constant additive noise f ( x ) = a x + a x + a , F ( x ) ≡ σ ∈ R . (6)Then we can actually use that h ξ i = 0 and get m ′ = h x ′ i = a h x i + a h x i + a = a m + a m + a . (7)3ence, we also need an equation for the moment m . Using Itˆo’s formula one finds thedifferential d( x ) = [2 xf ( x ) + σ ] d t + 2 xσ d W (8)and taking the expectation it follows that m ′ = 2 h a x + a x + a x i + σ + σ h xξ i = 2( a m + a m + a m ) + σ , (9)where h xξ i = 0 due to the martingale property of R t x ( s ) d W s . The key point is thatthe ODE for m depends upon m . The same problem repeats for higher moments andwe get an infinite system of ODEs, even for the simplified case considered here. For ageneric nonlinear SDE, the moment system is a fully-coupled infinite-dimensional systemof ODEs. Equations at a given order | j | = J depend upon higher-order moments | j | > J ,where | j | := P n j n .Another option to derive moment equations is to consider the Fokker-Plank (or for-ward Kolmogorov) equation associated to (1)-(2); see [44]. It describes the probabilitydensity p = p ( x, t | x , t ) of x at time t starting at x = x ( t ) and is given by ∂p∂t = − N X k =1 ∂∂x k [ pf ] + 12 N X i,k =1 ∂ ∂x i ∂x k [( F F T ) ik p ] . (10)Consider the case of additive noise F ( x ) ≡ σ , quadratic polynomial nonlinearity f ( x )and N = 1 = L as in (6), then we have ∂p∂t = − ∂∂x [( a x + a x + a ) p ] + σ ∂ p∂x . (11)The idea to derive equations for m j is to multiply (11) by x j , integrate by parts and usesome a-priori known properties or assumptions about p . For example, we have m ′ = h x ′ i = Z R x ∂p∂t d x = Z R − x ∂∂x [( a x + a x + a ) p ] d x + Z R x σ ∂ p∂x d x. If p and its derivative vanish at infinity, which is quite reasonable for many densities,then integration by parts gives m ′ = Z R [( a x + a x + a ) p ] d x = a m + a m + a (12)as expected. A similar calculation yields the equations for other moments. Using theforward Kolmogorov equation generalizes in a relatively straightforward way to otherMarkov process, e.g., to discrete-time and/or discrete-space stochastic processes; in fact,many discrete stochastic processes have natural ODE limits [9, 27, 89, 90]. In the contextof Markov processes, yet another approach is to utilize the moment generating functionor Laplace transform s
7→ h exp[i sx ] i (where i := √−
1) to determine equations for themoments. 4 .2 Kinetic Equations
A different context where moment methods are used frequently is kinetic theory [22, 83,93]. Let x ∈ Ω ⊂ R N and consider the description of a gas via a single-particle density ̺ = ̺ ( x, t, v ), which is nonnegative and can be interpreted as a probability density ifit is normalized; in fact, the notational similarity between p from Section 2.1 and theone-particle density ̺ is deliberate. The pair ( x, v ) ∈ Ω × R N is interpreted as positionand velocity. A kinetic equation is given by ∂̺∂t + v · ∇ x ̺ = Q ( ̺ ) , (13)where ∇ x = ( ∂∂x , . . . , ∂∂x N ) ⊤ , suitable boundary conditions are assumed, and ̺ Q ( ̺ )is the collision operator acting only on the v -variable at each ( x, t ) ∈ R N × [0 , + ∞ ) withdomain D ( Q ). For example, for short-range interaction and hard-sphere collisions [105]one would take for a function v G ( v ) the operator Q ( G )( v ) = Z S N − Z R N k v − w k [ G ( w ∗ ) G ( v ∗ ) − G ( v ) G ( w )] d w d ψ (14)where v ∗ = ( v + w + k v − w k ψ ), w ∗ = ( v + w + k v − w k ψ ) for ψ ∈ S N − and S N − denotes the unit sphere in R N . We denote velocity averaging by h G i = Z R N G ( v ) d v, (15)where the overloaded notation h·i is again deliberately chosen to highlight the similaritieswith Section 2.1. It is standard to make several assumptions about the collision operatorsuch as the conservation of mass, momentum, energy as well as local entropy dissipation h Q ( G ) i = 0 , h vQ ( G ) i = 0 , hk v k Q ( G ) i = 0 , h ln( G ) Q ( G ) i ≤ . (16)Moreover, one usually assumes that the steady states of (13) are Maxwellian (Gaussian-like) densities of the form ρ ∗ ( v ) = q (2 πθ ) N/ exp (cid:18) − k v − v ∗ k θ (cid:19) , ( q, θ, v ∗ ) ∈ R + × R + × R N (17)and that Q commutes with certain group actions [93] implying symmetries. Note that thephysical constraints (16) have important consequences, e.g., entropy dissipation impliesthe local dissipation law ∂∂t h ̺ ln ̺ − ̺ i + ∇ x · h v ( ̺ ln ̺ − ̺ ) i = h ln ̺Q ( ̺ ) i ≤ . (18)while mass conservation implies the local conservation law ∂∂t h ̺ i + ∇ x · h v̺ i = 0 (19)with similar local conservation laws for momentum and energy. The local conservationlaw indicates that it could be