State estimation of long-range correlated non-equilibrium systems: media estimation
SState estimation of long-range correlatednon-equilibrium systems: media estimation
Otto Pulkkinen
Fachrichtung Theoretische Physik, Universit¨at des Saarlandes, 66123 Saarbr¨ucken,GermanyE-mail: [email protected]
Abstract.
Non-equilibrium systems have long-ranged spatial correlations even faraway from critical points. This implies that the likelihoods of spatial steady stateprofiles of physical observables are nonlocal functionals. In this letter, it is shownthat these properties are essential to a successful analysis of a functional level inverseproblem, in which a macroscopic non-equilibrium fluctuation field is estimated fromlimited but spatially scattered information. To exemplify this, we dilute an out-of-equilibrium fluid flowing through random media with a marker, which can be observedin an experiment. We see that the hidden variables describing the random environmentresult in spatial long-range correlations in the marker signal. Two types of statisticalestimators for the structure of the underlying media are then constructed: a linearestimator provides unbiased and asymptotically precise information on the particledensity profiles, but yields negative estimates for the effective resistances of the mediain some cases. A nonlinear, maximum likelihood estimator, on the other hand, resultsin a faithful media structure, but has a small bias. These two approaches complementeach other. Finally, estimation of non-equilibrium fluctuation fields evolving in time isdiscussed.
Keywords : transport processes / heat transfer (theory), stationary states,disordered systems (theory), new applications of statistical mechanics a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r tate estimation of long-range correlated non-equilibrium systems: media estimation
1. Introduction
Interacting non-equilibrium systems exhibit spatial long-range correlations, in whichthe equal-time autocorrelation functions of physical observables show slower thanexponential decay as a function of the distance between points of measurement.Presence of conservation laws, spatial anisotropies and lack of detailed balance aretypical requirements for their emergence [1]. In some fluids, the couplings betweenhydrodynamic fields is the dominant cause for the correlations [2]. In contrast tothermodynamic equilibrium, proximity of a phase boundary is not needed.The long-range correlations show up in the fluctuations of observables on length-scales comparable to the system size. The statistics of these fluctuations are capturedby the likelihoods of macroscopic profiles, which often satisfy a large deviation principle.In those cases, the likelihood is characterized by a large deviation functional, ageneralization of the equilibrium concept of free energy, which measures the macroscopicfluctuations in terms of exponentially small probabilities [3]. As an example, thecloseness of a stationary but fluctuating fluid density field φ to an arbitrary function f in a volume V could be described by a large deviation functional F as P ( φ ∼ f ) ≈ e − V F ( f ) (1)with F attaining its minimum at the most likely profile.It has been shown that the non-equilibrium large deviation functionals are nonlocal ,in that the likelihood of a profile is not additive under the operation of uniting twosubsystems into a larger, joint system [4]. Instead, the probability of observing a givenprofile in a subdomain depends also on the form of the same profile elsewhere, i.e. onthe global structure. This is a consequence of spatial long-range correlations.Long-range correlations and nonlocal profile likelihoods suggest that it is possibleto analyze a functional level inverse problem : even small pieces of information on local,spontaneous fluctuations can be translated by an analytical and numerical machineryto the language of fluctuations on a global scale. Building such machinery is the topicof this letter. Working through a specific application of determining the structure ofrandom media using marker particle data, we see how to extract useful informationabout the fluctuation state of a random, macroscopic field from a weak signal.The problem of estimating the random media structure is attractive not onlybecause of its potential applications, but also because it offers a unique view to spatialcorrelations in non-equilibrium systems and their state estimation in general. First, wesee that hidden