Haut PDF Large Deviations in Single File Diffusion

Large Deviations in Single File Diffusion

Large Deviations in Single File Diffusion

Macroscopic fluctuation theory also provides a simple explanation of the long memory effects found in single- file, in which initial conditions continue to affect the posi- tion of the tracer, e.g., its variance, even in the long time limit [29, 30]. The statistical properties of the tracer po- sition are not the same if the initial state is fluctuating or fixed—this situation is akin to annealed versus quenched averaging in disordered systems [25]. For general inter- particle interactions, we derive an explicit formula for the variance of the tracer position in terms of transport coef- ficients and obtain new results for the exclusion process. We start by formulating the problem of tracer diffusion in terms of macroscopic fluctuation theory, or equiva- lently fluctuating hydrodynamics. The fluctuating den- sity field ρ(x, t) satisfies the Langevin equation [13]
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Tagged Particle in Single-File Diffusion

Tagged Particle in Single-File Diffusion

is related to φ(x) via a Legendre transform [44, 45]. In the present work we apply the macroscopic fluctuation theory (MFT) to single-file diffusion. The MFT was developed by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim [46–50] for calculating large deviation functions in classical diffusive systems; similar results were obtained in the context of shot noise in conductors [51, 52]. The MFT provides a significant step towards constructing a general theoretical framework for non-equilibrium systems [44, 53]. Over the past decade the MFT has been successfully applied to numerous systems [46–48, 53–70]. A perfect agreement between the MFT and microscopic calculations has been observed whenever results from both approaches were available. The MFT is a powerful and versatile tool, although the analysis is involved and challenging in most cases. The MFT allows one to probe large deviations of macroscopic quantities such as the total current. Intriguingly, large deviations of an individual microscopic particle in a single-file system can be captured by the MFT [29, 69]. The essential property of single-file systems—the fixed order of the particles— allows us to express the displacement of the tagged particle in a way amenable to the MFT treatment. The MFT is an outgrowth of fluctuating hydrodynamic [71]. Using a path integral formulation one associates a classical action to a particular time evolution of the system. The optimal path has the least action and the analysis boils down to solving the Hamilton equations corresponding to the least-action paths. The advantage is that within this formulation all the microscopic details of the
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Single-File Diffusion of Neo-Pentane Confined in the MIL-47(V) Metal-Organic Framework

Single-File Diffusion of Neo-Pentane Confined in the MIL-47(V) Metal-Organic Framework

Concluding Remarks In this work the dynamics of confined neo-pentane in the 1D-channel MIL-47(V) MOF was investigated by using Molecular Dynamics simulations with the consideration of a wide range of loading from 0.25 to 1.5 molecule/u.c. at 300 K. A single-file diffusion regime was simulated whatever the considered loading, thus confirming the experimental data previous reported by QENS measurements. A good agreement between the exper- imental and simulated single-file mobility factor was also obtained. We evidenced that this unusual diffusion behavior is un-correlated to the flexibility of the MIL- 47(V) and it is rather the result of the pore size effect that hinders the crossing of molecules in the channel. Moreover, from the calculated 2D-density plot for neo-pentane as well as the self Van Hove function and the structure factors we revealed the absence of translation jumps and the uncorrelated dynamics of neo-pentane between neighbors channels. Moreover, the diffusion was shown to mostly proceed via the mobility of single molecules rather than clusters. Eventually, we showed that while the translational diffusion decreases when the loading increases, the rotational diffusion coefficient remains constant which corresponds to a clear deviation to the Stokes Einstein relation. This work provides molecular insight into the nature of the gas transport in porous materials. This finding could be of great interest for the kinetic gases separation methods, gas storage and purification processes. It can contribute to a more basic understanding of the mechanisms that rule gas trans- port into porous catalysts, molecular sieves or nanofiltration through sub-nanoporous 3
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Single-file diffusion of particles in a box: Transient behaviors

