# Microscopic Structure of Shocks and Antishocks in the ASEP Conditioned on Low Current

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## Abstract

We study the time evolution of the ASEP on a one-dimensional torus with *L* sites, conditioned on an atypically low current up to a finite time *t*. For a certain one-parameter family of initial measures with a shock we prove that the shock position performs a biased random walk on the torus and that the measure seen from the shock position remains invariant. We compute explicitly the transition rates of the random walk. For the large scale behavior this result suggests that there is an atypically low current such that the optimal density profile that realizes this current is a hyperbolic tangent with a traveling shock discontinuity. For an atypically low local current across a single bond of the torus we prove that a product measure with a shock at an arbitrary position and an antishock at the conditioned bond remains a convex combination of such measures at all times which implies that the antishock remains microscopically stable under the locally conditioned dynamics. We compute the coefficients of the convex combinations.

### Keywords

Asymmetric symmetric exclusion process Large deviations Shocks Antishocks## 1 Introduction

The Asymmetric Simple Exclusion Process (ASEP) on \(\mathbb{Z}\) [1, 2] describes the Markovian time evolution of identical particles according to the following rules: (1) Each particle performs a continuous time simple random walk on \(\mathbb{Z}\) with hopping rate *c*_{r} to the right and *c*_{ℓ} to the left; (2) The hopping attempt is rejected when the site where the particles tries to move is occupied. Without loss of generality *c*_{r}>*c*_{ℓ} is assumed. An important family of invariant measures for this process are the Bernoulli product measures parametrized by the particle density *ρ*. The stationary current *j*^{∗} is then *j*^{∗}=(*c*_{r}−*c*_{ℓ})*ρ*(1−*ρ*). For *c*_{r}=*c*_{ℓ} (symmetric simple exclusion process) the dynamics is reversible and one has *j*^{∗}=0. The process may also be defined on a finite integer lattice with various types of boundary conditions which may lead to non-vanishing stationary currents even in the symmetric case. The presence of a stationary current indicates the lack of reversibility of the dynamics and therefore one is interested not only with its stationary expectation *j*^{∗}, but also its fluctuations.

Starting with the seminal papers [3, 4] the large deviation theory for the ASEP has been developed in considerable detail. Of particular interest in this context is the time-integrated current \(J_{k}(t)=J^{+}_{k}(t) - J^{-}_{k}(t)\) across a bond (*k*,*k*+1), where \(J^{+}_{k}(t)\) is the number of jumps of particles from *k* to *k*+1 up to time *t* and analogously \(J^{-}_{k}(t)\) is the number of jumps from *k*+1 to *k* up to time *t*, starting from some initial distribution of the particles. In a periodic system with *L* sites, i.e., on the torus \(\mathbb{T}_{L}:=\mathbb{Z}/L\mathbb {Z}\), a related quantity of interest is the time-integrated total current \(J(t) = \sum_{k\in\mathbb{T}_{L}} J_{k}(t)\) which is intimately related to the rate of entropy production [5, 6]. We denote its distribution by \(Z_{J}(t) := \operatorname{Prob}[J(t) = J]\) with \(J\in \mathbb{Z}\). One also considers the (local) mean current *j*_{k}(*t*)=*J*_{k}(*t*)/*t* and the (global) mean current density *j*(*t*)=*J*(*t*)/(*Lt*). For the Bernoulli product measure one has lim_{t→∞}*j*_{k}(*t*)=lim_{t→∞}*j*(*t*)=*j*^{∗}. The probability to observe for a long time interval *t* an untypical mean *j*≠*j*^{∗} is exponentially small in *t*. This is expressed in the large deviation property [7] *Z*_{J}(*t*)∝exp(−*f*(*j*)*Lt*) where *f*(*j*) is the rate function which plays a role analogous to the free energy in equilibrium statistical mechanics. Indeed, in complete analogy to equilibrium one introduces a generalized fugacity e^{s} with generalized chemical potential *s* and also studies the generating function *Y*_{s}(*t*)=〈e^{sJ(t)}〉=∑_{J}e^{sJ}*Z*_{J}(*t*). The cumulant function *g*(*s*)=lim_{t→∞}ln*Y*_{s}(*t*)/(*Lt*) is the Legendre transform of the rate function for the mean current, *g*(*s*)=max_{j}[*js*−*f*(*j*)]. The intensive variable *s* is thus conjugate to the mean current density *j*.

Recently a focus of attention has been on the spatio-temporal structure of the process conditioned on realizing a prolonged untypical behavior of the current. A convenient approach to study this rare and extreme behavior is to consider the process in terms of the conjugate variable *s*. Fixing some *s*≠0 corresponds to studying realizations of the process where the current fluctuates around some non-typical mean. We shall refer to this approach as grandcanonical conditioning, as opposed to a canonical condition where the current *J*(*t*) would be conditioned to have some fixed value *J*. For the weakly asymmetric exclusion process (WASEP) on \(\mathbb{T}_{L}\) where *c*_{r}−*c*_{ℓ}=*ν*/*L* is small, it turns out in the hydrodynamic limit that for any strictly positive *s* (i.e., for any current *j*>*j*^{∗}) the optimal macroscopic density profile *ρ*(*x*,*t*) that realizes such a deviation is time-independent and flat [8]. Thus on macroscopic scale the conditioned density profile is equal to the typical (unconditioned) density profile. However, a recent *microscopic* approach using a form of Doob’s *h*-transform has revealed [10, 11] that for large *s* the ASEP with arbitrary strength of the asymmetry *c*_{r}−*c*_{ℓ} exhibits interesting stationary correlations. Even more remarkably, the process undergoes a phase transition from the typical dynamics in the KPZ universality class with dynamical exponent *z*=3/2 to ballistic dynamics with *z*=1.