natural, similar to the SDE case above, to multiply the5inetic equation (13) by polynomials and then average. Let { m j = m j ( v ) } Jj =1 be a basisfor a J -dimensional space of polynomials M . Consider a column vector M = M ( v ) ∈ R J containing all the basis elements so that every element m ∈ M can be written as m = α ⊤ M for some vector α ∈ R J . Then it follows ∂∂t h ̺M i + ∇ x · h v̺M i = h Q ( ̺ ) M i (20)by multiplying and averaging. This is exactly the same procedure as for the forwardKolmogorov equation for the SDE case above. Observe that (20) is a J -dimensionalset of moment equations when viewed component-wise. This set is usually not closed.We already see by looking at the case M ≡ v that the second term in (20) will usuallygenerate higher-order moments. Another common situation where moment equations appear are network dynamical sys-tems. Typical examples occur in epidemiology, chemical reaction networks and socio-economic models. Here we illustrate the moment equations [73, 123, 136, 147] for theclassical susceptible-infected-susceptible (SIS) model [32] on a fixed network; for re-marks on adaptive networks see Section 4. Given a graph of K nodes, each node can bein two states, infected I or susceptible S . Along an SI -link infections occur at rate τ and recovery of infected nodes occurs at rate γ . The entire (microscopic) description ofthe system is then given by all potential configurations x ∈ R N = X of non-isomorphicgraph configurations of S and I nodes. Even for small graphs, N can be extremelylarge since already just all possible node configurations without considering the topologyof the graph are 2 K . Therefore, it is natural to consider a coarse-grained description.Let m I = h I i = h I i ( t ) and m S = h S i = h S i ( t ) denote the average number of infectedand susceptibles at time t . From the assumptions about infection and recovery rates weformally derive d m S d t = γm I − τ h SI i , (21)d m I d t = τ h SI i − γm I , (22)where h SI i =: m SI denotes the average number of SI -links. In (21) the first term de-scribes that susceptibles are gained proportional to the number of infected times therecovery rate γ . The second term describes that infections are expected to occur propor-tional to the number of SI -links at the infection rate τ . Equation (22) can be motivatedsimilarly. However, the system is not closed and we need an equation for h SI i . In ad-dition to (21)-(22), the result [147, Thm.1] states that the remaining second-order motifequations are given byd m SI d t = γ ( m II − m SI ) + τ ( m SSI − m ISI − m SI ) , (23)d m II d t = − γm II + 2 τ ( m ISI + m SI ) , (24)d m SS d t = 2 γm SI − τ m SSI , (25)6here we refer also to [73,123]; it should be noted that (23)-(25) does not seem to coincidewith a direct derivation by counting links [33, (9.2)-(9.3)]. In any case, it is clear thatthird-order motifs must appear, e.g., if we just look at the motif ISI then an infectionevent generates two new II -links so the higher-order topological motif structure doeshave an influence on lower-order densities. If we pick the second-order space of moments M = { m I , m S , m SI , m SS , m II } (26)the equations (21)-(22) and (23)-(25) are not closed. We have the same problems asfor the SDE and kinetic cases discussed previously. The derivation of the SIS momentequations can be based upon formal microscopic balance considerations. Another optionis write the discrete finite-size SIS-model as a Markov chain with Kolmogorov equationd x d t = P x, (27)which can be viewed as an ODE of 2 K equations given by a matrix P . One defines themoments as averages, e.g., taking h I i ( t ) := K X k =0 kx ( k ) ( t ) , h S i ( t ) := K X k =0 ( K − k ) x ( k ) ( t ) , (28)where x ( k ) ( t ) are all states with k infected nodes at time t . Similarly one can definehigher moments, multiply the Kolmogorov equation by suitable terms, sum the equationas an analogy to the integration presented in Section 2.2, and derive the moment equa-tions [147]. For any general network dynamical systems, moment equations can usuallybe derived. However, the choice which moment (or coarse-grained) variables to consideris far from trivial as discussed in Section 4. We have seen that moment equations, albeit being very intuitive, do suffer from thedrawback that the number of moment equations tends to grow rapidly and the exactmoment system tends to form an infinite-dimensional system given byd m d t = h ( m , m , . . . ) , d m d t = h ( m , m , . . . ) , d m d t = · · · , (29)where we are going to assume from now on the even