Derivation of the Fick’s law for the Lorentz model in a low density regime
We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick’s law, with the diffusion coefficient determined by the Green-Kubo formula.
G. Basile]email@example.com A. Nota]firstname.lastname@example.org F. Pezzotti]email@example.com M. Pulvirenti]firstname.lastname@example.org
- 1 Introduction
- 2 The model and main results
- 3 Proofs
- 4 The Hilbert expansions
- 5 The kinetic description
One of the most important and challenging problem in the rigorous approach to non-equilibrium Statistical Mechanics is the characterization of stationary nonequilibrium states exhibiting transport phenomena such as energy or mass transport, which are macroscopically described by Fourier’s and Fick’s law respectively. A simple microscopic model to validate the Fick’s Law is the Lorentz gas, namely a system of non interacting light particles in a distribution of scatterers, in contact with two mass reservoirs. One expects that under a suitable space-time scaling (hydrodynamical limit) the stationary mass current is proportional to the gradient of the density. However the rigorous proof of that is a difficult and still open problem.
In this paper we propose a contribution in this direction in a situation of low density. The system we study is the following. Consider the two-dimensional strip In the left and in the right of the boundaries, and respectively, there are two mass reservoirs constituted by free point particles at equilibrium at different densities , . Inside the strip there is a random distribution of hard disks of radius , distributed according to a Poisson law with density . Here is a small scale parameter and we let it go to zero. In the mean time is diverging in such a way that and . Therefore the scatterer configuration is dilute.
The light particles are flowing through the boundaries, from right with density and from left with density They are not interacting among themselves, but are elastically reflected by the obstacles. Their mean free paths vanish as , but not too quickly. More precisely they can vanish at most as in order to have a dilute configuration of scatterers.
We expect that there exists a stationary state for which
where is the mass current, is the mass density and is the diffusion coefficient. Formula (1.1) is the well known Fick’s law which we want to prove in the present context.
We underline preliminary that our result holds in a low-density regime. This means that we can use the linear Boltzmann equation as a bridge between our original mechanical system and the diffusion equation. This basic idea has been used in [ESY] [BGS-R] [BNP] to obtain the heat equation from a particle system in different contexts. It works once having an explicit control of the error in the kinetic limit, which suggests the scale of times for which the diffusive limit can be achieved. As a consequence the diffusion coefficient is given by the Green-Kubo formula for the kinetic equation at hand (namely linear Quantum Boltzmann for [ESY], linear Boltzmann for [BGS-R], linear Landau for [BNP]). In the present paper we work in a stationary situation for which we face new problems which will be discussed later on.
The idea of using the linear Boltzmann equation for the Lorentz gas in not new. In [LS] the authors consider exactly our system but with two thermal reservoirs at different temperatures at the boundaries. The aim was to study the energy flux in a stationary regime. However, as pointed out in [LS], due to the energy conservation of a single elastic collision, the energy is not diffused, there is no local equilibrium and hence the local temperature is not defined. As a consequence the Fourier’s law fails to hold, at least in the conventional sense.
This is the reason why we consider here the mass transport, being the heat equation for the mass density the unique hydrodynamical equation.
Our paper is organized as follows. The starting point is the transition from the mechanical system to the Boltzmann equation in a low density regime. We follow the classical analysis due to Gallavotti [G], complemented by an explicit analysis of the bad events preventing the Markovianity, in the same spirit of [DP], [DR] . This is necessary to reach a diffusive behavior on a longer time scale as in [BGS-R], [BNP].
Moreover we point out that our initial boundary value problem presents a new feature due to the presence of the first exit (stopping) time. This difficulty is handled by an extension procedure which essentially reduces our problem to the corresponding one in the whole space.
The transition from the mechanical system to the linear Boltzmann regime is presented in Section 5.
However we are interested in a stationary problem. This is handled, more conveniently, in terms of a Neumann series to overcome problems connected with the exchange of the limits , . To the best of our knowledge this is a new tool. This analysis is presented in Section 3. The basic idea is that the explicit solution of the heat equation and the control of the time dependent problem allow us to characterize the stationary solution of the linear Boltzmann equation and this turns out to be the basic tool to obtain the stationary solution of the mechanical system which is the basic object of our investigation.