For currents *below* the typical value (*s*<0) the large deviation approach for the WASEP yields a different phenomenon [8, 9]: For currents close to the typical current (but *j*<*j*^{∗}) the optimal macroscopic density profile *ρ*(*x*,*t*) is time-independent and flat as is the case for any *s*>0. However, below some critical value *s*_{c} (i.e. below some critical *j*_{c}<*j*^{∗}) the flat profile becomes unstable and a traveling wave of the form *ρ*(*x*−*vt*) develops. In the limit *ν*→∞ (expected to correspond to finite asymmetry in the ASEP) the optimal profile in this regime is predicted to have the form of a step function with two constant values *ρ*_{1} and *ρ*_{2}. This is a profile consisting of a shock discontinuity where the density jumps from *ρ*_{1} to *ρ*_{2}>*ρ*_{1} at a position *x*_{1}(*t*) and an antishock where the density jumps from *ρ*_{2} to *ρ*_{1} at position *x*_{2}(*t*). Hence, conditioning on a sufficiently large negative deviation of the current *j*<*j*_{c} induces a phase separation in the WASEP into a low density segment of length *r*=*x*_{1}−*x*_{2} with density *ρ*_{1} and a high density segment of length *L*−*r*=*x*_{2}−*x*_{1} with density *ρ*_{2} (positions and distances taken modulo *L*). A similar phase separated optimal profile, but without drift, has recently been obtained for atypically low activities *K*(*t*)=*J*^{+}(*t*)+*J*^{−}(*t*) in the SSEP [12].

In order to understand this phenomenon better we develop in this paper a microscopic approach for the ASEP for atypically low currents, but without restriction to the case of weak asymmetry. For a periodic system, as discussed above, one may expect in the long-time regime a traveling wave. In order to get insight into the microscopic structure of the traveling wave we focus on short times and address the question how for global conditioning a certain initial distribution which already has a shock evolves at some later *finite* time *t*. We also consider local conditioning of the current on a single bond. For this case we study the evolution for finite times *t* of a shock/antishock pair with densities *ρ*_{1,2}. Our rigorous analysis is an adaptation of the algebraic techniques developed in [13] and extended recently by Imamura and Sasamoto to study current moments of the ASEP [14]. This approach—which has its probabilistic origin in the self-duality of the ASEP [15, 16, 17]—constrains our approach to a specific family of initial shock measures which are certain functions of the hopping asymmetry. However, going from the weakly asymmetric case all the way to the totally asymmetric limit allows us to cover a wide range of scenarios.

The paper is organized as follows. In Sect. 2 we describe informally the well-established, but not so widely known tools required for studying grandcanonically conditioned dynamics. In Sect. 3 we introduce a new family of shock measures which are relevant for the grandcanonically conditioned time evolution of the ASEP. In Sect. 4 we state and prove the main result and in Sect. 5 we conclude with some further comments on a generalization of the present approach to more general shock measures and on the macroscopic large deviation results of [8].

## 2 ASEP Conditioned on a Low Current

*L*sites. For convenience we take

*L*even and denote sites (defined modulo

*L*) by an integer in the interval −

*M*+1,

*M*where

*M*:=

*L*/2. We shall identify jumps to the “right” with clockwise jumps. We introduce the hopping asymmetry

*ρ*by the fugacity

*z*

_{1,2}satisfying

### Condition S

This condition is motivated by properties of the representation of the quantum algebra *U*_{q}[*SU*(2)] which is a symmetry of the generator of the process with reflecting boundaries [13]. By convention we shall assume 1<*q*<∞, i.e. we consider preferred hopping to the “right” (clockwise) and we exclude the symmetric case *q*=1.

Grandcanonically conditioned Markov processes may be studied in the spirit of Doob’s *h*-transform [18], as was done for the ASEP conditioned on very large currents in [10, 11, 19], see below for the precise definition. To this end, we follow our earlier work [13] and we employ a matrix formulation to define the process. This formulation of Markovian dynamics, formalized in probabilistic terms for exclusion processes in [20] and described informally in detail in [2], allows for an alternative convenient formulation particularly of conditioned interacting particle systems, as discussed in [6]. For self-containedness we briefly explain the approach and introduce the necessary notation.

### 2.1 Matrix Formulation of the ASEP

*L*acting on cylinder functions

*f*(

*η*) of the configuration \(\eta\in\mathbb{V}\) is usually defined through the relation

*w*

_{η′,η}is the transition rate from a configuration

*η*to

*η*′. In particular for a probability measure

*μ*(

*t*) we have

_{μ}denotes expectation w.r.t.