information, in this case a static random environment, can bring aboutan effective particle-particle interaction which results in long-ranged spatial correlations.In fact, for the system under study, the correlations are entirely due to hidden variablesbecause of a special symmetry in the dynamics. The analysis is then transparent becauseestimation is based on a single type of correlation. Second, these long-range correlationsturn out to be of a very common form, given by a piecewise linear covariance. Suchcovariance functions can be found in a class of driven diffusive systems (includingexclusion and KMP processes [5]). Also the correlations induced by a coupling of tate estimation of long-range correlated non-equilibrium systems: media estimation
2. Media estimation
We study transport of particles through one-dimensional random media, which ismodeled by random single-particle transition rates between microscopic unit cells. Thesize of the unit cells is determined by the media correlation length, above which thetransition rates have independent statistics. We assume spatial homogeneity of themedia in a statistical sense. The rate at which particles move from the microscopic cell i to cell i + 1 and vice versa, is a random number v i , and these random numbers areindependent and identically distributed. The system is driven out of equilibrium by achemical potential difference at two boundary reservoirs. The transition rates v and v L at the boundaries are set equal to unity.The stationary states of particle transport in random media were studied in [6] for aclass of particle interactions. In these particle systems, the fugacity profile is a monotonefunction, which can be expressed in terms of partial resistances R j = 1 + (cid:80) j − i =1 v − i as φ j = φ − (cid:18) − R j R L (cid:19) + φ + R j R L , j = 1 , . . . , L, (2)where φ − and φ + are the fugacities at the left and right boundary reservoirs, respectively.The number of particles in a cell j is a function of φ j only. Thus the structure ofstationary density profiles on a macroscopic scale depends on the statistics of theresistances v − i . For finite expected resistances, the fugacity converges to a linear,deterministic function by the law of large numbers as the number of unit cells diverges.On the other hand, for E v − i = ∞ , percolation effects are important. In terms of amacroscopic spatial coordinate x ∈ [0 , φ ( x ) = φ − (cid:18) − R ( x ) R (1) (cid:19) + φ + R ( x ) R (1) , (3)where R is an α -stable non-decreasing L´evy process [7]. Here α + 1 ∈ (1 ,
2) is thepower-law exponent for the tail of the distribution for the resistances v − i . For α (cid:46) ρ ( x ) := E φ ( x ) = φ − (1 − x ) + φ + x. (4)On the other hand, for α small, the total resistance is dictated by just a few bottlenecks,which show up as large jumps in the fugacity profile, and as plateaus between the jumps tate estimation of long-range correlated non-equilibrium systems: media estimation C ( x, y ) := E φ ( x ) φ ( y ) − ρ ( x ) ρ ( y ) = (1 − α )( φ + − φ − ) ( x ∧ y ) (1 − x ∨ y ) , (5)where ∧ and ∨ denote the minimum and maximum of two numbers, respectively.We consider the limit of weak particle-particle interactions , in which case fugacityequals the particle density [6]. By equation (5), the correlations in the density profileare non-exponential, and mediated by the disordered media. They also vanish inequilibrium, i.e. when φ − = φ + . Thus static hidden variables in a non-equilibriumsystem can result in an effective interaction, and to long-range correlations.To obtain an image of the underlying media, we dilute the boundary reservoirswith marker particles. Only those can be observed in an experiment. For simplicity,the fraction of markers is taken to be the same in both reservoirs, but still φ − (cid:54) = φ + .For a very dilute marker, the numbers of observed marker particles in disjoint setsof macroscopic size within the media are asymptotically independent and Poissondistributed with the intensity of the distribution proportional the integral of the particledensity ‡ . P (cid:0) ∩ ki =1 { N ( A i ) = m i }| φ (cid:1) = k (cid:89) i =1 Λ( A i ) m i m i ! e − Λ( A i ) , Λ( A i ) = c (cid:90) A i φ ( x ) d x , (6)where A i ⊂ [0 ,