more general case h j = h j ( m , m , m , . . . )for all j . In some cases, working with an infinite-dimensional system of moments mayalready be preferable to the original problem. We do not discuss this direction furtherand instead try to close (29) to obtain a finite-dimensional system. The idea is to find amapping H , usually expressing the higher-order moments in terms of certain lower-ordermoments of the form H ( m , . . . , m κ ) = ( m κ +1 , m κ +2 , . . . ) (30)7or some κ ∈ J , such that (29) yields a closed systemd m d t = h ( m , m , . . . , m κ , H ( m , . . . , m κ )) , d m d t = h ( m , m , . . . , m κ , H ( m , . . . , m κ )) , ... = ...d m κ d t = h κ ( m , m , . . . , m κ , H ( m , . . . , m κ )) . (31)The two main questions are(Q1) How to find/select the mapping H ?(Q2) How well does (31) approximate solutions of (29) and/or of the original dynamicalsystem from which the moment equations (29) have been derived?Here we shall focus on describing the several answers proposed to (Q1). For a gen-eral nonlinear system, (Q2) is extremely difficult and Section 3.4 provides a geometricconjecture why this could be the case. In this section we focus on the SDE (1) from Section 2.1. However, similar principlesapply to all incarnations of the moment equations we have discussed. One possibility isto truncate [140] the system and neglect all moments higher than a certain order, whichmeans taking H ( m , . . . , m κ ) = (0 , , . . . ) . (32)Albeit being rather simple, the advantage of (32) is that it is trivial to implement anddoes not work as badly as one may think at first sight for many examples. A variationof the theme is to use the method of steady-state of moments by setting0 = h κ +1 ( m , m , . . . , m κ , m κ +1 , . . . ) , h κ +2 ( m , m , . . . , m κ , m κ +1 , . . . ) , ... = ... (33)and try to solve for all higher-order moments in terms of ( m , m , . . . , m κ ) in the algebraicequations (33). As we shall point out in Section 3.4, this is nothing but the quasi-steady-state assumption in disguise. Similar ideas as for zero and steady-sate moments can alsobe implemented using central moments and cumulants [140].Another common idea for moment closure principles is to make an a priori assump-tion about the distribution of the solution. Consider the one-dimensional SDE example( N = 1 = L ) and suppose x = x ( t ) is normally distributed. For a normal distributionwith mean zero and variance ν , we know the moments h x j i = ν j ( j − , if j is even, h x j i = 0 , if j is odd, (34)so one closure method, the so-called Gaussian (or normal) closure , is to set m j = 0 if j ≥ j is odd ,m j = ( m ) j/ ( j − j ≥ j is even.8 similar approach can be implemented using central moments. If x turns out to deviatesubstantially from a Gaussian distribution, then one has to question whether a Gaussianclosure is really a good choice. The Gaussian closure principle is one choice of a widevariety of distributional closures. For example, one could assume the moments of a lognormal distribution [35] instead x ∼ exp[˜ µ + ˜ ν ˜ x ] , ˜ x ∼ N (0 , , ⇒ h x j i = m j = exp (cid:20) j ˜ µ + 12 j ˜ ν (cid:21) (35)where ’ ∼ ’ means ’distributed according to’ a given distribution and N (0 ,
1) indicates thestandard normal distribution. Solving for (˜ µ, ˜ ν ) in (35) in terms of ( m , m ) yields amoment closure ( m , m , . . . ) = H ( m , m ). The same principle also works for discretestate space stochastic process, using a-prior distribution assumption. A typical exampleis the binomial closure [80] and mixtures of different distributional closure have also beenconsidered [84, 85]. In the context of moment equations of the form (20) derived from kinetic equations,a typical moment closure technique is to consider a constrained closure based upona postulated physical principle. The constraints are usually derived from the originalkinetic equation (13), e.g., if it satisfies certain symmetries, entropy dissipation and localconservation laws, then the closure for the moment equations should aim to capture theseproperties somehow. For example, the assumptionspan { , v , . . . , v N , k v k } ⊂ M (36)turns out to be necessary to recover conservation laws [93], while assuming that the space M is invariant under suitable transformations is going to preserve symmetries. However,even by restricting the space of moments to preserve certain physical assumptions, thisusually does not constraint the moments enough to get a closure. Following [93] supposethat the single-particle density is given by ̺ = M ( α ) = exp[ α ⊤ M ( v )] , m = m ( v ) ∈ M