Nonequilibrium phase transitions in finite arrays of globally coupled Stratonovich models: Strong coupling limit

A finite array of $N$ globally coupled Stratonovich models exhibits a continuous nonequilibrium phase transition. In the limit of strong coupling there is a clear separation of time scales of center of mass and relative coordinates. The latter relax very fast to zero and the array behaves as a single entity described by the center of mass coordinate. We compute analytically the stationary probability and the moments of the center of mass coordinate. The scaling behaviour of the moments near the critical value of the control parameter $a_c(N)$ is determined. We identify a crossover from linear to square root scaling with increasing distance from $a_c$. The crossover point approaches $a_c$ in the limit $N \to \infty$ which reproduces previous results for infinite arrays. The results are obtained in both the Fokker-Planck and the Langevin approach and are corroborated by numerical simulations. For a general class of models we show that the transition manifold in the parameter space depends on $N$ and is determined by the scaling behaviour near a fixed point of the stochastic flow.


Introduction
Arrays of stochastically driven nonlinear dynamical systems may exhibit nonequilibrium phase transitions of continuous or discontinuous type, for a recent review see [1], cf. also [2,3]. Concepts developed to describe equilibrium phase transitions such as symmetry or ergodicity breaking, order parameter, critical behaviour, critical exponents etc. have been successfully transfered to noise induced nonequilibrium phase transitions. The structure of the theory will be generically of mean eld type, if innite globally coupled arrays are studied which allows for a number of analytical results.
Remarkably, essential characteristics of phase transitions can already be found in the case of a single Stratonovich model. This is mainly due to the multiplicative nature of the noise. Models driven by additive noise do not show this peculiar property. The Langevin equation for the single-site Stratonovich model [4,5,6] reads where a is a control parameter, σ denotes the strength of the noise and W (t) is a Wiener process with autocorrelation W (t)W (s) = min (t, s). Equation (1) is interpreted in the Stratonovich sense as indicated by the symbol •. The Stratonovich model describes, e.g., the overdamped motion in a biquadratic potential U (x) = − a 2 x 2 + 1 4 x 4 where the control parameter is stochastically modulated, a → a + ξ t , with a Gaussian white noise ξ t .
The associated Fokker-Planck equation (FPE) describing the evolution of the probability density P (x, t) is Equation (2) has a weak stationary solution, a Dirac distribution δ(x) located at the common zero x = 0 of drift and diusion coecient, which is also a zero of the stochastic ow in Eq. (1). If the system is initially at x = 0 it will always stay there. Furthermore, there exist spatially extended strong stationary solutions determined up to a constant factor, P s (x) ∝ |x| 2a/σ 2 −1 exp{−(x/σ) 2 }. P s (x) will live on S + = [0, ∞) if the initial distribution lives on S + \0, and on S − = (−∞, 0], if the initial distribution lives on S − \0. The constant is determined such that the solution is normalized integrating over the support and can be interpreted as probability density, i.e. provided 2a/σ 2 > 0. For 2a/σ 2 ≤ 0 the normalization Z diverges since the integrand in (4) scales like |x| 2a/σ 2 −1 as x → 0. In this case it can be shown (see Appendix B) that a weakly normalized version converges to the known weak solution δ(x) and x = 0 is an absorbing xed point of the system. If fractions of the initial distribution of given weights live on S − , on S + , and on 0, all will keep their weight and evolve to the stationary probability densities living on their respective support as guaranteed by a H-theorem [7].