Single-file diffusion of particles in a box: Transient behaviors

ciently interacting with their neighbors, nor with the confining walls. Therefore, they undergo a ballistic flight at their thermal velocity √k B T /M. In Figs. 1 – 3 we plot the MSD measured in the simulations and the corresponding analytical expressions obtained from Eq. (5) and Eq. (6) . Although the calculations are done without any free parameter, the agreement is excellent, which shows that the dynamics of the confined particles can be explained by our model of a masses and springs chain. We will now bring some light on the qualitative behaviors of the particles by carefully studying the evolution of each normal modes according to the time.
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Large deviations of a tracer in the symmetric exclusion process

Large deviations of a tracer in the symmetric exclusion process

PACS numbers: 05.40.-a,05.60.-k The collective dynamics of a complex system can be probed by attaching a neutral tag to a particle, that does not alter its interactions with the environment, and by monitoring the position of the tagged particle in time. This technique is a powerful tool to study flows in ma- terial sciences, biological systems and even social groups (see e.g., [1–4] and references therein). The averaged tra- jectory of a tracer carries information on the overall mo- tion of the fluid whereas its fluctuations are sensitive to the statistical properties of the medium. The canonical example is the Brownian motion of a grain of pollen im- mersed in water at thermal equilibrium, and the simplest model for this diffusion is given by independent random walkers symmetrically hopping on a lattice; the position of any walker, as a function of time t, spreads as √ t. In presence of weak-interactions, diffusive behavior generi- cally prevails but the amplitude of the spreading, mea- sured by the diffusion constant, is a function of the total density of particles [1, 5, 6].
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Nonequilibrium Markov Processes Conditioned on Large Deviations

Nonequilibrium Markov Processes Conditioned on Large Deviations

We now come to the main point of this paper, which is to define a Markov process via the generalized Doob transform and prove its asymptotic equivalence with the conditioned process X t |A T = a. This equivalence is obtained, as just mentioned, by first proving the asymptotic equivalence of the path measure of the driven process with the canonical path measure, and by then proving the equivalence of the latter measure with the microcanonical path measure using known results about ensemble equivalence. Following these results, we discuss interesting properties of the driven process related to its reversibility and constraints satisfied by its transition rates (in the case of jump processes) or drift (in the case of diffusion). Some of these properties were announced in [ 72 ]; here we provide their full proofs in addition to derive new results concerning the reversibility of the driven process. Our main contribution is to treat the equivalence of the canonical and microcanonical path ensembles explicitly and derive conditions for this equivalence to hold. In previous works, the conditioned process is often assumed to be equivalent with the driven process and, in some cases, wrongly interpreted as the canonical path ensemble.
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Single-file diffusion of particles with long-range interactions: Damping and finite-size effects

Single-file diffusion of particles with long-range interactions: Damping and finite-size effects

(Received 18 March 2011; published 5 July 2011) We study the single file diffusion of a cyclic chain of particles that cannot cross each other, in a thermal bath, with long-ranged interactions and arbitrary damping. We present simulations that exhibit new behaviors specifically associated with systems of small numbers of particles and with small damping. In order to understand those results, we present an original analysis based on the decomposition of the particles’ motion in the normal modes of the chain. Our model explains all dynamic regimes observed in our simulations and provides convincing estimates of the crossover times between those regimes.
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Large and moderate deviations of realized covolatility

Large and moderate deviations of realized covolatility

In the bivariate case, Hayashi and Yoshida [5] considered the problem of estimating the covariation of two diffusion processes under a non-synchronous sampling scheme. They proposed an alternative estimator and they investigated the asymptotic distributions. In [2], the authors complement the results in [5] by establishing a second-order asymptotic expansion for the distribution of the estimator in a fairly general setup, including random sampling schemes and (possibly random) drift terms. Several further works have been realized when data on two securities are observed non-synchronously, see also [1]. Here we do not consider the asynchronous case. In the bivariate case we also mention the work of Mancini and Gobbi [9] which deal with the problem of distinguishing the Brownian covariation from the co-jumps using a discrete set of observations.
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On the Hamiltonian structure of large deviations in stochastic hybrid systems