1] are disjoint macroscopic sets. In other words, a steady state snapshotof marker particles is a Cox (or doubly stochastic Poisson point) process directed by arandom measure Λ with density c φ [8, 9]. In the following, we take c = 1.Given the relation (2) between the the partial resistances and the particle density,we see that r ( a, b ) = r ( a, b ; φ ) = φ ( b ) − φ ( a ) φ + − φ − (7)is the resistance of a macroscopic interval ( a, b ] relative to the total resistance of themedia (which can be determined from a flow experiment). In order to infer this quantityfor any a and b from a snapshot of the marker particles, i.e. the Cox process data athand, we need a reliable estimator for the density profile φ . Bayesian analysis showsthe minimum mean square error estimator (MMSE estimator) of φ given markers atlocations ( x , . . . , x n ) is [9, 10]ˆ φ ( x ) = E φ ( x ) e − (cid:82) [0 , φ ( y ) d y (cid:81) ni =1 φ ( x i ) E e − (cid:82) [0 , φ ( y ) d y (cid:81) ni =1 φ ( x i ) . (8)An obvious choice for the estimator of the relative resistance would then beˆ r ( a, b ) = r ( a, b ; ˆ φ ) . (9)However, expectations in expression (8) are not analytically tractable to the author’sknowledge (even the single-point moments E φ ( x ) n have quite complicated formulae [6]),and therefore good approximative estimators are needed. ‡ Mathematically, this is obtained by having a fraction of markers in each reservoir scale in inverseproportion to the number of unit cells. tate estimation of long-range correlated non-equilibrium systems: media estimation The MMSE estimator (8) is in general a nonlinear function of the observations. Inparticular, it lacks additivity under inclusion of new information. Nonlinear estimationcan be computationally costly because the estimator has to be recalculated completelyeach time new data becomes available. We next consider linear estimators of the formˆ φ L ( x ) = (cid:90) [0 , k ( x, y ) d N ( y ) , (10)where N is the counting measure for marker particles in a snapshot ( i.e. a Poissonrandom measure with random directing intensity measure Λ( A ) = (cid:82) A φ ( x ) d x ). Inparticular, (cid:82) [0 , k ( x, y ) d N ( y ) = (cid:80) ni =1 k ( x, x i ) for n observed particles at locations x i .The deterministic kernel k translates each observed particle to the language of globaldensity fluctuations separately.The original construction of a MMSE linear estimator for a general Cox process witha directing density is due to Grandell [11]. An uncomplicated derivation uses the factthat the space in which the estimator errors are measured, the space of square integrablefunctions L ( P ), is a Hilbert space. The trick is that the MMSE linear estimator is aprojection to the linear subspace spanned by functionals of the form (cid:82) [0 , f ( y ) d N ( y )[12]. Thus the error function φ − ˆ φ L has to be orthogonal to every such functional: E (cid:110) [ φ ( x ) − ˆ φ L ( x )] (cid:90) [0 , f ( y ) d N ( y ) (cid:111) = 0 . (11)A straightforward calculation § shows that the orthogonality relation is solved byˆ φ L ( x ) = ρ ( x ) + (cid:90) [0 , K ( x, y )[ d N ( y ) − ρ ( y ) d y ] (12)if the kernel K satisfies the integral equation K ( x, y ) ρ ( y ) + (cid:90) [0 , K ( x, z ) C ( z, y ) d z = C ( x, y ) . (13)A remarkable feature of the result (12),(13) is that only the mean ρ ( x ) and the covariance C ( x, y ) of the directing density are used in the construction. The information on highermoments and correlations is neglected. In this sense, linear estimation is a mean fieldapproximation reminiscent of Gaussian approximations.For media estimation with the mean and covariance of the particle density givenby equations (4) and (5), equation (13) for the kernel becomes a coupled pair of integralequations because the covariance function C ( x, y ) is piecewise defined. However, dueto piecewise linearity of C ( x, y ), differentiation of these equations twice with respect to y leads to two decoupled second order differential equations. Consequently, the linearmedia estimation has an explicit solution: K ( x, y ) = h ( x, y ) C ( x, x ) h ( x, x ) ρ ( x ) + (cid:82) h ( x, z ) C ( z, x ) d z , h ( x, y ) = g − ( x ∧ y ) g + ( x ∨ y ) , (14) § Observe that E d N ( x ) d N ( y ) = ( δ ( x − y ) E φ ( x ) + E φ ( x ) φ ( y )) d x d y . tate estimation of long-range correlated non-equilibrium systems: media estimation x x (a) (b) Figure 1.
Linear (dashed) and maximum likelihood (dash-dotted) estimators forsimulated marker density profiles (solid line) at (a) α = 0 . φ + = 30, and (b) α = 0 . φ + = 100. In both cases, φ − = 0. The black circles on shaded backgroundshow the observed marker particles. Insets: the kernel functions K ( x, .
