PACS numbers: 05-40.-a; 05-60.-k
Keywords: ASEP, functional Bethe Ansatz, large deviations
The one dimensional asymmetric simple exclusionprocess (ASEP) is one ofthe most simple examples of a classical interacting particles system exhibiting a non equilibrium steady state. This stochastic system has been studied much inthe past, both inthe mathematical [1, 2, 3] and physical literature [4, 5, 6, 7, 8, 9, 10, 11]. It consists of particles hopping locally on a one dimensional lattice, with an asymmetry between the forward and backward hopping rates. Theexclusion constraint prevents the particles from moving to a site already occupied by another particle. The asymmetry between the hopping rates models the action of an external driving field inthe bulk ofthe system, which maintains a permanent macroscopic currentinthe system. This current breaks the detailed balance and keeps the system out of equilibrium even inthe stationary state. The special case for which the particles hop forward and backward with equal rates is called the symmetric simple exclusionprocess (SSEP). It corresponds to a situation for which the detailed balance holds inthe bulk which means (inthe absence of boundary conditions breaking the forward-backward symmetry) that the system reaches equilibrium inthe long time limit. In this case, the system belongs to the universality class ofthe Edwards–Wilkinson (EW) equation [6, 12]. On the contrary, if the two hopping rates are different, detailed balance is broken and the system reaches inthe long time limit a non equilibrium steady state characterized by the presence of a currentof particles flowing through the system. In that case, the system belongs to the universality class ofthe Kardar–Parisi–Zhang (KPZ) equation [6, 13].
independent in a system of size L, we shall therefore calculate F k only for
k ≤ L − 1. Thus, we need to consider reduced words of length j ≤ L − 1. Let W be such a word, and I(W ) be the set of indices ofthe operators M i
that compose W ; our aim is to find the expression of W and to calculate its prefactor from equation (15). Because the rules (2, 4) do not suppress or add any new index, the following property is true : if a word W ′ appearing
m(t, x) = x ∧ (1 − x) ∧ t , t > 0 , x ∈ [0, 1] , (1.7) see Figure 3. The macroscopic stationary state is reached at the finite time t f = 1/2.
One would naturally expect KPZ fluctuations to occur around this growing front. Let us recall a famous result of Bertini and Giacomin in that direction. They consider the WASEP on the infinite lattice Z, with upwards asymmetry ǫ. Starting from a flat initial profile, the hydrodynamic limit grows uniformly in space and at constant speed. Bertini and Giacomin look at fluctuations around this hydrodynamic limit according to the fol- lowing scaling: ǫ in height, ǫ −2 in space and ǫ −4 in time. They show that the rescaled interface converges to the Hopf-Cole solution ofthe KPZ equation on R. We stress that this height-space-time scaling is rigid in order to observe KPZ fluctuations inthe exclu- sion process.
current is not affected as further particles are accomodated by growing the HD zones at the expense ofthe LD zones [14, 15]. These MF arguments capture the essential transport features well, as is shown by the simulation data on Fig.2 (and this remains true also for triangular and square lattices). Deviations arise, however, on thecurrent plateau (where MF underestimates thecurrent) and close to the transitions (where explicit simulations furthermore reveal the particle- hole asymmetry, which increases with connectivity c). The numerical MF algorithm does not provide a solution on the plateau, since the assumption of homogeneous segments does not hold, but it otherwise reproduces the theoretical results with great precision. In summary, despite its random nature, TASEP transport through a Bethe network may be understood in terms of a single effective vertex, similar to the Ising model on a Bethe lattice .
In this paper we consider the symmetric exclusionprocess with long jumps in contact with two reservoirs with different densities at the boundaries. We show that inthe non-equilibrium stationary state the average density current scales with the length N ofthe system as N −δ , 0 < δ < 1. We also show that the stationary density profile is described by the stationary solution of a fractional diffusion equation with Dirichlet boundary conditions. Observe that in a diffusive regime, δ = 1 and that the stationary profile is the stationary solution of a usual diffusion equation with Dirichlet boundary conditions. Similar conclusions to ours, as well as extensions to theasymmetric case, have been obtained in a non-rigorous physics paper by J. Szavits-Nossan and K. Uzelac ([ 32 ]). As a final remark of this introduction let us observe that in our paper, as well as in [ 32 ], the reservoirs are described by infinite reservoirs. This has the advantage to avoid a truncation ofthe long range transition probability p(·). However other reservoirs descriptions are possible but we conjecture that they could have a quantitative effect on the form ofthe stationary profile. Indeed, since the fractional Laplacian is a non-local operator, the fractional Laplacian with Dirichlet boundary conditions can be interpreted in several ways giving rise to different stationary solutions. The (microscopic) description used for the reservoirs fix the (macroscopic) interpretation ofthe fractional Laplacian with Dirichlet boundary conditions. In our case it is the so called “restricted fractional Laplacian” which appears. This sensitivity to the form ofthe reservoirs is due to the presence of long jumps and does not appear for theexclusionprocess with short jumps. This sensitivity has also been observed in models of (non interacting) Levy walks and inthe context of 1d superdiffusive chains of oscillators ([ 22 ]).
