Large deviations of a tracer in the symmetric exclusion process

TL;DR

This paper derives exact formulas for the large deviations and fluctuations of a tracer particle in a one-dimensional symmetric exclusion process, revealing detailed statistical properties in both equilibrium and non-equilibrium states.

Contribution

It provides the first exact characterization of the large deviation function and cumulant generating function for a tracer in the symmetric exclusion process.

Findings

Exact formulas for cumulant generating function and large deviation function.

Full statistical description of tracer position in equilibrium and non-equilibrium.

Special case results for equilibrium fluctuations with uniform density.

Abstract

The one-dimensional symmetric exclusion process, the simplest interacting particle process, is a lattice-gas made of particles that hop symmetrically on a discrete line respecting hard-core exclusion. The system is prepared on the infinite lattice with a step initial profile with average densities $\rho_{+}$ and $\rho_{-}$ on the right and on the left of the origin. When $\rho_{+} = \rho_{-}$, the gas is at equilibrium and undergoes stationary fluctuations. When these densities are unequal, the gas is out of equilibrium and will remain so forever. A tracer, or a tagged particle, is initially located at the boundary between the two domains; its position $X_t$ is a random observable in time, that carries information on the non-equilibrium dynamics of the whole system. We derive an exact formula for the cumulant generating function and the large deviation function of $X_t$, in the long time limit, and deduce the full statistical properties of the tracer's position. The equilibrium fluctuations of the tracer's position, when the density is uniform, are obtained as an important special case.

Equations

$$⟨X_t^2⟩=\frac{2(1-\rho )}{\rho }\sqrt{\frac{t}{\pi }}.$$

$$\text{Prob}\left(\frac{X_t}{\sqrt{4t}}=-\xi \right)\sim \exp [-\sqrt{t}\varphi (\xi )].$$

$$⟨e^{s{X}_t}⟩\sim e^{-\sqrt{t}C(s)}\mathrm{when}t\to \infty .$$

$$C(s)=\underset{\xi }{\min }\left(2s\xi +\varphi (\xi )\right).$$

$$\{\begin{array}{ll}\varphi (\xi )& =\mu (\xi ,{\lambda }^{\ast }),\\ \frac{\partial \mu (\xi ,{\lambda }^{\ast })}{\partial \lambda }& =0,\end{array}$$

$$\mu (\xi ,\lambda )=\sum _{n=1}^{\infty }\frac{(-\omega {)}^n}{n^{3/2}}A(\sqrt{n}\xi )+\xi \log \frac{1+{\rho }_{+}(e^{\lambda }-1)}{1+{\rho }_{-}(e^{-\lambda }-1)}.$$

$$A(\xi )$$

$$N(x,t)=N_t+\{\begin{array}{ll}+{\sum }_{y=1}^x{\eta }_y(t),& x>0,\\ 0,& x=0,\\ -{\sum }_{y=x+1}^0{\eta }_y(t),& x<0,\end{array}$$

$$\mathbb{P}[X_t\le x]=\mathbb{P}[N(x,t)>0].$$

$$⟨e^{\lambda N(x,t)}⟩\sim e^{-\sqrt{t}\mu (\xi ,\lambda )}$$

$$\varphi (\xi )-\varphi (-\xi )=2\xi \log \frac{1-{\rho }_{+}}{1-{\rho }_{-}}.$$

$$\frac{⟨X_t^4{⟩}_c}{\sqrt{4t}}=\frac{1-\rho }{\sqrt{\pi }{\rho }^3}[1-(4-(8-3\sqrt{2})\rho )(1-\rho )+\frac{12}{\pi }(1-\rho {)}^2]$$

$$\frac{⟨X_t⟩}{\sqrt{4t}}=-{\xi }_0,$$

$$2{\xi }_0{\rho }_{-}=({\rho }_{+}-{\rho }_{-}){\int }_{{\xi }_0}^{\infty }\text{erfc}(u)\mathrm{d}u.$$

$$\mathrm{Var}(X_t)=\frac{4K({\rho }_{+}-{\rho }_{-}{)}^{2}A({\xi }_0)\sqrt{t}}{({\rho }_{+}\text{erfc}({\xi }_0)+{\rho }_{-}\text{erfc}(-{\xi }_0){)}^2}$$

