Non-equilibrium Isothermal transformations in a temperature gradient from a microscopic dynamics
We consider a chain of anharmonic oscillators immersed in a heat bath with a temperature gradient and a time-varying tension applied to one end of the chain while the other side is fixed to a point. We prove that under diffusive space-time rescaling the volume strain distribution of the chain evolves following a non-linear diffusive equation. The stationary states of the dynamics are of non-equilibrium and have a positive entropy production, so the classical relative entropy methods cannot be used. We develop new estimates based on entropic hypocoercivity, that allow to control the distribution of the position configurations of the chain. The macroscopic limit can be used to model isothermal thermodynamic transformations between non-equilibrium stationary states.
Key words and phrases:Hydrodynamic limits, relative entropy, hypocoercivity, non-equilibrium stationary states, isothermal transformations, Langevin heat bath
2000 Mathematics Subject Classification:60K35,82C05,82C22,35Q79
Macroscopic isothermal thermodynamic transformations can be modeled microscopically by putting a system in contact with Langevin heat bath at a given temperature . In  a chain of anharmonic oscillators is immersed in a heat bath of Langevin thermostats acting independently on each particle. Macroscopically equivalent isothermal dynamics is obtained by elastic collisions with an external gas of independent particles with maxwellian random velocities with variance . The effect is to quickly renew the velocities distribution of the particles, so that at any given time it is very close to a maxwellian at given temperature. The chain is pinned only on one side, while at the opposite site a force (tension) is acting. The equilibrium distribution is characterized by the two control parameters (temperature and tension). The total length and the energy of the system in equilibrium are in general non-linear functions of these parameters given by the standard thermodynamic relations.
By changing the tension applied to the system, a new equilibrium state, with the same temperature , will be eventually reached. For large , while the heat bath equilibrates the velocities at the corresponding temperature at time of order 1, the system converges to this global equilibrium length at a time scale of order . In  it is proven that the length stretch of the system evolves in a diffusive space-time scale, i.e. after a scaling limit the empirical distribution of the interparticle distances converges to the solution of a non-linear diffusive equation governed by the local tension. Consequently this diffusive equation describes the non-reversible isothermal thermodynamic transformation from one equilibrium to another with a different tension. By a further rescaling of the time dependence of the changing tension, a so called quasi-static or reversible isothermal transformation is obtained. Corresponding Clausius equalities/inequalities relating work done and change in free energy can be proven.
The results of  summarized above concern isothermal transformations from an equilibrium state to another, by changing the applied tension. In this article we are interested in transformations between non-equilibrium stationary states. We now consider the chain of oscillators immersed in a heat bath with a macroscopic gradient of temperature: each particle is in contact with thermostats at a different temperature. These temperatures slowly change from a particle to the neighboring one. A tension is again applied to the chain. In the stationary state, that is now characterized by the tension and the profile of temperatures , there is a continuous flow of energy through the chain from the hot thermostats to the cold ones. Unlike the equilibrium case, the probability distribution of the configurations of the chain in the stationary state cannot be computed explicitly.
By changing the applied tension we can obtain transitions from a non-equilibrium stationary state to another, that will happen in a diffusive space-time scale as in the equilibrium case. The main result in the present article is that these transformations are again governed by a diffusive equation that takes into account the local temperature profile. The free energy can be computed according to the local equilibrium rule and its changes during the transformation satisfy the Clausius inequality with respect to the work done. This provides a mathematically precise example for understanding non-equilibrium thermodynamics from microscopic dynamics.
The results in  where obtained by using the relative entropy method, first developed by H.T.Yau in  for the Ginzburg-Landau dynamics, which is just the over-damped version of the bulk dynamics of the oscillators chain. The relative entropy method is very powerful and flexible, and was already applied to interacting Ornstein-Uhlenbeck particles in the PhD thesis of Tremoulet  as well as many other cases, in particular in the hyperbolic scaling limit for Euler equation in the smooth regime [13, 4]. This method consists in looking at the time evolution of the relative entropy of the distribution of the particle with respect to the local Gibbs measure parametrized by the non-constant tension profile corresponding to the solution of the macroscopic diffusion equation. The point of the method is in proving that the time derivative of such relative entropy is small, so that the relative entropy itself remains small with respect to the size of the system and local equilibrium, in a weak but sufficient form, propagates in time. In the particular applications to interacting Ornstein-Uhlenbeck particles [14, 9], the local Gibbs measure needs to be corrected by a small recentering of the damped velocities due to the local gradient of the tension.
The relative entropy method seems to fail when the stationary measures are not the equilibrium Gibbs measure, like in the present case. The reason is that when taking the time derivative of the relative entropy mentioned above, a large term, proportional to the gradient of the temperature, appears. This term is related to the entropy production of the stationary measure. Consequently we could not apply the relative entropy method to the present problem.
