Second order Boltzmann-Gibbs principle for polynomial functions and applications

# Second order Boltzmann-Gibbs principle for polynomial functions and applications

Patrícia Gonçalves Patrícia Gonçalves Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,1, 1049-001 Lisboa, Portugal. 33email: patricia.goncalves@math.tecnico.ulisboa.ptMilton Jara Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil. 66email: mjara@impa.brMarielle Simon Équipe MEPHYSTO, Inria Lille – Nord Europe, 40 avenue du Halley, 59650 Villeneuve d’Ascq, France. 99email: marielle.simon@inria.fr    Milton Jara and Marielle Simon Patrícia Gonçalves Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,1, 1049-001 Lisboa, Portugal. 33email: patricia.goncalves@math.tecnico.ulisboa.ptMilton Jara Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil. 66email: mjara@impa.brMarielle Simon Équipe MEPHYSTO, Inria Lille – Nord Europe, 40 avenue du Halley, 59650 Villeneuve d’Ascq, France. 99email: marielle.simon@inria.fr
###### Abstract

In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in gj2014 (). The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of super-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in GubJar (); GubPer (), in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.

###### Keywords:
Boltzmann-Gibbs principle, equilibrium fluctuations, stochastic Burgers equation, Ornstein-Uhlenbeck process.

## 1 Introduction

The classical Boltzmann-Gibbs principle, introduced in TB (), states that the space-time fluctuations of any local field associated to a conservative model can be written as a linear functional of the conservative field, denoted here by . A second-order Boltzmann-Gibbs principle has been introduced in gj2014 () in order to investigate the first-order correction of this limit, in which case it is given by a quadratic functional of the conservative field .

In gj2014 () the proof of that result was based on a multiscale analysis as done in G (), assuming that the underlying particle system is of exclusion type and for which a spectral gap inequality holds. Then, it has been extended to other dynamics, like, for example, zero-range models in GJS (). We give here a new proof of that second-order Boltzmann-Gibbs principle, without requiring a spectral gap inequality. The latter was a crucial ingredient in both gj2014 (); GJS (). More precisely, the multiscale analysis was exposed in two main steps: the first one is reminiscent from the well-known one-block estimate, which consists in replacing a local function by its average on a microscopic block; the second one is reminiscent from the well-known two-blocks estimate and consists in a key iterative bound to replace the aforementioned average on a microscopically big block by an average on a macroscopically small block.

Here we look at specific local fields whose additive functionals can be written as polynomials. We follow step by step the multiscale analysis argument, after decomposing suitably the polynomials, in such a way that there is no need to apply a spectral gap inequality. In FGS (), Franco, Gonçalves and Simon already improved the proof of this second-order Boltzmann Gibbs principle in order to fit exclusion processes with one slow bond, for which the arguments of gj2014 (); GJS () do not apply.

In addition, here we prove the convergence of the fluctuation field . Provided that the second order Boltzmann-Gibbs principle is satisfied, we can formulate some simple consequences from it. The first one, is that for super-diffusive systems – for example, the asymmetric simple exclusion process – the density fluctuation field, when properly centered and re-scaled, does not evolve up to a certain time scale. For diffusive systems – for example, weakly asymmetric rates as considered in gj2014 () – it is known that the sequence of processes is tight. From our Botzmann-Gibbs principle we can also prove that any of the limit points of is an energy solution of the stochastic Burgers equation as done in gj2014 (). In order to characterize the convergence for this type of processes we notice that very recently, Gubinelli and Perkowski GubPer () obtained the uniqueness of energy solutions as defined in GubJar (). From our estimates it is simple to check that the limit points of are concentrated on energy solutions of the stochastic Burgers equation in the sense of GubPer (), from which the convergence of the sequence follows.

To sum up, in this paper we complete the result of gj2014 () and we extend the field of its applications to new interacting particle systems. There are still other models that could be solved by our new approach, like kinetically constrained exclusion processes and zero-range processes, for which the spectral gap inequality is not known. This is a subject for future work.