$$K=\frac{{\rho }_{+}^3+{\rho }_{-}^3-3{\rho }_{+}^2{\rho }_{-}-3{\rho }_{+}{\rho }_{-}^2+4{\rho }_{+}{\rho }_{-}}{({\rho }_{+}+{\rho }_{-})({\rho }_{+}-{\rho }_{-}{)}^2}-\frac{A(\sqrt{2}{\xi }_0)}{\sqrt{2}A({\xi }_0)}.$$

$$\varphi (\xi )={\left\{\sqrt{{\rho }_{+}\Xi (\xi )}-\sqrt{{\rho }_{-}\Xi (-\xi )}\right\}}^2$$

$$\varphi (x_1,\dots ,x_n;t)=⟨{\tau }^{N_{\tau }(x_1,t)}\dots {\tau }^{N_{\tau }(x_n,t)}⟩$$

$$⟨{\tau }^{n{N}_{\tau }(x,t)}⟩={\tau }^{-n\frac{x}{2}}{\tau }^{n(n-1)/2}\prod _{i=1}^n\left(1-\frac{r_{-}}{{\tau }^i{r}_{+}}\right)$$

$$\times \int \cdots \int \prod _{i<j}\frac{z_i-z_j}{z_i-\tau z_j}\prod _{i=1}^n\frac{F_{x,t}(z_i)}{(1-\frac{z_i}{\tau {\theta }_{+}})(z_i-{\theta }_{-})}\mathrm{d}{z}_i$$

$$F_{x,t}(z)={\left(\frac{1+z}{1+z/\tau }\right)}^x{e}^{-\frac{q(1-\tau {)}^{2}z}{(1+z)(\tau +z)}t}.$$

$$\sum _{j=0}^n(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})⟨{\tau }^{(n-j)N_{\tau }}⟩=⟨(1-{\tau }^{N_{\tau }}{)}^n⟩={ϵ}^n⟨N^n⟩+o({ϵ}^n)$$

$$⟨(1-{\tau }^{N_{\tau }}{)}^n⟩=\sum _{k=0}^n{\mu }_{n,k}(ϵ)J_k{ϵ}^k$$

$$\frac{⟨N(x,t{)}^n{⟩}_c}{\sqrt{t}}\sim \sum _{l=1}^n\frac{{\alpha }_{n,l}(r_{+},r_{-})}{\sqrt{l}}\Xi (-\sqrt{l}\xi )-2{\alpha }_{n,l}(1,0)\xi {\rho }_{+}^l$$

$$\frac{{\alpha }_{n,l}(a,b)}{(l-1)!}=(-1{)}^l\sum _{\begin{array}{c}\sum j{l}_j=n\\ \sum l_j=l\end{array}}\frac{n!}{{\prod }_{j=1}^n{l}_j!}\prod _{j=1}^n{\left(\frac{a+(-1{)}^{j}b}{j!}\right)}^{l_j}.$$

$$X_t={ϵ}_1+\dots {ϵ}_{t/dt}$$

$$⟨e^{s{X}_t}⟩=⟨e^{s{ϵ}_i}{⟩}^{t/dt}\sim e^{-tC(s)}$$

$$C(s)=q(1-e^{-s})-p(e^s-1).$$

$$Prob\left(\frac{X_t}{t}=-\xi \right)\simeq \exp [-t\varphi (\xi )]$$

$$\varphi (\xi )=p+q-\sqrt{{\xi }^2+4pq}+\xi \ln \frac{\sqrt{{\xi }^2+4pq}+\xi }{2q}.$$

$$Prob\left(X_t>x\right)$$

Full text

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 material sciences, biological systems and even social groups (see e.g., Chaikin and Lubensky (2000); Kärger and Ruthven (1992); Tsien (1998); Schadschneider et al. (2010) and references therein). The averaged trajectory of a tracer carries information on the overall motion 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 immersed 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 $\sqrt{t}$. In presence of weak-interactions, diffusive behavior generically prevails but the amplitude of the spreading, measured by the diffusion constant, is a function of the total density of particles Spohn (1991); Krapivsky et al. (2010); Chaikin and Lubensky (2000).