Finally the transition from Boltzmann to the diffusion equation is classical and ruled out by the Hilbert expansion method which is presented in Section 4. This step is discussed in detail, not only for completeness, but also because we need an apparently new analysis in , for the time dependent problem (needed for the control of the Neumann series) and a analysis for the stationary problem.
2. The model and main results
Let be the strip . We consider a Poisson distribution of fixed hard disks (scatterers) of radius in and denote by their centers. This means that, given , the probability density of finding obstacles in a bounded measurable set is
where and .
A particle in moves freely up to the first instant of contact with an obstacle. Then it is elastically reflected and so on. Since the modulus of the velocity of the test particle is constant, we assume it to be equal to one, so that the phase space of our system is .
We rescale the intensity of the obstacles as
where, from now on, is fixed and is slowly diverging as . More precisely we make the following assumption.
As , diverges in such a way that
We denote by the probability density (2.1) with replaced by . will be the expectation with respect to the measure restricted on those configurations of the obstacles whose centers do not belong to the disk of center and radius .
For a given configuration of obstacles , we denote by the (backward) flow with initial datum and define , , as the first (backward) hitting time with the boundary. We use the notation to indicate the event such that the trajectory , , never hits the boundary. For any the one-particle correlation function reads
where and the boundary value is defined by
with the density of the uniform distribution on and . Here denotes the horizontal component of the velocity . Without loss of generality we assume . Since , from now on we will absorb it in the definition of the boundary values . Therefore we set
Here we allow overlapping of scatterers, namely the Poisson measure is that of a free gas. It would also be possible to consider the Poisson measure restricted to non-overlapping configurations, namely the Gibbs measure for a systems of hard disks in the plane. However the two measures are asymptotically equivalent and the result does hold also in the last case.
Note also that the dynamics is well defined only almost everywhere with respect to .
We are interested in the stationary solutions of the above problem. More precisely for any solves
The main result of the present paper can be summarized in the following theorem.
For sufficiently small there exists a unique stationary solution for the microscopic dynamics (i.e. satisfying (2.5)). Moreover, as
where is the stationary solution of the heat equation with the following boundary conditions
The convergence is in .
Some remarks on the above Theorem are in order. The boundary conditions of the problem depend on the space variable only through the horizontal component. As a consequence, the stationary solution of the microscopic problem, as well as the stationary solution of the heat equation, inherits the same feature. This justifies the convergence in instead of in . The explicit expression for the stationary solution reads
Next we discuss the Fick’s law by introducing the stationary mass flux
and the stationary mass density
Note that is the total amount of mass flowing through a unit area in a unit time interval. Although in a stationary problem there is no typical time scale, the factor appearing in the definition of , is reminiscent of the time scaling necessary to obtain a diffusive limit.
Theorem 2.2 (Fick’s law).
Observe that, as expected by physical arguments, the stationary flux does not depend on the space variable. Furthermore the diffusion coefficient is determined by the behavior of the system at equilibrium and in particular it is equal to the diffusion coefficient for the time dependent problem.
Remark (The scaling).
We have formulated our result in macroscopic variables . Another point of view is to argue in terms of microscopic variables.
Let us set our problem in these variables denoted by . This means that the radius of the disks is unitary while the strip, seen in micro-variables, is .
To deal with a low density situation, we rescale the density as , where is gently diverging. Note that in the usual Boltzmann-Grad limit At times of order , one particle has an average number of collisions of order . At larger times, namely of order , we expect a diffusive behavior. Actually this emerges from the linear Boltzmann equation (see equation (3.10) and Proposition 3.2 below) which is derived from the microscopic dynamics through the scaling and .
In this paper we consider a two dimensional case but our techniques apply in higher dimensions as well since in this case the pathological events are less likely. Moreover we consider the easier geometrical setting. However we believe that there are no serious obstructions to extend our results to more general geometries.
In this section we prove Theorems 2.1 and 2.2, postponing the technical details to the next sections. In order to prove Theorem 2.1 our strategy is the following. We introduce the stationary linear Boltzmann equation
where is the linear Boltzmann operator defined as
and the outward normal to the hard disk (see Figure 1). Here is the impact parameter, namely with the angle of incidence.
Since the boundary conditions depend on the space variable only through the horizontal component, the stationary solution inherits the same feature, as well as and .
The strategy of the proof consists of two steps. First we prove that there exists a unique which converges, as , to given by (2.8). See Proposition 3.3 below. Secondly we show that there exists a unique asymptotically equivalent to . See Proposition 3.6 below. This result is achieved by showing that the memory effects of the mechanical system, preventing the Markovianity, are indeed negligible.