s.t. m ( v ) = α ⊤ M ( v ) (37)for some moment densities α = α ( x, t ) ∈ R J . Using (37) in (20) leads to ∂∂t h M ( α ) M i + ∇ x · h v M ( α ) M i = h Q ( M ( α )) M i . (38)Observe that we may view (38) as a system of J equations for the J unknowns α . Hence,one has formally achieved closure. The question is what really motivates the exponentialansatz (37). Introduce new variables η = h M ( α ) M i and define a function H ( η ) = −h M ( α ) i + α ⊤ η (39)and one may show that α = [D η H ]( η ). It turns out [93] that H ( η ) can be computed bysolving the entropy minimization problemmin ̺ {h ̺ ln ̺ − ̺ i : h M ̺ i = η } = H ( η ) , (40)9here the constraint h M ̺ i = η prescribes certain moments; we recall that M = M ( v )is the fixed vector containing the moment space basis elements and the relation α =[D η H ]( η ) holds. From a statistical physics perspective, it may be more natural toview (40) as an entropy maximization problem [70] by introducing another minus sign.Therefore, the choice of the exponential function in the ansatz (37) does not only guar-antee non-negativity but it was developed as it is the Legendre transform of the so-calledentropy density ̺ ̺ ln ̺ − ̺ so it naturally relates to a physical optimization prob-lem [93].To motivate further why using a closure motivated by entropy corresponds to certainphysical principles, let us consider the ’minimal’ moment space M = span { , v , . . . , v N , k v k } (41)The closure ansatz (37) can be facilitated using the vector M ( v ) = (1 , v , . . . , v N , k v k )but then [94] the ansatz is related to the Maxwellian density (17) since ρ ∗ ( v ) = exp[ α ⊤ M ( v )] , α = (cid:18) ln (cid:18) q (2 πθ ) / (cid:19) − k v ∗ k θ , v ∗ θ , − θ (cid:19) ⊤ (42)but Maxwellian densities are essentially Gaussian-like densities and we again have a Gaussian closure . Using a Gaussian closure implies that the moment equations becomethe Euler equations of gas dynamics, which can be viewed as a mean-field model nearequilibrium for the mesoscopic single-particle kinetic equation (13), which is itself a limitof microscopic equations for each particle [23, 142].Taking a larger moment space M one may also get the Navier-Stokes equation as alimit [93], and this hydrodynamic limit can even be justified rigorously under certainassumptions [51]. This clearly shows that moment closure methods can link physicaltheories at different scales. Since there are limit connections between the microscopic level and macroscopic momentequations, it seems plausible that starting from an individual-based network model, onemay motivate moment closure techniques. Here we shall illustrate this approach for theSIS-model from Section 2.3. Suppose we start at the level of first-order moments and let M = { m I , m S } . To close (21)-(22) we want a map m SI = H ( m I , m S ) . (43)If we view the density of the I nodes and S nodes as very weakly correlated randomvariables then a first guess is to use the approximation m SI = h SI i ≈ h S ih I i = m S m I . (44)Plugging (44) into (21)-(22) yields the mean-field SIS model m ′ S = γm I − τ m S m I ,m ′ I = τ m S m I − m I . (45)10he mean-field SIS model is one of the simplest examples where one clearly sees thatalthough the moment equations are linear ODEs, the moment-closure ODEs are fre-quently nonlinear . It is important to note that (44) is not expected to be valid for allpossible networks as it ignores the graph structure. A natural alternative is to consider m SI = h SI i ≈ m d h S ih I i = m d m S m I , (46)where m d is the mean degree of the given graph/network. Hence it is intuitive that (44)is valid for a complete graph in the limit K → ∞ [136].If we want to find a closure similar to the approximation (44) for second-order mo-ments with M as in (26), then the classical choice is the pair-approximation [54, 75, 76] m abc ≈ m ab m bc m b , a, b, c ∈ { S, I } (47)which just means that the density of triplet motifs is given approximately by countingcertain link densities that form the triplet. In (47) we have again ignored pre-factorsfrom the graph structure such as the mean excess degree [33,73]. As before, the assump-tion (47) is neglecting certain correlations and provides a mapping( m SSI , m