The Stratonovich model exhibits a strong ergodicity breaking [8] depending on the control parameter a, since the state space decomposes into regions where the system cannot reach one region if it has started in a dierent one. For a ≤ 0 the only stationary solution is δ(x), i.e. the xed point x = 0 of the stochastic dynamics is absorbing. Additionally, for a > 0 we have the spatially extended solution (3) living on S ± depending on the initial distribution. This is reected by the mean value Obviously, x ± can serve as an order parameter and shows a critical behavior x ± ∼ ± √ π σ (a − a c (1)) β as a → a c (1) = 0 with β = 1. Note that also the location of the maximum of the spatially extended density undergoes a bifurcation, x max The critical behaviour of an array of innitely many globally coupled Stratonovich models has been thoroughly investigated in [9]. The scaling of higher moments was considered in [10], see also [11]. The stationary probability density is the solution of a nonlinear Fokker-Planck equation which depends on the order parameter. The scaling behaviour of the order parameter is analytically determined, x ± ∼ ±(a − a c (∞)) β as a → a c (∞) with a c (∞) = −σ 2 /2 and β = sup{1/2, σ 2 /(2D)}, where D is the strength of the harmonic coupling between the systems [9]. The strong coupling limit, D → ∞, of an innite array of globally coupled systems was analytically treated already in the pioneering paper [12], cf. also [13,14].
In this paper we investigate nonequilibrium phase transitions in nite arrays of globally coupled Stratonovich models in the strong coupling limit. We introduce center of mass and relative coordinates and exploit that for strong coupling there is a clear separation of time scales. The relative coordinates relax very quickly to zero and the system behaves as a single entity described by the center of mass coordinate R t . Thus, we can adiabatically eliminate the relative coordinates. The stationary probability density of the center of mass coordinate p s (R) is analytically calculated for a class of nonlinear systems and a scheme to determine the transition manifold in the parameter space is developed. For nite arrays of Stratonovich models the mean value R of the center of mass coordinate is computed analytically. Near a critical value of the control parameter a the stochastic system shows a scaling behaviour similar to the order parameter of the single Stratonovich model with the same critical exponent β = 1 but with a dierent a c (N ) which is also given analytically. Keeping a nite small distance to a c (N ) we recover for N → ∞ the known result of the self consistent theory [9] with critical exponent β = 1/2, see above. For nite N we identify a crossover value of the control parameter a (N ). For a c (N ) < a a (N ) we have a linear scaling as for N = 1 whereas for a a (N ) a square root behaviour as for N → ∞ is observed. The analytical results are coroborrated by numerical simulations.
Recently, nite arrays of (non-) linear stochastic systems have been investigated also by Muñoz et al. [10], and by Hasegawa [15].
Muñoz et al. tried to obtain for multiplicative noise characteristics of the probability density of the mean eld for nite N . They argued that the Langevin equation for the mean eld variable is of similar form as the Langevin equation for a single system. Assuming that the multiplicative driving noise and the local eld variable are uncorrelated, they inferred the scaling behaviour of the variance of an eective multiplicative noise with N , and of the critical value a c (N ) of the control parameter. They also predicted a crossover from a critical exponent β = 1 near a c (N ) to the critical exponents for N → ∞ for larger distances to a c (N ). Note that in [10] the Langevin equation was treated in the Ito-sense which leads to a shift of the critical control parameter compared to the same equation in the Stratonovich-sense.
Hasegawa considered nite systems with additive and multiplicative noise using his augmented moment method which is applicable for small noise strength. He emphasized that multiplicative noise and the local eld variable are not uncorrelated in contrast to the assumption in [10] and demonstrated some consequences of such a simplication.
Our approach, though similar in spirit to [10], is controllable, valid in leading order for strong coupling D, and provides explicit analytic results which are conrmed by independent numerical simulations. It may serve as a starting point to calculate next order corrections ∼ 1/D.
The paper is organized as follows. In the next section we consider two harmonically coupled Stratonovich models and show that for strong coupling the center of mass coordinate R is the relevant degree of freedom. The mean value of R shows a critical behavior which is analytically characterized. Section 3 deals with a class of N globally coupled systems of general kind. For strong coupling we compute analytically the stationary probability distribution p s (R) after eliminating the relative coordinates. Further, we determine the transition manifold in the parameter space where p s (R) undergoes a transition from a delta-distribution to a spatially extended solution. In Sec. 4 we specialize to the case of N globally coupled Stratonovich models and determine the critical behaviour of the order parameter and of higher moments of R for strong coupling. Conclusions are drawn and a summary is given in Section 5. In Appendix A we introduce the concept of weak normalization for the case that a spatially extended solution of the stationary FPE cannot be normalized in the naive sense. Appendix B shows that the Langevin approach both in Stratonovich-and in Ito-interpretation leads to the same results as the Fokker-Planck approach used in the main part of the paper.