If the interactions induce long-range correlations either in space or time direction, or if the environment is out of equilibrium (by carrying some internal currents), the motion of a tagged particle can exhibit unusual statistical properties such as anomalous diffusion and/or non-Gaussian fluctuations. For example, a tracer trapped in a linear array of convection rolls spreads only as $t^{1/3}$ with time Young et al. (1989); Bouchaud and Georges (1990). Correlations are usually enhanced in low dimensional systems such as narrow quasi-one-dimensional channels, in which the order amongst the particles is preserved because of steric hindrance. For such a single-file motion, the typical displacement $X_{t}$ of a tracer at large times grows as $t^{1/4}$ , which is much slower than the usual $\sqrt{t}$ law, regardless of the precise form of the interaction. However, collective diffusion of local density fluctuations remains normal and behaves as $\sqrt{t}$. Similarly, the time-integrated current at a given location of a single-file channel also displays $t^{1/4}$-fluctuations. This anomalous single-file diffusion has been demonstrated in various experiments involving different types of physical systems such as zeolites, capillary pores, carbon nanotubes or colloids Hodgkin and Keynes (1955); Lea (1963); Kukla et al. (1996); Chou and Lohse (1999); Wei et al. (2000); Kollmann (2003); Lin et al. (2005). Single-file diffusion is also discussed in numerous theoretical papers, at various levels of physical intuition Levitt (1973); *Percus1974; *Richards1977; *Alexander1978; *VanBeijeren1983; *Majumdar1991; *Rodenbeck1998; *Leibovich2013; *Illien2013; *Lizana2014 or mathematical rigor Harris (1965); *Spitzer1970; *Arratia1983; *Spohn1990; Spohn (1991); Liggett (2004).

One of the simplest models in non-equilibrium statistical physics is the Symmetric Exclusion Process (SEP) Spitzer (1970), a lattice gas of particles performing symmetric random walks in continuous time and interacting by hard-core exclusion: each particle attempts to hop with rate unity from its location to an empty neighboring site; double occupancy of a site is forbidden. Thanks to the wealth of analytical knowledge accumulated during the last few decades, this process and its variants, are used as paradigms in non-equilibrium statistical mechanics Katz et al. (1984); Lebowitz et al. (1988); Derrida (2007, 2011); Krapivsky et al. (2010). In a one dimensional lattice, the SEP is a pristine model of a single-file diffusion, amenable to quantitative analysis. In equilibrium case with uniform density $\rho$, the variance of the position $X_{t}$ of a tagged particle initially located at the origin is given, in the long time limit, by Arratia (1983); Spohn (1991)

[TABLE]

It has also been proved that the rescaled position $\frac{X_{t}}{t^{1/4}}$ satisfies a central limit theorem and converges to a fractional Brownian motion with Hurst index 1/4 Arratia (1983); Peligrad and Sethuraman (2008).

The full distribution of $X_{t}$ and its higher cumulants are, however, not known. The tracer, being immersed in fluctuating environment, far from equilibrium, can display large and non-typical excursions. Such rare events are quantified by a large deviation function Touchette (2009); Dembo and Zeitouni (2009). Large deviation functions appear as appropriate candidates for macroscopic potentials under non-equilibrium conditions. Moreover, the fluctuation theorem, which is one of the few exact results valid far from thermodynamic equilibrium, is stated as a property of large deviation functions Gallavotti and Cohen (1995); Lebowitz and Spohn (1999). It emphasizes the role of microscopic time-reversal symmetry for macroscopic fluctuations. In present day statistical physics, large deviations play an increasingly important role Derrida (2007, 2011); Jona-Lasinio (2014); Bertini et al. (2015).

Recently, the large deviation principle for the tracer position has been proved rigorously Sethuraman and Varadhan (2013): when $t\to\infty$ there exists a large-deviation function $\phi(\xi)$, such that

[TABLE]

Note the prefactor $\sqrt{t}$ in the exponent; for non-interacting particles, the prefactor would be $t$ Sup . Alternatively, one studies the characteristic function of $X_{t}$, which behaves as

[TABLE]

The Taylor expansion of the cumulant generating function $C(s)$ with respect to $s$ generates all the cumulants of $X_{t}$. The functions $C(s)$ and $\phi(\xi)$ are related by Legendre transform Touchette (2009); Dembo and Zeitouni (2009):

[TABLE]

Each of these functions carries information on the long time behavior of the process. Although the SEP has been studied for more than 40 years, analytic formulas for these functions are not yet known.