ISI ) = H ( m II , m SS , m SI ) = (cid:18) m SS m SI m S , m SI m SI m S (cid:19) (48)and substituting (48) into (23)-(25) yields a system of five closed nonlinear ODEs. Manyother paradigms for similar closures exist. The idea is to use the interpretation of themoments and approximate certain higher-order moments based upon certain assump-tions for each moment/motif. In the cases discussed here, this means neglecting certain correlation terms from random variables. At least on a formal level, this is approach isrelated to the other closures we have discussed. For example, forcing maximum entropymeans minimizing correlations in the system while assuming a certain distribution forthe moments just means assuming a particular correlation structure of mixed moments. All the moment closure methods described so far, have been extensively tested in manypractical examples and frequently lead to very good results; see Section 4. However,regarding the question (Q2) on approximation accuracy of moment closure, no completelygeneral results are available. To make progress in this direction I conjecture that a high-potential direction is to consider moment closures in the context of geometric invariantmanifold theory. There is very little mathematically rigorous work in this direction [143]although the relevance [31, 116] is almost obvious.Consider the abstract moment equations (29). Let us assume for illustration purposes11hat we know that (29) can be written as a systemd m d t = h ( m , m , . . . , m κ , m κ +1 , m κ +2 , . . . ) , d m d t = h ( m , m , . . . , m κ , m κ +1 , m κ +2 , . . . ) , ... = ...d m κ d t = h κ ( m , m , . . . , m κ , m κ +1 , m κ +2 , . . . ) . d m κ +1 d t = ε h κ +1 ( m , m , . . . , m κ , m κ +1 , m κ +2 , . . . ) . d m κ +2 d t = ε h κ +2 ( m , m , . . . , m κ , m κ +1 , m κ +2 , . . . ) . ... = ... (49)where 0 < ε ≪ h is of order O (1) as ε →
0. Then (49) is a fast-slow system [71, 88] with fastvariables ( m κ +1 , m κ +2 , . . . ) and slow variables ( m , . . . , m κ ). The classical quasi-steady-state assumption [132] to reduce (49) to a lower-dimensional system is to take0 = d m κ +1 d t , m κ +2 d t , · · · . (50)This generates a system of differential-algebraic equations and if we can solve the alge-braic equations0 = h κ +1 ( m , m , . . . ) , h κ +2 ( m , m , . . . ) , · · · (51)via a mapping H as in (30) we end up with a closed system of the form (31).The quasi-steady-state approach hides several difficulties that are best understoodgeometrically from the theory of normally hyperbolic invariant manifolds, which is wellexemplified by the case of fast-slow systems. For fast-slow systems, the algebraic equa-tions (51) provide a representation of the critical manifold C = { ( m , m , . . . ) : h j = 0 for j > κ , j ∈ N } . (52)However, it is crucial to note that, despite its name, C is not necessarily a manifold butin general just an algebraic variety. Even if we assume that C is a manifold and we wouldbe able to find a mapping H of the form (30), this mapping is generically only possible locally [40,88]. Even if we assume in addition that the mapping is possible globally, thenthe dynamics on C given by (30) does not necessarily approximate the dynamics of thefull moment system for ε >
0. The relevant property to have a dynamical approximationis normal hyperbolicity , i.e., the ’matrix’ (cid:18) ∂h j ∂m l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) C , j, l ∈ { κ + 1 , κ + 2 , . . . } (53)has no eigenvalues with zero real parts; in fact, this matrix is just the total derivativeof the fast variables restricted to points on C but for moment equations it is usuallyinfinite-dimensional. Even if we assume in addition that C is normally hyperbolic,which is a very strong and non-generic assumption for a fast-slow system [71, 88], then12he dynamics given via the map H is only the lowest-order approximation . The correctfull dynamics is given on a slow manifold C ε = { ( m κ +1 , m κ +2 , . . . ) = H ( m , m , . . . , m κ ) + O ( ε ) } (54)so H is only correct up to order O ( ε ). This novel viewpoint on moment closure shows whyit is probably quite difficult [60] to answer the approximation question (Q2) since for ageneral nonlinear system, the moment equations will only admit a closure via an explicitformula locally in the phase space of moments. One has to be very lucky, and probablymake very effective use of special structures [46, 117] in the dynamical system, to obtainany global closure. Local closures are also an interesting direction to pursue [14]. Historically, applications of moment closure can at least be traced back to the classi-cal Kirkwood closure [79] as well as statistical physics applications, e.g., in the Isingmodel [77]. The Gaussian (or normal) closure has a long history as well [155]. In me-chanical applications and related nonlinear vibrations questions, stochastic mechanicsmodels have been among the first where moment closure