Fick’s Law for the Lorentz Model in a weak coupling regime
Abstract
In this paper we deal with further recent developments, strictly connected to the result obtained in BNPP (). We consider the Lorentz gas out of equilibrium in a weak coupling regime. Each obstacle of the Lorentz gas generates a smooth radially symmetric potential with compact support. We prove that the macroscopic current in the stationary state is given by the Fick’s law of diffusion. The diffusion coefficient is given by the GreenKubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.
1 Introduction
The understanding of transport phenomena of nonequilibrium thermodynamics starting form the microscopic dynamics is one of the most challenging problem in statistical mechanics.
Nonequilibrium stationary states describe the state of a mechanical system driven and maintained out of equilibrium. Their main characteristic is that they commonly exhibit transport phenomena. They sustain steady flows (e.g. energy flow, particles flow or momentum flow) and the usually conserved quantities ( mass, momentum and energy) flow in response to a gradient. For instance the heat flow and the mass flow appear in response to a temperature gradient and a concentration gradient respectively. These processes are well described by phenomenological linear laws, the Fourier’s and Fick’s law respectively.
In the current literature there are very few rigorous results concerning the derivation of the these phenomenological laws from a microscopic model (see for instance [LS], [LS1], [LS2]). A contribution in this direction is the validation of the Fick’s law for the Lorentz model in a low density situation which has been recently proven in BNPP (). To consider the system out of equilibrium, in BNPP (), they study the Lorentz gas in a bounded region in the plane and couple the system with two mass reservoirs at the boundaries. More precisely they consider the slice in the plane. In the left half plane there is a free gas of light particles at density , in the right half plane there is a free gas of light particles at density which play the role of mass reservoirs. The light particles are not interacting among themselves. Inside there is a Poisson distribution of intensity of hard core scatterers. The light particles flow through the boundaries and are elastically reflected by the scatterers. For this model they prove the existence of a stationary state for which
(1) 
where is the mass current, is the mass density and is the diffusion coefficient. Formula (1) is the well known Fick’s law whose validity has been proven in BNPP (). We remind that according to the lowdensity regime considered they can use the linear Boltzmann equation as a bridge between the original mechanical system and the diffusion equation. This strategy works since they provide an explicit control of the error in the kinetic limit which suggests the scale of times for which the diffusive limit can be achieved. The result is presented in a two dimensional setting but it holds in dimension higher than two. The two dimensional case is the most interesting to analyze since the pathologic configurations preventing the Markovianity on a kinetic scale are harder to estimate in this case.
We may wonder if the same result could be achieved if we slightly modify the model. We consider the same geometry described above but inside now we have a Poisson distribution of scatterers which are no longer hard cores. We assume that each obstacle generates a smooth, radial, shortrange potential. In the same spirit as in BNP (), ESY (), we scale the range of the interaction and the density of the scatterers according to
(2) 
with and .
The scaling (2) means that the kinetic regime describes the system for kinetic times (i.e. ). Observe that when the limiting cases and correspond respectively to the low density limit and the weakcoupling limit. In the intermediate scale between the low density and the weakcoupling regime the kinetic equation that appears in the limit is the linear Landau equation. One can go further to diffusive times provided that is not too large. The intermediate level of description between the mechanical system and the diffusion equation is given by the linear Landau equation with a divergent factor in front of the collision operator. Since the scale of time for which the system diffuses should not prevent the Markov property, there is a constraint on . More precisely there exists a threshold , emerging from the explicit estimate of the set of pathological configurations producing memory effects, s.t. for , the microscopic solution of the time dependent problem converges to the solution of the heat equation in the limit . We refer to BNP (), Section 6, for further details. The result mentioned above concerns the time dependent problem. In this paper we deal with the stationary problem and provide a rigorous derivation of Fick’s law of diffusion for this model. We prove that 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. We underline that in order to obtain the stationary solution of the microscopic dynamics we need to characterize the stationary solution of the linear Landau equation. To handle this problem we will use the analysis of the time dependent problem and the explicit solution of the heat equation.
2 The model and main results
Let be the strip . We consider a Poisson distribution of fixed 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
(3) 
where and . 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. More precisely we make the following assumption.
Assumption 1
We set , the parameter is such that as ,
(4) 
namely .
Accordingly, we denote by the probability density (3) with replaced by . will be the expectation with respect to the measure .
We now introduce a radial potential such that

,

and is strictly decreasing in .
We rescale the intensity of the interaction potential as
Then the Equations of motion are
(5) 
For a given configuration of obstacles , we denote by the (backward) flow, solution of (5), 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 oneparticle correlation function reads
(6) 
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
(7) 
We are interested in the stationary solutions of the above problem. More precisely solves
(8) 
The main result of the present paper can be summarized in the following theorem.
