Radically Elementary Analysis of an Interacting Particle System at an Unstable Equilibrium

We investigate an interacting particle system consisting of two types of particles located at a finite point-lattice. The particles randomly change their type and neighboring particles randomly interchange positions. The system seems to remain at equilibrium for a substantial amount of time until it suddenly, at a critical time T , leaves equilibrium along what seems to be a deterministic trajectory. The analysis reveals, however, that the trajectories are determined randomly, but only by the systems behavior at very early times, much prior to T. In the nonstandard model used, the system randomly 'chooses' the trajectory in an infinitesimal interval [0, ε], ε ≈ 0, but this choice only becomes visible in the interval [T − ε, T]. The underlying reason for this behavior is revealed by a decomposition of the systems trajectories with respect to an eigenbasis (g k) k∈K of the discrete Laplace operator. It shows that after an initial random period the system's dynamics behaves, coordinate-wise, like t → e (λ+µ k)(t−T) υ k (ω), where λ is unlimited ('infinitely large'), µ k g k = g k and υ k (ω) denotes a random quantity. The hyperfinite result obtained is translated into a standard limit theorem.


Introduction
Interacting particle systems have been a prospering field of mathematical studies in a standard setting (Griffeath [11] and Liggett [15]) as well as a nonstandard one (Helms and Loeb [12], and Albeverio, Fenstad, Høegh-Krohn and Lindstrøm [1,Chapter 7]), the most prominent being the Ising model.
The model under consideration is presented within a nonstandard setting.It shares with the Ising model the property of being a Markovian lattice model and that there exist two states for each particle, or equivalently that there are two particle types, or particles and holes.It differs, however, in that a large number of particles occupies one position at a time.In this regard, it possesses similarities with discrete-time zero-range processes (in the sense of Evans and Hanney [10]) or reaction diffusion processes (in the sense of Chen [7, Section 13.2]).
The system's dynamics is at first defined only if particles of both types are present at any position.
We investigate the evolution starting in the unique unstable equilibrium of a corresponding deterministic system (briefly discussed in Remark 5.5).We are only interested in the way the system leaves this equilibrium.This can equally well be investigated within any extension of the original system.Thus we extend the system's dynamics in a mathematically appropriate way to arbitrary (negative, real valued) quantities of particles.For the sake of simplicity, we describe the extended dynamics by the deviation of the pointwise particle concentration from the equilibrium.The system's evolution can be divided into three periods.The first and the third period are very short compared to the second one.During the first and second period the system stays infinitesimally close to the unstable equilibrium, and during the third period it drives with high velocity away from this initial state.
In the first period the system's evolution is particularly governed by stochasticity.In the second and third one each path of the system stays infinitesimally close to a deterministic trajectory. 2 Thus the system's behavior in periods two and three is approximately described by a probability distribution on a family of deterministic trajectories.The effect of stochasticity in periods two and three, therefore, originates approximately from a random choice of a deterministic trajectory made during period one, while the additional influence of randomness during periods two and three is rather negligible.
To obtain an intuition for the system's behavior, suppose that we are unable to recognize infinitesimal differences.Then the system seems to stay in equilibrium during periods one and two.In period three we observe that the system drives away from the unstable equilibrium along a randomly chosen, but deterministic trajectory.We know, however, that the system has already come to the random decision for this particular trajectory during period one.
The deterministic trajectories associated with the system are solutions of a linear system of first order infinitesimal difference equations y t+δt = Ly t , where the linear transformation L is diagonalizable with respect to an eigenbasis of the discrete Laplace operator.Stochastically the system shows a Gaussian behavior: Projections of the system's random-state onto orthogonal eigenvectors of the Laplacian are approximately independent, approximately normally distributed random variables.The variances of these variables increase geometrically with time.The velocity of the increase depends on the corresponding eigenvalues of the Laplacian.This leads to a preference of low frequencies and represents a certain degree of coherence induced by stochasticity, although the term 'stochastic coherence' seems usually to be associated only with nonlinear systems (e.g.Sagues, Sancho and Garcia-Ojalvo [23]).
We are interested in the system's behavior for large numbers of particles.This is within standard mathematics expressed by limit theorems.Largeness can however be directly expressed within a nonstandard framework.In such a setting hyperfinite collections are large compared to standard finite ones.It is further possible to obtain from results concerning the hyperfinite situation corresponding limit results in standard mathematical terms.In this way Lindeberg type limit theorems have been proved in Weisshaupt [28].Following this idea we characterize the system's dynamics for hyperfinite particle-collections first (Theorem 6.5), and apply afterward transfer and the permanence principle to obtain a corresponding standard limit result (Theorem 7.9).The article follows Nelson's axiomatic approach IST [18] to nonstandard analysis.It is radically elementary in the sense that it is based on (hyper)-finite probability spaces and the IST-axioms of idealization and transfer, while the IST-axiom of standardization is not used in the whole article.Only in the formulation of Corollary 7.11 do we make use of uncountable probability spaces, since the standard limit object involved can not be defined on a finite probability space.For this reason we also included appendix B that connects our internal concepts to standard measure theoretic ones.Note however, that appendix B is still radically elementary in the sense that it only uses idealization and transfer to establish this connection.