In this letter, we shall present an exact formula for the large-deviation function $\phi(\xi)$ in (2) of the tracer position in the SEP. As an initial condition, we prepare a step density profile with an average density $\rho_{+}$ on the right of the origin and $\rho_{-}$ on the left. (See the right figure in Fig. 2.) In a parametric representation, $\phi(\xi)$ is given by

[TABLE]

where the second equation defines implicitly $\lambda^{*}=\lambda^{*}(\xi)$, and $\mu(\xi,\lambda)$ is

[TABLE]

Here $\omega=r_{+}(e^{\lambda}-1)+r_{-}(e^{-\lambda}-1)$ with $r_{\pm}=\rho_{\pm}(1-\rho_{\mp})$ and

[TABLE]

where the complementary error function is defined by $\text{erfc}{(z)}=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-u^{2}}{\rm d}u$. This is the central result in this letter. For $\xi=0$, we have $A(0)=1/\sqrt{\pi}$ and $\mu(0,\lambda)$ reduces to the expression found in Derrida and Gerschenfeld (2009a); *Gerschenfeld2009 for the current fluctuations in the SEP at the origin. Our formula (6) generalizes it and leads us to a complete analytic description of the statistical properties of the tracer in the long time limit. The figure is also easily drawn, see Fig 1.

To explain the meaning of $\mu$ and the derivation of our formula, we first recall the set-up of the asymmetric simple exclusion process (ASEP), see Fig. 2. The position of a particle is labeled by an integer $x\in\mathbb{Z}$. Particles hop to the right and to the left with rates $p$ and $q$, respectively. The asymmetry parameter is $\tau=p/q$ with $0\leq\tau\leq 1$. The symmetric model, which is the main target of our study, corresponds to $p=q=\tau=1$. We adopt the convention that a current flowing from right to left is counted positively Tracy and Widom (2008); A. Borodin, I. Corwin, and T. Sasamoto (2014). The initial condition is the step density profile with $\rho_{+}$ and $\rho_{-}$. Typically, we have $\rho_{+}\geq\rho_{-}$. The stationary case corresponds to $\rho_{+}=\rho_{-}=\rho$. We emphasize that the initial profile displays randomness: statistical averages will be taken both over the dynamics and the initial conditions.

The tracer is defined to be the particle in the region $x>0$, which is initially the closest to the origin. (For quantities in the long time, which we are interested in this article, this is equivalent to putting the tracer at the origin at $t=0$. ) Its position at time $t$ is denoted by $X_{t}$.

In order to study the position of the continuously moving tracer, it is useful to relate $X_{t}$ to a local observable. Let $N_{t}$ denote the integrated current through the bond $(0,1)$ for the duration $[0,t]$; i.e., $N_{t}$ is equal to the total number of particles having hopped from 1 to 0 minus the total number of particles having hopped from 0 to 1 during the time interval $[0,t]$. We define the following quantity Schütz (1997); Imamura and Sasamoto (2011)

[TABLE]

where $\eta_{x}(t)=1$(resp. 0) when the site $x$ is occupied (resp. empty) at time $t$. We recall that the observable $h(x,t)=N(x,t)-x/2$ is the local height function appearing when the ASEP is mapped to a growth process Krapivsky et al. (2010); Halpin-Healy and Zhang (1995); Prähofer and Spohn (2002).

Using particle number conservation, one can verify Sup that the tagged particle position $X_{t}$ and $N(x,t)$ satisfy

[TABLE]

This identity will allow us to relate the statistical properties of the tracer $X_{t}$ with those of the height $N(x,t)$ and, in particular, to express the cumulant generating function and the large deviation function of $X_{t}$ in terms of the corresponding quantities for $N(x,t)$.

In the long time limit, the characteristic function of $N(x,t)$ behaves as

[TABLE]

with $\xi=-\frac{x}{\sqrt{4t}}$ and its cumulants are obtained by expanding $\mu(\xi,\lambda)$ with respect to $\lambda$. This $\mu(\xi,\lambda)$ is nothing but the one in (6). From the identity (9), we see Sup that the large deviation functions of $X_{t}$ is given through the characteristic function of $N(x,t)$ as $\phi(\xi)=\max_{\lambda}\mu(\xi,\lambda)$, which is equivalent to (5).

We now investigate some properties of the above formulas and extract some concrete results from them. We also retrieve and generalize some results previously known in certain particular cases.