Theorem 2.1
For sufficiently small there exists a unique stationary solution for the microscopic dynamics (i.e. satisfying (8)). Moreover, as
(9) 
where is the stationary solution of the heat equation with the following boundary conditions
(10) 
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
(11) 
where is the horizontal component of the space variable . We note that in order to prove Theorem 2.1 it is enough to assume that i.e.
The stronger Assumption 1 is needed to prove Theorem 2.2 below.
Next, to discuss the Fick’s law, we introduce the stationary mass flux
(12) 
and the stationary mass density
(13) 
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)
We have
(14) 
as . The convergence is in and is given by the GreenKubo formula
(15) 
Moreover
(16) 
where the convergence is in and
(17) 
where is the linear profile (11).
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.
3 Proofs
In order to prove Theorem 2.1 our strategy is the following. We introduce the stationary linear Landau equation
(18) 
where and is the Laplace Beltrami operator on the circle of radius , namely . Moreover we introduce the stationary linear Boltzmann equation
(19) 
where and L is the linear Boltzmann operator defined as
(20) 
with
(21) 
and is the unit vector bisecting the angle between the incoming velocity and the outgoing velocity as specified in Figure 1.
Since the boundary conditions depend on the space variable only trough the horizontal component, the stationary solution and inherit 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 (11). See Proposition 4 below. Secondly we show that there exists a unique asymptotically equivalent to . See Proposition 7 below. This result is achieved using two steps. The first one concerns the convergence of towards , the stationary solution of the linear Boltzmann equation, by showing that the memory effects of the mechanical system, preventing the Markovianity, are indeed negligible. The second one concerns the grazing collision limit which guarantees the asymptotic equivalence of and .
Let be the solution of the problem
(22) 
We can write 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
Observe that solves
(23) 
We set . Let be the semigroup whose generator is the operator , i.e. . Hence
We observe that , solution of (18), satysfies, for
so that we can formally express as the Neumann series
(24) 
We now establish existence and uniqueness of by showing that the Neumann series (28) converges. In order to do it we extend the action of the semigroup to the space , namely
for any Here is the characteristic function of and is the extension of the semigroup to the whole space . For the sake of simplicity from now on we set
As we proved in BNPP (), the same technique works for , solution of the following Boltzmann equation
(25) 
The solution of the problem (25) has the following explicit representation
(26) 
with defined in (7). 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 (21). As before , , is the first (backward) hitting time with the boundary. We remind that .
We set
Observe that solves
(27) 
Let be the Markov semigroup associated to the second sum in (26), hence . Moreover , solution of (19), satysfies, for
so that we can formally express as the Neumann series
(28) 
Proposition 1
There exists such that for any and for any we have
(29) 
As a consequence there exists a unique stationary solution satisfying (18).
To prove Proposition 1 we also need the following result
Proposition 2
For every
(30) 
Proof
We look at the evolution of , namely
(31) 
where . We observe that we can write (31) as
(32) 
Hence we can consider , in (32), as a source term. Recalling that
we set
with . Integrating with respect to and using symmetry arguments we obtain
Observe that . (See Figure 1). We remind that the scattering angle
and (see DR (), Section 3, for further details). Moreover
is the diffusion coefficient of the Landau equation, , hence
Therefore
(33) 
which vanishes for .
Proposition 3
Let . For any
(34) 
The proof is essentially the same of Proposition (2), and to let it work we observe that we need the extension procedure discussed in BNPP (), Section 5, for .
Proof (Proof of Proposition 1)
From Proposition 2.1 in BNPP (), for any , we have
(35) 
Therefore for small enough
(36) 
Finally, since , by (28) we get
The last step is the proof of the convergence of to the stationary solution of the diffusion problem
(37) 
with the diffusion coefficient given by the GreenKubo formula (15). We remind that the stationary solution to the problem (37) has the following explicit expression
(38) 
where .
By using the Hilbert expansion technique in we can prove
Proposition 4
For the proof we refer to BNPP (), Section 4.2. This concludes our analysis of the Markov part of the proof.
Recalling the expression (6) for the oneparticle correlation function , we introduce a decomposition analogous to those ones used for and , namely
(40) 
and
(41) 
so that
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 (8)) by setting, for ,
and we can formally express the stationary solution as the Neumann series
(42) 
To show the convergence of the series (42) and hence existence of we first need the following Propositions.
Proposition 5
Proposition 6
For every
(44) 
where .
See Section 5 and Section 6 in BNP (), and Section 5 in BNPP () for the proof. As a corollary we can prove
Proposition 7
For sufficiently small there exists a unique stationary solution satisfying (8). Moreover
(45) 
where .