ZRP conditioned on an atypical current

Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current

Ori Hirschberg, David Mukamel, Gunter M. Schütz Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel Department of Physics, Technion, 3200003 Haifa, Israel Institute of Complex Systems II, Theoretical Soft Matter and Biophysics, Forschungszentrum Jülich, 52425 Jülich, Germany Interdisziplinäres Zentrum für komplexe Systeme, Universität Bonn, Brühler Str. 7, 53119 Bonn, Germany ori.hirschberg@ph.technion.ac.il david.mukamel@weizmann.ac.il g.schuetz@fz-juelich.de , ,
July 5, 2019
Abstract

We study the asymmetric zero-range process (ZRP) with sites and open boundaries, conditioned to carry an atypical current. Using a generalized Doob -transform we compute explicitly the transition rates of an effective process for which the conditioned dynamics are typical. This effective process is a zero-range process with renormalized hopping rates, which are space dependent even when the original rates are constant. This leads to non-trivial density profiles in the steady state of the conditioned dynamics, and, under generic conditions on the jump rates of the unconditioned ZRP, to an intriguing supercritical bulk region where condensates can grow. These results provide a microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly asymmetric case: It turns out that the predictions of MFT remain valid in the non-rigorous limit of finite asymmetry. In addition, the microscopic results yield the correct scaling factor for the asymmetry that MFT cannot predict.

Keywords: Zero-range processes, Large deviations in non-equilibrium systems, Stochastic particle dynamics (Theory)

ams:
82C22, 60F10, 60K35

1 Introduction

There has been considerable progress in recent years in the understanding of nonequilibrium, current-carrying systems [1, 2, 3, 4, 5]. Recently, much effort has been devoted to the study of fluctuations which result in atypical currents in such systems [1, 5], where equilibrium concepts of entropy and free energy could be generalized to a nonequilibrium setting. Following the seminal papers [6, 7], the non-typical large-deviation behaviour of Markovian stochastic lattice gas models has come under intense scrutiny in the framework of what is now known as macroscopic fluctuation theory (MFT) [5]. In particular, the asymmetric simple exclusion process (ASEP) [8, 9] has been studied in great detail under the condition that the dynamics exhibits a strongly atypical current for a very long interval of time. Among the questions of considerable importance is the optimal density profile that realizes such a rare large deviation. Using MFT it was found for the weakly asymmetric simple exclusion process, where the hopping bias is of order , that for an atypically low current a dynamical phase transition occurs in the case of periodic boundary conditions: Above a critical non-typical current the optimal macroscopic density profile is flat, while rare events below that critical current are typically realized by a traveling wave [10]. This phenomenon was subsequently been studied numerically with Monte-Carlo simulations [11, 12]. Non-flat optimal profiles were obtained also for open boundary conditions [13].

More recently, the space-time realizations of large-deviation events in the ASEP with finite (strong) hopping bias were studied on the microscopic scale. This was done by conditioning the time-integrated current in a finite time interval to attain some atypical value. Furthermore, the full conditioned probability distribution on a finite lattice has been examined. For low non-typical current, the microscopic structure of a fluctuating traveling wave and an antishock has been identified [14, 15, 16]. Recently duality was used to extend the microscopic approach to show that the traveling wave may exhibit a microscopic fine structure consisting of several shocks and antishocks [17]. These findings are consistent with the macroscopic results for the optimal density profile [10, 13], but provide much more information in that the complete time-dependent measure was obtained for the microscopic dynamics.

Using a type of Doob’s -transform [18, 19, 20, 21] one may also study the large time limit on microscopic lattice scale. This transformation generates effective dynamics which make the rare large deviation behaviour of the original dynamics typical. For the ASEP with periodic boundary conditions and large non-typical current, inaccessible to MFT, it was found that a different dynamical phase transition involving a change of dynamical universality class occurs: Instead of the well-known generic universality class of the KPZ equation with dynamical exponent for the typical dynamics [22] one has a ballistic universality class with dynamical exponent where fluctuations spread much faster [23, 24, 25]. In fact, it turns out that these results follow predictions from conformal field theory and are thus expected to be universal [26]. Moreover, using the -transform technique, long-range effective interactions which make these rare events typical were obtained [23].