Organization of the Article
In Section 3 we describe the basic dynamics of the interacting particle system and indicate how this dynamics relates to the extended dynamics defined in Section 5. We further outline the main result and discuss the outline in some detail.We briefly indicate how our simple interacting particle systems may relate to more complex systems in chemical reaction kinetics.Finally we discuss the main proof-steps.Section 4 introduces some fundamental notions and results in nonstandard analysis like infinitesimals, uniform S-continuity, near intervals and the symbol .We further introduce the discrete Laplace operator and its eigenbasis, which becomes in Section 5 the fundamental tool for the investigation of the extended dynamics.Finally the concepts of conditional probability, partially defined random variable, stochastic process and approximately normally distributed variable are introduced.
In Section 5 we introduce the extended model in a mathematically self contained way not relying on Section 3, however without the motivation and explanation already given before.The main purpose of Section 5 is to obtain a description of the extended dynamics in coordinates with respect to the eigenbasis of the discrete Laplacian introduced in Section 4.
In Section 6 we prove the main results of this article in their internal form (Theorem 6.5 and Theorem 6.9).Both theorems describe the coordinate-wise deviation (with respect to the eigenbasis of the discrete Laplace operator) of the system from deterministic trajectories.While Theorem 6.5 deals with the case of a small (standard finite) number of available particle-positions, the Theorem 6.9 is concerned with the hyperfinite case.The proof of Theorem 6.5 is based on Theorem A.7, the description of the system's dynamics obtained in Section 5 and the Doob inequality (stated as Proposition C.3). Theorem 6.9 is a consequence of Theorem 6.5 and the axion of idealization.Section 7 finally turns Theorem 6.5 into the standard limit Theorem 7.9.For this purpose the mathematical objects in the preceding sections have to be replaced by standard sequences.The relations fulfilled by the nonstandard elements of these sequences coincide with the relations fulfilled by the objects of the preceding sections.To obtain standard limit theorems we translate these relations into assertions concerning the limits of these sequences.It turns out that this is possible without the use of the standardization-axiom.
Appendix A is concerned with the internal central limit theorems A.5 and A.7.Under the hypotheses of these theorems the concatenation of a group homomorphism into the real numbers with the final state of certain Markov chains (on abelian groups) is approximately normally distributed.The proof of Theorem A.5 exploits the relationship between infinitesimal diffusion processes and the diffusion equation in analogy with Weisshaupt [28], while Theorem A.7 is just a modification of Theorem A.5 obtained by a time transform.We regard these theorems-as well as their proofs-as interesting in their own right. of 'convergence toward a N(0, id) distributed random variable', while Appendix C collects miscellaneous results.
The article is largely self-contained.It only makes use of some very elementary results from nonstandard analysis (Remark 4.3), elementary facts concerning discrete Fourier analysis and the discrete Laplacian (also collected in Section 4), the well-known Doob inequality (displayed for the readers convenience at the end of Appendix C) and a consequence (Proposition B.4) of the Cramér-Wold device.We do not make use of other auxiliary results.We especially state and prove in Appendix A a central limit theorem along the lines of [28] that is fundamental for the proof of our main results.Note however that it would have been possible to apply the martingale central limit theorem (Bhattacharya and Majumdar [4, Section 5.4, Proposition 4.1]) to prove our main results instead.

Description of the basic dynamics
The particle systems under consideration consist of a constant finite number N of particles described by their position and their type.At a given time-point t a particle possesses a position x in the finite point lattice H := hZ/Z with 1/h ∈ N and is either of type A or of type B.
We suppose for arbitrary x ∈ H that the number of particles located at x is independent of time and equals hN ∈ N. We further assume that particles of the same type are only distinguished by their position, but are otherwise indistinguishable.Thus at any time t the system is completely described by the spatial distribution of type-A or type-B particles.
Using a nonstandard framework it is convenient to model time by near intervals [ 0 . . .T ], i.e., hyperfinite-and thus discrete-subsets of [0, T] introduced in Definition 4.4, and to denote small time steps corresponding to the spacings of points in near intervals by δt.
In a small piece of time δt, one particle may change its type and two neighboring particles may interchange their position.Which particles interchange and if there is any interchange at all is a uniformly distributed pure random event independent of the particle-configuration.The probability that a particular particle at position x changes its type also depends on the configuration, at position x.The influence of randomness on the system is expressed by random elements ω in some hyperfinite space Ω.
We describe the random evolution of our particle system by consecutive reaction and diffusion steps.We suppose that the reaction steps take place in the time intervals (t, t + δt/2] while the diffusion steps follow in (t + δt/2, t + δt], with the time-points t being elements of the discrete set [ 0 . . .T ].Instead of t + δt/2 we write t + .In a reaction step a particle may change its type, while in a diffusion step two particles may interchange.It is sufficient to describe the interchange of particles of different types in the diffusion step, since we are unable to observe the interchange of particles of the same type. Let N A,t (ω) ∈ {0, . . ., hN} H denote the number of type-A particles at time t under the random influence ω at different positions x ∈ H before the reaction step.Let further N A,t + (ω) ∈ {0, . . ., hN} H denote the number of type-A particles at time t + under the random influence ω at different positions x ∈ H before the diffusion step.The evolution of the system can be described by the functions t → N A,t and t → N A,t + , with t ∈ [ 0 . . .T ].We note that the evolution of the system can equivalently be described by the number of type-B particles given by hN − N A,t and hN − N A,t + .We further let j,k = 1 if j = k and j,k = 0 for j = k and define functions e x : H → {0, 1} by e x (y) := x,y and 1I K : Z → {0, 1} by 1I K (x) := sup k∈K k,x .
Considering particles of type A only and regarding particles of type B as holes (free space that may be occupied by particles of type A), our dynamical system is described by hopping of particles to neighboring positions3 (instead of an interchange of particles) and the overall particle number is not conserved any more.It shares these properties with discrete-time zero-range processes with non-conservation of particle numbers in the sense of [10] or reaction-diffusion processes in the sense of [7,Section 13.2].Fluid limits of reaction-diffusion processes have been considered in [7,Chapter 16] and Boldrighini, De Masi and Pellegrinotti [6].Condensation phenomena for zero-range process with non-conservation of particle numbers have been investigated in Angel, Evans, Levine and Mukamel [3].The dynamics considered in all these instances differ from ours in at least three points: Unstable equilibria (similar to ours) are not investigated (and thus obtained results are entirely different), hopping rates of particles do not depend on the occupation number at a neighboring site and reaction-rates are not 'infinitely' large compared to diffusion-rates.
We describe the reaction and the diffusion steps in more detail:

Reaction step:
For N A,t (ω) = n A,t ∈ {1, . . ., hN − 1} H we let i.e., the number of type-A particles remains unchanged or changes at exactly one position by ±1.This formalizes the fact that in one reaction step at most one particle in the system reacts, i.e., changes its type.
The conditional probabilities (Definition 4.15) for these reactions/changes are given by: By equation ( 3) the probability that one of the type-B particles located at position x reacts to a type-A particle is proportional to the number of type A particles located at x, while by equation (4) the same statement holds true with the particle-types interchanged.

Diffusion step:
i.e., the system remains unchanged or the number of type-A particles at some position x decreases by 1 while the number of type-A particles at position x − h or x + h increases by 1.This formalizes the interchange of a type-A particle at some position x with a type-B particle at a neighboring position.
The probabilities for this interchange of particles are given by: i.e., the probability that one of the type A particles located at position x interchanges with a type B particle at a neighboring position is proportional to the number of type A particles located at x and proportional to the concentration of type B particles located at the neighboring position.

Heinz Weisshaupt
We are interested in the system's dynamics when the reaction rate λ in equations ( 3) and ( 4) is large compared to 1, i.e., we are interested in situations when reactions occur much more frequently than interchanges of particles.

Extended dynamics
Note that the reaction and diffusion steps have till now only been defined if An extension of this system's dynamics is introduced in Section 5.It is based on random variables Ξ t and − 1 2 as long as N A,t (ω), N A,t + (ω) are defined.Note that the random variables Ξ t and Ξ t + model the deviations of generalized concentrations of type-A particles from 1/2.They may take on arbitrary values in R H .Some of these values do not correspond to actual particle numbers and can not be interpreted as actual particle concentrations.However, up to the random time the original and the extended system are indistinguishable.The fact that the effects we are interested in are caused while t ≤ τ ω ensures that the extended dynamics captures the behavior of the particle system.

Outline of the main result
We now outline the main result of the paper.A stronger coordinate-wise version is provided by Theorem 6.5.The following outline as well as Theorem 6.5 are formulated within a nonstandard setting.A corresponding formulation as a limit theorem is provided by Theorem 7.9.
3.1 Theorem Suppose that the particle number N is hyperfinite 4 and that h and thus H = hZ/Z is standard.Let the reaction rate λ be such that (8) e 2λT = 4hN for some limited T ∈ (0, ∞).
Let the length of the time steps δt be a constant δ independent of t and sufficiently small.Suppose that the initial state of the system is given by Ξ 0 = 0 and that the evolution of the dynamical system is governed by Definition 5.3.Then there exists an approximately N(0, id) distributed random variable Γ T : Ω → R H and a jointly diagonalizable family (Φ −t ) t∈[ 0...T ] of linear mappings Φ −t : R H → R H (Definition 6.2) such that for any unlimited ν ∈ (0, ∞) with ν/λ ∈ ( 0 . . .T ) infinitesimal, such that T − ν/λ ∈ ( ν/λ . . .T ) and any standard ε > 0 Further for any standard ε > 0 3.2 Remark Theorem 3.1 is a consequence of Theorem 6.5 and Corollary 6.7.A proof is given in Section 6.