Two coupled Stratonovich systems
We consider a pair of particles with coordinates x 1 (t) and x 2 (t) in a biquadratic potential which are coupled harmonically with positive coupling strength D and each subjected to independent Gaussian white noise of strength σ. The system of Langevin equations reads where W i (t) denotes independent Wiener processes with W i (t)W j (s) = δ i,j min (t, s). In contrast to Eq. (1) no exact solution of system (6) is known.

5
The joint probability density P (x 1 , x 2 ; t) is governed by the FPE where, adopting the notation of [21], denote drift and diusion coecients, respectively. One can show that the system (7) exhibits no detailed balance. Hence, there is no easy way to obtain analytically the stationary solution P s .
For strong coupling, however, a systematic analytical approach is possible. With increasing coupling strength the particles become tightly glued together and move as a single entity. Therefore it appears natural to introduce center of mass and relative coordinates. Simulations of Eq. (6) show that indeed the stationary distribution of the relative coordinatep s (r) becomes very sharp for large values of D, cf. Fig. 1. The stationary distribution of the center of mass p s (R) shows a behaviour which is similar to the distribution of a single Stratonovich model. For large values of a we have a monomodal distribution which vanishes at the boundaries of the support, cf. Fig. 1. For small values of a the distribution p s (R) diverges as R → 0 in a normalizable way, cf. Fig. 2. For even smaller values of a all trajectories x i (t) approach zero and the distributions of both r and R are δ-distributions. Accordingly, the mean value R undergoes a continuous transition at a critical value of a. In the following we analytically calculate p s (R) and R and its scaling characteristics in the strong coupling limit, D → ∞. We introduce the center of mass coordinate R(t) The symbols show data from 5 × 10 6 realizations generated by a stochastic Runge-Kutta algorithm [16]. Initial values were chosen in the positive sector. The lines are guides to the eye. and the relative coordinate r(t) by with the inverse transformation With the Langevin equations (6) then transform to where the transformed Wiener processes W i (t) are dened as The FPE associated to (14,15) governing the probability density of center of mass and relative coordinates P (R, r; t) reads where the Fokker-Planck operators are Note that only L r depends on D. In the strong coupling limit D → ∞ the relative coordinate vanishes, r t → 0, on a very fast time scale of the order 1/D, cf. Eq. (15). Hence, the stationary probability density factorizes to P s (R, r) = p s (R)δ(r) with a Dirac distribution for the relative coordinate. In this case there is no ow related to the relative coordinate r, i.e. L r P = L rR P ≡ 0, since for any suitable function ϕ Integrating Eq. (17) with respect to r yields in the stationary case Similarly as for the single Stratonovich model, there is always a weak solution δ(R).
For initial values x i (0) > 0 ∀i (or x i (0) < 0 ∀i) the spatially extended solution of Eq. (22) lives on the support S + = [0, ∞) (or on S − = (−∞, 0]) and can be normalized provided a > a c (2) = −σ 2 /4. For a ≤ a c (2) the weakly normalized version of the spatially extended solution converges to δ(R). Thus, we have There is a strong ergodicity breaking when a crosses a c (2). The mean value R ± calculated with (23) is and 3. General N-site systems

Adiabatic elimination of relative coordinates
In the following we demonstrate that the strategy sketched above can be generalized for a class of N coupled systems. We consider with i = 1, . . . , N and where f and g are smooth (with no singularities) and twice dierentiable chosen such that the stochastic process x(t) = {x i (t), i = 1, . . . , N } has natural boundaries at innity [4,20]. Both f and g may depend on a ddimensional set of control parameters a. D > 0 is the coupling strength of the harmonic attraction. Note that we have absorbed a factor σ, the strength of the noise, in the function g.