*μ*. In particular, taking

*f*to be the indicator function \({\bf1}_{\eta}\) on a fixed configuration

*η*yields the

*master equation*

*μ*(

*η*;

*t*) of finding the configuration

*η*at time

*t*. A stationary distribution, i.e., an invariant measure satisfying d

*μ*/d

*t*=0 is denoted by

*μ*

^{∗}and a stationary probability of a configuration

*η*is denoted

*μ*

^{∗}(

*η*).

*η*a column vector |

*η*〉 which together with the transposed vectors 〈

*η*| form an orthogonal basis of a complex vector space with inner product 〈

*η*|

*η*′〉=

*δ*

_{η,η′}. Here

*δ*

_{η,η′}is the Kronecker symbol which is equal to 1 if the two arguments are equal and zero otherwise. Therefore a measure can be written as a probability vector

*μ*(

*η*;

*t*)=〈

*η*|

*μ*(

*t*)〉. The bra-ket notation for vectors is an elegant tool borrowed from quantum mechanics.

*t*≥0, the master equation (6) then takes the form

*H*

_{η,η′}of the matrix

*H*are the (negative) transition rates

*w*

_{η,η′}and the diagonal entries

*H*

_{η,η}are the sum of all outgoing transition rates

*w*

_{η′,η}from configuration

*η*.

*s*|

*s*|

*H*=0, i.e., the row vector 〈

*s*| is a left eigenvector of

*H*with eigenvalue 0. A right eigenvector with eigenvalue 0 is an invariant measure |

*μ*

^{∗}〉.

*f*〉=∑

_{η}

*f*(

*η*)

*μ*(

*η*) of a function

*f*(

*η*) are obtained by taking the scalar product \(\langle s| \hat{f} |\mu \rangle\) of the diagonal matrix

*η*〉〈

*η*| is used as a shorthand for |

*η*〉⊗〈

*η*| as is standard in the quantum mechanics literature. Moreover, we introduce for

*t*≥0 the non-diagonal matrices

*μ*(0) the useful identity

*t*

_{i+1}≥

*t*

_{i}≥0 one has

*L*and

*H*we shall somewhat loosely refer also to

*H*as generator of the process, since we can identify −

*H*with usual generator acting on cylinder functions

*f*by its action to the left on \(\langle s| \hat {f} H\).

*A*

^{⊗k}denotes the

*k*-fold tensor product of

*A*. The superscript

*T*indicates transposition. A configuration

*η*∈{0,1}

^{L}will be represented by the vector \(|\eta \rangle\in\mathbb{C}^{2^{L}}\) which is defined in the following manner

*i*=−

*M*+1,…,

*M*, |

*η*(

*i*)〉=(0,1)

^{T}, if the

*i*-th site in the configuration

*η*contains a particle, and |

*η*(

*i*)〉=(1,0)

^{T}, otherwise. Observe that for any configuration

*η*, the corresponding vector |

*η*〉 has 1 at one of its components and 0 at all others.

*H*for the ASEP we introduce the three 2-by-2 matrices

*M*+

*k*)-th position, counting from the left to the right. We also introduce Open image in new window and Open image in new window.

*σ*

^{−}=(1,0), \((1,0)\hat{n}=(1,0)\sigma^{-}=(0,1)\sigma ^{+}=(0,0)\); and analogously,

*σ*

^{+}(0,1)

^{T}=(1,0)

^{T}, \(\sigma^{-}(1,0)^{T}=\hat{n}(0,1)^{T}=(0,1)^{T}\), \(\hat{n}(1,0)^{T}=\sigma^{+}(1,0)^{T}=\sigma^{-}(0,1)^{T}=(0,0)\). These identities and the representation (15) yield the following properties of \(\sigma^{\pm}_{k}\) and \(\hat{n}_{k}\): If a configuration

*η*does not have a particle at the site

*k*then \(|\eta^{\prime}\rangle:=\sigma^{-}_{k}|\eta \rangle\) corresponds to the configuration that coincides with

*η*on \(\mathbb{T}_{L}\setminus\{k\}\) and has a particle at

*k*; if to the contrary

*η*has a particle at

*k*then \(\sigma^{-}_{k}|\eta \rangle=0\), i.e., the vector with all components equal to 0. If a configuration

*η*has a particle at the site

*k*then \(|\eta^{\ast}\rangle:=\sigma^{+}_{k} |\eta \rangle\) corresponds to the configuration that coincides with

*η*on \(\mathbb{T}_{L}\setminus\{k\}\) and does not have a particle at

*k*; if to the contrary

*η*does not have a particle at

*k*then \(\sigma^{+}_{k}|\eta \rangle=0\). Using the equality \((\sigma^{+}_{k})^{T}=\sigma^{-}_{k}\) and the above notations, we have that if

*η*does not have a particle at

*k*then \(\langle\eta|\sigma ^{+}_{k}=\langle\eta^{\prime}|\) and \(\langle\eta|\sigma^{-}_{k}=0\), while if

*η*has a particle at

*k*then \(\langle\eta|\sigma^{+}_{k}=0\) and \(\langle\eta |\sigma^{-}_{k}=\langle\eta^{\ast}|\). Accordingly, \(\sigma^{-}_{k}\) and \(\sigma^{+}_{k}\) are called

*the particle creation/annihilation operators*. The operator \(\hat{n}_{k}\) is called

*the number operator*; when applied to |

*η*〉, it returns |

*η*〉, if there was a particle at the site

*k*, and results in 0 otherwise, i.e. we have Similarly, \(\hat{v}_{k}\) is the vacancy operator.

