Entropy of non-equilibrium stationary measures of boundary driven TASEP

# Entropy of non-equilibrium stationary measures of boundary driven TASEP

Cédric Bernardin Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07 - France.
e-mail: cedric.bernardin@ens-lyon.fr
Patrícia Gonçalves Departamento de Matemática, PUC-RIO, Rua Marquês de São Vicente, no. 225, 22453-900, Rio de Janeiro, Rj-Brazil and CMAT, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal.
e-mail: patricia@mat.puc-rio.br and patg@math.uminho.pt
and  Claudio Landim IMPA, Estrada Dona Castorina 110, CEP 22460 Rio de Janeiro, Brasil and CNRS UMR 6085, Université de Rouen, Avenue de l’Université, BP.12, Technopôle du Madrillet, F76801 Saint-Étienne-du-Rouvray, France.
e-mail: landim@impa.br
###### Abstract.

We examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs-Shannon entropy of the non equilibrium stationary state converges to the Gibbs-Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.

###### Key words and phrases:
Non-equilibrium stationary states, phase transitions, large deviations, quasi-potential, boundary driven asymmetric exclusion processes

## 1. Introduction

Nonequilibrium stationary states (NESS) maintained by systems in contact with infinite reservoirs at the boundaries have attracted much attention in these last years. In analogy with the usual Boltzmann entropy for equilibrium stationary states, we introduced in [3] the entropy function of NESS and we computed it explicitly in the case of the boundary driven symmetric simple exclusion process. In the present paper we extend this work to the boundary driven totally asymmetric simple exclusion process (TASEP) and we show that the entropy function detects phase transitions.

The boundary driven asymmetric simple exclusion process is defined as follows. Let and . The microstates are described by the vectors where for , if the site is occupied and if the site is empty. In the bulk of the system, each particle, independently from the others, performs a nearest-neighbor asymmetric random walk, where jumps to the right (resp. left) neighboring site occur at rate (resp. rate ), with the convention that each time a particle attempts to jump to a site already occupied, the jump is suppressed in order to respect the exclusion constrain. At the two boundaries the dynamics is modified to mimic the coupling with reservoirs of particles: if the site is empty (resp. occupied), a particle is injected at rate (resp. removed at rate ); similarly, if the site is empty, a particle is injected at rate (resp. removed at rate ). For any sites , we denote by (resp. ) the configuration obtained from by the exchange of the occupation variables and (resp. by the change of into ). The boundary driven (nearest neighbor) asymmetric simple exclusion process is the Markov process on whose generator is given by

 L=L0+L−+L+,

where act on functions as follows

 (L0f)(η)=N−1∑x=−N{pηx(1−ηx+1)+qηx+1(1−ηx)}[f(σx,x+1η)−f(η)],(L−f)(η)=c−(η−N)[f(σ−Nη)−f(η)],(L+f)(η)=c+(ηN)[f(σNη)−f(η)]

with given by

 c−(η)=α(1−η−N)+γη−N,c+(η)=δ(1−ηN)+βηN.

The density of the left (resp. right) reservoir is denoted by (resp. ) and can be explicitly computed as a function of (resp. ). For simplicity we will focus only on the totally asymmetric simple exclusion process (TASEP) which corresponds to or . Furthermore, if we take , , . If , we take , and . Since the reservoirs induce a flux of particles from the right to the left. On the other hand the bulk dynamics produces a mean displacement of the particles with a drift equal to . For both effects cooperate to push the particles to the left and we call the corresponding system the cooperative TASEP. If the two effects push the particles in opposite directions and we call the corresponding system the competitive TASEP.

The unique non-equilibrium stationary state of the boundary driven TASEP is denoted by . In the case , is given by the Bernoulli product measure on . In the non-equilibrium situation, the steady state has a lot of non-trivial interesting properties. The phase diagram for the average density is well known and one can distinguish three phases: the high-density phase (HD) for which , the low density phase (LD) for which and the maximal current phase (MC) where , see [8]. The transition lines between these phases are second order phase transitions except for the boundary in the competitive case where the transition is of first order. On this line, the typical configurations are shocks between LD phase with density at the left of the shock and HD phase with density at the right of the shock. The position of the shock is uniformly distributed along the system and the average profile is given by . This is summarized in Figure 1.

The entropy function of introduced in [3] is the function defined by

 S(E)=limδ→0limN→+∞12N+1log(∑η∈ΩN1{∣∣∣12N+1log(μss,N(η))+E∣∣∣≤δ})

if the limit exists.