In this paper, we study similar questions, regarding the emergence of atypical currents, in another prototypical model of nonequilibrium systems — the zero range process (ZRP) [27, 28]. The steady state of the ZRP is exactly soluble, and thus it has often been used in studies of out-of-equilibirum features. The ZRP is a stochastic lattice gas model where each lattice site can be occupied by an arbitrary number of particles. Particles move randomly to neighbouring sites with a rate that depends only on the occupation number of the departure site and not on the state of the rest of the lattice. We consider a one-dimensional lattice of sites with open boundaries where the system exchanges particles with external reservoirs. The hopping events may be biased in one direction.

The precise definition of the model is given below. As an introduction we summarize some well-known features of the ZRP. The stationary state of the model with periodic boundary conditions, where the total particle number is conserved, may exhibit a condensation transition above a critical density. In this scenario, the existence of which depends on the form of the hopping rates , all sites but one are occupied by a particle number that fluctuates around the critical density, while one randomly selected site carries all the remaining excess particles, whose number is of the order of the lattice size (for a review see [28]). On the other hand, for open boundary conditions where the total particle number fluctuates due to random injection and absorption of particles at the two boundaries of the system, the scenario is different: For boundary rates for which a stationary distribution exists condensation never occurs. Only in the non-stationary case of very strong injection a dynamical condensation phenomenon occurs: The boundary sites can become supercritical, in which case the occupation number of these sites grows indefinitely [29]. This gives rise to a non-stationary condensate-like structure, but, in further contrast to the usual stationary bulk condensate, this can happen for any choice of bounded hopping rates and, moreover, the phenomenon is strictly limited to the boundary sites.

This brief outline of the properties of the ZRP concerns typical behaviour of the stochastic dynamics. It is the purpose of this work to study on microscopic scale the behaviour of the ZRP in a regime of strongly atypical behaviour of the particle current. This is of interest for several reasons. First, it serves not only as a test for macroscopic fluctuation theory (MFT) [5], but also probes its validity in the regime of strong asymmetry where MFT is not rigorous. Second, we will be able to compute effective microscopic interactions that make atypical behaviour typical. Third, it will transpire that unexpected and novel condensation patterns can occur even when the unconditioned dynamics do not exhibit condensation.

Current large deviations of finite duration have been investigated for the ZRP in the context of the breakdown of the Gallavotti-Cohen symmetry for the current distribution in a ZRP with open boundary conditions [30, 31]. It turns out that the failure of the Gallavotti-Cohen symmetry argument, which is based on a very general time-reversal property of stochastic dynamics [32, 33], can be related to the formation of “instantaneous condensates” [30]. These condensates were investigated recently in terms of Doob’s -transform for the ZRP with a single site [34]. It was shown that in some parameter regimes, Doob’s -transform fails to represent the effective dynamics that make the large deviation typical.

The microscopic large-deviation properties of the ZRP in the regime of atypical currents that persist for a very long time have not yet been explored. In this paper we address this problem. Moreover, we do not limit ourselves a single site; rather, we consider the full problem of open boundaries with any number of sites. It turns out that Doob’s -transform can be computed exactly from a product ansatz for the lowest eigenvector. Thus we are able to calculate the effective interactions that make the current large deviations typical. This turns out to be a ZRP with space-dependent hopping bias. Interestingly, the effective process satisfies detailed balance if and only of one conditions on vanishing macroscopic current. These results are outside the scope of MFT.