The tracer’s large deviation function $\phi(\xi)$ satisfies a version of the Fluctuation Theorem Gallavotti and Cohen (1995); Lebowitz and Spohn (1999),

[TABLE]

The Fluctuation Theorem is a symmetry relation that originates from an underlying time-reversal invariance. It implies, in particular, that the Einstein relation is true for the SEP Ferrari et al. (1985). The proof of (11) is based on the fact that $\lambda^{*}(-\xi)=\log\frac{r_{-}}{r_{+}}\ -\lambda^{*}(\xi)$ Sup . We also note that, while the fluctuation theorems have been established mainly for a large system in the infinitely late time, ours is for a system on the infinite lattice and for a large time.

Explicit formulae for the first few cumulants of $X_{t}$ can be obtained by substituting an expression of $\phi(\xi)$ in (5) into (4). For a stationary initial condition, $\rho_{+}=\rho_{-}=\rho$, we have calculated the first few cumulants: the variance is given by (1) and at the fourth order, we find

[TABLE]

(the subscript $c$ indicates a cumulant), in agreement with calculations based on the Macroscopic Fluctuation Theory (MFT) Krapivsky et al. (2014). Considering the MFT is a description at the level of hydrodynamics, this coincidence provides a highly nontrivial check of the MFT. The procedure can be carried out to higher orders in $s$ Sup .

For non-equilibrium initial conditions, $\rho_{+}>\rho_{-}>0$, the tracer drifts away from the origin as

[TABLE]

where $\xi_{0}$ is the unique solution of

[TABLE]

Solving (5) around $\xi_{0}$ leads to the variance of the tracer

[TABLE]

with

[TABLE]

In the special case $\rho_{-}=0$, the tracer is the left-most particle of a SEP expanding in a half-empty space and finding the distribution of $X_{t}$ becomes identical to a problem in extreme value statistics. From the above expressions, it can be shown that $\langle X_{t}\rangle\sim\sqrt{t\log t}$ and $\mathrm{Var}(X_{t})\sim\frac{t}{\log t}$. The tracer follows a Gumbel law, which is well-known to appear for independent walkers, in spite of interaction effects in the SEP Arratia (1983); Sabhapandit (2007).

In the low density limit $\rho_{-},\rho_{+}\ll 1$, the SEP becomes equivalent to an ensemble of reflecting Brownian particles Harris (1965). This system can be viewed as independent Brownian motions that exchange their labels when they collide and has been solved exactly using various techniques. Retaining only the first order terms in $\rho_{\pm}$ in the formula (6), and using (5), we obtain the large deviation of a tracer in the reflecting Brownian limit:

[TABLE]

where $\Xi(\xi)=\int_{\xi}^{\infty}\text{erfc}{(u)}{\rm d}u$. This generalizes the known result in the uniform case $\rho_{+}=\rho_{-}$ Rödenbeck et al. (1998); Krapivsky et al. (2014); Hegde et al. (2014); Derrida and Sadhu (2015). A figure of this large deviation function is also drawn in Fig. 1. By comparing to the one for the SEP, the effect of interaction among particles of the SEP is clearly seen.

In the last part of this work, we outline the derivation of the main formula (6). The strategy is to find exact expressions for all the moments of $N(x,t)$ and then construct the cumulant generating function $\mu(\xi,\lambda)$. The time evolution equations for the moments of $N(x,t)$ form a hierarchy of coupled differential equations that must be solved simultaneously. This seems to be a daunting task.

Our strategy is to make a detour though the ASEP, with $\tau<1$, for which the observable $N_{\tau}(x,t)$, defined in (8), satisfies a remarkable self-duality property Schütz (1997); Imamura and Sasamoto (2011); A. Borodin, I. Corwin, and T. Sasamoto (2014). For $x_{1}<x_{2}<\ldots<x_{n}$, $n$-point correlations of the type

[TABLE]

follow the same dynamical equations as the ASEP with a finite number $n$ of particles located at $x_{1},\ldots,x_{n}$. Using the fact that the ASEP with $n$ particles is solvable by Bethe Ansatz, these $\tau$-correlations can be expressed as a multiple contour integral in the complex plane Tracy and Widom (2008); Borodin and Corwin (2014); A. Borodin, I. Corwin, and T. Sasamoto (2014). For the step initial condition with the densities $\rho_{\pm}$, we can write

[TABLE]

with $r_{\pm}$ defined below (6), $\theta_{\pm}=\rho_{\pm}/(1-\rho_{\pm})$ and

[TABLE]