Observe that coincides with the large deviations function of the random variables under the probability measure . Therefore, the (concave) Legendre transform of the entropy function ,

 P(θ):=Pp,qρ−,ρ+(θ)=infE≥0{θE−S(E)}, (1.1)

that we call the pressure, is by the Laplace-Varadhan theorem simply related to the cumulant generating function of the random variables , i.e.

 P(θ)=−limN→+∞12N+1log(∫e(2N+1)(1−θ)YN(η)μss,N(dη))=−limN→+∞12N+1log⎛⎝∑η∈ΩN(μss,N(η))θ⎞⎠. (1.2)

In the equilibrium case , denoting by

 φ=log(ρ1−ρ)∈R (1.3)

the corresponding chemical potential, it is easy to show that the entropy function is given by

 Sρ,ρ(E)=−s(−E+log(1+eφ)φ) (1.4)

where . The pressure is then given by

 P(φ,θ):=θlog(1+eφ)−log(1+eφθ). (1.5)

In the non-equilibrium case , since has not a simple form, the computation of the entropy function is much more difficult. It has been proved in [3] that if a strong form of local equilibrium holds (see Section 5 for a precise definition), then the entropy function can be expressed in a variational form involving the non-equilibrium free energy and the Gibbs-Shannon entropy :

 S(E)=supρ∈M{S(ρ);V(ρ)+S(ρ)=E}, (1.6)

where the set of density profiles is defined in (2.1) and the Gibbs-Shannon entropy of the profile is defined by

 S(ρ)=−12∫1−1s(ρ(x))dx. (1.7)

The interval composed of the such that is called the energy band. The bottom and the top of the energy band are defined respectively by

 E−:=infρ∈M{S(ρ)+V(ρ)}andE+:=supρ∈M{S(ρ)+V(ρ)} (1.8)

The non-equilibrium free energy is the large deviation function of the empirical density under . Its value does not depend on nor but only on the sign of , and we denote it by if and by if . The explicit computation of this functional has been obtained first in [8] and generalized to other systems in [2]. Similarly, the entropy (resp. pressure) of the competitive TASEP is denoted by (resp. ) and the entropy (resp. pressure) of the cooperative TASEP by (resp. ). It follows easily from (1.1) and (1.6) that

 P(θ)=infρ∈M{θ(V(ρ)+S(ρ))−S(ρ)}.

This formula can also be obtained starting from (1.2) and using the local equilibrium statement as it is done in [3] for the entropy function.

In this paper we compute explicitly and (resp. Theorem 2.2 and Theorem 3.2) and and (resp. Theorem 2.3 and Theorem 3.3). From those results we deduce several interesting consequences (see Theorem 2.4 and Theorem 3.4):

• We recover some results of [1] for the TASEP, showing that the Gibbs-Shannon entropy of the non-equilibrium stationary state of the TASEP is the same, in the thermodynamic limit, as the Gibbs-Shannon of the local Gibbs equilibrium measure, see Theorems 2.4 and 3.4. In this case, the local Gibbs equilibrium measure is , namely, the Bernoulli product measure where is the one-site Bernoulli measure on with density and is the stationary profile.

• For the competitive TASEP, contrarily to what happens for the boundary driven symmetric simple exclusion process ([3, 6]), the fluctuations are Gaussian with the same variance as the one given by the local equilibrium state.

• For the cooperative TASEP, the same occurs if or if . But in the MC phase , the fluctuations are not Gaussian. This is reminiscent of [5, 8] where it is shown that the fluctuations of the density are non-Gaussian 111The non-Gaussian part of the fluctuations can be described in terms of the statistical properties of a Brownian excursion ([5])..

Our last results concern the presence of phase transitions 222We refer the interested reader to [10] for more informations about the implications of these facts from a physical viewpoint. for the competitive and the cooperative TASEP. For the cooperative TASEP the function is a continuously differentiable concave function on its energy band but has linear parts. As a consequence the pressure function is a concave function with a discontinuous derivative. The function may also have a linear part due to the fact that the entropy does not necessarily vanish at the boundaries of the energy band. If , then the function is a smooth concave function on its energy band, but does not vanish at the top of the energy band. Consequently the pressure function is concave with a linear part on an infinite interval. If or , the entropy function has a discontinuity of its derivative at some point in the interior of the energy band but vanishes at the boundaries of the energy band. Then, the pressure function has a linear part on a finite interval.