Discussion
Equation (8) relates the reaction rate λ, the overall particle-number N , the number of available positions 1/h and the approximate time T it takes till an effect becomes visible.If ln(1/h) is small compared to ln(N) (as it is under the hypothesis that h is standard and N is hyperfinite), then T equals approximately ln(N)/2λ and the influence of h on T is negligible.For times smaller than T − ν/λ the system stays by (10) infinitesimally close to 0, while for times larger than ν/λ it shows by (9) already an approximately deterministic behavior.The system 'approximately decides' in the first time period [ 0 . . .ν/λ ) for some deterministic trajectory (y δt exhibits an analogy between the first order approximation and ( 27), i.e., the reaction steps of our interacting particle system can-in conditional expectation-be viewed as infinitesimal steps in a first order approximation (at ξ = 0) of the dynamics of (11).
Thus the interacting particle system under consideration may be considered as a linearization of interacting particle systems modeling the spatio-temporal behavior of more complex chemical reactions.We do not further dwell on the question of linearization of more complex models in this article.

Remark
Before we start with our introduction to nonstandard analysis, the formulation of the exact hypotheses for our extended model etc., we outline the main steps of our investigation that lead to the proof of our main results, the Theorems 6.5 and 3.1.
The Hypotheses 5.1 and 5.3 give us the stochastic model under consideration.It is a discrete time Markov process (Ξ 0 , . . ., Ξ t , Ξ t + , . . ., Ξ T ).However, by the use of nonstandard analysis, our model may be considered as quasi-continuous.
By Proposition 5.4 and Definition 5.6 we split the short term evolution of our process Ξ into a conditional deterministic and a pure random part, summarized in Remark 5.7 by the formula: with Σ t and Σ t + random variables possessing expectation 0.
In Proposition 5.9 the conditional covariance of the projections of Σ onto directions η 1 and η 2 is investigated.In Proposition 5.11 the same is done for the conditional variance . These investigations lead to the insight that mutually orthogonal projections of the random variables Σ t show almost independent behavior, while the variables Σ t + are rather small.Consequently it seems obvious to expand the system with respect to an orthonormal basis.Since the dynamics involves (id + δt h ) Ξ t + an eigenbasis of the Laplace operator should be a good choice.
Thus we describe the systems dynamics in Remark 5.13 with respect to such an eigenbasis (g k ) k∈K as we obtain by recursion in Proposition 6.11 that By rescaling the random variables Σ s := Σ s,k g k by linear transformations Φ s (with approximately inverse transformations Φ −s , introduced in Definitions 6.2 and 6.3) the equality above can also be expressed by (see Proposition 6.11): δΓ s and δΓ s := Φ s ( Σ s ).
Note that the operators Φ −t are defined in such a way, that-for δt = δ independent of t and λ 2 δ infinitesimally small-we obtain (compare with Proposition 6.4) The Proposition 5.17 is obtained from the Propositions 5.9 and 5.11 via the Propositions 5.14 and 5.15.Proposition 5.17 shows that the random variables Σ t,k and Σ t,j are for k = j almost independent, possess expectation 0 and possesses approximately a variance of λ 2 δt hN .From this we derive the formula (77) for the conditional variances of δΓ s .We show in Lemma 6.14 that This is applied (in the proof of Lemma 6.15) to sum the conditional variances of δΓ s given by equation (77) in Proposition 6.12.Since the random variables δΓ s are for t = s independent, we know from Theorem A.7 that the random variables Γ t are for sufficiently large t approximately normally distributed.Altogether we obtain by the scaling e 2λT = 4hN(1+ ) that Γ t ∼ N(0, id K ) for any t in [ ν/λ . . .T ] with ν unlimited.
It finally remains to prove that the path of our stochastic process Ξ stay almost surely infinitesimally close to 0 on the near interval [ 0 . . .T − ν/λ ] and that they follow almost surely the deterministic trajectories [Φ −t • Γ T ](ω) on the near interval [ ν/λ . . .T ], i.e., to prove formulas (66) and (65) in Theorem 6.5 (and thus (10) and ( 9) in Theorem 3.1).This aim is achieved by application of the Doob inequality and use of the linear transformations Φ −t in the second step of the proof of Lemma 6.15 and at the end of Section 6.While (66) and ( 10) bound the absolute deviation of stochastic paths from 0, the inequalities (65) and ( 9) bound the relative deviations of stochastic paths from deterministic trajectories.
Note that (65) and ( 12) together imply that t → Ξ t,k (ω) behaves for almost all ω shows approximately exponential growth with rate λ + µ k .
So, to understand the main ideas of the article, one has to decompose the system's dynamics with respect to an eigenbasis (g k ) k∈K of the Laplacian ∆, to admit formula (54), to have a look at the derivation of (77) from (54) in the proof of Proposition 6.12, and the derivation of Lemma 6.14.Going trough the first part of the proof of Lemma 6.15 one concludes (80)-( 83) from Proposition 6.12, Lemma 6.14 and Theorem A.7.One proceeds with the second part of the proof of Lemma 6.15 that shows (65).Theorem 6.5 finally follows by some further simple computations.