Here follows an outline of this paper. In Section 2 we present our main results: the second order Boltzmann-Gibbs principle and the convergence of the sequence of fluctuation fields for systems evolving in different regimes of time. We also give a quick review on the notion of stationary energy solutions of the stochastic Burgers equation and we explain how to obtain the convergence of the fluctuation field to the energy solution of the stochastic Burgers equation, starting from the second-order Boltzmann-Gibbs principle. Section 3 is devoted to applications of this principle to several models. In Sections 4 and 5 we give the complete proof of the main tool, namely the second-order Boltzmann-Gibbs principle, for degree two polynomial functions and higher degree functions, respectively.

## 2 Framework and statement of the results

In this section we introduce the notation and the main results of this paper. In order to make the presentation as general as we can, we consider the interacting particle systems evolving in a certain time scale and we detail all the assumptions that we need for our method to work.

### 2.1 The microscopic dynamics

Let be a scaling parameter and fix . We are interested in the evolution of a Markov process in the accelerated time scale , defined through its infinitesimal generator . This process belongs to the class of conservative one-dimensional interacting particle systems, with state space . For instance, if the model is of exclusion type, then (so that there is at most one particle per site), whereas for hamiltonian oscillators, or (the dynamics being on positions and velocities).

We need three assumptions for our method to work. The first one involves the invariant measures, more precisely:

###### Assumption 2.1 (Invariant measures)

We assume that the process has a family of invariant measures denoted by , where represents the range of values for the parameter. These measures are associated to the conserved quantity: which we call density. For any we assume that

1. is a product measure on ;

2. is invariant by translation, so that for any ;

3. has finite first moments,

 ∫Ω|η(x)|kνρ(dη)<+∞,for k=2,3,4.
###### Remark 1

Assumptions and above imply that is invariant with respect to the change of variables , for any , with given by

 ηz,z+1(x)=⎧⎨⎩η(z+1);x=z,η(z);x=z+1,η(x);x≠z,z+1.

In fact, we do not need to require to be product, but it should be at least invariant under the permutation of nearest neighbouring coordinates, and translation invariant. In that case, we would also have to assume that, for any and , the following bound holds:

 ∫(∑x∈Zv(x)τxf(η))2νρ(dη)≤C∑x∈Zv2(x),

where, for , we denote by the translated operator that acts on a function as , and is the configuration obtained from by shifting: for , .

We denote by the centered variable and the variance:

 χ(ρ):=∫Ω(η(x)−ρ)2νρ(dη).

Let us fix once and for all . The generator can be decomposed in into its symmetric and antisymmetric parts, more precisely we write

 Ln=An+Sn,

where

 Sn=(Ln+L∗n)/2andAn=(Ln−L∗n)/2,

with being the adjoint of in . By the conservation law, for any , there exists a function defined on such that

 Lnη(x)=jnx−1,x(η)−jnx,x+1(η)

and is called the instantaneous current of the system at the bond . To fix notation we denote

 Snη(x)=jn,sx−1,x(η)−jn,sx,x+1(η)%andAnη(x)=jn,ax−1,x(η)−jn,ax,x+1(η),

so that . We denote by the Dirichlet form associated to the Markov process, which is defined on local functions as

 Dn(f)=−∫Ωf(η)Lnf(η)νρ(dη)=−∫Ωf(η)Snf(η)νρ(dη).

The second assumption that we need is the following:

###### Assumption 2.2 (Dirichlet form)

There exists a bounded function with such that the Dirichlet form reads as where

 Inz,z+1(f):=∫Ωζnz,z+1(η)(∇z,z+1f(η))2νρ(dη), (1)

with , .

Assumption 2.2 may look restrictive but is actually valid for many models of interest. For example, lattice gas dynamics, either symmetric or asymmetric, with positive jumps rates, fall into this category (see Sections 3.1 and 3.1 below).

The path space of right-continuous and left-limits trajectories with values in is denoted by . For any initial probability measure on , we denote by the probability measure on induced by and the Markov process . If we denote and its expectation by .

Our process of interest is the density fluctuation field, defined on functions in the Schwartz space , as

 Ynt(H)=1√n∑x∈ZH(xn)(ηntna(x)−ρ).

