# Lower Current Large Deviations for Zero-Range Processes on a Ring

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## Abstract

We study lower large deviations for the current of totally asymmetric zero-range processes on a ring with concave current-density relation. We use an approach by Jensen and Varadhan which has previously been applied to exclusion processes, to realize current fluctuations by travelling wave density profiles corresponding to non-entropic weak solutions of the hyperbolic scaling limit of the process. We further establish a dynamic transition, where large deviations of the current below a certain value are no longer typically attained by non-entropic weak solutions, but by condensed profiles, where a non-zero fraction of all the particles accumulates on a single fixed lattice site. This leads to a general characterization of the rate function, which is illustrated by providing detailed results for four generic examples of jump rates, including constant rates, decreasing rates, unbounded sublinear rates and asymptotically linear rates. Our results on the dynamic transition are supported by numerical simulations using a cloning algorithm.

## Keywords

Zero-range process Large deviations Current fluctuations Condensation## 1 Introduction

The large deviation behaviour of dynamic observables has been a topic of major recent research interest in driven diffusive systems. Most studies, as summarized in a recent review [1], focus on the particle current as one of the most important characteristics of nonequilibrium systems in one dimension. In general, current fluctuations are studied from a microscopic or macroscopic point of view. For the first perspective, algebraic techniques are implemented to calculate eigenvalues and eigenvectors of an exponential tilted version of the generator of a stochastic lattice gas. In this way, the rate function of the large deviations of the current is calculated as a Legendre–Fenchel transform of the greatest eigenvalue of the tilted generator. These methods were successfully applied to the asymmetric simple exclusion process (ASEP) [2, 3], also in combination with the matrix product ansatz [4], and to zero-range processes (ZRP) [5, 6, 7]. The statistics of the current and symmetry properties of the rate function can also be understood in the framework of the fluctuation theorem [8]. However, the symmetry relation stemming from the fluctuation theorem, also called Gallavotti–Cohen symmetry, breaks down in high current regimes for some condensing systems [9, 10]. Almost all previous studies focus on open boundary conditions, with only few available for periodic boundary conditions [11, 12], where microscopic results are difficult to obtain due to temporal correlations [13].

From the macroscopic point of view, one of the most powerful frameworks introduced in recent years is the macroscopic fluctuation theory (MFT) (see [14] and references therein), whose more general rigorous description is based on empirical flows [15, 16]. This is able to provide, as a result of a variational principle, the time evolution of the most likely density profile which typically gives rise to a given fluctuation. It turns out that it can be hard to solve the variational problem and an expression for the density profiles has only been obtained for some specific models [1, 14].

In general, macroscopic approaches rely on a hydrodynamic description of the process in terms of a mass conservation law. Lower current deviations, that is fluctuations of the current below its typical value, are usually realized by phase separated states for systems with concave flux function such as the exclusion process. These states can be described as weak solutions of the conservation law on a hydrodynamic level, while upper large deviations of the current are associated to hyperuniform states with long-range correlations [17, 18]. The connection between hydrodynamics and large deviations is provided by the well-known concept of entropy production in weak solutions that exhibit shocks [19]. Using all possible entropy functionals, this can be used to identify a unique entropic solution to the hydrodynamic equation describing the typical behaviour. For non-entropic solutions the entropy production can provide the large deviation rate function for observing such a non-typical profile, if the correct thermodynamic entropy is used [20]. This connection has been proved rigorously for the ASEP [21, 22], giving rise to the so-called Jensen–Varadhan theory. In [23], this has been applied heuristically to obtain a macroscopic derivation of the rate function for lower current deviations, which coincide with results based on exact microscopic computations and are in agreement with MFT predictions.