Somewhat surprisingly the exact results show that bulk condensation in the conditioned open system may occur, as is the case for periodic boundary conditions under typical dynamics. However, in contrast to typical behaviour (and to naive expectation), the results suggest that a whole lattice segment may become supercritical rather than just a single site. The segment location and length are fixed by the current on which one conditions rather than randomly fluctuating as one has for the condensate position in the periodic case with typical dynamics.

Our exact results are valid for any asymmetry, including the weakly asymmetric case. This allows for a comparison with the predictions of the macroscopic fluctuation theory which we apply to the ZRP with weak asymmetry. The microscopic results show that the MFT results remain intact in the limit of finite asymmetry, a limit in which the validity of the macroscopic approach is a-priori questionable. Moreover, we obtain the scaling factor for the asymmetry that cannot be computed from the MFT.

The paper is organized as follows: In Sec. 2 we introduce the model and define the conditioned dynamics. In Sec. 3 we derive the exact microscopic results for the -transform. This yields the effective dynamics and the spatial condensation patterns of the conditioned process. In Sec. 4 we follow the macroscopic approach for the weakly asymmetric case to compute the optimal macroscopic profile that realizes a current large deviation.

2 Open ZRP conditioned on an atypical current

2.1 Definition of the model

In the bulk of the lattice a particle from site hops to site with rate , and to site with rate . The parameters and determine the asymmetry of the process: The hopping is symmetric when and asymmetric otherwise. For the function we have , as particles cannot leave an empty site, otherwise is arbitrary. By definition the th bond is between sites and for . Bond 0 () represents the link between the lattice and a left (right) boundary reservoir that is not modeled explicitly. At the left boundary, particles enter with rate from the left reservoir and exit with rate . Similarly, at the right boundary particles enter with rate from the right boundary reservoir and exit with rate (Fig. 1).

Figure 1: Some possible particle hopping events in the ZRP with sites and open boundary conditions.

The dynamics of the zero-range process can be conveniently represented using the quantum Hamiltonian formalism [9, 35]. In this approach one defines a probability vector where is the canonical basis vector of associated with the particle configuration and the probability measure on the set of all such configurations. For a single site in this tensor product the configuration with particles on that site is represented by the basis vector which has component 1 at position and zero elsewhere. By definition obeys the normalization condition with the summation vector

(1)

and the orthogonality condition . Within this formalism the Markovian time evolution of the ZRP is represented by the Master equation

(2)

with

(3)

where and are infinite-dimensional particle creation and annihilation matrices

(4)

and is a diagonal matrix with the th element given by . The subscript in and indicates that the respective matrix acts non-trivially on site of the lattice and as unit operator on all other sites. We also introduce the local particle number operator with the diagonal particle number matrix and note the useful identity

(5)

for any non-zero complex number . The global particle number operator commutes with the bulk part of , expressing particle number conservation of the bulk jump processes.

According to (2) the probability vector at time is given by

(6)

Using and for the one-site summation vector one verifies that the global summation vector (1) is a left eigenvector of with zero eigenvalue. This expresses conservation of probability through the relation . A stationary distribution, denoted , is a right eigenvector of with eigenvalue zero.

In [29] it is shown that the stationary distribution of the ZRP is given by a product measure

(7)

where the marginal distribution is the single-site probability vector with components

(8)

The empty product is defined to be equal to 1 and is the local partition function

(9)

The fugacities in the steady state are given by

(10)

The mean density at site is related to the fugacity by (see (8)–(9))

(11)

The absence of detailed balance is reflected in a steady-state current

(12)

Notice that the existence of a steady state is determined by the radius of convergence of . If the form of the transition rates results in some outside this radius of convergence, then a stationary distribution does not exist. One expects relaxation to the local marginal distribution wherever the radius of convergence is finite, and a growing condensate on the other sites [29].