*μ*(0) on {0,1}

^{L}and its vector representation |

*μ*(0)〉 as defined above we denote by |

*μ*(

*t*)〉 the vector representation of the distribution of ASEP at time

*t*, starting from

*μ*(0). We point out that in the tensor basis a product measure is represented by a tensor product of the single-site marginals. For the summation vector we have the tensor representation 〈

*s*|=(1,1)

^{⊗L}.

### 2.2 Grandcanonical Conditioning on an Atypical Current

*H*defined above is for the hopping dynamics only, it does not include the evolution of the integrated current

*J*(

*t*). Since

*J*(

*t*) can take any integer value

*J*, the state space for the full process is \(\{0,1\}^{L}\times\mathbb{Z}\). In order to construct the matrix representation we choose as basis for this space the set of product vectors |

*η*〉⊗|

*J*〉 with \(J\in\mathbb{Z}\). (Notice that the symbols 〈⋅| and |⋅〉 can be vectors in different vector spaces, as will be clear from the form of the argument.) Defining (infinite-dimensional) operators

*A*

^{±}through the relation

*A*

^{±}|

*J*〉=|

*J*±1〉 we obtain following Ref. [6] the generator

*η*according to which jump has occurred, but also change the value of total integrated current

*J*accordingly. An initial configuration is represented by a tensor vector |

*η*,

*J*〉:=|

*η*〉⊗|

*J*〉. We shall take

*J*(0)=0 as initial value of

*J*(

*t*). Hence an initial measure for the full process has the form |

*μ*(0)〉⊗|0〉. In order to compute expectation values we define the extended summation vector \(\langle\hat{s}| := \langle s|\otimes\langle s'|\) where 〈

*s*′| is the infinite-dimensional summation vector for the counting variable

*J*.

*J*(

*t*) having reached some fixed value

*J*at time

*t*≥0. The conditional distribution

*μ*(

*η*,

*J*;

*t*) of the ASEP under this conditioning has the form

*s*,

*J*|:=〈

*s*|⊗〈

*J*| with \(J\in\mathbb{Z}\). We write the conditional expectation of some function

*f*of the occupation numbers of the ASEP as

*J*of the current, but rather in expectations for an ensemble where the current is allowed to fluctuate around some generally atypical value. To this end we follow standard procedure and define a “grand canonical” current ensemble

*η*as

*Y*

_{0}(

*t*)=1 and that therefore |

*μ*

_{0}〉=|

*μ*〉 is the usual (unconditioned) measure and 〈

*f*(

*t*)〉

_{0}is the usual (unconditioned) expectation of

*f*. We shall refer to the latter quantity as the “typical” expectation. For

*s*≠0 we call corresponding quantities atypical.

*A*

^{±}, see [6]. One has for the associated unnormalized probability vector

^{L}obtained from

*G*by substituting the counting operators

*A*

^{±}by the

*c*-numbers e

^{±s}. Notice that \(\tilde{H}(0)=H\) is the generator of the ASEP constructed above. For

*s*≠0 the matrix \(\tilde{H}(s)\) does not conserve probability, but nevertheless has an intuitive probabilistic interpretation. It gives a weight e

^{±s}to each transition in a particular realization of the process. Hence we shall refer to \(\tilde{H}(s)\) as the weighted generator. The generalized chemical potential

*s*parametrizes the mean current of this weighted process. Thus we arrive at the central object of our interest, which is the conditional distribution

*s*which indicates the

*s*-dependence. (Grandcanonical) conditioned expectations at time

*t*are then computed as follows:

In the same spirit one can investigate the space time structure of the process under the condition that the integrated current across some fixed bond (*k*,*k*+1) has attained a certain value. Going to the grandcanonical conditioning leads to a matrix \(\tilde{H}^{(k)}(s)\) where only the hopping terms for the bond (*k*,*k*+1) have the weights e^{±s}. As discussed in the introduction we refer to this setting as *local conditioning*, as opposed to the *global conditioning* involving the global current. Notice that the global time-integrated current *J*(*t*) is (trivially) extensive in system size *L* and hence its generating function does not have a good limit for *L*→∞. Below we shall choose under global conditioning the coefficient *s* to be order 1/*L* as this yields the generating function for the current density *j*. This quantity has finite expectation 〈*j*〉 even for *L*→∞.

## 3 Shock Measures

We shall consider the evolution of two distinct types of shock measures. First we recall the definition of a shock measure with shock at site *m* in the infinite integer lattice \(\mathbb{Z}\) as a Bernoulli product measure with marginal fugacity *z*_{1} up to site *m*−1 and fugacity *z*_{2}>*z*_{1} from site *m* onwards [13]. We denote these measures by \(\mu^{+}_{m}\). Likewise one can define an antishock measure \(\mu^{-}_{n}\) for the infinite lattice with antishock at site *n*, i.e., fugacities *z*(*k*)=*z*_{2} for *k*≤*n* and *z*(*k*)=*z*_{1} for *k*>*n*.