In this paper, we extend the Jensen–Varadhan approach to study lower current deviations for ZRPs which have a concave current-density relation. We focus on totally asymmetric dynamics with periodic boundary conditions, for which only few results exist so far. The ZRP was originally introduced in [24] and it has simple stationary distributions of factorized form [25] which allow for a detailed stationary analysis. At the same time ZRPs can exhibit a condensation transition in homogeneous systems due to particle interactions when the density exceeds a critical value [26, 27]. This has been studied in detail in recent years (see e.g. [28, 29, 30] and references therein), and has seen many applications [31, 32, 33], as well as rigorous mathematical work (see e.g. [34] and references therein). Here we focus on densities below the critical value, but we establish a dynamic transition for certain ZRPs where for sufficiently small currents the large deviations are dominated by condensed profiles rather than profiles arising from the Jensen–Varadhan approach. Our main result is a complete characterization of the rate function for lower current deviations for general totally asymmetric ZRPs with concave flux function.

The remainder of the paper is structured as follows. In Sect. 2, we define stochastic lattice gases in terms of generators and we define current conditioning in the context of large deviation theory. We introduce four generic classes of ZRPs with concave flux function, which we will analyze throughout the paper using specific examples of jump rates. In Sect. 3 we present a general formulation of the Jensen–Varadhan approach for ZRPs, and compare corresponding cost functions for large deviation events to those of condensed states. Section 4 contains a detailed study of generic examples of ZRPs introduced in Sect. 2 which cover several cases of possible behaviour, two of which exhibit the dynamic transition.

## 2 Definitions and Setting

### 2.1 TAZRP on a Ring

*f*, see [25] for details on infinite lattices. As usual, we denote by \(\eta ^{x,x+1}\) the configuration obtained from \(\eta \) after a particle jumps from site

*x*to \(x+1\), i.e. \(\eta ^{x,x+1}_y =\eta _y -\delta _{y,x} +\delta _{y,x+1}\). To avoid degeneracies and for later convenience we assume that the rates are in fact defined by a smooth function \(u:{\mathbb {R}}\rightarrow [0,\infty )\) with

*grand-canonical measures*,

*fugacity*[24, 25]. The mass function of the single site marginal with respect to the counting measure \(d\eta \) on \(X_L\), is given by

*grand-canonical partition function*, and the measures \(\nu _\phi \) exist for all \(\phi \ge 0\) such that \(z(\phi )<\infty \). We denote by \(\phi _c \in (0,\infty ]\) the radius of convergence of \(z(\phi )\), which we assume to be strictly positive. A convenient sufficient condition to ensure this, is that the jump rates are asymptotically bounded away from 0, i.e. \(\liminf _{k\rightarrow \infty } u(k)>0\) (see e.g. [34]).

### 2.2 Current Large Deviations

*L*and

*N*the ZRP is a finite-state, irreducible Markov chain on \(X_{L,N}\), and a general approach in [15, 16] implies a large deviation principle (LDP) for the empirical current (13) in the limit \(t\rightarrow \infty \). The authors establish an LDP for general empirical densities and flows on path space, and the particle current is a continuous and in fact linear function of the empirical flow. Then using the contraction principle (see e.g. [35, 36]) and linearity they were able to show that the current \(\mathcal {J}^{L} (t)\) satisfies an LDP with a convex rate function. We denote the associated rate function by \(I^L\), and following the usual compact formulation for LDPs (see e.g. [36]) on the level of logarithmic equivalence we have for all lower deviations \(j\le J(\rho )\)

*j*for \(s\le t\). For a discussion of examples where conditioning does not lead to time-homogeneous behaviour see e.g. [40].

In analogy to results for exclusion processes [3], we will see that if the system does not exhibit condensation (\(\rho _c = \infty \)) then typical realizations of lower current deviations for large *L* are dominated by phase separated states which are non-entropic weak solutions of the hydrodynamic limit of the ZRP (see Sect. 2.4) with two spatially separated regions at different densities. Since the phase boundaries move at non-zero speed we will refer to these as travelling wave profiles, which may exist only in a limited range of conditional currents. Outside this range, or for systems with finite critical density (\(\rho _c < \infty \)), condensed states may dominate the current large deviation, where a finite fraction of particles concentrates on a single, fixed lattice site.

### 2.3 Generic Examples

In the following, we will discuss some examples of TAZRPs which obey (17) and will be used throughout to illustrate our results. This includes models with bounded and unbounded jump rates.