Note that, by Dynkin’s formula, for

 Mnt(H):=Ynt(H)−Yn0(H)−∫t0naLn(Yns(H))ds

is a martingale. Let us define

A simple computation shows that the integral part of can be written as

 Int(H):=∫t0na−1√n∑x∈Z∇nH(xn)(jnx,x+1(ηnsna)−Enρ[jnx,x+1(η)])ds. (2)

Finally, our last assumption is related to the decomposition of the current. For a function , let us define the centered variable

 ¯ψ(η):=ψ(η)−Enρ[ψ(η)].
###### Assumption 2.3 (Instantaneous current)

There exists a local function and a constant , such that for every ,

 ¯jn,sx,x+1(η)=τxh(η)−τx+1h(η)and¯jn,ax,x+1(η)=C¯η(x)¯η(x+1)+τxg(η),

where is a local function such that for all

 limn→∞Enρ[(∫t0na−1√n∑x∈Z∇nH(xn)τxg(ηnsna)ds)2]=0. (3)
###### Remark 2

The last condition (3) may look strong, but what me mean here is that the error produced by is small compared to the one induced by the degree two polynomial , as it is the case for example for polynomials of degree greater or equal than 3 (see Theorem 5.1 below).

According to the Assumption 2.3 and by a summation by parts, we can rewrite in the following way:

 ∫t0na−2√n∑x∈ZΔnH(xn)τxh(ηnsna)ds+C∫t0na−1√n∑x∈Z∇nH(xn)¯ηnsna(x)¯ηnsna(x+1)ds, (4)

plus a term which is negligible in and given in (3). We notice that the first claim of the previous assumption is satisfied by models which are of gradient type. Since for the models of interest , to treat the term on the left hand side of (4) one can use the classical Boltzmann-Gibbs principle introduced in TB () and the treatment of the term on the right hand side of (4) is the main purpose of this paper. More precisely, we look at the first-order correction for the usual limit projection of space-time fluctuations of the latter specific field. We focus on the additive functional of and show how its fluctuations can be written as a linear functional of the conservative field plus a quadratic functional of this same field. The crucial point on the proof of this result relies on sharp quantitative bounds on the error that we are able to obtain when we perform the aforementioned replacement.

### 2.2 The second-order Boltzmann-Gibbs Principle

In the following, we simply write for , for the sake of clarity. For any square summable, we denote:

 ∥v∥22,n:=n−1∑x∈Zv2(x)<∞. (5)
###### Theorem 2.1 (Second-order Boltzmann-Gibbs principle)

There exists a constant such that, for any and , and for any function :

 Enρ[(∫t0∑x∈Zv(x){¯ηsna(x)¯ηsna(x+1)−(→ηLsna(x))2+χ(ρ)L}ds)2]≤Ct{Lna−1+tnL2}∥v∥22,n (6)

where

 →ηL(x)=1Lx+L∑y=x+1¯η(y).

Last result can be extended to higher degree polynomials, provided that higher moments are finite: more precisely, if one wants to replace in (6) the function with a polynomial of degree , then condition in Assumption 2.1 has to be replaced by

 ∫Ω|η(x)|kνρ(dη)<+∞,for k=2,...,2d.

This generalization will be the main purpose of Section 5 below. Before that, let us present various of its applications.

### 2.3 Consequences of the Boltzmann-Gibbs Principle

#### Super-diffusive systems

In this section we consider systems which fulfill the assumptions above and that evolve super-diffusively so that . Recall from above that

 Mnt(H)=Ynt(H)−Yn0(H)−∫t0na−2√n∑x∈ZΔnH(xn)τxh(ηnsna)ds+C∫t0na−1√n∑x∈Z∇nH(xn)¯ηnsna(x)¯ηnsna(x+1)ds, (7)

plus a term which is negligible in and given in (3).

Since and is a local function, a simple computation shows that the first time integral above vanishes in , as goes to infinity. For the second one, we note that by the simple inequality , the second order Boltzmann-Gibbs principle stated above and the Cauchy-Schwarz inequality, we can show, as in G (), that for it also vanishes in , as goes to infinity. More details will be given ahead when we apply this result to some concrete examples. As a consequence we conclude the triviality of the fluctuations stated in the next theorem.