The FPE for the joint probability density P ( Using the shorthands f i = f (x i ), g i = g(x i ), and g i = ∂ x i g i , drift coecient and diusion matrix are given by It is advantageous to introduce center of mass and relative coordinates {R, r}, r = {r k , k = 2, . . . , N }, by the linear transformations The inverse transformation is given by Observing the rules for linear transformations we have Drift and diusion coecients in the new coordinates are given by, cf. also [21], where y and z stand for the new coordinates R, r k , and r l , respectively. Again, the FPE determining P (R, r; t) has the form ∂ t P = LP , with L = L R + L r + L rR where Explicitly, the new drift and diusion coecients are All arguments in f i , g i , and g i have to be expressed by (R, r), see Eqs. (32,33). Note that only D r k depends on the coupling strength D explicitly. For large times the probability density P (R, r; t) converges to a stationary probability density, cf. [7], determined by LP s (R, r) = 0. For D → ∞ this enforces which has a weak solution P s (R, r) = p s (R)δ(r).
In the strong coupling limit all uctuations of the relative coordinates vanish. The system is concentrated on the center of mass and moves stochastically as a whole, combined particle. The probability density of the center of mass p s (R) can be determined by integrating P s (R, r) over all relative coordinates. Performing this integration we obtain from the stationary FPE provided that the boundary terms associated with the relative coordinates vanish.
In the strong coupling limit we have P s (R, r) ∝ δ(r), (47) holds in any case and leads to where L R is given by (37). From (40) and (42) we infer drift and diusion for r = 0 as The spatially extended strong solution of (48) is given by provided that the normalization constant Z is nite. Whether or not this is the case depends on the scaling behaviour of the functions f (R) and g(R) near a common zero R 0 which, if existing, will build a boundary of the support. This is explained in detail in the next subsection. Equation (50) shows that in the strong coupling limit the diusion coecient D R,R scales like σ 2 /N , cf. also [10,15]. For the innite system and nite noise strength σ the stationary probability density of the center of mass p s (R) is a Dirac measure located at one of the attractive zeros of the drift coecient (49), depending on the initial conditions.
For D → ∞ all particles are strongly correlated. The variance of the coordinate x i (R, r) of an arbitrary system i calculated with P s (R, r) = p s (R)δ(r) is Due to the strong correlations, the variance of the center of mass scales like N 0 in contrast to the case of uncorrelated systems where the central limit theorem predicts a scaling like N −1 .

Determination of the transition manifold
There will be a strong ergodicity breaking if the state space decomposes into dierent regions with the property that one region will not be accessible if we start in a dierent one [8].
For multiplicative noise, zeros of the stochastic ow separate the state space into mutually non-accessible regions. If we place the system initially on such a zero, i.e. on a xed point of the stochastic dynamics, it will stay there forever. Accordingly, the FPE has a weak solution, a δ-distribution living on that xed point. If any trajectory in the neighborhood 'asymptotically' reaches the xed point (the xed point is absorbing), there will be no spatially extended probability density in this neighborhood. The spatially extended stationary solution of the FPE cannot be normalized in the naive sense. The weak normalization procedure leads to the weak solution.
If trajectories cannot reach the xed point, the stationary solution of the FPE will be normalizable and we will have a spatially extended probability density living on the support bounded by the xed point. This properties can be exploited to determine the transition manifold in the parameter space.
We suppose that f (R, a) and g(R, a) near a common zero R 0 have the following scaling behaviour where m f , m g > 0. Near R 0 we have for the stationary solution (51) of the reduced FPE (48) For m f − 2m g > −1 the integral in (55) gives a contribution ∝ ε m f −2mg+1 at the upper boundary which vanishes for ε → 0 so that in leading order p s (R 0 + ε) ∝ |ε| mg(N −2) . The exponent m g (N − 2) is negative only for N = 1. In this case, if m g 1 the singularity of p s at R = R 0 is not normalizable and we have only a weak stationary solution p s (R) = δ(R − R 0 ). Note that for N 2 coupled systems of this kind the singularity of p s does not occur.