A previous method was developed by Guo, Papanicolaou and Varadhan in  for over-damped dynamics. In this approach the main step in closing the macroscopic equation is the direct comparison of the coarse grained empirical density in the microscopic and macroscopic space scale. They obtain first a bound of the Dirichlet form (more precisely called Fisher information) from the time derivative of the relative entropy with respect to the equilibrium stationary measures. This bound implies that the system is close to equilibrium on a local microscopic scale, and that the density on a large microscopic interval is close to the density in a small macroscopic interval (the so called one and two block estimates, see  chapter 5).
In the over-damped dynamics considered in , the Dirichlet form appearing in the time derivative of the relative entropy controls the gradients of the probability distributions with respects to the position of the particles. In the damped models, the Dirichlet form appearing in the time derivative of the relative entropy controls only the gradients on the velocities of the probability distribution of the particles. In order to deal with damped models a different approach for comparing densities on the different scales was developed in , after the over-damped case in , based on Young measures. Unfortunately this approach requires a control of higher moments of the density that are difficult to prove for lattice models. Consequently we could not apply this method either in the present situation.
The main mathematical novelty in the present article is the use of entropic hypocoercivity, inspired by . We introduce a Fisher information form associated to the vector fields , defined by (2.27). By computing the time derivative of this Fisher information form on the distribution at time of the configurations, we obtain a uniform bound . This implies that, at the macroscopic diffusive time scale, velocity gradients of the distribution are very close to positions gradients. This allows to obtain a bound on the Fisher information on the positions from the bound on the Fisher information on the velocities. At this point we are essentially with the same information as in the over-damped model, and we proceed as in . Observe that the Fisher information we introduce in (2.27) is more specific and a bit different than the distorted Fisher information used by Villani in , in particular is more degenerate. On the other hand the calculations, that are contained in appendix D are less miraculous than in , and they are stable enough to control the effect of the boundary tension and of the gradient of temperature. This also suggests that entropic hypocoercivity seems to be the right tool in order to obtain explicit estimates uniform in the dimension of the system.
Adiabatic thermodynamic transformations are certainly more difficult to be obtained from microscopic dynamics, for some preliminary results see [13, 4, 1, 10]. Equilibrium fluctuations for the dynamics with constant temperature can be treated as in . The fluctuations in the case with a gradient of temperature are non-equilibrium fluctuation, and we believe that can be treated with the techniques of the present article together with those developed in the over-damped case in .
Large deviations for the stationary measure also require some further mathematical investigations, but we conjecture that the corresponding quasi-potential functional () is given by the free energy associated to the local Gibbs measure, without any non-local terms, unlike the case of the simple exclusion process.
The article is structured in the following way. In section 2 we define the dynamics and we state the main result (Theorem 2.1). In section 3 we discuss the consequences for the thermodynamic transformations from a stationary state to another, the Clausius inequality and the quasi-static limit. In section 4 are obtained the bounds on the entropy and the various Fisher informations needed in the proof of the hydrodynamic limit. In section 5 we show that any limit point of the distribution of the empirical density on strain of the volume is concentrated in the weak solutions of the macroscopic diffusion equation. The compactness, regularity and uniqueness of the corresponding weak solution, necessary to conclude the proof, are proven in the first three appendices. Appendix D contains the calculations and estimates for the time derivative of the Fisher information .
2. The dynamics and the results
We consider a chain of coupled oscillators in one dimension. Each particle has the same mass, equal to one. The configuration in the phase space is described by . The interaction between two particles and is described by the potential energy of an anharmonic spring. The chain is attached on the left to a fixed point, so we set . We call the interparticle distance.
We assume V to be a positive smooth function, that satisfy the following assumptions:
there exists a constant such that:
there exists a constant such that:
In particular these conditions imply for some constant . Notice that this conditions allows potentials that may grow like for large , with .
Energy is defined by the following Hamiltonian function:
The particle dynamics is subject to an interaction with an environment given by Langevin heat bath at different temperatures . We choose as slowly varying on a macroscopic scale, i.e. for a given smooth strictly positive function , such that .
The equations of motion are given by
Here are -independent Wiener processes, is the coupling parameter with the Langevin thermostats. The time is rescaled according to the diffusive space-time scaling, i.e. is the macroscopic time. The tension changes at the macroscopic time scale (i.e. very slowly in the microscopic time scale). The generator of the diffusion is given by
where is the Liouville generator
while is the operator
2.1. Gibbs measures
For constant, and homogeneous, the system has a unique invariant probability measure given by a product of invariant Gibbs measures :
where is the energy of the particle :
The function is the Gibbs potential defined as:
Notice that, thanks to condition (2.1), is finite for any and any . Furthermore it is strictly convex in .