On the Hamiltonian structure of large deviations in stochastic hybrid systems

most biological applications, one is not interested in the internal discrete state of the system, that is, one only observes the statistical behavior of the continuous variable x(t). For example, x(t) could represent the membrane voltage of a single neuron [33] or the synaptic current in a population of neurons [3]. Faggionato et al. [14, 13] explicitly calculated the rate function (3.4) for a restricted class of stochastic hybrid systems, whose stationary density is exactly solvable. One major constraint on this class of model is that the vector field of the piecewise deterministic system has non- vanishing components within a given confinement domain. However, this constraint does not hold for biological systems such as ion channels [22, 33, 5], gene networks [23, 30], and neural networks [3]. (Faggionato et al. were motivated by a model of molecular motors that is exactly solvable. Such a solvability condition also breaks down for molecular motors when local chemical signaling is taken into account [31].) 3.2. Statement of main theorem
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On the Hamiltonian structure of large deviations in stochastic hybrid systems

On the Hamiltonian structure of large deviations in stochastic hybrid systems

1. Introduction There are a growing number of problems in biology that involve the coupling between a piecewise deterministic dynamical system in R d and a time-homogeneous Markov chain on some discrete space Γ [ 5 ], resulting in an a type of stochastic hybrid system (SHS) known as a piecewise deterministic Markov process (PDMP) [ 11 ] ‡ . One important example at the single-cell level is given by membrane voltage fluctuations in neurons due to the stochastic opening and closing of ion channels [ 17 , 10 , 24 , 20 , 9 , 39 , 44 , 35 , 6 , 1 ]. Here the discrete states of the ion channels evolve according to a continuous-time Markov process with voltage-dependent transition rates and, in-between discrete jumps in the ion channel states, the membrane voltage evolves according to a deterministic equation that depends on the current state of the ion channels. In the limit that the number of ion channels goes to infinity, one can apply the law of large numbers and recover classical Hodgkin-Huxley type equations. However, finite-size effects can result in the noise-induced spontaneous firing of a neuron due to channel fluctuations. Another major example is a gene regulatory network, where the continuous variable is the concentration of a protein product and the discrete variables represents the activation state of the gene [ 25 , 2 , 32 , 36 , 38 ]. A third example concerns a stochastic formulation of synaptically-coupled neural networks that has a mathematical structure analogous to gene networks [ 3 , 5 ].
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Upper large deviations for Branching Processes in Random Environment with heavy tails

Upper large deviations for Branching Processes in Random Environment with heavy tails

the upper rate function when the reproduction laws have at most geometric tails, which excludes heavy tails. Exceptional growth of BPREs can be due to an exceptional environment and/or to exceptional reproduction in some given environment. In this paper, we focus on large deviation probabilities when the offspring distributions may have heavy tails and the exceptional reproduction of a single individual can now contribute to the large deviation event. This leads us to consider new auxiliary power series and higher order derivatives of generating functions for the proof.
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EXCEEDINGLY LARGE DEVIATIONS OF THE TOTALLY ASYMMETRIC EXCLUSION PROCESS

EXCEEDINGLY LARGE DEVIATIONS OF THE TOTALLY ASYMMETRIC EXCLUSION PROCESS

scale behaviors are also observed in random matrix theory (e.g., the asymmetric tails of the Tracy-Widom distributions, c.f., [ TW94 ]), and in stochastic scalar conservation laws [ Mar10 ]. The existence of two-scale large deviations can be easily understood in the context of the exclusion processes. Recall that, for the TASEP, h(t, x) records the total number of particle passing through x. The lower deviation N 1 h(N, 0) < h(1, 0) − α (with α > 0) can be achieved by slowing down the Poisson clock at x = 0. Doing so creates a blockage and decelerates particle flow across x = 0. Such a situation involves slowdown of a single Poisson clock for time t ∈ [0, N ], and occurs with probability exp(−O(N )). On the other hand, for the upper deviation 1
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Large deviations for the dynamic $\Phi^{2n}_d$ model