For m f − 2m g = −1 the integral gives a logarithmic contribution (A f /A 2 g ) ln |ε| which leads to If the exponent in (56) is smaller than −1 the density p s (R) will diverge for R → R 0 and will not be normalizable in a naive way. The weak normalization procedure leads to a Dirac measure located at R 0 . If the exponent is larger than −1 the density p s (R) will be normalizable and we will have a spatially extended probability density. The transition manifold A c in the control parameter space R d is determined by the condition that the exponent in (56) is −1, 13 In the rst case p s (R) is normalizable and we have a spatially extended stationary probability density. In the latter case the weak normalization procedure yields a Dirac measure at R 0 . A change in the sign of A f induced by tuning a control parameter is associated with a change of the stability of the xed point R 0 of the deterministic ow f (R) and leads to a signicant alteration of the ergodic properties. In a vicinity of R 0 the behaviour of the stochastic system is dominated by the deterministic ow. The transition manifold does not depend on the system size and the amplitude A g in contrast to (57). If the system lives on the d − 1 dimensional transition manifold (58), that is A f = 0, the scaling of the deterministic ow is not described by (53) but by f (R 0 + ε) ∼ B f ε n f with n f > m f . The systems behaviour, now depending on B f , A g , n f , m g and N , could be classied in more detail repeating the above procedure.

N coupled Stratonovich models
Now we return to our specic example and consider N globally coupled Stratonovich models in the strong coupling limit. For drift and noise function we have f (R; a) = aR − R 3 and g(R; σ) = σR .
The common zero of f and g is R 0 = 0 with m f − 2m g = −1, and A f = a, A g = σ.
Inserting this in (57) we obtain an explicit representation of the transition curve in the 2-dimensional parameter space, Given the noise strength σ we have which reproduces the results for N = 1, for N = 2 (see above), and for N → ∞. For initial values x i > 0 (or x i < 0) ∀i the stationary distribution for the center of mass (51) lives on S + or S − , respectively, and is given by Keeping N nite, the rst moment scales as a → a c (N ) like dimensional transition manifold (58), that is A f = 0, the scaling of the deterministic flow is not described by (53) but by f (R 0 + ε) ∼ B f ε nf with n f > m f . The systems behaviour, now depending on B f , A g , n f , m g and N , could be classified in more detail repeating the above procedure.

IV. N COUPLED STRATONOVICH MODELS
Now we return to our specific example and consider N globally coupled Stratonovich models in the strong coupling limit. For drift and noise function we have f (R; a) = aR − R 3 and g(R; σ) = σR . (59) The common zero of f and g is R 0 = 0 with m f − 2m g = −1, and A f = a, A g = σ. Inserting this in (57) we obtain an explicit representation of the transition curve in the 2-dimensional parameter space, Given the noise strength σ we have which reproduces the results for N = 1, for N = 2 (see above), and for N → ∞. Figure 3 compares results For initial values x i > 0 (or x i < 0) ∀i the stationary distribution for the center of mass (51) lives on S + or S − , respectively, and is given by Similar to the single Stratonovich model, t qualitative change in the shape of the spatially probability density. The maximum of p s (R) u a bifurcation at a max c = σ 2 (1/N − 1/2). Figu pares for different system sizes the asymptotic r with histograms obtained by simulations for lar a > a c (N ) the nth moment of the center of ma evaluated as Keeping N finite, the first moment scales as a like since Γ(z) ∼ 1/z as z → 0 [21]. Note that also t moments R n scale linear with a − a c (N ). Keeping a finite distance to a c (N ) we obtain ∞, observing Γ(z + 1/2)/Γ(z) = √ which reproduces the result in [9] for D > σ 2 moments of order n scale with β = n/2. We define the crossover value a (N ) by a c (N ))N/σ 2 = 1. For a a (N ) = −σ 2 /2 + (3 we have a linear scaling as for N = 1 wh a a (N ) a square root behaviour as for N observed, cf. Fig. 5.