###### Theorem 2.2 (Trivial Limit)

For any , the sequence of processes converges in distribution with respect to the Skorokhod topology of , as , to the process given on by .

#### Diffusive systems

In this section we consider systems which fulfill the assumptions above and that evolve diffusively so that . Recall (7) and note that if we add a weak asymmetry to the system given by , for then, the last integral in the martingale decomposition reads as

 ∫t0na−1−γ√n∑x∈Z∇nH(xn)¯ηnsn2(x)¯ηnsna(x+1)ds.

In this case, as a consequence of the second order Boltzmann-Gibbs principle stated above, one can show a crossover on the fluctuations which depends on the strength of the asymmetry.

###### Theorem 2.3 (Crossover fluctuations)

The sequence of processes converges in distribution with respect to the Skorokhod topology of , as , to

• an Ornstein-Uhlenbeck process as in (8), if ,

• an energy solution of the stochastic Burgers equation as in (10), if .

Before proceeding any further, we explain in detail what do we mean by energy solution of the stochastic Burgers equation, and we state the uniqueness result of GubPer (); DieGubPer () regarding these solutions.

#### Energy solutions of the stochastic Burgers equation

Let us describe the concept of energy solutions of the stochastic Burgers equation. Fix . Let be the Schwartz space of distributions and the space of continuous paths in . We say that a process with trajectories in has zero quadratic variation if the real-valued process has zero quadratic variation for any test function . Let and be a space-time standard white noise. Let us denote by the -norm of , that is:

 ∥H∥2L2(R):=∫RH2(x)dx.
###### Definition 1

We say that a pair of stochastic processes with trajectories in is controlled by the Ornstein-Uhlenbeck process

 ∂tYt=νΔYt+√2νσ2∇Wt (8)

if:

1. for each fixed time , the -valued random variable is a white noise of variance ,

2. and the process has zero quadratic variation,

3. for each , the process

 Mt(H):=Yt(H)−Y0(H)−∫t0Ys(νΔH)ds−At(H)

is a Brownian motion of variance with respect to the natural filtration of ,

4. the reversed processes also satisfy (iii).

If , then is the unique martingale solution of the Ornstein-Uhlenbeck equation (8). The interest of the notion of controlled processes, is that it allows to define some non-trivial functions of the process . Let be an approximation of the identity and . Then we define the process as

 Bεt(H):=∫t0∫R(Ys∗ιε(x))2∇H(x)dxds,

where denotes the convolution operator. The following proposition has been proved in GubJar (); GubPer (); DieGubPer ():

###### Proposition 1

Let be controlled by the Ornstein-Uhlenbeck process given in (8). Then the limit

 Bt(f)=limε→0Bεt(f) (9)

exists in and it does not depend on the choice of the approximation of the identity . Moreover, the distribution-valued process defined in this way has zero quadratic variation.

This proposition gives a possible way to define the square of the distribution-valued process . This definition can be used to pose the Cauchy problem for the stochastic Burgers equation.

###### Definition 2

Let . We say that a stochastic process is a stationary controlled solution of the stochastic Burgers equation

 ∂tYt=νΔYt+λ∇Y2t+√2νσ2∇Wt (10)

if:

1. there exists a process of zero quadratic variation such that is controlled by the Ornstein-Uhlenbeck equation (8),

2. for any .

The importance of this definition comes from the fact that it gives uniqueness in law for the process (we refer to DieGubPer (); GubPer () for a proof):

###### Proposition 2

Any two stationary controlled solutions and of the stochastic Burgers equation have the same law.

In the context of interacting particle systems, another notion of solution is more suitable. We say that a process with trajectories in is stationary if the -valued random variable is a white noise of variance for any .

Recall that is an approximation of the identity. For each and each consider the process as in (9). For , let us define .