25) (solid line)and K ( x, .
75) (dashed). where the auxiliary functions g + and g − are given in terms of modified Bessel functions I and K (see [13] for definition and properties) as g + − ( x ) = 2 (cid:112) ρ ( x ) (cid:110) I (cid:16) (cid:112) (1 − α ) ρ ( x ) (cid:17) − I (2 (cid:113) (1 − α ) φ + − ) K (2 (cid:113) (1 − α ) φ + − ) K (cid:16) (cid:112) (1 − α ) ρ ( x ) (cid:17) (cid:111) . (15)Figure 1 shows the linear estimator for density profiles obtained from simulationsof α -stable processes for two disorder strengths α . In both cases, the insets show thekernel K , i.e. the effect of a single particle. As the true density profiles get closer tothe expected profile as α →
1, also the effect a single observed particle on the estimatorgets smaller, and the function K ( x, y ) thus gets less peaked around y . This is due to unbiasedness of the linear estimator, E ˆ φ L ( x ) = ρ ( x ). Remarkable is also the asymmetryof the peaks, which is a consequence of the boundary conditions imposed on the densityprofiles.Figure 1 (b) shows that the linear estimator yields non-monotone density profiles,which lead to negative resistance estimates ˆ r L ( a, b ) = r ( a, b ; ˆ φ L ), defined through (7).The problem is not serious at large marker densities: for 0 ≤ φ − := σ − γ < σ + γ =: φ + , √ γ E (cid:2) (ˆ r L ( a, b ) − r ( a, b )) (cid:3) → √ − α (cid:16)(cid:112) σ + b + σ − (1 − b ) + (cid:112) σ + a + σ − (1 − a ) (cid:17) σ + − σ − ) (16)as γ → ∞ , in that the estimation error vanishes as the inverse square root of the markerdensity. The proof is based on the previous Hilbert space techniques, in combinationwith asymptotic formulae for the modified Bessel functions.The constrained optimization problem of finding a linear monotone estimatorseems much more difficult to solve than the unconstrained one. However, the negativeresistances can also be avoided by truncating the negative values of the linear resistance tate estimation of long-range correlated non-equilibrium systems: media estimation r L . This also reduces the estimation error. Alternatively, one can inspect theincrements of the best monotone approximation to ˆ φ L in the uniform topology [14],ˆ φ mon , L ( x ) = 12 (cid:18) sup ≤ y ≤ x ξ ( y ) + inf x ≤ z ≤ ξ ( z ) (cid:19) , ξ ( x ) = φ − ∨ ˆ φ L ( x ) ∧ φ + . (17)Next we introduce a monotone estimator that complements the linear one. In order to find an inherently monotone estimator, we neglect the information that wehave on the statistics of the density profile increments and look for a non-decreasing function that maximizes the likelihood function L ( φ, ( x i )) = e − (cid:82) [0 , φ ( x ) d x n (cid:89) i =1 φ ( x i ) , (18)under the assumption that φ − ≤ φ ( x ) ≤ . . . ≤ φ ( x n ) ≤ φ + . The problem has beendiscussed for non-negative functions by numerous authors (see e.g. [15, 16]). However,a heuristic construction is hard to find in the literature, and since we have to includethe extra condition φ − ≤ φ ( x ), we provide a derivation.Suppose one fixes the values ψ i = φ ( x i ) in expression (18). Then the likelihoodmaximizing function is the one that minimizes the integral in the same expression,which is nothing but the lower step function (the subscript ML stands for maximumlikelihood) ˆ φ ML ( x ) = φ − for 0 = x ≤ x ≤ x ,φ − ∨ ψ i ∧ φ + for x i ≤ x ≤ x i +1 , i = 1 . . . , n ,φ + for x = x n +1 = 1 . (19)The true problem is to find the optimal values for ψ i . By inspecting the log-likelihoodlog L ( φ, ( x i )) = − log φ − + n (cid:88) i =0 [log ψ i − ψ i ∆ x i ] , (20)where ∆ x i = x i +1 − x i , we see that if ∆ x i − ≥ ∆ x i for every i = 1 , . . . , n , the optimalchoices are ψ i = 1 / ∆ x i .Suppose that the required monotonicity of the gap sequence (∆ x i ) is broken atparticle i , in that ∆ x i − < ∆ x i . Clearly one needs to level the values ψ i − and ψ i insuch a way that the sequence is monotone again; ψ i − ,i = ψ i = ψ i − . This common valueappears as log ψ i − ,i − ψ i − ,i (∆ x i − + ∆ x i ) in the log-likelihood, so the