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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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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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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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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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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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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(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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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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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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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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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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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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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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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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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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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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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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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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