The study of non-equilibrium processes will begin in Section II. We shall use as a leitmotiv for non-equilibrium, the picture of rod (or pipe) in contact with two reservoirs at different temperatures, or at different electrical (chemical) potentials (see Figure 8). This simple picture will allow us to formulate some ofthe basic questions that have to be answered in order to understand non-equilibrium physics. Thecurrent theory of non-equilibrium processes requires the use of some mathematical tools, such as large-deviation functions, that are introduced, through various examples (Independent Bernoulli variables, random walk...), in Section II A; in particular, we explain how the thermodynamic Free Energy is connected to the large deviations ofthe density profile of a gas enclosed in a vessel. In Section II B, we show the relations between the large-deviation function and cumulantsof a random variable. Section II C is devoted to the very important concept of generalized detailed balance, a fundamental remnant ofthe time-reversal invariance of physics, that prevails even in situations far from equilibrium. Then, in Section II D, the Fluctuation Theorem is derived for Markov system that obey generalized detailed balance.
The aim ofthe present work is to derive analytical results for thecurrent statistics in ASEP with forward and backward jumps (sometimes called the partially asymmetricexclusionprocess) from the Bethe Ansatz. We overcome the technical difficulty that hindered the solution ofthe Bethe equations inthe general case by reducing them to an effective one variable problem thanks to a suitable reformulation, akin to the so-called functional Bethe Ansatz. This one variable equation can be interpreted as a purely algebraic question involving a divisibility condition between two polynomials. In this work, we use this formalism to derive the expressions ofthe mean value ofthecurrent and its variance. Our technique can be used to calculate thecurrent cumulant to any desired order.
The crossover between these two regimes is characterized by the system size L and the observation time T going both to infinity with T ∼ L 3/2 . This corresponds to the
scale of relaxation to stationarity, with the dynamical exponent 3/2 of one-dimensional KPZ universality. We compute the large scale limit ofthe fluctuations ofthecurrentin a specific model, the totally asymmetric simple exclusionprocess (TASEP) [19, 20], conditioned on special initial and final states. Our main result (11) is very similar to an equilibrium expectation value, with a sum over discrete realizations of a scalar field ϕ in a linear potential.
scale behaviors are also observed in random matrix theory (e.g., theasymmetric tails ofthe Tracy-Widom distributions, c.f., [ TW94 ]), and in stochastic scalar conservation laws [ Mar10 ].
The existence of two-scale large deviations can be easily understood inthe context oftheexclusion 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
During the last two decades, substantial progress has been made towards a statistical theory of non-equilibrium systems [1–6]. Large deviation functions, that encode atypical fluctuations of a physical observable, are likely to be the best candidates to generalize the traditional thermodynamic potentials. Moreover, it has been proved that large deviations functions display symmetry prop- erties, called ‘Fluctuation Theorems’, that remain valid far from equilibrium . These remarkable relations im- ply linear response theory inthe vicinity of equilibrium. Hence, the determination of large deviations in a non- equilibrium system, whether theoretically, numerically, or experimentally, is a question of fundamental impor- tance [7–14].
When α = 1/2, a sharp estimate on τ can be obtained directly from spectral considerations (Section 5.1), but when α ∈ [0, 1/2) we need a refinement ofthe strategy used in [LL18]: The first step (Proposition 10) is to obtain a control on the position ofthe leftmost particle which matches the lower bound provided by the hydrodynamic limit. This requires a new proof since the argument used in [LL18] is not sharp enough to cover all biases. The second step is to use contractive functions once the system is at macroscopic equilibrium, this is sufficient to treat most cases. A third and new step is required to treat the case when the bias b N of order log N/N or smaller: as we are working under the assumption (10) we only need to treat this case when k = N o(1) . In this third step we use diffusive estimates to control the hitting time of zero for the function f N,k (0) (h ∧ t ) − f N,k (0) (h ∨ t ) where f (0) was introduced in Subsection 3.4.
In particular, R (w n ) 2 dν ρ ≤ C(q, ρ)n 2α−2 and for 0 < α < 1, R (w n ) 2 dν ρ → 0
as n → ∞, and the lemma is trivial (there is no need of time integration). When 1 ≤ α < 2 an extra argument is needed. The idea is the following. For sites x near the origin, jumps of opposite sign cancel each other, as inthe case of a diffusive system. Since the transition rate p(·) has infinite second moment, these cancellations do not hold for large x. Introduce a truncation around 0 of size n γ .
in contact with reservoirs [ 21 , 22 , 46 ]. It is well established that superdiffusive systems are
much more sensitive to the reservoirs and boundaries than diffusive systems but quantitative informations, like the form ofthe singularities ofthe profiles at the boundaries, are still missing. In Chapter 3 , motivated by these studies, we consider the boundary driven exclusionprocess with long jumps whose distribution is inthe form of ( 1.3.1 ) with 1 < γ < 2, which may be considered as a substitute to Lévy flights in bounded domains with reservoirs when Lévy flights are moreover interacting. As we will see, the main operator emerging from the microscopic dynamic is a non-local operator, namely, the regional fractional Laplacian. For that reason we recall the definition and basic properties ofthe regional fractional Laplacian. Details can be found in [ 37 , 9 ].