2.2 Grandcanonical conditioning

By ergodicity, the stationary current is the long-time mean of the time-integrated current across a bond , i.e. of the number of positive particle jumps from site to minus the number of negative particle jumps from site to up to time . Because of bulk particle number conservation the stationary current does not depend on . It is of great interest to consider not only the mean , but also the fluctuations of the time-averaged current , in particular its large deviations. The distribution has the large deviation property [33]. For particle systems with finite state space the large deviation function does not depend on the site and satisfies the Gallavotti-Cohen symmetry [32, 33] and this is widely believed to be generic. However, it has been pointed out that neither statement remains valid for the ZRP with open boundaries if one considers sufficiently large atypical currents. In order to get deeper insight into this breakdown we study here the ZRP conditioned to produce a strong atypical current. However, instead on enforcing a particular (integer) value of the time-integrated current, we consider a “grandcanonical” conditioning defined in terms of a Legrende transform with conjugate parameter . This corresponds to an ensemble of ZRP-histories where the current is allowed to fluctuate around some atypical mean. We refer to this ensemble as the -ensemble.

Grandcanonically conditioned Markov processes may be studied in the spirit of Doob’s -transform [18, 19], as was done for the ASEP conditioned on very large currents in [14, 23, 24], see also [20, 21] for recent discussions. This microscopic approach also provides a construction to compute effective interactions for which the atypical behaviour becomes typical, see [14, 23] for applications to the ASEP. Besides the conceptual insight gained in this way into extreme behaviour, this is potentially interesting from a practical perspective: By adjusting interactions between particles, one obtains a means to make desirable, but normally rare, events frequent.

Following [36, 37] this construction is done by first defining a weighted generator where the operators that correspond to a jump to the right (left) are multiplied by a factor (). Then one considers the lowest left eigenvector of , i.e., the eigenvector to the lowest eigenvalue of , which we denote by . We also introduce the diagonal matrix which has the components of the lowest eigenvector on its diagonal. Since all off-diagonal elements of are non-positive we appeal to Perron-Frobenius theory and argue that all components of can be chosen to be real and non-zero. Thus is invertible and we have

(13)

This allows us to introduce the grandcanonical Doob’s -transform

(14)

Notice that by construction has non-positive off-diagonal elements and, moreover, by (13), . Hence is the generator of some Markov process, which we shall refer to as the effective generator, with jump rates . We denote by the invariant measure of the effective process . The steady state of this Markov process provides the steady state of the grandcanonically biased model. It is easy to prove that where are the components of the lowest right eigenvector of the weighted generator . In order to avoid heavy notation we suppress in the following the dependence on if there is no danger of confusion.

2.3 Local conditioning in the ZRP

Under the condition that the mean integrated current across some fixed bond fluctuates around a certain value parameterized by we obtain, following Ref. [37], the weighted generator

(15)

Here

(16)

We refer to this setting as local conditioning. Notice that . We denote the lowest left eigenvector of by . Specifically for we drop the subscript i.e., we write

Define the partial number operator . From (5) one concludes . This implies for the left eigenvector

(17)

where the lowest eigenvalue is independent of and

(18)

with . This yields effective dynamics

(19)

We conclude that the effective dynamics does not depend on and we can focus on conditioning on a current across bond , i.e. between the left reservoir and site 1 of the lattice.

3 Microscopic density profiles and condensation

3.1 Left eigenvector

The left eigenvector of was computed in [30]. For self-containedness we repeat here the essential steps, which are based on a product ansatz with the diagonal matrix where are the diagonal one-site number operators. Then

One sees that is a left eigenvector if the following equations are satisfied:

(22)
(23)
(24)

We define the hopping asymmetry . The ansatz

(25)

yields and therefore solves (22).