Type I shock measures for the torus \(\mathbb{T}_{L}\) are defined in analogy to these shock measures as follows:

### Definition 3.1

*n*≠

*m*is a Bernoulli product measure with fugacities

*c*

_{r}>

*c*

_{ℓ}we have

*z*

_{2}>

*z*

_{1}and therefore

*ρ*

_{2}>

*ρ*

_{1}. Hence in the definition (3.1) the first index marks the site after which the high density region

*P*

_{high}begins, which extends up to site

*n*, counted modulo

*L*in the principal domain {−

*M*+1,…,

*M*}. We call site

*m*the

*microscopic position of the shock*and site

*n*the

*microscopic position of the antishock*in the shock measure of type I. In vector representation we have with \(A^{+}_{m,n} = (1+z_{1})^{2M+m-n}(1+z_{2})^{n-m}\) and \(A^{-}_{m,n} = (1+z_{2})^{2M-m+n}(1+z_{1})^{m-n}\). Tensor products with exponent 0 are defined to be absent. We remark that in terms of densities

*ρ*

_{i}=

*z*

_{i}/(1+

*z*

_{i}) these vectors read The unnormalized (!) restriction of \(\mu^{\mathrm{I}}_{m,n}\) to the sector with

*N*particles is denoted by \(\mu^{\mathrm{I},N}_{m,n}\), i.e.,

*N*particles is defined by Shock measures of type I are a particular microscopic realization of a shock/antishock pair in a macroscopic step function density profile with density

*ρ*

_{2}in the interval [

*x*,

*y*) of rescaled coordinates

*m*→

*x*,

*n*→

*y*(modulo 1) under suitable rescaling of space. For

*M*→∞,

*n*→∞ and

*m*fixed we recover the Bernoulli shock measures \(\mu^{+}_{m}\) of [13] for the ASEP defined on \(\mathbb{Z}\) fugacities

*z*(

*k*)=

*z*

_{1}for

*k*≤

*m*and

*z*(

*k*)=

*z*

_{2}for

*k*>

*m*. In a similar fashion one can recover an antishock measure \(\mu^{-}_{n}\) by taking the thermodynamical limit such that

*n*remains fixed and both

*M*and

*m*are taken to infinity.

*n*=

*M*we shall drop the second subscript and write \(\mu^{\mathrm {I}}_{m} := \mu^{\mathrm{I}}_{m,M}\) and similarly for the vectors and the projections on

*N*particles. For

*n*=

*M*and

*m*=±

*M*the shock measures reduce to the usual Bernoulli product measures which we denote by

*y*=

*ρ*/(1−

*ρ*)∈[0,∞) is the fugacity. In particular,

We shall make use of the following transformation property.

### Lemma 3.2

*Let*|

*μ*

_{y}〉

*be the vector representation of the Bernoulli product measure with fugacity*

*y*

*for*

*L*

*sites and*\(\hat{N}_{m} = \sum_{k=m+1}^{M} \hat{n}_{k}\)

*be the partial number operator*.

*Then*∀

*z*

_{1},

*z*

_{2}∈(0,∞)

*and*−

*M*≤

*m*≤

*M*

*and for fixed particle number*

*N*

### Proof

*N*sites can be interchanged with the number operator and that the projected vector is an eigenstate with eigenvalue

*N*of the number operator \(\hat{N}\). □

*z*

_{1}. In particular, (39) may be written

*m*=−

*M*we have the analogue of (40) for the sector of fixed

*N*:

Next we introduce a new one-parameter family of shock measures which—to our knowledge—has not yet been considered in the literature.

### Definition 3.3

*M*+1≤

*m*≤

*M*−1 a shock measure \(\mu^{\mathrm{II}}_{m}\) of type II on \(\mathbb{T}_{L}\) with shock at position

*m*is a Bernoulli product measure with space-dependent fugacities

*N*particles are denoted \(\mu^{\mathrm{II},N}_{m}\).

Shock measures of type I and type II are related by a similarity transformation. This is the content of the following lemma.

### Lemma 3.4

*Assume Condition*S

*and let*

*Then for*−

*M*≤

*m*≤

*M*

*we have*

*with*

*Moreover*,

*for fixed particle number*

*N*

### Proof

Observing that \(z_{1} q^{2\frac{2M}{L}} = z_{1} q^{2} = z_{2}\) the proof works like for Lemma 3.2, but using also that not only the projector *P*_{N} on *N* particles and the number operator \(\hat{N}\) are diagonal in the basis used throughout this work, but also the transformation *U*. Hence all three operations can be arbitrarily interchanged. □

*c*,

*E*through e

^{2E}:=

*q*

^{2}, e

^{−2Ec}:=

*z*

_{1}. Using Condition S gives

*z*

_{2}=e

^{−2E(c−1)}and we can then express \(\mu^{\mathrm{II}}_{m}\) as a Bernoulli product measure with densities

*q*of order 1 the variation of density between neighboring sites is of order 1/

*L*, except between site

*m*and

*m*+1 where there is a jump of order 1. On the other hand, for strong asymmetry with

*E*of order

*L*the density jumps from nearly 0 to nearly 1 at the shock position

*m*and there is a sharp transition from nearly 1 to nearly 0, extending over a few lattice sites in the vicinity of

*k*

^{∗}=−

*cL*(modulo

*L*). Here “nearly” means up to corrections that are exponentially small in

*L*. This sharp transition is the microscopic analogue of an antishock in the type II shock measure which exists only for strong asymmetry. For small asymmetry the density profile is almost linear, see Fig. 1.