Preliminaries
The notation and argumentation used in this article is supplied by the axiomatic system IST (see Nelson [18] or F and M Diener [8] and Kanovei and Reeken [13, Chapter 3]) that provides, beside the binary ZFC-predicate ∈, also an unary predicate st(.) called standard.The results and arguments used in this paper however remain valid in other approaches to nonstandard analysis as well.
The reader familiar with a model theoretic approach (as found in Robinson [21], Stroyan and Bayod [24] or Lindstrøm [16]), or the axiom system HST [13, Chapter 1], has to keep in mind that the plain term set is used synonymously with the term internal set and that we work within one single model.We do not use a * -operation and denote by N and R the standard sets of all natural and real numbers, i.e., the sets N and R contain standard as well as nonstandard elements.
The reader new to nonstandard analysis is advised to have a look at the first pages of [18] or [8] to make himself familiar with the notions of standard, internal and external formula, the principles of transfer and idealization and some elementary consequences thereof.
To keep notation simple we write ∀ st xφ(x) instead of ∀x(st(x) ⇒ φ(x)) and ∃ st xφ(x) instead of ∃x(st(x) ∧ φ(x)).Given a set M we use x ∈ M as shorthand for x ∈ M ∧ st(x) and x ∈ M as shorthand for x ∈ M ∧ ¬st(x).
4.1 Notation Let (X, .) be a normed space.We say that x ∈ (X, .) is limited and write x +∞ if ∃ st n ∈ N such that x < n; otherwise, we say that x is unlimited.In the case that (X, . x is infinitely small we write x ≈ x .Thus if x is infinitely small we write x ≈ 0. We say that x ∈ R is appreciable if it is limited but not infinitesimal.We call a set hyperfinite if it is finite and of unlimited (=hyperfinite) cardinality.Note that all the concepts introduced above are external.
We state some elementary results and definitions that can be obtained in IST without the axiom of standardization.

Definition
Let (Y, .) be a normed space.We say that the sequence Let Z be a subset of a normed space (X, .).We say that a function f :

Remark
A standard sequence (x n ) n∈N S-converges if and only if there exists a standard x ∞ such that (x n ) n∈N converges (in the usual ZFC-based sense) to x ∞ .A standard function f is uniformly S-continuous if and only if it is uniformly continuous in the usual sense.Both assertions follow from the permanence principle (e.g.Van den Berg [5, Chapter IV, Section 1]) and transfer.Further a bounded standard function is limited.

Definition
and the distance of consecutive elements is infinitesimally small.We denote by t + δt ∈ [ t 0 . . .T ] the successor of t ∈ [ t 0 . . .T ] with respect to the usual ordering ≤ on [ t 0 . . .T ].We say that the near interval [ 0 . . .T ] is equally spaced if δt is a constant δ independent of t and call δ the spacing of the near interval [ 0 . . .T ].

Remark
It is convenient to use in some steps (Lemma 6.14) of the proof of Theorem 6.5 (and thus also in its statement) an equally spaced near interval.However, we use general near intervals in the formulation of Theorem A.7 and some other results, since such a formulation may turn out to be useful for further applications.Note that if we speak of a near interval [ t 0 . . .T ] we presuppose the limitedness of t 0 and T .

Notation
The domain and the range of a function F is denoted by dom(F) and ran(F).We further introduce the symbol and the notations ≤ and = .We use them to handle calculations with non explicitly stated infinitesimal quantities, which simplify our notational effort.Let F(x) and G(y) denote functions of the variables x and y.We define The symbol is used in the same manner if the character ≤ in ( 13) is replaced by the character =, i.e., Note that = is not symmetric.(For example, we have ρ = for any infinitesimal ρ, but = ρ for any ρ ∈ R.) Our definitions imply that For a different definition of the symbol leading to the same use in calculus see Koudjeti and Van den Berg [14].

Discrete Fourier Analysis and the Laplacian
We will make use of the following well-known results from discrete Fourier analysis.For more information on the topic of discrete Fourier Analysis consult Terras [25] or Luong [17].

Definition
4.12 Remark Note that the functions g k provided by Definition 4.7 are the eigenvectors of h , i.e., h g k = µ k g k .Further ( 14) This is most easily seen using the identity e 2πikx = cos(2πkx) + i sin(2πkx) and calculating

Remark
For standard k ∈ K(h)-and thus especially for any k ∈ K(h) provided that h is standard-we have −∞ µ k ≤ 0.  P(dom(X)) .Note that dom(X) = Ω implies that P(X = x) = P(X = x).Given a function f with ran(X) ⊆ dom(f ) and ran(f ) ⊆ R J (with J an arbitrary set), we let E[f • X] = x P(X = x)f (x).In the case that dom(X| Y=y ) = ∅ we define by P(X = x|Y = y) := P(X| Y=y = x) the conditional probability that X = x under the hypothesis that Y = y.
and it is a martingale if M = R J (for some arbitrary set J ) and 4.17 Definition Given a topological space X , we denote by (C b (X ), .∞ ) the space of all bounded continuous functions f : X → R endowed with the .∞ -norm defined by f ∞ := sup x∈X |f (x)|.We further denote by C n b (R) the space of all n-times differentiable functions from R to R such that all derivatives (including the 0 th ) are continuous and bounded functions.We let

Definition
Let J be a finite set.We let S * (R J ) denote the family of all functionals ψ * : R J → R of the from ψ * (ξ) = j∈J ψ j ξ|e j , with ψ j ∈ R and j∈J ψ 2 j = 1.