The contour of $z_{i}$ include $-1,\tau\theta_{+}$ and $\{\tau z_{j}\}_{j>i}$ but not $-\tau,\theta_{-}$; integrations are performed from $z_{n}$ down to $z_{1}$, see Fig. 3. This contour formula is a generalization of the $\rho_{-}=0$ case studied in A. Borodin, I. Corwin, and T. Sasamoto (2014). See also a recent related work Aggarwal . The symmetric limit, $\epsilon=1-\tau\to 0$, is performed using the identity

[TABLE]

that relates the $\tau$-moments of $N_{\tau}(x,t)$ in the ASEP to the $n$-th moment of the observable $N(x,t)$ in the SEP. Each term on the left-hand side is given by a complex contour integral, that has to be expanded with respect to $\epsilon$. This is achieved first by evaluating the residues of the contour integrals at the poles in the vicinity of $\theta_{+}$, leading to a formula in the form,

[TABLE]

where the combinatorial coefficients $\mu_{n,k}(\epsilon)$ contain the contributions of the residues and the $J_{k}$’s are $k$-fold integrals localized around the origin. Then, explicit recursive relations for the $\mu_{n,k}(\epsilon)$’s are found and large time asymptotics of the $J_{k}$’s are extracted. This allows us, finally, to obtain a formula for the $n$-th moment of $N(x,t)$ and for its $n$-th cumulant. The expressions for the cumulants are given by

[TABLE]

where $\Xi(\xi)$ was defined below (14) and

[TABLE]

Taking the generating function of the cumulants leads to (6). The full details of the derivation will be given in InP .

In this work, we obtain the exact formula for the large deviations of a tracer in the one dimensional symmetric simple exclusion process. This formula yields all the cumulants of the tracer position, in the long time limit. This answers a problem that has eluded solution for years Harris (1965); Sethuraman and Varadhan (2013). Our results are valid both when the system is at equilibrium with uniform density, and when the system is out of equilibrium, starting with a step density profile, the tracer being initially located at the boundary between the two domains of unequal density. Some of our formulas for the cumulants are prone to experimental tests, e.g. using colloidal particles Wei et al. (2000). They can also be used as benchmarks for numerical methods to evaluate large deviations, such as the one proposed in C. Giardina, J. Kurchan and L. Peliti (2006).

The derivation of the central formula (6) uses the powerful mathematical arsenal of integrable probabilities developed to solve the one-dimensional Kardar-Parisi-Zhang (KPZ) equation, the ASEP and related asymmetric models Sasamoto and Spohn (2010); Imamura and Sasamoto (2007, 2012); Corwin (2012); A. Borodin, I. Corwin, and T. Sasamoto (2014); Borodin and Corwin (2014). Generalizations of the ideas and techniques in this article will allow us to reveal various intricate properties of the SEP and related symmetric models, which would have been difficult with other means.

Infinite systems out of equilibrium keep in general the memory of the initial conditions Leibovich and Barkai (2013). For the models in the KPZ universality class, it has been well established that different initial conditions can lead to different statistical laws in the long time limit Prähofer and Spohn (2002); Sasamoto (2008); Corwin (2012); Takeuchi et al. (2011). This must also be true in the tagged particle problem in the SEP and one would like to study more general set-ups than the step profile. In particular, instead of taking averages over an ensemble of fluctuating initial step profiles and over the dynamics (annealed case), one could start with a deterministic initial configuration and average only over the history of the process (quenched case). For the latter case, even less is known Derrida and Gerschenfeld (2009b); Krapivsky et al. (2015); Derrida and Sadhu (2015) compared with the former, but new progress is expected to be achieved by extending our approach, combined with results for the ASEP, e.g. Ortmann et al. (2016).

Finally, we would like to relate our derivation to the macroscopic fluctuation theory (MFT) Derrida (2007); Bertini et al. (2015), one of the most promising approaches to study systems far from equilibrium. The MFT is based on a variational principle, that determines the optimal path that produces a given fluctuation, leading to two coupled nonlinear Euler-Lagrange equations. For reflecting Brownian particles, these equations can be linearized and solved, leading to the large deviations of the tracer Krapivsky et al. (2015). However, for the symmetric exclusion process, the MFT equations are, for the moment, intractable. Our exact calculations may give some hint to solve the MFT equations for this non-linear case.