## 4 Microscopic Conditioned Evolution of Shocks

### 4.1 Main Results

We are now in a position to state and prove our main results on the time evolution of the ASEP under conditioned dynamics, viz. for type II shocks under global conditioning, and for type I shocks under local conditioning. Throughout this section we assume Condition S to be satisfied.

### Theorem 4.1

(Shock evolution under global conditioning)

*Consider the ASEP on*\(\mathbb{T}_{L}\)

*with*

*N*

*particles and hopping rates*

*c*

_{ℓ}

*and*

*c*

_{r}

*to the left and to the right respectively*.

*Let the initial distribution at time*

*t*=0

*be given by the type II shock measure*\(\mu^{\mathrm{II},N}_{m}\)

*and let*\(\mu ^{\mathrm{II},N}_{m}(t)\)

*denote the grandcanonically conditioned distribution of the ASEP at time*

*t*

*under global conditioning with weight*e

^{sL}=

*c*

_{ℓ}/

*c*

_{r}.

*Then for*\(m \in\mathbb{T}_{L} \)

*and any*

*t*≥0

*where*

*with the modified Bessel function*

*I*

_{n}(⋅)

*and*\(\tilde{c}_{r} = c_{r} q^{-\frac{2N}{L}}\)

*and*\(\tilde{c}_{\ell}= c_{\ell}q^{\frac{2N}{L}}\).

We remark that *p*_{t}(*l*|*m*) is the probability that a particle that performs a continuous-time simple random walk on \(\mathbb{T}_{L}\) with hopping rates \(\tilde{c}_{r}\) to the right and \(\tilde{c}_{\ell}\) to the left, respectively, is at site *l* at time *t*, starting from site *m*. Hence the interpretation of (52) is that at any time *t*≥0 the distribution of this process is a convex combination of shock measures of a *conserved microscopic structure equal to that of the initial one*. The probabilistic interpretation of the weights in this convex combinations allows us to say that the shocks (or better, the shock positions) perform continuous time random walks on \(\mathbb{T}_{L}\) with hopping rates \(\tilde{c}_{r}\) (to the right) and \(\tilde{c}_{\ell}\) (to the left). Seen from the shock position the measure is invariant. As opposed to the macroscopic results of [8] our results provide exact information for any *t*≥0 and for finite lattice distances. Notice however, that our results require the choice e^{sL}=*c*_{ℓ}/*c*_{r} with arbitrary asymmetry *q*, while the phenomena observed in [8] are valid for any *s*, but require weak asymmetry.

### Proof of Theorem 4.1

*m*. We recall (8) and get for the unnormalized measure

*i*=

*m*. We add \(-(c_{r} - c_{\ell}) (\hat{n}_{i+1} - \hat{n}_{i})\) to each term in (58) and \(-(c_{r} - c_{\ell}) (\hat{n}_{m+1} - \hat{n}_{m})\) to (59). Since the sum of these terms is zero we arrive at which we write in the form \(\tilde{H}^{(m)} = \tilde{H}^{(m)}_{b} + B^{(m)}\) with bulk term \(\tilde{H}^{(m)}_{b} = \sum^{\prime}_{i} \tilde{h}^{b}_{i}\) given by (60) and transformation term

*B*

^{(m)}given by (61).

*V*

_{m}and the number operators \(\hat{n}_{i}\) are both diagonal and hence commute. With the projector property \(\hat{n}_{k}^{2} = \hat{n}_{k}\) one has \(q^{-2N/L} V_{m+1} = (1 + (q^{-2}-1) \hat{n}_{m+1} ) V_{m}\) and \(q^{2N/L} V_{m-1} = (1 + (q^{2}-1) \hat{n}_{m}) V_{m}\). Therefore \(c_{r} q^{-2N/L} V_{m+1} + c_{\ell}q^{2N/L} V_{m-1} - (c_{r}+c_{\ell}) V_{m} = - (c_{r}-c_{\ell}) (\hat {n}_{m+1} - \hat{n}_{m}) V_{m}\). Putting these results together yields

*t*≥0.

*p*

_{0}(

*l*|

*m*)=

*δ*

_{l,m}. The theorem thus follows from translation invariance

*p*

_{t}(

*l*+

*r*|

*m*+

*r*)=

*p*

_{t}(

*l*|

*m*) \(\forall r \in\mathbb{Z}\) and periodicity

*p*

_{t}(

*l*+

*pL*|

*m*)=

*p*

_{t}(

*l*|

*m*) \(\forall p \in\mathbb{Z}\) of the transition probability. □

### Corollary 4.2

*The unique stationary grandcanonically conditioned measure*

*is the uniform mixture of the initial shock measures*\(\mu^{\mathrm {II},N}_{l}\), \(l\in\mathbb{T}_{L}\).