We say that satisfies an energy estimate if there is a finite constant such that

 E[(Bεs,t(f)−Bδs,t(f))2]≤κε(t−s)∥∇f∥2L2(R) (11)

for any , for any and any . The following proposition has been proved in gj2014 (), but it is also a consequence of the second order Boltzmann-Gibbs Principle stated above.

###### Proposition 3

Let be a process with trajectories in . Assume that is stationary and it satisfies an energy estimate. Then the process given by is well defined and it satisfies the estimate

 E[(Bt(H)−Bs(H))2]≤~κ|t−s|3/2∥∇H∥2L2(R)

for some finite constant , for any and .

This proposition gives an alternative way to make sense of the nonlinear term of the stochastic Burgers equation.

###### Definition 3

We say that a process with trajectories in is a stationary energy solution of the stochastic Burgers equation if:

1. for each the -valued random variable is a white noise of variance ,

2. the process satisfies an energy estimate,

3. for any the process

 Yt(H)−Y0(H)−∫t0Ys(νΔH)ds+λBt(H)

is a Brownian motion of variance ,

4. the reversed process also satisfies (iii).

In GubJar (); GubPer (); DieGubPer (), stationary controlled solutions of the stochastic Burgers equation are actually called energy solutions. This is due to the following result:

###### Proposition 4

Let be a stationary process satisfying an energy estimate. Then the process constructed in Proposition 3 has zero quadratic variation. In particular, the notions of stationary energy solutions and stationary controlled solutions of the stochastic Burgers equation are equivalent.

###### Proof

First we observe that

 E[n∑i=1(Bt(i+1)/n(H)−Bti/n(H))2]≤~κ∥∇H∥2L2(R)n−1/2.

This shows that a stationary energy solution is also a stationary controlled solution. But in fact, stationary controlled solutions satisfy stronger energy bounds than (11), see Section 2.3 of GubPerEBP ().

## 3 Applications to interacting particle systems

### 3.1 Exclusion processes

#### The WASEP and the stochastic Burgers equation

For this model we have and the infinitesimal generator is given by

 Lnf(η)=∑x∈Z(12+b2nγ)η(x)(1−η(x+1))∇x,x+1f(η)+(12−b2nγ)η(x)(1−η(x−1))∇x,x−1f(η),

where , see for example gj2014 (). The dynamics conserves the total number of particles and the invariant measures are given by a family of Bernoulli product measures parametrized by the density which are translation invariant, since for any , . Notice that every moment of this measure is finite, so that Assumption 2.1 holds. One can easily check that the Dirichlet form does write on the form (1) with

 ζn0,1(η) =(12+b2nγ)η(0)(1−η(1))+(12−b2nγ)η(1)(1−η(0)) =(η(0)−η(1))2+b2nγ[η(0)(1−η(1))−η(1)(1−η(0))].

Moreover one gets

Therefore, in this case , and

 g(η)=−b2nγ(1−2ρ)(¯η(x)+¯η(x+1)).

To simplify the exposition we take (hence ), nevertheless we notice that by a Galilean transformation, which removes the transport velocity to the system, one could redefine the density fluctuation field and take other values of , for more details we refer the reader to, for example, G (). In that case, the integral part of the martingale (2) can be written as

 ∫t0na−22√n∑x∈ZΔnH(xn)¯ηsna(x)ds+∫t0bna−1−γ2√n∑x∈Z∇nH(xn)¯ηsna(x)¯ηsna(x+1)ds.

Here the interesting time scale is the diffusive one, namely , so that the previous expression can be written as

 ∫t012Yns(ΔnH)ds+∫t0bn12−γ2∑x∈Z∇nH(xn)¯ηsna(x)¯ηsna(x+1)ds.

Now we sketch the proof of Theorem 2.3 in this case. By Theorem 2.1, with , together with Young’s inequality and a Cauchy-Schwarz inequality, the variance of the term on the right hand side of last expression is bounded above by , which vanishes, as , if . From this it can be shown (see gj2014 ()) that for , the limiting process is an Ornstein-Uhlenbeck process. Nevertheless, for , by Theorem 2.1 with , the term on the right hand side of last expression can be written as

plus a term that vanishes in , as and . From this one can show (see gj2014 () and Subsection 2.3) that for , the limiting process is the unique energy solution of the stochastic Burgers equation, as stated in Theorem 2.3.