The free energy of the equilibrium state is given by the Legendre transform of :
The corresponding convex conjugate variables are the equilibrium average length
and the tension
2.2. The hydrodynamic limit
We assume that for a given initial profile the initial probability distribution satisfies:
for any continuous test function . We expect that this same convergence happens at the macroscopic time :
where satisfies the following diffusive equation
We say that is a weak solution of (2.18) if for any smooth function on such that and we have
In appendix C we prove that the weak solution is unique in the class of functions such that:
Let the inhomogeneous Gibbs measure
Let the density, with respect to , of the probability distribution of the system at time t, i.e. the solution of
where is the adjoint of with respect to , i.e. explicitly
Define the relative entropy of with respect to as:
We assume that the initial density satisfy the bound
We also need some regularity of : define the hypercoercive Fisher information functional:
where , and . We assume that
with growing less than exponentially in . We will show in Appendix D that for any we have .
Furthermore we assume that
for any continuous test function .
Furthermore a local equilibrium result is valid in the following sense: consider a local function such that for some positive finite constants we have the bound
where is the local support of . Let the length of , and let be the shifted function, well defined for , and define
3. Non-equilibrium thermodynamics
We collect in this section some interesting consequences of the main theorem for the non-equilibrium thermodynamics of this system. All statements contained in this section can be proven rigorously, except for one that will require more investigation in the future. The aim is to build a non equilibrium thermodynamics in the spirit of [3, 2]. The equilibrium version of these results has been already proven in .
As we already mentioned, stationary states of our dynamics are not given by Gibbs measures if a gradient in the temperature profile is present, but they are still characterized by the tension applied. We denote these stationary distributions as non-equilibrium stationary states (NESS). Let us denote the density of the stationary distribution with respect to .
It is easy to see that
In fact, since and
In order to study the transition from one stationary state to another with different tension, we start the system at time with a stationary state with tension , and we change tension with time, setting for . The distribution of the system will eventually converge to a stationary state with tension . Let be the solution of the macroscopic equation (2.19) starting with . Clearly , as .
3.1. Excess Heat
The (normalized) total internal energy of the system is defined by
It evolves as:
is the (normalized) work done by the force up to time , while
is the total flux of energy between the system and the heat bath (divided by ). As a consequence of theorem 2.1 we have that
where , the total macroscopic length at time . While for the energy difference we expect that
where is the average energy for , i.e.
with . Unfortunately (3.5) does not follow from (2.33), since (2.31) is not satisfied. Consequently at the moment we do not have a rigorous proof of (3.5). In the constant temperature profile case, treated in , this limit can be computed rigorously thanks to the use on the relative entropy method  that gives a better control on the local distribution of the energy.
Since as , it follows that
and the energy change will become
where is the limit of (3.4), which is called excess heat. So equation (3.6) is the expression of the first principle of thermodynamics in this isothermal transformation between non–equilibrium stationary states. Here isothermal means that the profile of temperature does not change in time during the transformation.
3.2. Free energy
Define the free energy associated to the macroscopic profile :
Correspondingly the free energy associated to the macroscopic stationary state is:
A straightforward calculation using (2.19) gives
and after the time limit
i.e. Clausius inequality for NESS. Notice that in the case constant, this is just the usual Clausius inequality (see ).
3.3. Quasi-static limit and reversible transformations
The thermodynamic transformation obtained above from the stationary state at tension to the one at tension is an irreversible transformation, where the work done on the system by the external force is strictly bigger than the change in free energy.
In thermodynamics the quasi-static transformations are (vaguely) defined as those processes where changes are so slow such that the system is in equilibrium at each instant of time. In the spirit of  and , these quasi static transformations are precisely defined as a limiting process by rescaling the time dependence of the driving tension by a small parameter , i.e. by choosing . Of course the right time scale at which the evolution appears is and the rescaled solution satisfy the equation
By repeating the argument above, equation (3.10) became:
By the same argument used in  for constant, it can be proven that the last term on the right hand side of (3.12) converges to as , and that for almost any and . Consequently in the quasi-static limit we have the Clausius equality
This implies the following equality for the heat in the quasi-static limit:
analogous of the equilibrium equality .
4. Entropy and hypercoercive bounds
In this section we prove the bounds on the relative entropy and the different Fisher informations that we need in the proof of the hydrodynamic limit in section section 5. These bounds provide a quantitative information on the closeness of the local distributions of the particles to some equilibrium measure.
In order to shorten formulas, we introduce here some vectorial notation. Given two vectors , define
We also use the notations
Observe that with this notations we can write
where and the denotes the usual scalar product in . Then we define the following Fisher informations forms on a probability density distribution (with respect to ):
Let the solution of the forward equation (2.22). Then there exist a constant such that
Taking the time derivative of the entropy we obtain:
So that, using (2.23), we have
Recall that , then the time integral of the second term on the RHS of (4.6) gives
By the entropy inequality, for any , using the first of the conditions (2.1),
By integration by part and Schwarz inequality, for any we have
By our assumptions on and assumption (2.3) on , we have that for some constant depending on and ,