It would be interesting to see how these results extend to other asymmetric systems for which the quasi-potential has been explicitly computed ([2]). The form of the entropy function obtained for the TASEP is relatively simple but follows from long computations. We did not succeed in giving a simple intuitive explanation to the final formulas obtained. We also notice that extending these results to a larger class of systems would require to prove the strong form of local equilibrium for them in order to get (1.6). This seems to be a difficult task.

The paper is organized as follows. In Section 2 we obtain the entropy and the pressure functions for the competitive TASEP and deduce some consequences of these computations. In Section 3 we obtain similar results for the cooperative TASEP. The local equilibrium statement is proved in Section 5. Technical parts are postponed to the Appendix.

## 2. Competitive TASEP

In this section we derive the variational formula for the entropy function (1.6) for the competitive TASEP. Denote by , the mobility of the system, that is is defined by . The chemical potential corresponding to is denoted by and satisfies , see (1.3).

We consider the set equipped with the weak topology and as the set

 M={ρ∈L∞([−1,1]):0≤ρ≤1} (2.1)

which is equipped with the relative topology. Denote by the stationary density profile. We recall that for , i.e. , for , i.e. and if . Let so that if . Let

 Φ={φ:=φy:x∈[−1,1]→φ−1{−1≤x

For we define the functional

 H(ρ,φ)=12∫1−1[(1−ρ(x))φ(x)−log(1+eφ(x))]dx. (2.2)

Then the quasi-potential of the competitive TASEP is given ([8], [2]) by

 V+(ρ)=−S(ρ)+infφ∈ΦH(ρ,φ)−¯V+

where

 ¯V+=−S(¯ρ)+infφ∈ΦH(¯ρ,φ)=log(minρ∈[ρ−,ρ+]χ(ρ)).

Let us also introduce and .

For each , and we define

 ξ0:=log(1+eφ+)−log(1+eφ−)φ+−φ−∈(0,1),^ξ0:=log(1+eφ+)+log(1+eφ−)φ+−φ−. (2.3)

### 2.1. Energy bands

In this section we determine the energy band of the competitive TASEP. This is summarized in Figure 2.

###### Proposition 2.1.

The bottom of the energy band is given by

 E−+∞=−¯V+−log(1+eφ0),

where and the top of the energy band is given by

 E++∞=−¯V++⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩\vspace0.2cmφ−log(1+eφ+)−φ+log(1+eφ−)φ+−φ−,ρ−≤12≤ρ+,\vspace0.2cm−log(1+eφ+),ρ−<ρ+≤12,\vspace0.2cm−log(1+e−φ−),12≤ρ−<ρ+.

### 2.2. Entropy

Now we compute the entropy function. We introduce

 (2.4)

which corresponds to

 W(φ−,φ+)=φ−log(1+eφ+)−φ+log(1+eφ−)φ+−φ− (2.5)

and coincides with the first coordinate of one of the (possible) two intersection points of the curves and , where is defined in (1.4).

###### Theorem 2.2.

The restriction of the entropy function on the energy band is given by

 S+(E)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Sρ0,ρ0(−(E+¯V+)),ρ−≤12≤ρ+,Sρ−,ρ−(−(E+¯V+))1{(E+¯V+)≤W(ρ−,ρ+)}+Sρ+,ρ+(−(E+¯V+))1{(E+¯V+)>W(ρ−,ρ+)},ρ−<ρ+≤12,Sρ−,ρ−(−(E+¯V+))1{(E+¯V+)≥W(ρ−,ρ+)}+Sρ+,ρ+(−(E+¯V+))1{(E+¯V+)

and is a concave function. Therefore, when , its derivative is continuous on the energy band, but in the remaining cases is continuous except where .

The supremum in the definition of , see (1.6), for is realized for a unique profile whose value is given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩u¯ρ,ρ−≤12≤ρ+,uρ−1{(E+¯V+)≤W(ρ−,ρ+)}+uρ+1{(E+¯V+)>W(ρ−,ρ+)},ρ−<ρ+≤12,uρ−1{(E+¯V+)≥W(ρ−,ρ+)}+uρ+1{(E+¯V+)

where for any , the profile is the constant profile equal to .

### 2.3. Pressure

We recall that the pressure function is defined as the Legendre transform of the entropy function :

 P+(θ)=infE≥0{θE−S+(E)}.

We introduce the two parameters , where is defined in (2.3).