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Why don’t ALL committees reviewing protocols for funding reject those that cannot be applied to real people inthe real world? Of course, “internal validity” is necessary, and while these may reduce type 1 and type 2 errors (false positives and false negatives), research is also open to what might be called type 3 errors. In these situations, we get the “right” answer but it’s to the wrong question – unlike work that leads to type 4 problems: the wrong answer to the wrong question.
The validity of our results should be tested in other contexts. The challenges that indigenous people face in Peru are shared with other indigenous populations in Latin America, and evidence for the internalization of ethno-racial discrimination in aspiration formation may be found in other countries. But the prevalence of a contemporaneous hierarchy mostly based on socio-economic status could be specific to Peru, where the concept of ethnic identity is particularly fluid. Peru has a low level of politicization of ethnic cleavages, and by comparison with Mexico, Guatemala, Bolivia and Ecuador, has few important social movements based on ethnic identity (Sulmont, 2011). Peru’s low levels of mobilization on the basis of ethnic identity may be associated with a lack of resonance ofthe ethnic group notion among indigenous people themselves.
Generalization The expression ofthe cumulant-generating function of multiinformation inthe case of multivariate normal distributions is quite simple. This simplicity is mostly a consequence ofthe stability ofthe mul- tivariate normal family to most operations performed here: (i) the product ofthe marginals is also a multivariate normal distribution (this is strongly related to the fact that independence and uncorrelatedness are equivalent for multivariate normal distributions); (ii) the ratio ofthe joint distribu- tion to its marginals takes a simple form that is again closely related to the multivariate normal family; and (iii) the exponentiation of t times the log of this ratio still has the form of a multivariate normal distribution. It would be of interest to determine such a simplicity would still hold in more general settings such as more general distribution families and more general functions ofthe ratio f X (X)/ Q f X i (X i ).
is purely based on Meissner expulsion.
The particular choice of orientation ofthe Pb stripes on top ofthe Al film has been made merely as a way to illustrate the proof of principle of our device. The observed, current- direction-dependent, vortex trapping could also be used to rectify vortex motion. 12,15,16 In addition, more controllable systems can be obtained by combining two independent current sources along the x and y directions. 17–19 In this case a vortex transistor can be developed where large voltage drops due to vortex motion along the y direction can be controlled with small forces or gating inthe x direction. We expect the observed, current-direction-dependent, vortex-trapping mechanism can be magnified by making more repulsive cages with higher T c superconductors or magnetic templates with
The second mechanism derives from the fact that being indigenous is associated with other forms of disadvantage, such as being poor or living in a rural environment. These ‘external constraints’ largely result from the colonial period (1514–1821) when the Spaniards introduced discriminatory practices and developed extractive institutions in Peru. These institutions concentrated power, land ownership and access to education inthe hands of a small elite. By contrast, indigenous people have been confined to the poorest parts of society, with limited access to education and other opportunities to develop their human capital, which has impeded their entrance to the modern sector and their political participation. These external constraints may be the main determinant of aspiration failure, as they limit access to information and to opportunities to invest inthe future. For example, indigenous children are more likely to live in remote areas, where information about occupational opportunities and access to quality education are limited. They may receive less support for their education from their parents, who are themselves poor. As a result, they may stop aspiring to high levels of education, and to prestigious occupations that can only be reached with family support. In addition, they are often growing up in poorer neighbourhoods. The peers visible in their ‘aspiration window’ are more likely to have occupations providing low socio-economic status. On the external channel hypothesis, indigenous children may not aspire to become doctors because they know their chance of continuing on to further study is limited, partly because their parents would not have the funds to pay for their studies. With the internal channel hypothesis, they will not aspire to become doctors because they think that a doctor has to be ‘white’ or that they are not smart enough to succeed at medical school.
The research relies on a very rich data set, the Young Lives data, where 678 children and their caregiver were interviewed three times, when children were 8, 12 and 15 years old.
The study feeding this policy brief focussed on the older cohort, surveyed at 8, 12 and 15 years old. This cohort included 714 children in 2002, 685 children in 2006, and 678 children in 2009. Children are defined as indigenous if the first language of one of their parents (mother, father or caregiver) learned as a child is Quechua, Aymará or a language ofthe Amazon. Aspirations are measured from the answer ofthe children to the question about what they want to be when they grow up. Aspirations are classified according to the socio-economic status related to the occupation desired by the child. To test whether the ‘internal channel’ hypothesis is verified, we estimate with OLS and probit model the level of aspiration by introducing ethnic group and proxies of external constraints as explanatory variables. Indeed, if ethnic belonging determines specific behaviour and decision-making as result ofthe internalization of discriminatory values (‘internal channel’ hypothesis), we expect that being indigenous negatively and significantly affects aspiration, once external constraints are taking into account. To identify the causal effect of aspiration on educational outcomes, we have adopted an identification strategy based on an instrumental variable.