Large deviations for the dynamic $\Phi^{2n}_d$ model

We are dealing with the validity of a large deviation principle for a class of reaction- diffusion equations with polynomial non-linearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale ǫ and δ(ǫ), respectively, with 0 < ǫ, δ(ǫ) << 1. We prove that, under the assumption that ǫ and δ(ǫ) satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension.
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Stochastic 2D hydrodynamical type systems: Well posedness and large deviations

Stochastic 2D hydrodynamical type systems: Well posedness and large deviations

that due to the bilinear term which arises in hydrodynamical models, the control of any moment of the V ′ norm for time increments of the solution is not as good as that in [31]. On the other hand, giving up the compactness conditions allows us to cover the important class of hydrodynamical models in unbounded domains. The payoff for this is some H¨older condition in time for the diffusion coefficient (see condition (C4) below). The paper by W. Liu [25] does not assume the compactness embeddings in the Gelfand triple and uses another way to obtain the compactness of the set of solutions in some set of time continuous functions. This approach is based on the rather strong approximation hypothesis which involves some compact embeddings (see Hypotheses (A4) and (A5) in [25]). The technique introduced by W. Liu might be used in our framework, in order to avoid the extra time regularity of σ, at the expense of some compact approximation condition of diffusion term. The corresponding proof is quite involved and we do not adapt it to keep down the size of the paper; we rather focus on the main technical problems raised by our models, even in the simple case of a time-independent diffusion coefficient. Note that even in the case of monotone and coercive equations, the diffusion coefficient considered in [31] (resp. [25]) cannot involve the gradient in order to satisfy condition (H5) (resp. (A4)).
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Large deviations for the Boussinesq Equations under Random Influences

Large deviations for the Boussinesq Equations under Random Influences

I and then some Freidlin-Wentzell inequality, which states continuity of the process with respect to the noise except on an exponentially small set. Another classical method in proving LDP for evolution equations is to establish both some exponential tightness and exponentially good approximations for some approximating sequence where the diffusion coefficient is stepwise constant. These methods require some time H¨older regularity that one can obtain when the diffusion coefficient is controlled in term of the L 2 -norm of the solution, but not in the framework we will use here,
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Transverse single-file diffusion near the zigzag transition

Transverse single-file diffusion near the zigzag transition

In this paper we have studied the transverse fluctuations in quasi-1D systems of particles in a thermal bath, near the zigzag transition. We have demonstrated that close to the zigzag threshold, the transverse fluctuations exhibit the typical SFD behavior, with a MSD that scales as the square root of time. This subdiffusive behavior traces back to strong interparticle correlations, and is observed on both sides of the transition. In contrast with the longitudinal fluctuations, for which the correlations are due to the noncrossing condition, the transverse motions of the particles are only correlated close to the zigzag transition. Their dynamics is closely linked to the overdamped modes in the vicinity of the soft mode that appears at the transition, and replaces the zero frequency mode due to translational invariance for longitudinal motion [ 5 , 15 ].
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Dynamical properties of single-file diffusion

Dynamical properties of single-file diffusion

The advantage of our approach is its generality, viz. it applies to all single-file systems. The microscopic details of the system, e.g. the inter-particle interaction and the size of the particles, appear at the macroscopic scale only in terms of two transport coefficients D(ρ) and σ(ρ). It is still extremely difficult to solve the Euler- Lagrange equations for an arbitrary single-file system, essentially the only fully solvable case corresponds to a single-file system composed of Brownian particles. Less complete results, however, can be established for an arbitrary single-file system. Specifically, we shall describe a perturbation scheme which allows, in principle, to compute the multi- time correlations order by order. Using this method we derive general formulas (5)–(6) for the two-time correlation function.
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A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs

A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs

2010 Mathematics Subject Classification: 60K35, 82C20 1. Introduction 1.1. Basic notations, the models and a first look at the main question. Large systems of interacting diffusions with mean field type interactions have been an important research topic in the mathematical community at least since the 60’s. The program of identifying the emerging behavior for n → ∞, where n is the number of interacting units, has been fully developed under suitable regularity and boundedness assumptions on the coefficients defining the system. In particular, Law of Large Numbers, Central Limit Theorems and Large Deviation Principles have been established (see for example [26, 21, 27, 7, 5]). A number of important issues remain unsolved, like the generalization to singular interactions (e.g. [16]) or understanding the delicate issue of considering at the same time large n and large time (e.g. [18]). But another direction in which mathematical results are still very limited is about relaxing the complete graph assumption for the interaction network – complete graph is just a different wording for mean field – and going towards more heterogeneous interaction networks. This is an issue that emerges in plenty of applied disciplines and giving a proper account of the available literature would be a daunting task: so we limit ourselves to signaling the recent survey [25] which contains an extended literature.
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Equilibrium large deviations for mean-field systems with translation invariance

Equilibrium large deviations for mean-field systems with translation invariance

systems with translation invariance Julien Reygner A BSTRACT . We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean-Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of ap- propriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.
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Large Deviations for a matching problem related to the ∞ -Wasserstein distance

Large Deviations for a matching problem related to the ∞ -Wasserstein distance

2 i , α 2 i+1 )} ≤ max{δ µ (a, c), δ µ (c, b)} + ε which leads to the announced inequality by taking ε → 0. 3. Let a, b ∈ S(µ) be such that a ↔ b. By definition of ↔ for every n ≥ 2 and every ¯ α ∈ Rn(a, b) we have d(a, b) ≤ maxi∈{1,...,n−1} d(αi, αi+1) hence d(a, b) ≤ δ µ (a, b). It follows that if a ↔ b we have δ µ (a, b) = d(a, b) since δ µ (a, b) ≤ d(a, b) is always true. Conversely assume that a, b ∈ S(µ) are such that we have δ µ (a, b) = d(a, b) thus d(a, b) ≤ δ µ (a, b). It follows from the definition of δ µ that for every integer n ≥ 2 and every ¯ α ∈ Rn(a, b) we have d(a, b) ≤ maxi∈{1,...,n−1} d(αi, αi+1) which by definition means a ↔ b.  Proof of Lemma 1.10 First we assume that S(µ) is connected. Then it is well- chained i.e. for every a, b ∈ S(µ) such that a 6= b and every ε > 0 there exist an integer n ≥ 2 and ¯ α ∈ Rn(a, b) such that for every 1 ≤ i ≤ n − 1 we have d(αi, αi+1 ) ≤ ε, see e.g. 8.2 of Chapter I in Whyburn ( 1942 ). Thus δ µ (a, b) = 0 for every a, b ∈ S(µ) hence, according to the third point of Lemma 4.1 , a ↔ b if and only if d(a, b) = 0 whence Zµ = {0}. Now we prove that if Zµ = {0} then S(µ) is connected. It is sufficient to prove that S(µ) is well-chained since, according to e.g. 9.21 of Chapter I in Whyburn ( 1942 ) any metric, compact, well-chained space is connected. Actually we shall prove that if Zµ = {0} then necessarily for every a, b ∈ S(µ) we have δ µ (a, b) = 0 since this is obviously equivalent to the well-chained condition. To this end we proceed by contradiction and assume that we have Zµ = {0} and that there exists some a, b ∈ S(µ) such that δ µ (a, b) > 0. We shall construct (u, v) ∈ S(µ) 2 such that δ µ (u, v) = d(u, v) = δ µ (a, b) > 0 in contradiction with Zµ = {0}.
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