*L*, the system phase separates into a fluid phase, which is homogeneously distributed as \(\nu _{\phi _c}\), and a condensed phase or condensate, where a finite fraction of \((\rho -\rho _c )L\) particles concentrates on a single lattice site (see e.g. [28, 42, 43]). The interesting feature for this paper is that in addition to the density, also the range of admissible currents \(j\le J(\rho )\) by travelling wave profiles is bounded as explained in Sect. 4.4. The partition function \(z(\phi )=\, _2 F_1 (1,1;1+b;\phi ):= \sum _{n=0}^\infty \frac{(1)_n (1)_n}{(1+b)_n} \frac{\phi ^n}{n!}\) can be written in terms of hypergeometric functions \(_2 F_1\) [42] using the Pochhammer symbol \((a)_n =\prod _{k=0}^{n-1} (a+k)\), which leads to similar expressions for for the convex function \(R(\phi )\) and will be useful for numerical computations later.

*z*to simplify the numerics in this case. \(J(\rho )\) turns out to be concave for all \(\rho \ge 0\) and behaves asymptotically as \(J(\rho )\simeq u(\rho )\simeq (1+\rho )^\gamma /\gamma \) as \(\rho \rightarrow \infty \).

### 2.4 Hydrodynamics and the Jensen–Varadhan Functional

*entropy*, with corresponding

*entropy flux*\(g\left( \rho \right) \) such that

This result has been applied in [3] heuristically in a different scaling. For fixed, large system size *L*, lower current deviations for the asymmetric exclusion process on a ring are realized by phase separated travelling wave step profiles with two densities \(\rho _1 <\rho _2\), which are uniquely determined by the total mass and conditional current. The probabilistic cost to realize such a profile does not depend on system size since only the non-entropic down shock has to be stabilized. This cost is equal to the entropy production across the reversed stable shock given by \(\mathcal {F} (\rho _1 ,\rho _2 )\), which is also equal to \(-\mathcal {F} (\rho _2 ,\rho _1 )\) by obvious symmetry in (33).

## 3 General Results

Even though they are only proved for the asymmetric exclusion process, the results in [20, 21, 22] depend only on the hyperbolic scaling limit and are of a general nature that can, at least heuristically, be applied directly to other particle systems. Therefore we assume that the same formalism used for the exclusion process in [3] applies to the ZRPs we consider here, since we assume that they also have concave flux functions \(J(\rho )\).

*j*using a Jensen–Varadhan approach, similar to that used in [3] for the exclusion process. We denote this cost by \(E_\mathrm{tw}(j)\) (see (43)). Secondly, if the process can exhibit condensation under the stationary measures (i.e. \(\rho _c < \infty \)) we will see that such a large deviation in the current are sometimes more efficiently realised by condensed states. We denote the large deviation cost associated with realising a current \(j< J(\rho )\) by a condensed state by \(E_c(j)\) (see (50)). Our main result is that for any TAZRP with concave flux function the large deviation rate function (16) in the limit \(L\rightarrow \infty \) is given by

*j*. Details on applying this to different examples and finite-size corrections for large

*L*will be discussed in Sect. 4, in the following we provide definitions and general results for travelling wave and condensed profiles.

### 3.1 Travelling Wave Profiles

Important general properties of (42) are the following. \(F(\phi _1 ,\phi _2 )\) is decreasing in \(\phi _1\) and increasing in \(\phi _2\), and it is anti-symmetric, i.e. \(F(\phi _1 ,\phi _2 )=-F(\phi _2 ,\phi _1 )\). Therefore \(F(\phi ,\phi )=0\), which corresponds to 0 cost for vanishing step size, and it is positive for \(\phi _2>\phi _1\). In all examples we have studied *F* is also convex and has concave level lines, but we are not able to show this in general. In our examples, *F* is also a smooth function on its domain of definition which is either \([0,\phi _c )^2\) or \([0,\phi _c ]^2\) in case of a condensing system with \(\phi _c <\infty \). This is always the case as long as \(\ln z\) is smooth.