#### The ASEP and the time invariance of the density fluctuation field

For this model we have and the infinitesimal generator is given by

 Lnf(η)=∑x∈Zpη(x)(1−η(x+1))∇x,x+1f(η)+(1−p)η(x)(1−η(x−1))∇x,x−1f(η),

with , see for example G (). As above, the dynamics conserves the total number of particles and the invariant measures are the Bernoulli product measures parametrized by the density . Assumption 2.2 on the Dirichlet form holds with

 ζn0,1(η) =pη(0)(1−η(1))+(1−p)η(1)(1−η(0)) =η(0)(1−η(1))+(p−1)[η(0)(1−η(1))−η(1)(1−η(0))].

We also have

 ¯jnx,x+1(η)=pη(x)(1−η(x+1))−(1−p)η(x+1)(1−η(x))−(2p−1)χ(ρ).

A simple computation shows that can be written as

 ¯jnx,x+1(η)= −(2p−1)¯η(x)¯η(x+1)−((1−p)(1−ρ)+pρ)(¯η(x+1)−¯η(x)) +(2p−1)(1−2ρ)¯η(x). (12)

As for the WASEP, we simplify the exposition by assuming , so that the previous expression reads as

 −(2p−1)¯η(x)¯η(x+1)−12(¯η(x+1)−¯η(x)),

and therefore , and . Performing a summation by parts and by the Cauchy-Schwarz inequality, the integral part of the martingale (2) can be written as

 −(2p−1)∫t0na−1√n∑x∈Z∇nH(xn)¯ηsna(x)¯ηsna(x+1)ds, (13)

plus a term which is negligible in if . For this model, the interesting time scale is a longer time scale than the hyperbolic one, so that we take , with . By Theorem 2.1 the variance of the previous term can be estimated doing the following estimates. To fix notation we denote the previous integral by . By summing and subtracting inside the sum above and by the inequality we have that

 ≤CEnρ[(∫t0nα√n∑x∈Z∇nH(xn)[¯ηsna(x)¯ηsna(x+1)−(→ηLsna(x))2+χ(ρ)L]ds)2] +CEnρ[(∫t0nα√n∑x∈Z∇nH(xn)[(→ηLsna(x))2−χ(ρ)L]ds)2]. (14)

By (6) the first expectation is bounded by

 CHn2αn{tLnα+t2nL2}.

Now, we treat the remaining expectation. By splitting the sum over intervals of size , by the independence under of and whenever , and by the Cauchy-Schwarz inequality, (14) can be bounded from above by

 t2n2αnL∑x∈Z(∇nH(xn))2∫Ω[(→ηLsna(0))2−χ(ρ)L]2νρ(dη)≤CHt2n2αL.

Putting together the previous two estimates, optimizing over , taking and , we see that the previous errors vanish as , if . Therefore we obtain the result of (G, , Theorem 2.6), which we recall here:

###### Theorem 3.1 (G ())

Fix . For any and ,

 limn→∞Enρ[(∫t0nα√n∑x∈ZH(xn)¯ηsn1+α(x)¯ηsn1+α(x+1)ds)2]=0.

From this, one can show Theorem 2.2 which says that up to the time scale the limiting density field does not evolve in time. This result is not optimal, it is conjectured that the temporal invariance of the field should go up to the time scale (see, for example, (S, , Chapter 5)).

#### Weakly asymmetric speed change exclusion processes

The speed change exclusion processes have been investigated for example in LV (); S (). Here we look at their weakly asymmetric version so that and the infinitesimal generator is given by

 Lnf(η)=∑x∈Zcx,x+1(η)(12+b2nγ)η(x)(1−η(x+1))∇x,x+1f(η)+∑x∈Zcx,x−1(η)(12−b2nγ)η(x)(1−η(x−1))∇x,x−1f(η),

where , and the rate functions satisfy the translation invariance property: there is such that . For our approach to work, we need to assume:

1. (Gradient) There exists a local function such that, for any ,

 η(0)(1−η(1))