The authors are grateful to S. Mallick for a careful reading of the manuscript and to P. L. Krapivsky for interesting discussions. We thank the JSPS core-to-core program ”Non-equilibrium dynamics of soft matter and information” which initiated this work. Parts of this work were performed during stays at ICTS Bangalore and at KITP Santa Barbara. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The works of T.I and T.S. are also supported by JSPS KAKENHI Grant Numbers JP25800215, JP16K05192 and P25103004, JP14510499, JP15K05203, JP16H06338 respectively.

I Large deviation function for non-interacting particles

When particles do not interact the motion of a tracer is a single-body problem. We recall the calculation of the large deviation function of a single particle on a one-dimensional lattice, hopping to the right and to the left with rates $p$ and $q$, respectively. We use the same notations as in the main text. Discretizing time in infinitesimal steps $dt$, we write the position $X_{t}$ of the particle at time $t$ as

[TABLE]

where the increments $\epsilon_{i}$ are independent and can take three values, +1, -1 and 0 with probabilities $pdt$, $qdt$ and $1-(p+q)dt$, respectively. Because of independence, we have

[TABLE]

with

[TABLE]

The Large Deviation Function, defined as

[TABLE]

is obtained from $C(s)$ by Legendre Transform and is given by

[TABLE]

Note that the rate for large deviations is proportional to the time $t$ as expected in a non-interacting system (whereas for the SEP we have a $\sqrt{t}$ prefactor to the large deviations).

II Proof of the relation (9) between $X_{t}$ and $N(x,t)$

We first define $Q(x,t)$ the time-integrated current that has flown through the bond $(x,x+1)$ between time 0 and $t$. The total current $Q(x,t)$ is equal to the total number of particles that have jumped from $x+1$ to $x$ minus the total number of particles that have jumped from $x$ to $x+1$ during the time interval $(0,t)$ (because we take the convention that a current flowing from right to left is counted positively).

By particle conservation, we find that the relation between $Q(x,t)$ and $N(x,t)$ is given by:

• •

for $x=0$, $Q(0,t)=N(0,t)=N_{t}$.

• •

For $x>0$, Q(x,t)=Q(0,t)+\sum_{y=1}^{x}\big{(}\eta_{y}(t)-\eta_{y}(0)\big{)}, or equivalently $N(x,t)=Q(x,t)+\sum_{y=1}^{x}\eta_{y}(0)$.

• •

For $x<0$, $N(x,t)=Q(x,t)-\sum_{y=x+1}^{0}\eta_{y}(0)$.

Consider a site $x>0$, located to the right of $X_{0}$, the initial position of the tracer. For the tracer $X_{t}$ to be to the right of $x$, it is necessary that all the particles that were initially between $X_{0}$ and $x$ have crossed the bond $(x,x+1)$ from left to right (including the tracer itself). This means that the total current $Q(x,t)$ has to be less than $-\sum_{i=X_{0}}^{x}\eta_{i}(0)=-\sum_{i=1}^{x}\eta_{i}(0)$ [here we use the fact the tracer is defined to be the particle which at $t=0$ is the closest to the origin from the right; therefore all sites between 1 and $X_{0}-1$ are empty at $t=0$]. We conclude that for $x\geq X_{0}$,

[TABLE]

A similar reasoning allows us to show that for $x<X_{0}$,

[TABLE]

The two identities (S2-S3) imply the relation (6) of the main text.

III Relation between $\phi(\xi)$ and $\mu(\xi,\lambda)$

Here we show that $\phi(\xi)$, the large deviation function of tracer’s position can be written in terms of $\mu(\xi,\lambda)$, the characteristic function of $N(x,t)$, as

[TABLE]

For this purpose we introduce $\Phi(\xi,q)$, the large deviation function of $N(x,t)$, given by

[TABLE]

with $\xi=-\frac{x}{\sqrt{4t}}.$ From (S3) and (S5), we find

[TABLE]

Note that the two functions $\Phi$ and $\mu$ are Legendre transforms of each other

[TABLE]

Thus from (S6) and (S7), we get (S4).