Our results are analytically derived for the st pling limit in a controllable approach. We note the critical and the crossover value of the co rameter are in accordance with the values pro different grounds in [10] for weak and intermedi provided the shift due to the Ito interpretation u is taken into account.

V. CONCLUSIONS
In this paper we have determined the charact a continuous nonequilibrium phase transition i 7 Figure 3: Transition point a c vs system size N for σ 2 = 0.1. Simulation results for D = 81 obtained by maximizing the linear correlation coecient [22] (gray bullets) and by short time relaxation [23] (black bullets) are both in good agreement with the asymptotic result (61) for D → ∞ (solid line). The histograms are obtained from 10 6 realizations generated by a stochastic Euler method [26]. On the right we show only results for N = 8 for a c < a = −0.42 < a max c ; here 5 × 10 6 realizations were generated by a stochastic Runge-Kutta algorithm [16]. σ 2 = 1.
since Γ(z) ∼ 1/z as z → 0 [24]. Note that also the higher moments R n scale linear with a − a c (N ).
Our results are analytically derived for the strong coupling limit in a controllable approach. We note that both the critical and the crossover value of the control parameter are in accordance with the values proposed on dierent grounds in [10] for weak and intermediate noise, provided the shift due to the Ito interpretation used there is taken into account. We have analytically determined both the critical value of the control parameter a c (N ) and the scaling behaviour of the order parameter and of higher moments. With increasing distance from a c (N ) a crossover from linear to square root behaviour is found. For N → ∞ the known scaling behaviour is reproduced. The formal results, i.e., the computation of the stationary distribution of the center of mass coordinate (up to a quadrature) and the determination of the transition manifold are given for a general class of systems.
Our approach may serve as a starting point to calculate next order corrections in 1/D. In a multiscale analysis we have to take into account that for finite but large D the distribution of the relative coordinates is, though very sharp, of finite width.
The observation that a solution of the stationary Fokker-Planck equation which is not normalizable in a naive way converges to the weak solution if weakly normalized is certainly of interest in a broader context. (64) for N = 1 (dash-dotted line), 2 (dashed line), and 100 (solid line); σ 2 = 1. The symbols show averages over 2 × 10 6 realizations generated by a stochastic Runge-Kutta scheme [16] for N = 2 (circles) and 100 (squares); σ 2 = 1 and D = 100. The dotted straight lines have slope 1 (left) and slope 1/2 (right), respectively. The arrows indicate the crossover points a (N ) − a c (N ).

Conclusions
In this paper we have determined the characteristics of a continuous nonequilibrium phase transition in a nite array of globally coupled Stratonovich models in the limit of strong coupling D → ∞. In this limit there is a clear separation of the time scales governing the evolution of the center of mass coordinate and the relative coordinates: The characteristic time of the relative coordinate scales as 1/D and thus becomes short in the strong coupling limit. The slow center of mass coordinate enslaves the fast relative coordinates, its mean value serves as order parameter. This allows a controllable and consistent treatment both in the Fokker-Planck and the Langevin description which is corroborated by numerical simulations. The reduction of a high-dimensional problem to a problem of low dimension is of course inspired by generalizations of slaving and adiabatic elimination techniques and the concept of center manifolds to stochastic systems developed in a dierent context [29,30,28], cf. also [31,32].
We have analytically determined both the critical value of the control parameter a c (N ) and the scaling behaviour of the order parameter and of higher moments. With increasing distance from a c (N ) a crossover from linear to square root behaviour is found. For N → ∞ the known scaling behaviour is reproduced. The formal results, i.e., the computation of the stationary distribution of the center of mass coordinate (up to a quadrature) and the determination of the transition manifold are given for a general class of systems.