*F*and

*G*in a given example, the minimizer in (43) is often a local minimizer in the interior of the domain and can be found as a solution to the following system of equations

**Properties of the travelling wave profile**For the constant rate example illustrated in Fig. 1, picking \(\phi _1 =0\), it is clear that all currents \(0\le j\le J(\rho )\) are admissible for the constraint (40) \(G\left( 0,\phi _2\right) =\rho \frac{\phi _2}{R\left( \phi _2\right) }=0\), since \(\phi _2 /R(\phi _2 )=1-\phi _2 \rightarrow 0\) as \(\phi _2\rightarrow 1\). As is illustrated in Fig. 3, the smallest current

*j*admissible by travelling wave profiles is in general given by

*j*is possible due to a bounded range of densities in condensing systems (e.g. with rates (22)), where \(j_{min}=\phi _c \frac{\rho }{\rho _c}\), or if \(R(\phi )\) is asymptotically linear, as is the case for the system with rates (24), where \(j_{min}=\rho \).

*j*, \(\phi _2\) is uniquely determined by \(\phi _1\). Therefore, for any admissible

*j*with \(\phi _1\le j\le J(\rho )\) the solution of the constraint (40) implicitly defines a function

*j*for systems with \(j_{min} >0\), and for non-accessible currents \(j<j_{min}\) the function (46) is not defined. This applies to the examples in Sects. 4.3 and 4.4 and is discussed there in detail. At the left boundary for \(\phi _1 =0\) the value of \(\bar{\phi }_2(0)>0\) is the positive solution to

*x*(39) as well as the speed of profile

For all the examples we studied it further turns out that \(\bar{\phi }_2(\phi _1)\) is convex, and with convexity of \(F(\phi _1 ,\phi _2 )\) and resulting concave level lines, this leads to a unique minimum of the cost *F* along the curve \((\phi _1,\bar{\phi }_2(\phi _1))\) as is illustrated in Fig. 1 (right) for the constant rate process. This minimum could be located inside the domain of definition, or located at the boundary \(\phi _1 =0\) or \(\phi _2 =\phi _c\) in the case \(\phi _c <\infty \). The location of minima for different \(j<J(\rho )\) is shown by a full red line in Fig. 1 (right). For the typical current \(j=J(\rho )\) no condition on the system is imposed and the optimal pair is given by \(\phi _1 =\phi _2 =J(\rho )\).

Since we assume non-linearity and concavity of the function \(J(\rho )\), it is clear from Fig. 3 that \(j_{min} <J(\rho )\) and there are currents at least close to the typical one which are admissible by travelling wave profiles. Furthermore, due to smoothness of the constraint curve (40) and the Jensen–Varadhan functional (42), and due to anti-symmetry of the latter, the travelling wave cost function (43) is continuous and \(E_{tw} (J(\rho ))=0\) at the typical value for the current. Therefore \(E_{tw} (j)\) itself is a proper rate function for the current, and in many cases \(I(j)=E_{tw} (j)\).

### 3.2 Condensed States

*R*(

*j*) and all the excess mass \((\rho -R(j))L\) being located on one single (fixed) lattice site. In general, when conditioning on a low current

*j*, a stable condensed state is obtained when the current out of the condensate matches the current \(j<J(\rho )\) in the bulk phase of the system. The condensate acts as a boundary reservoir, the exit rate of which has to be slowed down from a value of order \(u\big ( (\rho -R(j))L\big )\) to

*j*, to assure the right incoming current into the bulk. Then the cost to maintain a stable condensate corresponds to the cost of slowing down a Poisson process across one bond (see e.g. [5])

*u*(

*n*) by an average value, but with our regularity assumptions (2) on

*u*this is correct to leading order in

*L*. Condensed phase separated profiles are illustrated in Fig. 2. Note that opposed to travelling wave profiles, the range of admissible currents for condensed states is always given by the full interval \(\left[ 0,J\left( \rho \right) \right) \).

For unbounded rates *u*, \(E^L_c (j)\) diverges as \(L\rightarrow \infty \) of order \(u\big ( (\rho -R(j))L\big )\). However, travelling wave profiles always yield costs \(E_{tw} (j)\) which are independent of the system size *L* (see (43)) for \(j_\mathrm{min}< j < J(\rho )\). For such systems the current rate function (36) is therefore given by \(I(j)=E_{tw} (j)\) for all \(j>j_{min}\), and condensed profiles may only contribute in systems with bounded jump rates or if \(j_{min} >0\) in which case not all currents are admissible by travelling wave profiles. An example of the latter is given by asymptotically linear jump rates (24), which is discussed in detail in Sect. 4.3.