IV Proof of the fluctuation relation

We start with the parametric representation of the large deviation function (eq. (5) in the main text). The function $\phi(\xi)$, for a given value of $\xi$ is given by $\phi(\xi)=\mu(\xi,\lambda^{*})$ where $\lambda^{*}(\xi)$ is such that $\frac{\partial\mu(\xi,\lambda^{*})}{\partial\lambda}=0$, which, using the formula (6) for $\mu(\xi,\lambda)$, is equivalent to

[TABLE]

First, we note under the change $\lambda\rightarrow\log\frac{r_{-}}{r_{+}}\ -\lambda$,

[TABLE]

Using these properties and $A(\xi)=A(-\xi)$ to (S8), we deduce that

[TABLE]

Therefore we have

[TABLE]

Using again the formula (6) and the fact that $\omega$ is invariant, we find that this expression is equal to

[TABLE]

where we have written $\lambda$ for $\lambda^{*}(\xi)$ and $\lambda^{\prime}$ for $\lambda^{*}(-\xi)$. After simplification this expression reduces to

[TABLE]

thus proving the Fluctuation Theorem.

V Calculation of the higher cumulants of $X_{t}$

To extract the cumulants of the $X_{t}$ from $\mu(\xi,\lambda)$, it is useful to use a parametric representation for the cumulant generating function $C(s)$:

[TABLE]

The first two equations define implicitly two functions $\xi(s)$ and $\lambda(s)$, that, after substitution in the third equation, provide us with the cumulant generating function $C(s)$.

V.1 Equilibrium case with uniform density

For a stationary initial condition, $\rho_{+}=\rho_{-}=\rho$, the functions $\xi(s)$ and $\lambda(s)$ vanish when $s\to 0$. The strategy is to write power-series expansions w.r.t. $s$ for these two functions

[TABLE]

In order to calculate the unknown coefficients $x_{i}$’s and $l_{i}$’s, we substitute these expansions in (S11) and in (S12). At each order $n$, we obtain two inhomogeneous linear equations for $x_{n}$ and $l_{n}$, the r.h.s of which involve higher powers of $x_{i}$’s and $l_{i}$’s with $i<n$. This system can be solved systematically to any desired order. This can be done by hand up to $n=4$ and for higher orders by using a symbolic computation software (such as Mathematica). Substituting the series truncated at order $n$ for $\xi(s)$ and $\lambda(s)$ in $C(s)$ using (S13) gives the values of the first $n$ cumulants. All odd cumulants vanish and the first few even cumulants are given by the following expression. At order 2:

[TABLE]

At order 4:

[TABLE]

At order 6:

[TABLE]

V.2 Non-equilibrium case with $\rho_{+}\neq\rho_{-}$

When the system starts with an initial step-profile, it remains out-of-equilibrium and the tracer drifts away from the origin as shown by (12) in the main text. In order to calculate the cumulants we must again solve the system (S11), (S12) and (S13). Here, when $s\to 0$, we still have $\lambda\to 0$ but $\xi\to\xi_{0}$ where $\xi_{0}$ satisfies (13) in the main text. The procedure explained above for the equilibrium case can still be applied but if one just wants to calculate the variance of the tracer’s position it is simpler to use the parametric representation of the large deviation function: $\phi(\xi)=\mu(\xi,\lambda^{*})$ where $\lambda^{*}(\xi)$ is such that $\frac{\partial\mu(\xi,\lambda^{*})}{\partial\lambda}=0$ (eq. (8) in the main text).

The large deviation function is strictly positive for $\xi\neq\xi_{0}$ and vanishes at $\xi_{0}$, i.e., $\mu(\xi_{0},\lambda^{*}(\xi_{0}))=0$. It is elementary to check from the formula (6) in the main text that $\lambda^{*}(\xi_{0})=0$.

To calculate the variance of $X_{t}$, we perform a second order expansion of $\phi(\xi_{0}+\epsilon)$ with respect to $\epsilon$. Writing $\lambda^{*}(\xi_{0}+\epsilon)\simeq 1+a\epsilon$, we determine $a$ such that $\partial_{\lambda}\mu(\xi_{0}+\epsilon,\lambda^{*}(\xi_{0}+\epsilon)))$ vanishes at first order in $\epsilon$. We find that $a=-U/V$ with

[TABLE]

and

[TABLE]

Substituting this result in $\phi(\xi_{0}+\epsilon)\simeq\mu(\xi_{0}+\epsilon,1+a\epsilon)$ gives the correct dominant term in the large deviation function at order $\epsilon^{2}$:

[TABLE]

From this Gaussian limiting form, we deduce that

[TABLE]

which, after rearranging the terms, leads to the formula for the variance of the tracer, given in the main text:

[TABLE]

with

[TABLE]