###### Theorem 2.3.

The pressure function is given by:

• If then

 P+(θ)={\vspace0.2cmP(φ−,−θ)−θ¯V+,θ≥θ−0,\vspace0.2cmP+(θ0)+E++∞(θ−θ−0),θ<θ−0.
• If then

 P+(θ)={\vspace0.2cmP(φ+,−θ)−θ¯V+,θ≥θ+0,\vspace0.2cmP+(θ0)+E++∞(θ−θ0),θ<θ+0.
• If then

 P+(θ)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩\vspace0.2cmP(φ−,−θ)−θ¯V+,θ≥θ−0,\vspace0.2cmP(φ+,−θ)−θ¯V+,θ≤θ+0,\vspace0.2cmP+(θ+0)+P+(θ−0)−P+(θ+0)θ−0−θ+0(θ−θ+0),θ∈(θ+0,θ−0).
• If then

 P+(θ)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩\vspace0.2cmP(φ+,−θ)−θ¯V+,θ≥θ+0,\vspace0.2cmP(φ−,−θ)−θ¯V+,θ≤θ−0,\vspace0.2cmP+(θ−0)+P+(θ+0)−P+(θ−0)θ+0−θ−0(θ−θ−0),θ∈(θ−0,θ+0),

where is given by (1.5).

It follows that the function is a concave continuously differentiable function with some linear parts.

The proof of this theorem is postponed to Appendix B.

### 2.4. Consequences

Let and be the associated chemical potential, see (1.3). Let us first observe that the equation of the tangent to the curve of at is given by

 Y=−E−log(χ(ρ)).

This is the unique point where the tangent has a slope equal to . Since is a concave function, the curve of is strictly below the tangent apart from the point . Moreover, , i.e. is to the right of the point where the function has its maximum.

This permits to show that is a non negative convex function which vanishes for a unique value of equal to . We recall that is the large deviations function of the random variables under .

From this we recover the result of Bahadoran ([1]) in the case of the TASEP. We also extend some of the results of [6] to the asymmetric simple exclusion process.

###### Theorem 2.4.

In the thermodynamic limit, the Gibbs-Shanonn entropy of the non-equilibrium stationary state defined by

 S(μss,N)=∑η∈ΩN[−μss,N(η)log(μss,N(η))]

is equal to the Gibbs-Shanonn entropy of the local equilibrium state, i.e.

 limN→+∞S(μss,N)2N+1=S(¯ρ).

Moreover, the corresponding fluctuations are Gaussian with a variance equal to the one provided by a local equilibrium statement, i.e.

 σ=S′′¯ρ,¯ρ(S(¯ρ))=12∫1−1χ(¯ρ)[(−s)′(¯ρ(u))]2du.

## 3. Cooperative TASEP

In this section we present the main results of the article in the case of the cooperative TASEP. We start by deriving the variational formula for the entropy function (1.6) for the cooperative TASEP. Let be the set

 F:={φ∈C1([−1,1]):φ(±1)=φ±,φ′>0}.

and recall that represents the mobility and is given by .

The quasi-potential of the cooperative TASEP ([8], [2]) is defined by

 V−(ρ)=−S(ρ)+supφ∈FH(ρ,φ)−¯V− (3.1)

where is defined in (1.7), is defined in (2.2) and

 ¯V−=log(maxρ∈[ρ−,ρ+]χ(ρ)).

### 3.1. Energy bands

In this section we determine the energy band of the cooperative TASEP. This is summarized in Figure 5.

###### Proposition 3.1.

The bottom of the energy band is given by

 E−−∞=−¯V−+⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−log(2),ρ−≤12≤ρ+,−log(1+e−φ+),ρ−<ρ+≤12,−log(1+eφ−),12≤ρ−<ρ+,

and the top of the energy band by is given by

 E+−∞=−¯V−−log(1+e−φ0)

where .

### 3.2. Entropy

We are now in position to state the main result of this section.

###### Theorem 3.2.

The restriction of the entropy function on the energy band is given by

 S−(E)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−(E+¯V−)1{(E+¯V−)≤s(ρ0)}+Sρ0,ρ0(−(E+¯V−))1{(E+¯V−)>s(ρ0)},ρ−≤12≤ρ+,Sρ+,ρ+(−(E+¯V−))1{(E+¯V−)s(ρ−)},ρ−<ρ+≤12,Sρ−,ρ−(−(E+¯V−))1{(E+¯V+)s(ρ+)},12≤ρ−<ρ+.

Moreover, the function is concave, its derivative is continuous on the energy band but its second derivative