*u*is bounded and has a limit, we have \(\phi _c =\lim _{k\rightarrow \infty } u(k)<\infty \) and for diverging system size the condensed cost converges to a finite value

*j*just below \(J(\rho )\) and \(E_{tw} (J(\rho ))=0\). Therefore the rate function is always dominated by travelling wave profiles for

*j*sufficiently close to \(J(\rho )\), and condensed profiles can only be relevant for lower values of

*j*where the description in (49) and (50) is valid.

*j*and \(\rho \) this fixes a particular pair \(\left( \phi _1^c,\phi _2^c\right) \) on the constraint curve (46) which does not necessarily minimize (42). From the phase separation conditions (37) and (38), we have

*L*. Note also that the speed (48) of such profiles vanishes

In case \(\rho _c < \infty \) we will see in Sect. 4.4 that the rate function \(I_L(j)\) can be given by the lower convex hull of the condensed and travelling wave costs as in (36).

## 4 Large Deviation Results for Different Models

In this section, we determine the optimal travelling wave profiles for different types of jump rates introduced in Sect. 2.3, finding explicit or numerical solutions to the minimization (44) for travelling wave profiles, which turn out to be unique in all cases as long as the conditioned current *j* is admissible. This unique solution depends on the parameters *j* and \(\rho \), and is denoted \(\left( \phi _1^{o} ,\phi _2^{o}\right) \) in the following and also referred to as the optimal pair or fugacities. In light of (36), we compare the resulting cost (43) with the condensed cost (49) to derive the large deviation rate function for the current *I*(*j*), and also include remarks on finite size versions \(I^L (j)\) where appropriate.

### 4.1 Constant Rate TAZRP

*F*has concave level lines, which leads to unique optimal pairs \(\left( \phi _1^{o} ,\phi _2^{o}\right) \). Using the above explicit expressions, the first equation in the system (44) can be simplified to the implicit relation

### 4.2 Unbounded Sublinear Rates

*x*of the high density phase vanishes in the limit \(j\rightarrow 0\) as well as the speed \(v_s\) of the profile. Continuity of the Jensen–Varadhan functional

*F*allows us to commute limits, and formally we get

*L*is approximately given by (49), which implies

As can be seen from Fig. 5, the cost for condensed profiles for all fixed \(j>0\) is again higher than the one for travelling wave profiles for large enough system size. Therefore the limiting rate function is simply \(I(j)=E_{tw} (j)\) and (35) holds. For finite systems with fixed large *L*, however, the condensed cost \(E_c^L (j)\) is eventually lower than \(E_{tw} (j)\) for small enough *j*, and is a concave function of *j*. This leads to a linear part of the rate function \(I^L (j)\) for small *j* indicating a mixture between travelling wave and completely condensed profiles where all particles are trapped on a single site. This feature is a rather persistent finite size effect illustrated by dashed lines in Fig. 5 (right). Note that the very small systems shown in the plot only contain of the order of 1 or 2 particles and are just intended for illustration. Low enough deviations in larger systems are not accessible numerically, so the crossover is hard to observe in simulations.

### 4.3 Asymptotically Linear Rates

*L*as

*L*-independent cost dominate the rate function and we have

*Lt*instead of

*t*holds in the limit \(L\rightarrow \infty \), which is illustrated in Fig. 6 together with (70).