Definition
Let J be a finite set and let X : Ω → R J .We say that the random variable X is approximately N(0, id) distributed on R J and write X ∼ N 0, id J or simply X ∼ N(0, id) if Note however that the finite set J is supposed to be standard in Proposition B.6, while this is not the case in Definition 4.22.

Remark
Note that X ∼ N(0, id) is equivalent with which is further equivalent with 5 The Model 5.1 Hypothesis Suppose that N ∈ N, λ ≈ ∞ and that h ∈ (0, 1] is such that
Let further (g k ) k∈K(h) denote the eigenbasis of the discrete Laplace operator, introduced in Definition 4.7, and denote by µ k the eigenvalue corresponding to g k .23) and ( 25) coincides via

Hypothesis Let
forms a Markov Chain.We specify this Markov chain by its transitions from Ξ t to Ξ t + and Ξ t + to Ξ t+δt given by random variables Q t and Q t + respectively, i.e., we suppose that Ξ 0 (ω) = 0 ∈ R H independent of ω and let (22) Ξ t + := Ξ t + Q t and Ξ t+δt : If ξ t ∈ X and P(Ξ t = ξ t ) > 0 we let Q t ∈ 0, − ex hN , + ex hN x ∈ H Ω be such that ( 23) x ∈ H Ω be such that If 5.4 Proposition Suppose that the Hypotheses 5.1 and 5.3 are fulfilled.For ξ t , ξ t + ∈ R H with P(Ξ t = ξ t ) > 0, P(Ξ t + = ξ t + ) > 0 we have: Proof In the case that ξ t ∈ R H \ X equation ( 27) is immediately derived from (24), since in this case p + λ,t (ξ, x) = p − λ,t (ξ, x).If ξ t ∈ X then (27) holds since by ( 23) Thus (27) holds for any ξ t ∈ R H .In the case that ξ t + ∈ R H \ X equation ( 28) is a consequence of (26).Finally if ξ t + ∈ X we obtain from ( 25) that . Thus (28) has been shown for any ξ t + ∈ R.

Remark
We may associate with the stochastic dynamical system fulfilling the Hypotheses 5.1 and 5.3 a deterministic system given by: (29) ξ t + = ξ t + δtλ ξ t and ξ t+δt = ξ t + + δt h ξ t + While the increments of the stochastic system are given by Q t and Q t + the increments of the deterministic system ( 29) coincide (by Proposition 5.4) in the cases that and . Note further that ξ = 0 is an equilibrium of the associated deterministic system, i.e., ξ t = 0 ⇒ ξ t+δt = 0.For standard h and unlimited λ the equilibrium ξ = 0 is unique and unstable, since ξ t+δt 2 ≥ 1 + δt λ 2 ξ t 2 .This last fact follows from an expansion of the dynamics with respect to the eigenbasis (g k ) k∈K(h) of ∆ h provided by 6 13 and 1/h ∈ N) and λ ≈ +∞,

Definition
To investigate Ξ t further we define: 5.7 Remark Note that the equalities (a) and (b) in ( 30) and (31) follow from ( 27) and ( 28) respectively.From ( 22), ( 30) and (31) we obtain that: 8 Remark As a consequence of Definition 5.6 we obtain for all ξ t , ξ t + ∈ R H with Further for all ω ∈ Ω we obtain from Definition 5.6 and Definition 5.3 that: Radically elementary analysis of a particle system 21 5.9 Proposition Suppose that the Hypotheses 5.1 and 5.3 are fulfilled and suppose that η 1 , η 2 ∈ S(R H ) , ξ t ∈ R H and P(Ξ t = ξ t ) > 0. Then For Σ t given by (30) we obtain in the case that ξ t ∈ X (which implies that ξ t is limited) and also in the case ξ t ∈ R H \X.
Proof (35) holds by (27) and because the finite sums involved in the calculation of the conditional expectation E[.|Ξ t = ξ t ] and the inner product .| η 1 interchange.We prove (36) for ξ t ∈ X first.From ( 23) we obtain (37) and thus further that We derive (36) by calculating with (a) a consequence of (30), equality (b) concluded from (35), (38) and equality (c) implied by (20) and the fact that ξ t ∈ X is limited.
(Equality (a) follows by the interchange of finite sums, equality (b) follows from (28) and equality (c) from the fact that h is symmetric, i.e., h acts on R H as a self-adjoint operator.)5.11 Proposition Suppose that the Hypotheses 5.1 and 5.3 are fulfilled.Let η ∈ R H and let Σ t + be given by (31).Suppose that ξ t + ∈ X and P(Ξ with η : R/Z → R denoting a differentiable extension of η and η denoting the derivative of η .
Proof We calculate for Note that inequality (a) follows from (25), equality (b) from (39) and inequality (c) from C.2.

5.
12 Definition Given a random variable X : Ω → R H we define random Fourier coefficients X k for k ∈ K(h) by series expansion of X with respect to the basis (g k ) k∈K(h) , i.e., we let (The random variables Γ t are introduced in Definition 6.3.)5.13 Remark From (32) and Definition 5.12 we obtain that (42) and from (32), Definition 5.12 and Remark 4.12 that From ( 42) and ( 43) we obtain that (44) 5.14 Proposition Suppose that the Hypotheses 5.1 and 5.3 are fulfilled, that Σ t and Σ t + denote the random variables introduced in Definition 5.6 and that the subscript k refers to coordinates with respect to the series expansion introduced in Definition 5.12.