*The approach to stationary is exponential with a relaxation time that grows asymptotically with system size proportional to*

*L*

^{2}.

This corollary is a simple consequence of the fact that the shock position performs a continuous-time random walk on \(\mathbb{T}_{L}\).

### Theorem 4.3

(Shock/antishock evolution under local conditioning)

*Let the initial distribution of the ASEP with*

*N*

*particles be given by the type I shock measure*\(\mu^{\mathrm{I},N}_{m}\)

*satisfying Condition*S

*with shock at bond*

*m*

*and antishock at bond*

*M*

*and let*\(\tilde{\mu}^{\mathrm{I},N}_{m}(t)\)

*denote the unnormalized grandcanonically conditioned measure at time*

*t*

*under local conditioning at bond*

*M*

*with weight*e

^{s}=

*c*

_{ℓ}/

*c*

_{r}.

*Then we have*∀

*t*≥0

*with*

*p*

_{t}(

*l*|

*m*)

*defined in*(4.1)

*and*\(\delta=\frac {1+z_{1}}{1+q^{2}z_{1}}\).

### Proof

*B*

_{M}given by (59) is the generator for the grandcanonically conditioned evolution under local conditioning at bond (

*M*,−

*M*+1) with weight e

^{s}=

*c*

_{ℓ}/

*c*

_{r}. Therefore \(\mathrm{d}/\mathrm{d}t |\tilde{\mu}^{\mathrm{I},N}_{m} \rangle = - \tilde{H}^{M} |\tilde{\mu}^{\mathrm{I},N}_{m} \rangle\). Notice next that

*U*=

*WV*

_{M}where

*U*is defined in (47) and

*V*

_{M}is defined in (54). Therefore we may write \(\tilde{H}^{M} = U^{-1} \tilde{H} U\) and we have by Theorem 4.1 in conjunction with Lemma 3.4

*m*=

*M*we have Straightforward computation yields

*Y*

_{M}/

*Y*

_{−M+1}=[(1+

*z*

_{2})/(1+

*z*

_{1})]

^{L−1}.

From (70)–(73) one finds that the transformed measure \(|\tilde{\nu}^{\mathrm{I},N}_{m} \rangle(t) = \delta^{m} q^{2Nm/L} |\tilde{\mu}^{\mathrm{I},N}_{m} \rangle(t)\) satisfies the system of linear ODE’s (62). Hence, by the same arguments as used in Theorem 4.1 we obtain (68). □

### Corollary 4.4

*The normalized grandcanonically conditioned measure at time*

*t*

*of Theorem*4.3 \(\mu^{\mathrm {I},N}_{m}(t)\)

*is of the form*

*with a normalization constant*\(C(t) = \sum_{l=-M+1}^{M}p_{t}(l |m) \delta^{l-m} q^{\frac{2N(l-m)}{L}} \langle s | \mu^{\mathrm{I},N}_{l} \rangle\)

*that is strictly positive and finite for all*

*t*≥0.

*In particular*,

*with*\(C^{\ast}:= \sum_{l=-M+1}^{M} \delta^{l-m} q^{\frac{2N(l-m)}{L}} \langle s | \mu^{\mathrm{I},N}_{l} \rangle\),

*is the unique stationary conditioned limiting measure*.

The properties of *C*(*t*) are obvious from the fact it is a transition probability for a random walk with unnormalized positive initial distribution and that the random walk propagator on \(\mathbb{T}_{L}\) satisfies lim_{t→∞}*p*_{t}(*l*|*m*)=1/*L*\(\forall l,m \in\mathbb{T}_{L}\). The uniqueness of the measure follows from the Perron-Frobenius theorem for the weighted generator. □

We remark that due to the boundary conditions (72), (73) there is no simple random walk interpretation of the shock position in a finite system under locally conditioned dynamics. However, the system of Eqs. (70), (72), (73) is straightforwardly diagonalized by lattice Fourier transformation and yields an exponential relaxation with asymptotically finite relaxation time approaching \(1/(\tilde{c}_{r} + \tilde{c}_{\ell}- c_{r} - c_{\ell})\) as system size *L* tends to infinity.

## 5 Final Remarks

Our results are exact results on the lattice scale and for finite times. With regard to the macroscopic large-deviation theory of [8] Theorem 4.1 suggests that there exists a value of *s*(*ν*) such that the optimal macroscopic density profile obtained from the large-deviation theory of [8] in the limit *ν*→∞ is of the form *ρ*(*x*,*t*)=1/2[1−tanh(*E*(*x*+*a*−*vt*))] with speed *v*=*c*_{r}exp(−2*ρs*)−*c*_{ℓ}exp(2*ρs*) and a jump discontinuity at some point *x*^{∗}(*t*) that also travels with speed *v*. The stationary solution in Theorem 4.3 for local conditioning with weight e^{s}=*c*_{ℓ}/*c*_{r} indicates the existence of an antishock that remains microscopically sharp and does not fluctuate. This behavior is in sharp contrast to the typical behavior of the ASEP where an antishock evolves into a rarefaction wave.