*L*we also have to compare to the option of slowing down the jump rate at all lattice sites which is always of order

*L*and therefore irrelevant in other examples. This cost is approximately given by

*j*large enough (depending on the parameter

*d*), and as \(j\rightarrow 0\) we have \(E_c^L (0)=L\rho <LJ(\rho )=E_i^L (0)\). This is illustrated in Fig. 6 (bottom row) for two parameter values \(d>0\). This crossover enters the rate function of the modified LDP with speed

*tL*. In this scaling, the cost of travelling wave profiles is

*L*-independent expressions given in (69) and (71). For

*d*large enough the rate function is simply linear between \(j=0\) and \(j=\rho \) and independent of \(e_i (j)\), whereas \(e_i (j)\) dominates an increasing part of the convex hull for decreasing

*d*. For the degenerate limiting case of independent particles with \(d=0\) we have \(J(\rho )=\rho \) and therefore \(e_{tw} (j)=\infty \) for all \(j<J(\rho )\) and it does not contribute to the rate function. Then (73) is given by the cost \(e_i (j)\) of slowing down the clock of the process on all sites, or equivalently slowing down all independent particles as is expected in this case (see Fig. 6, top right).

It is currently out of reach to numerically confirm the extensive behaviour of the rate function for \(j\le j_{min}\) for \(d>0\) in reasonably large systems, but our heuristics is consistent with the case of independent particles with \(d=0\), for which the rate function is exact. The cases in Fig. 6 (top left) for very small system sizes are numerically accessible but contain only between 1 and 3 particles, and are only shown for illustration. We do not expect the rate function measured in such systems to coincide with the lower convex hull of the costs since our theoretical arguments only apply for large enough *L*.

### 4.4 Condensing TAZRP

*x*and the speed \(v_s\) of the profile, as shown in Fig. 7 (right). It also leads to a kink in the cost curve \(E_{tw} (j)\) at \(j=j^B\). This kink is hard to observe numerically for interesting parameter values and not of particular interest as \(E_{tw} (j)\) remains a convex function.

*F*and get from (42)

*b*. This is the maximum of the cost curve \(E_{tw} (j)\) attained at \(j=j_{min}=\rho (b-2)\) shown in Fig. 8 for two different values of \(\rho \). As in the constant rate case (63), the limiting condensed cost is given by the simple expression \(E_c (j)=1-j+j\ln j<\infty \) independently of all system parameters and valid for all \(j\in [0,J(\rho )]\). Depending on the parameters \(b>2\) and \(\rho <\rho _c\), the costs \(E_{tw} (j)\) and \(E_c (j)\) may or may not intersect, as is illustrated in Fig. 8. In fact, for any fixed \(b>2\), there exists \(\rho \) small enough such that \(E_{tw}\left( j\right) \leqslant E_c\left( j\right) \) for all \(j \in \left[ j_{min},J\left( \rho \right) \right] \). To obtain the largest such \(\rho \), we can compare (77) with the condensed cost at \(j=j_{min}\) to obtain the condition

*j*in the affine region of the rate function \(J(\rho )\), the large deviation is realized by a temporal mixture between travelling wave and condensed profiles in analogy to classical phase separation phenomena (see e.g. [36, 50]). The dynamical phase transition is confirmed by numerical results presented in the next subsection, which require a detailed consideration of finite size corrections to the above arguments.

### 4.5 Numerical Results for the Condensing TAZRP

We numerically approximate the scaled cumulant generating function \(\lambda (k)\) given in (18) using a cloning algorithm approach (see e.g. [38]), which is explained in Appendix 1. The finite-size rate function \(I^L\) is then approximated by numerically performing the Legendre–Fenchel transform (19) of the generated data. The results for the ZRP with rates (22) with \(b=3.5\) and density \(\rho = 0.25\), are shown in Fig. 9 (left), and agree well with our theoretical prediction after finite size corrections. The finite-size cost functions \(E_c^L (j)\) and \(E_\mathrm{tw}^L(j)\) are defined using the canonical current density relation \(J_{L,N} = \langle u \rangle _{L,N}\) with \(N=[\rho L]\) as given in (10), in place of the limiting current \(J(\rho )\). It is well known that \(J_{L,N} =Z_{L,N-1}/Z_{L,N}\), and it can be computed exactly using the recursion \(Z_{L,N} =\sum _{k=0}^N w(k)\, Z_{L,N-k}\) for the partition function (see e.g. [51] and references therein). For finite *L*, the maximum current is larger than the limiting value, \(\phi _c^L >\phi _c =1\), and the current is known to significantly differ from its limiting behaviour above the critical density [51]. Inversion of this function defines the density \(R^L (\phi )\) as a function of the current. This leads to a finite-size version of the Jensen–Varadhan functional (42) \(F^L (\phi _1 ,\phi _2 )\) and of the constraint function \(G^L (\phi _1 ,\phi _2 )\), which are used as in (43) to define a finite-size version of \(E_{tw}^L (j)\). The density \(R^L (j)\) is also used in (49) to define a finite-size corrected version of \(E_c^L\). The resulting finite size corrections to the predicted rate function are significant, as shown in Fig. 9 (right).