Our approach may serve as a starting point to calculate next order corrections in 1/D. In a multiscale analysis we have to take into account that for nite but large D the distribution of the relative coordinates is, though very sharp, of nite width.
The observation that a solution of the stationary Fokker-Planck equation which is not normalizable in a naive way converges to the weak solution if weakly normalized is certainly of interest in a broader context.

Appendix A. Weak normalization
The FPE for multiplicative noise has two types of stationary solutions: weak solutions, i.e. Dirac-distributions living on the zeros of the stochastic ow and spatially extended strong solutions which live on a support which is bounded by zeros of the stochastic ow or by natural boundaries at innity. Under certain conditions the spatially extended solution may diverge at a zero of the stochastic ow so strongly that it is not normalizable and therefore cannot be considered as a probability density. Here we introduce the concept of weak normalization and show that in the latter case the weakly normalized solution converges to the Dirac distribution living on that zero.
We assume that the unnormalized solutionP s (x) lives on [x 0 , b) where x 0 is a zero of the stochastic ow and scales for x → x 0 as The normalization integral diverges at the lower boundary and scales like Introducing a test function ϕ(x) which can be expanded near x 0 as ϕ(x) = ϕ(x 0 ) + ϕ (x 0 )(x − x 0 ) + . . . we have as Now we obtain for the hereby dened weakly normalized probability density P w which implies that P w s (x) = δ(x − x 0 ).

Appendix B. Langevin approach
In Sections 2 and 3 we used the Fokker-Planck approach in the center of mass and relative coordinates to calculate a c (N ) for D → ∞. In this limit the relative coordinates r k → 0, and it is easy to calculate the reduced stationary probability density of the center of mass coordinate p s (R). We determined a c (N ) such that p s (R) is a Dirac measure at R = 0 for a < a c (N ) and it is spatially extended for a > a c (N ).
The same result can be obtained in the Langevin approach, both in Stratonovich and Ito-interpretation as explained in the following for the special case N = 2. The generalization to N > 2 is straightforward.
We exploit that for large D the characteristic time scale of the relative coordinate r(t) is 1/D so that r(t) becomes very fast. Then the (slow) center of mass coordinate R(t) feels only the average of the fast process r(t) and it is justied to replace in the Stratonovich-Langevin equation (14) for R the terms associated with r by their average, since for D → ∞, r → 0 we have r 2 = 0. However, the second average r(t) • d W 2 (t) is not zero as one could naively expect. With the help of the Furutsu Novikov theorem [33,34] we obtain r(t) • ξ 2 (t) = t −∞ ds ξ 2 (t) ξ 2 (s) δr(t) δ ξ 2 (s) Note that the averages here are with respect to realizations of ξ 2 . We now observe that the resulting equation for R does not depend on ξ 2 , therefore R = R, and obtain from which the threshold a c (2) = −σ 2 /4 follows.
The system (14,15) in Stratonovich sense can be written in a compact form as dρ = f (ρ)dt + j=1,2 g (j) (ρ) • d W j (t), where ρ = (R, r) T . The equivalent Ito system is dρ = (f + 1/2 j ∂ ρ g (j) g (j) )dt + j g (j) d W j (t), where the drift term is modied by the Ito shift; ∂ ρ g (j) is the shorthand of a Jacobian, cf. e.g. [35]. For our system we have g (1) = σ/ √ 2 ρ and g (2) = σ/ √ 2 (r, R) T . The Ito shift amounts to σ 2 /2 ρ so that the equivalent Ito version of (14) reads Again, R feels only the average of the terms associated with the fast process r, we have r 2 = 0 but now also the second average vanishes since in the Ito calculus r(t)d W 2 (t) = r(t) d W 2 (t) = 0 which results in This is indeed the Ito eqivalent to Eq. (B.3) which can be seen observing that in the single variable case the Ito shift is simply 1/2 g g = σ 2 /4 R. For arbitrary N the same procedure leads to a c (N ) = −(σ 2 /2) (1 − 1/N ) as obtained above.