Proposition
Suppose that the Hypotheses 5.1 and 5.3 are fulfilled, that Σ t and Σ t + denote the random variables introduced in Definition 5.6 and that the subscript k refers to coordinates with respect to the series expansion introduced in Definition 5.12.

Definition
Suppose that Σ t and Σ t + denote the random variables introduced in Definition 5.6 and that the subscript k refers to coordinates with respect to the series expansion introduced in Definition 5.12.Let 5.17 Proposition Suppose that the Hypotheses 5.1 and 5.3 are fulfilled, that Σ t,k and Σ t denote the random variables introduced in Definition 5. 16.Then for all ω ∈ Ω and all ξ t ∈ R with P(Ξ t = ξ t ) > 0 we have: 52) is a consequence of (34), −∞ µ k ≤ 0 (Remark 4.13) and Definition 5.16, while (53) follows from (33) and Definition 5.16 by We finally obtain (54) since with (a) a consequence of −∞ µ k ≤ 0 (Remark 4.13), (b) a consequence of Proposition 5.15 and (c) a consequences of λ ≈ ∞ and standardness of j, k ∈ Z. 6 The main Theorem 6.1 Remark We use-throughout section 6-the random variables Σ t introduced in Definition 5.16 and the random coefficients of the series expansions with respect to (g k ) k∈K(h) introduced in Definition 5.12.

Definition
We define linear operators Φ −t : R H → R H by: with (g k ) k∈K(h) the eigenbasis of the discrete Laplace operator ∆ h introduced in Definition 4.7 and µ k the respective eigenvalue of ∆ h that corresponds to g k .

Definition
Let Φ s : R H → R H be the linear operator given by and note that for t > s 6.4 Proposition Suppose that Hypothesis 5.1 holds, let [ 0 . . .T ] be an equally spaced near interval with spacing δ and let λ 2 δ ≈ 0. Then for s ∈ [ 0 . . .T ] we obtain 59) is a consequence of (56) and Proposition C.1, while (60) is a consequence of Definition 6.3, λ ≈ ∞ and the fact that by Remark 4.13 we have −∞ µ k ≤ 0.
We display now the main theorem in the case that the number 1 h ∈ N of positions occupied by particles in our dynamical system is standard.
6.5 Theorem Suppose that the Hypotheses 5.1 and 5.3 are fulfilled. 8Let T , N , λ and h be such that Let the near interval [ 0 . . .T ] be equally spaced and let the spacing δ be such that Let Γ T be the random variable introduced in Definition 6.3.Then Let ν ≈ ∞ be such that and let Φ −t be the linear operators introduced in Definition 6.2.Then for any k ∈ K(h) (66) 6.6 Remark Note that in the case that T is appreciable (61) and ( 20) imply (62).Note that λ ≈ ∞ and (62) imply λδ ≈ 0. Further (56) implies that the trajectories 6.7 Corollary Under the hypotheses of Theorem 6.5 (ν ≈ ∞ etc.) we obtain for any standard ε > 0 that (67) Derivation of Corollary 6.7 from Theorem 6.5 We calculate From (63), Remark 4.20 and ν ≈ ∞ we obtain that (69) and from (68) and (69) we obtain that (67) holds.

Heinz Weisshaupt
since for all standard ε > 0 with (a) a consequence of ( 69) and ( 67) and (b) a consequence of ν ≈ ∞ and the standardness of h.
6.8 Remark By the axiom of idealization Theorem 6.5 extends to the hyperfinite situation: 6.9 Theorem There exists a χ ∈ N such that Theorem 6.5 still holds if the hypothesis ( 19) is replaced by Derivation of Theorem 6.9 from Theorem 6.5 By Remark 4.24 formula (63) says that for any standard ε ∈ (0, ∞) and for any standard f ∈ C b (R) we have that: Since Theorem 6.5 holds for any h ∈ {1/n | n ∈ N} we obtain that (71) holds for any By an application of the idealization axiom of IST we obtain that (71) holds for any (h, I.e., (71) holds for any standard ε ∈ (0, ∞), for any standard f ∈ C b (R) and any h ∈ {1/n | χ ≥ n ∈ N}, and thus (63) holds for any h that fulfills (70).That there exists a χ ∈ N such that (65) and (66) hold for any h that fulfills N 1/h ≤ χ is obtained by application of idealization in an analogous manner.To complete the proof of the theorem simply let χ := min( χ, χ).
6.10 Remark Before we prove Theorem 6.5 we prove Proposition 6.11 that expresses the system's dynamics with respect to Γ t , Proposition 6.12 that provides some information concerning δΓ s , Lemma 6.14 that gives a formula for summing the variances of the δΓ s and Lemma 6.15 that proves (65) and prepares for the final steps in the proof of Theorem 6.5.