Let be the solution of the problem
Then has the following explicit representation
with defined in (2.4). Given , denotes the trajectory whose position and velocity are
The transitions are obtained by means of a scattering with an hard disk with impact parameter via (3.3). As before , , is the first (backward) hitting time with the boundary. We remind that
In formula (3.5) results as the sum of two contributions, one due to the backward trajectories hitting the boundary and the other one due to the trajectories which never leave . Therefore we set
where and are respectively the first and the second sum on the right hand side of (3.5). Observe that solves
We denote by the Markov semigroup associated to the second sum, namely
with In particular
We observe that , solution of (3.1), satisfies, for
so that we can formally express as the Neumann series
Note that is a fixed point of the map solution to (3.4). Hence belongs to a periodic orbit, of period , of the flow . But this orbit consists of a single point because the Neumann series, being convergent, identifies a single element. This implies that is constant with respect to the flow (3.4) and hence stationary.
We now establish existence and uniqueness of by showing that the Neumann series (3.7) converges. In order to do it we need to extend the action of the semigroup to the space , namely
for any Here is the characteristic function of .
There exists such that for any and for any we have
As a consequence there exists a unique stationary solution satisfying (3.1).
To prove Proposition 3.1 we have first to exploit the diffusive limit of the linear Boltzmann equation in a setting and in the whole space. We introduce the solution of the following rescaled linear Boltzmann equation
with is a smooth function of the variable only (local equilibrium).
We can prove
Let be the solution of (3.10), with an initial datum . Then, as , converges to the solution of the heat equation
where is given by the Green-Kubo formula
The convergence is in
Proof of Proposition 3.1.
We can rewrite (3.8) as
where is the first component of . Note that the insertion of is due to the constraint . Therefore
We denote by a mollified version of , namely , , and . Therefore
The series on the right hand side of (3.13) defines a function which solves
where is the solution of (3.11) with initial datum . Here and in the sequel denotes a positive function vanishing with . On the other hand
Therefore for small enough
We are using (3.13) for .
Finally, since , by (3.7) we get
Remark ( vs. ).
The control of the Neumann series (3.7) in a setting seems quite natural. This is provided by the bound (3.9). It basically means that for a time the probability of a backward trajectory to fall out of is strictly positive. To prove rigorously this rather intuitive fact, we use Proposition 3.2 and explicit properties of the solution of the heat equation. The price we pay is to develop an Hilbert expansion analysis (see Section 4.1) which is, however, interesting in itself. On the other hand the use of the well known version of Proposition 3.2 requires a control of the Neumann series which seems harder, weaker and less natural.
The last step is the proof of the convergence of to the stationary solution of the diffusion problem
By using again the Hilbert expansion technique (this time in ) we can prove
The proof is postponed to Section 4.2.
This concludes our analysis of the Markov part of the proof.
Recalling the expression (2.3) for the one-particle correlation function , we introduce a decomposition analogous to the one used for , namely
Here is the contribution due to the trajectories that do leave at times smaller than , while is the contribution due to the trajectories that stay internal to . We introduce the flow such that
and remark that is just the dynamics ”inside” . In particular
To detect the stationary solution for the microscopic dynamics we proceed as for the Boltzmann evolution (see (2.5)) by setting, for ,
and we can formally express the stationary solution as the Neumann series
To show the convergence of the series (3.19) and hence existence of we first need the following two Propositions.
Let . For any
where solves (3.6).
The proof of the above two Propositions is postponed to Section 5. As a corollary we can prove
For sufficiently small there exists a unique stationary solution satisfying (2.5). Moreover
We prove the existence and uniqueness of the stationary solution by showing that the Neumann series (3.19) converges, namely
In fact, since
with , for sufficiently small (remind that as ). This guarantees the existence and uniqueness of the microscopic stationary solution .
In order to prove we compare the two Neumann series representing and ,
As regard to the second sum on the right hand side of (3.25) we have
One could try to characterize and in terms of the long (macroscopic) time asymptotics of and . The trick of expressing both stationary states by means of Neumann series avoids the problem of controlling the convergence rates, as , with respect to the scale parameter .
We conclude by proving Theorem 2.2 which actually is a Corollary of the previous analysis.