The evolution of the shock initial measures as a one-particle random walk indicates a self-duality relation of the conditioned ASEP with periodic boundary conditions, which is absent for the unconditioned dynamics. From a Bethe-ansatz perspective it is interesting to note that (52) can be understood as an exact resummation of a whole family of special Bethe eigenvectors where the Bethe roots are carried by the particles themselves [9, 19]. Furthermore, results from [13] for the microscopic dynamics of more than one shock, each satisfying Condition S, and properties of the underlying representation theory of the quantum group *U*_{q}[*SU*(2)] for periodic systems [22] suggests that one may study the behavior of shock/antishock measures with several shocks under global and local conditioning. One then expects the evolution of *n* shocks to be given by the dynamics of *n* interacting particles. The results of [14] might then provide information about more general solutions of the large deviation theory and about their fluctuations.

## Notes

### Acknowledgements

G.M.S. thanks FAPESP for financial support and IME at the University of São Paulo for kind hospitality. V.B. acknowledges financial support of FAPESP and CNPq.

### References

- 1.Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999) MATHCrossRefGoogle Scholar
- 2.Schütz, G.M.: Exactly solvable models for many-body systems far from equilibrium. In: Domb, C., Lebowitz, J. (eds.) Phase Transitions and Critical Phenomena, vol. 19, pp. 1–251. Academic Press, London (2001) CrossRefGoogle Scholar
- 3.Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys.
**107**, 599–634 (2002) MathSciNetADSMATHCrossRefGoogle Scholar - 4.Bertini, L., De Sole, A., Gabrielli, D., Jona Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys.
**107**, 635–675 (2002) ADSMATHCrossRefGoogle Scholar - 5.Lebowitz, J.L., Spohn, H.: A Gallavotti–Cohen–type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys.
**95**, 333–365 (1999) MathSciNetADSMATHCrossRefGoogle Scholar - 6.Harris, R.J., Schütz, G.M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech. P07020 (2007) Google Scholar
- 7.Derrida, B., Lebowitz, J.L.: Exact large deviation function in the asymmetric exclusion process. Phys. Rev. Lett.
**80**(2), 209–213 (1998) MathSciNetADSMATHCrossRefGoogle Scholar - 8.Bodineau, T., Derrida, B.: Distribution of current in non-equilibrium diffusive systems and phase transitions. Phys. Rev. E
**72**, 066110 (2005) MathSciNetADSCrossRefGoogle Scholar - 9.Simon, D.: Bethe ansatz for the weakly asymmetric simple exclusion process and phase transition in the current distribution. J. Stat. Phys.
**142**, 931–951 (2010) ADSCrossRefGoogle Scholar - 10.Popkov, V., Simon, D., Schütz, G.M.: ASEP on a ring conditioned on enhanced flux. J. Stat. Mech. P10007 (2010) Google Scholar
- 11.Popkov, V., Schütz, G.M.: Transition probabilities and dynamic structure factor in the ASEP conditioned on strong flux. J. Stat. Phys.
**142**(3), 627–639 (2011) MathSciNetADSMATHCrossRefGoogle Scholar - 12.Lecomte, V., Garrahan, J.P., Van Wijland, F.: Inactive dynamical phase of a symmetric exclusion process on a ring. J. Phys. A
**45**, 175001 (2012) MathSciNetADSCrossRefGoogle Scholar - 13.Belitsky, V., Schütz, G.M.: Diffusion and coalescence of shocks in the partially asymmetric exclusion process. Electron. J. Probab.
**7**(11), 1–21 (2002) MathSciNetGoogle Scholar - 14.Imamura, T., Sasamoto, T.: Current moments of 1D ASEP by duality. J. Stat. Phys.
**142**(5), 919–930 (2011) MathSciNetADSMATHCrossRefGoogle Scholar - 15.Schütz, G.M.: Duality relations for asymmetric exclusion process. J. Stat. Phys.
**86**(5/6), 1265–1287 (1997) ADSMATHCrossRefGoogle Scholar - 16.Giardinà, C., Kurchan, J., Redig, F., Vafayi, K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys.
**135**, 25–55 (2009) MathSciNetADSMATHCrossRefGoogle Scholar - 17.Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for Q-TASEP and ASEP. arXiv:1207.5035 [math.PR] (2012)
- 18.Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 1. Foundations. Wiley, New York (1994) MATHGoogle Scholar
- 19.Simon, D.: Construction of a coordinate Bethe ansatz for the asymmetric simple exclusion process with open boundaries. J. Stat. Mech. P07017 (2009) Google Scholar
- 20.Lloyd, P., Sudbury, A., Donnelly, P.: Quantum operators in classical probability theory: I. “Quantum spin” techniques and the exclusion model of diffusion. Stoch. Process. Appl.
**61**(2), 205–221 (1996) MathSciNetMATHCrossRefGoogle Scholar - 21.Feller, W.: Introduction to the Probability Theory and Its Applications, vol. I, 3rd edn. Wiley, New York (1968) Google Scholar
- 22.Pasquier, V., Saleur, H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B
**330**, 523–556 (1990) MathSciNetADSCrossRefGoogle Scholar