*k*, rather than conditioning the path distribution on a current

*j*. Both parameters a conjugate, and the average current

*j*(

*k*) for a given value of

*k*is given by \(\partial _k \lambda (k)\). Affine regions of the rate function

*I*correspond to discontinuous derivatives of \(\lambda (k)\), and cannot be explored by the cloning algorithm. On finite systems these effects are smoothed out somewhat, which leads to data points from the simulations also in the affine regions of the rate function. From simulations with a cloning ensemble it is not possible to directly observe temporal mixtures, which realize such large deviation events for the original ZRP conditioned on a current

*j*in the affine region of the rate function. The slight systematic error visible in Fig. 9 is due to a generic sampling bias, which is caused by finite observation times leading to under-estimation of the probability for small values of

*j*, and an over-estimation for values of

*j*close to \(J(\rho )\).

## 5 Conclusion and Outlook

We study lower current large deviations for general TAZRP with concave flux functions \(J(\rho )\), which can be realized by phase separated density profiles. Travelling wave profiles related to non-entropic hydrodynamic shocks are identified as the universal typical realization at least for small deviations from the typical current. These shocks can be stabelized by local changes in the dynamics and lead to rate functions which are independent of the system size, which have been studied before for the exclusion process. The range of accessible currents for these profiles may be limited, and we established a dynamical phase transition where large deviations for low currents are realized by condensed profiles. In this case the rate function is determined by slowing down the exit process out of the condensate which is again independent of the system size in the case of bounded rates. The transition is caused by two basic mechanisms (summarized in Fig. 3); firstly, the range of densities in travelling wave profiles is bounded by the critical density in condensing ZRPs, this leads to a minimal accessible current of \(j_{min}=\rho /\rho _c\). Secondly, the ratio of limiting current and density appearing in (45) may be bounded due to an asymptotically linear current density relation. In this case the rate function for condensed states is extensive in the system size. We have studied these cases in detail for typical examples of jump rates, together with other generic models with bounded and unbounded rates which do not exhibit a dynamic transition. In this way we cover all qualitative cases of concave flux functions which gives a complete picture of the large deviations for lower current deviations formulated in (35) and (36) in the limit of diverging system size. For condensing systems large deviations of the current may be realized by a temporal mixture leading to a convex rate function, which we have confirmed by numerical simulations using a cloning algorithm in Sect. 4.4. For finite systems, other strategies beyond travelling waves or condensed profiles may play a role as is illustrated for asymptotically linear rates in Sect. 4.3.

For future works it would be desirable to complement our analysis with exact results derived from a microscopic approach, analogously to results for open boundary systems [7], and to investigate how the dynamic transition can be understood in the framework of macroscopic fluctuation theory. While directly analogous results can be derived for upper large deviations when the flux function is convex, it would be interesting to see if general flux functions can at least partially be covered by our approach, or how it extends to partially asymmetric dynamics. As summarized e.g. in [34], more general Misanthrope processes also provide interesting candidates to study dynamic transitions for current large deviations. Condensed states may require a possibly modified structure, while travelling wave profiles depend only on the hydrodynamic behaviour of the process and are expected to apply in great generality.

## Notes

### Acknowledgements

AP acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC), Grant No. EP/L505110/1 and by the Italian National Group of Mathematical Physics (GNFM-INdAM) Progetto-Giovani-2016 *Statistical Mechanics for Deep Learning*. Moreover, some special thanks go to Yevgeny Vilensky for having shared his PhD thesis with us. We are grateful for inspiring discussions with colleagues, in particular Thomas Rafferty.

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