new optimizerreads ψ i − ,i = 2 / (∆ x i − + ∆ x i ). If after that ψ j − ≤ ψ i − ,i ≤ ψ j +1 is violated, onemakes the same adjustment for a necessary number of values to the left and to the rightfrom the particle at x i . The log-likelihood involving adjustments of ψ j and ψ k with j ≤ i ≤ k has a term log ψ k +1 − ji − ψ i (∆ x j + . . . + ∆ x k ), which leads to an optimaldensity ψ i = ( k + 1 − j ) / ( x k +1 − x j ) on that plateau. The condition that the adjustmentprocess ends at index j at left and at k at right is1∆ x j − ≤ k + 1 − jx k +1 − x j ≤ x k +1 ⇔ k + 1 − ( j − x k +1 − x j − ≤ k + 1 − jx k +1 − x j ≤ k + 2 − jx k +2 − x j , (21) tate estimation of long-range correlated non-equilibrium systems: media estimation ψ i = max ≤ j ≤ i min i ≤ k ≤ n k + 1 − jx k +1 − x j . (22)The maximum likelihood (ML) estimator, given by (19) and (22), is plotted in figure1. The estimated density profiles consist of discontinuities at marker particle positionsand of plateaus in between them. This makes the ML estimator flexible as compared tothe linear estimator and therefore particularly suited for estimation at small values of α . For the same reason, the corresponding relative resistance measure does not have acontinuous density as in linear estimation, but a spike train just like the original media.The lack of information on the disorder strength α , however, poses some problemsat low marker densities. Especially for α close to one, the estimated profiles are typicallyfar away from the expected profile ρ ( x ), and the ML estimator is outperformed by thelinear estimator (see figure 1 (a)).Contrary to the linear estimator for the density, the ML estimator is biased . Iteasily underestimates the density at very low marker densities because in absence ofmarker particle observations, ˆ φ ML ( x ) = φ − for x <
3. Discussion
We have shown that long-range range correlations present in non-equilibrium systemscan be used to extract information on an underlying fluctuation field even from verylimited information. We applied the methods from statistical state estimation theory toparticle transport in disordered media and estimated the media structure from a dilutemarker signal. Linear estimation turned out to deliver unbiased but not necessarilypositive estimates for the effective resistances of the media. This could be remedied bytruncation or by monotonizing the density estimator. On the other hand, a nonlinear,maximum likelihood estimator yielded positive resistances but has an inherent bias,which is severe only at very small marker densities. Although quantitative results aboutthe asymptotic error of the estimates could be derived in the linear case only, numericalsimulations indicate fast convergence of the ML estimator.The state-estimation of a non-equilibrium system from marker data was based onthe assumption that the marker particles in a stationary flow are described by a Coxprocess. Furthermore, the directing density of the process was taken to be of the sameform as the total particle density. This assumption can be broken in systems with stericeffects, such as particle-particle exclusion, in which case one has to be able to tell, fromwhich reservoir the particles originated from.In driven systems without quenched disorder, the long-range correlations are oftenweak, in that the amplitude of the correlations decays as a function of the system size. Itis only the macroscopic correlations of a suitably rescaled fluctuation field that persist inthe scaling limit [17, 5]. This is in contrast to non-vanishing correlations in the densityprofiles in random media. Probably a stronger marker signal is required for a successfulestimation in case of weak correlations. tate estimation of long-range correlated non-equilibrium systems: media estimation
Acknowledgments
The author would like to thank Damien Simon for a discussion. This study was fundedfrom a DFG grant BE 2478/2-1 and by the DFG Research Training Group GRK 1276.
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