Proposition
Suppose that the linear operators Φ −t are given by Definition 6.2 and the random variables δΓ t and Γ t by Definition 6.3.Then or equivalently Proof By Definition 5.16 the recursion (44) becomes From Ξ 0 = 0 and (74) we obtain by recursion that the coordinate wise system's dynamics is given by (72), while equation ( 73) is just a reformulation of (72) using the Definition 6.3 and especially (57).
7 Reformulation as a standard limit theorem 7.1 Remark In this section we formulate a limit result (Theorem 7.9).This limit theorem is still formulated within the realm of finite probability spaces.Its corollary 7.11 makes however use of random variables that are N(0, id) distributed in the usual ZFC based sense (not in our IST based approximate sense).Such random variables can not be defined on a finite or countable probability space.For general measure theoretic probability theory adequate for dealing with random variables on uncountable spaces we refer the reader to Dudley [9].

Remark
We reformulate parts of Theorem 6.5 as a limit theorem in standard mathematical terms.To do this we have to consider sequences of interacting particle systems instead of a single system.We therefore replace the mathematical objects N ,λ, δ , Ω, [ 0 . . .T ], X, Ξ, Σ Γ, Γ and Φ introduced in the sections 5 and 6 by sequences β the eigenbasis of the discrete Laplace operator ∆ h introduced in Definition 4.7 and µ k the respective eigenvalue of ∆ h that corresponds to g k .
7.6 Definition Define the sequence (Γ β T β ) β∈N of random variables Γ β T β in analogy with the definitions given in the sections 5 and 6 starting with Ξ β instead of Ξ and replacing objects like Σ, Φ etc. in the consecutive Definitions 5.6, 5.12, 5.16 and 6.3 by consecutively defined objects Σ β , Φ β etc.

Hypothesis
For any sequence and Proof The theorem is a statement of ZFC.By an application of transfer we suppose without loss of generality that all objects named in the theorem (including h) are standard.Thus, by Proposition 7.8, Hypotheses 7.3 implies that Hypothesis 7.7 holds.The Hypotheses 7.4 and 7.7 imply together with standardness of h that for any β ∈ N the hypotheses of Theorem 6.5 are fulfilled with N , λ, δ and [ 0 . . .T ] replaced by N β , λ β , δ β and [ 0 . . .T β ], respectively.Thus also the conclusions of Theorem 6.5 are fulfilled with the respective replacements, i.e., i.e., (97) implies (98).Since (93) implies (97)-by standardness of the involved sequences-and (98) implies (by the permanence principle) (94), the formula (94) has been derived.The proof of (95) is similar to the proof of (94) and thus omitted.It remains to prove (92).Formula (96) is by Remark 4.24 equivalent with
when it stays infinitesimally close to 0, and during the third time period [ T − ν/λ . . .T ], when it takes on appreciable values.By Remark 6.6 the deterministic trajectory (y t (ω)) t∈[ ν/λ...T ] fulfills y t+δt = Ly t with L the linear transformation given by Lg k = (1 + µ k δt)(1 + λδt)g k , where (g k ) k∈K denotes an eigenbasis of the discrete Laplace operator and µ k denotes the eigenvalue corresponding to g k .Our investigations are partially motivated by the following simple chemical reaction system: (11) 2A + B → 3A and 2B + A → 3B, Let [A] and [B] denote the concentrations of the chemical species A and B. Suppose that [A] + [B] = 1 and that the kinetic constant of both reactions equals 2λ.Introduce further the variable ξ = [A] − 1 2 .Then the kinetic equation of the reaction system is given by dξ dt = λξ − 4λξ 3 .A first order approximation of this kinetic equation at the unstable equilibrium ξ = 0 gives dξ dt = λξ .Replacing dξ by E[Q t |Ξ = ξ t ] and dt by Heinz Weisshaupt

4. 10 Remark
Note that the family (g k ) k∈K(h) of functions g k ∈ R H defined in 4.7 forms an orthonormal bases of R H with respect to .| ., i.e., g k | g l = k,l .This is most easily seen using the identities cos(2πkx) = e 2πikx + e −2πikx /2 and sin(2πkx) = e 2πikx − e −2πikx /2i and that for k, l ∈ K(h) we have h x∈H e 2πi(k−l)x = k,l and h x∈H e 2πi(k+l)x = (k+l mod 1/h),0 4.11 Definition Let 1/h ∈ N and let H := hZ/Z.Define the discrete Laplace operator

4. 15
Definition (Compare with [19, Chapters 1 and 2]) Let X and Y be random variables.Let P(X = x) := P(X=x) The conditional expectation E(f •X|Y = y) is defined by replacing the probabilities in the definition of the expectation above by conditional probabilities.Given a function F with ran(Y) ⊆ dom(F) we use P(X = x|Y) = F • Y as a shorthand notation for (∀y ∈ ran(Y)) (dom(X| Y=y ) = ∅ =⇒ P(X = x|Y = y) = F(y)).

4. 23 Remark
Definition 4.22 is partially justified by Proposition B.6 in Appendix B.

5. 10 Remark
Suppose that we are given a function η ∈ R H . Then (39) x∈H e x+h − e x |η 2 = x∈H e x−h − e x |η 2 and