techniques for stochastic pro-cesses have become standard tools [18,69] including the idea to just discard higher-ordermoments [125]. By now, moment closure methods have permeated practically all naturalsciences as evidenced by the classical books [2,151]. For SDEs, moment closure methodshave not been used as intensively as one may guess but see [13].For kinetic theory, closure methods also have a long history, particularly startingfrom the famous Grad 13-moment closure [52, 146], and moment methods have becomefundamental tools in gas dynamics [145]. One particularly important application forkinetic-theory moment methods is the modelling of plasmas [59, 127]. In general, it isquite difficult to study the resulting kinetic moment equations analytically [30, 37] butmany numerical approaches exist [56, 95, 102, 149]. Of course, the maximum entropyclosure we have discussed is not restricted to kinetic theory [137] and maximum entropyprinciples appear in many contexts [1, 19, 24, 26, 124].One area where moment closure methods are employed a lot recently is mathematicalbiology. For example, the pair approximation [73] and its variants [10] are frequentlyused in various models including lattice models [36, 41, 42, 101, 108, 131], homogeneousnetworks [118,130] and many other network models [5,122,126,154]. Several closures havealso included higher-order moments [67,74] and truncation ideas are still used [15,16,61].Applications to various different setups for epidemic spreading are myriad [61, 62]. Atypical benchmark problem for moment methods in biology is the stochastic logisticequation [6, 99, 100, 110, 111, 113, 139]. Furthermore, spatial models in epidemiology andecology have been a focus [91, 98, 114, 115]. There are several survey and comparisonpapers with a focus on epidemics application and closure-methods available [17,104,107,123]. There is also a link from mathematical biology and moment closure to transportand kinetic equations [63, 64], e.g., in applications of cell motion [65]. Also physicalconstraints, as we have discussed for abstract kinetic equations, play a key role in biology,e.g., trying to guarantee non-negativity [62].13nother direction is network dynamics [119], where moment closure methods havebeen used very effectively are adaptive, or co-evolutionary, networks with dynamics ofand on the network [53, 54]. Moment equations are one reason why one may hopeto describe self-organization of adaptive networks [20] by low-dimensional dynamicalsystems models [86]. Applications include opinion formation [78, 109] with a focus onthe classical voter model [120,141,152]; see [29] for a review of closure methods applied tothe voter model. Other applications are found again in epidemiology [55, 87, 97, 133, 134,148, 156] and in game theory [28, 39, 45]. The maximum entropy-closure we introducedfor kinetic equations has also been applied in the context of complex networks [128] andspatial network models in biology [121]. An overview of the use of the pair approximation,several models, and the relation to master equations can be found in [49]. It has also beenshown that in many cases low-order or mean-field closures can still be quite effective [50].On the level of moment equations in network science, one has to distinguish betweenpurely moment or motif-based choices of the space M and the recent proposal to useheterogeneous degree-based moments. For example, instead of just tracking the momentof a node density, one also characterizes the degree distribution [48] of the node via newmoment variables [34]. Various applications of heterogeneous moment equations havebeen investigated [96, 135].Another important applications are stochastic reaction networks [7,8,150], where themean-field reaction-rate equations are not accurate enough [92]. A detailed computationof moment equations from the master equation of reaction-rate models is given in [38]. Ina related area, turbulent combustion models are investigated using moment closure [11,81,106,112,129]. For turbulent combustion, one frequently considers so-called conditionalmoment closures where one either conditions upon the flow being turbulent or restrictsmoments to certain parts of phase space; see [82] for a very detailed review.Further applications we have not focused on here can be found in genetics [4], client-server models in computer science [57, 58], mathematical finance [138], systems biol-ogy [47], estimating transport coefficients [25], neutron transport [21], and radiativetransport problems [43, 144]. 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