The left boundary equation (23) yields

(26)

which gives

(27)

On the other hand, the right boundary equation (24) yields

(28)

Plugging this into leads to

(29)

and

(30)

For the lowest eigenvalue we get

from which one finds

(32)

3.2 Right eigenvector

Following [30] we make a product ansatz also for the right eigenvector: where is the unnormalized vector with components

(33)

This yields

(34)
(35)
(36)

One sees that is a right eigenvector if the following equations are satisfied:

(37)
(38)
(39)

with some constant . The ansatz

(40)

yields and therefore solves (22) with . Using the boundary recursions one then obtains

(41)

and

(42)

Notice that in terms of the local fugacities one has

(43)

3.3 Effective dynamics

The transformation (5) yields for the bulk hopping terms

and for the boundaries

and

Therefore the effective dynamics is given by

It is intriguing that this is a driven ZRP with a spatially varying driving field . This space-dependence will be present even in the case of non-interacting particles with . Therefore, conditioning on an atypical local current can be realized by an effective process with a space-dependent driving field.

The stationary distribution of the effective dynamics is a product state

(48)

where is the probability vector with components

(49)

Here

(50)

is the local fugacity given by (25) together with (29), (30) and (40) together with (41) (42). The normalization is the local analogue of the grand-canonical partition function, given by (9) as before. The density at site is related to the fugacity as in the unconditioned ZRP by Eq. (11). For example, for non-interacting particles , and therefore the density is simply . It is remarkable that one obtains a non-trivial density profile even for non-interacting particles.

The stationary current is

(51)

Plugging in the expressions yields

(52)

3.4 Examples

We now explore in more depth the results of the previous subsections for some specific choices of the model parameters.

3.4.1 Barrier-free reservoirs

To somewhat reduce the parameter space of the model, it is natural to consider

(53)
(54)

i.e., the hopping rates out of the first and last site are the same as the bulk hopping rates. We further simplify the notation by reexpressing the parameters and in terms of reservoir fugacities:

(55)
(56)

where is the fugacity of the left reservoir and is that of the right reservoir. The interpretation is that in the original (unconditioned) process, particles jump with the same rate between reservoir and boundary sites as inside the bulk of the chain. We therefore refer to this choice of rates as “barrier-free reservoirs”.

A few further notations will prove useful in what follows. First, we express the hopping asymmetry via

(57)

It is further convenient to characterize the boundary reservoirs by chemical potentials , and to define parameters , and as follows

(58)
(59)
(60)

Finally, we define a relative position along the lattice

(61)

3.4.2 Partially asymmetric ZRP with barrier-free reservoirs

With the choice (53)–(56) the parameters take the simpler form

(62)
(63)

This leads to

(64)

The effective hopping rates can now be read off Eq. (3.3). They corresponds to a ZRP in which the bulk bias to jump to the left and right is site dependent: a particle hops from site to with rate and from site to with rate where

(65)

and

(66)

These rates correspond to a space-dependent local field

(67)

It is interesting to note that the effective bulk hopping rates, and thus the effective field, are independent of the reservoir fugacities.

At the left boundary (site 1), according to (3.3), particles are effectively injected with rate and removed with rate , where are given in (65) and (66). Similarly at the right boundary (site ) the effective injection and removal rates are and , respectively. Thus, the effective boundary dynamics remains of the barrier-free form with the same reservoir fugacities, and only the effective space-dependent hopping rates depend on the current-conditioning.

We proceed to examine the typical fugacity profile during the atypical current event. To this end, we first calculate the right eigenvector

(68)

Thus, the fugacity profile is

(69)

with

(70)

The fugacity profile (69) along with the effective driving field are plotted in figure 2 for strongly asymmetric dynamics, and in figure 3 for weakly asymmetric dynamics.

Figure 2: The fugacity profile (left) and the effective field (right) for an asymmetric ZRP with , , , , and . Several values of are plotted including the typical profile of the unconditioned process (). The fugacity profile and the effective field are both flat, except for boundary layers of size sites.
Figure 3: The fugacity profile (left) and the effective field (right) for a weakly-asymmetric ZRP with , , , , and . Several values of are plotted including the typical profile of the unconditioned process (). The solid lines correspond to the macroscopic result, Eqs. (109) and (123).