$O(N)$ fluctuations and lattice distortions in 1-dimensional systems

TL;DR

This paper investigates how 1D Lennard-Jones chains exhibit large-scale fluctuations and lattice distortions that challenge traditional hydrodynamic descriptions and energy flow assumptions.

Contribution

It provides a novel interpretation of boundary-driven 1D systems showing $O(N)$ fluctuations and distortions, highlighting limitations of standard hydrodynamics.

Findings

Observation of $O(N)$ lattice distortions in 1D chains

Standard hydrodynamics does not apply due to large fluctuations

Proposed modifications to energy flow models

Abstract

Statistical mechanics harmonizes mechanical and thermodynamical quantities, via the notion of local thermodynamic equilibrium (LTE). In absence of external drivings, LTE becomes equilibrium tout court, and states are characterized by several thermodynamic quantities, each of which is associated with negligibly fluctuating microscopic properties. Under small driving and LTE, locally conserved quantities are transported as prescribed by linear hydrodynamic laws, in which the local material properties of the system are represented by the transport coefficients. In 1-dimensional systems, on the other hand, the transport coefficients often appear to depend on the global state, rather than on the local state of the system at hand. We interpret these facts within the framework of boundary driven 1-dimensional Lennard-Jones chains of $N$ oscillators, observing that they experience non-negligible $O(N)$ lattice distortions and fluctuations. This implies that standard hydrodynamics and certain expressions of energy flow do not apply in these cases. One possible modification of the energy flow is considered.

Tables

Table 1. Table 1: Relative variation δ 𝛿 \delta of the flux J i subscript 𝐽 𝑖 J_{i} for N = 64 𝑁 64 N=64 particles with first and second nearest neighbors interactions. T L = 1 subscript 𝑇 𝐿 1 T_{L}=1 while T R subscript 𝑇 𝑅 T_{R} takes eight different values. δ 1 subscript 𝛿 1 \delta_{1} is computed averaging over 2 ⋅ 10 9 ⋅ 2 superscript 10 9 2\cdot 10^{9} time steps, δ 2 subscript 𝛿 2 \delta_{2} over 4 ⋅ 10 9 ⋅ 4 superscript 10 9 4\cdot 10^{9} time steps.

$T_R$	${\delta }_1$	${\delta }_2$
1.1	0.0240	0.0199
1.5	0.0091	0.0077
2	0.0142	0.0145
4	0.0480	0.0481
8	0.0831	0.0829
16	0.1060	0.1062
32	0.1199	0.1201
64	0.1229	0.1232

Table 2. Table 2: Relative variation δ 𝛿 \delta of the average fluxes ⟨ J i ⟩ delimited-⟨⟩ subscript 𝐽 𝑖 \langle J_{i}\rangle defined by Eq.( 11 ). Chains with N = 64 𝑁 64 N=64 , N = 128 𝑁 128 N=128 and with N = 256 𝑁 256 N=256 particles, with nearest neighbors interactions are considered. Averages are computed over 2 ⋅ 10 9 ⋅ 2 superscript 10 9 2\cdot 10^{9} time steps. T L = 1 subscript 𝑇 𝐿 1 T_{L}=1 , while T R subscript 𝑇 𝑅 T_{R} takes eight different values.

$T_R$	$N=64$	$N=128$	$N=256$
1.1	0.0240	0.0117	0.0110656
1.5	0.0091	0.0297	0.0317283
2	0.0142	0.0534	0.0555437
4	0.0480	0.0817	0.104345
8	0.0831	0.0659	0.0907829
16	0.1060	0.0683	0.0485491
32	0.1199	0.1560	0.0643797
64	0.1229	0.2306	0.195046

Equations

$$V_1(r)=ϵ\left[{\left(\frac{a}{r}\right)}^{12}-2{\left(\frac{a}{r}\right)}^6\right],$$

$$V_2(s)=ϵ\left[{\left(\frac{2a}{s}\right)}^{12}-2{\left(\frac{2a}{s}\right)}^6\right],$$

$$F_1(r)=\frac{\partial V_1}{\partial r}(r),F_2(s)=\frac{\partial V_2}{\partial s}(s),$$

$$m{\ddot{x}}_1=F_1(x_1)-F_1(x_2-x_1)-{\xi }_1{\dot{x}}_1,$$

$$m{\ddot{x}}_i=F_1(x_i-x_{i-1})-F_1(x_{i+1}-x_i),i=2,...,N-1,$$

$$m{\ddot{x}}_N=F_1(x_N-x_{N-1})-F_1(x_{N+1}-x_N)-{\xi }_N{\dot{x}}_N,$$

$${\dot{\xi }}_1=\frac{1}{{\theta }_L^2}\left(\frac{m{\dot{x}}_1^2}{T_L}-1\right),{\dot{\xi }}_N=\frac{1}{{\theta }_R^2}\left(\frac{m{\dot{x}}_N^2}{T_R}-1\right),$$

$$m{\ddot{x}}_1=F_1(x_1)-F_1(x_2-x_1)+F_2(x_1+a)-F_2(x_3-x_1)-{\xi }_1{\dot{x}}_1,$$

$$m{\ddot{x}}_2=F_1(x_2-x_1)-F_1(x_3-x_2)+F_2(x_2)-F_2(x_4-x_2)-{\xi }_2{\dot{x}}_2,$$

$$m{\ddot{x}}_i=F_1(x_i-x_{i-1})-F_1(x_{i+1}-x_i)+F_2(x_i-x_{i-2})-F_2(x_{i+2}-x_i),i=3,\dots ,N-2,$$

$$m{\ddot{x}}_{N-1}=F_1(x_{N-1}-x_{N-2})-F_1(x_N-x_{N-1})+F_2(x_{N-1}-x_{N-3})-F_2(x_{N+1}-x_{N-1})-{\xi }_{N-1}{\dot{x}}_{N-1},$$

$$m{\ddot{x}}_N=F_1(x_N-x_{N-1})-F_1(x_{N+1}-x_N)+F_2(x_N-x_{N-2})-F_2(x_{N+2}-x_N)-{\xi }_N{\dot{x}}_N,$$

$$\begin{array}{rl}{\dot{\xi }}_l& =\frac{1}{{\theta }_L^2}\left(\frac{m{\dot{x}}_l^2}{T_L}-1\right),l=1,2,\\ {\dot{\xi }}_l& =\frac{1}{{\theta }_R^2}\left(\frac{m{\dot{x}}_l^2}{T_R}-1\right),l=N-1,N.\end{array}$$

$$T_i=⟨\frac{{p_i}^2}{m}⟩,i=1,...,N.$$

$$\begin{array}{rl}{J}_i=& \frac{1}{2}(x_{i+1}-x_i)F_1(x_{i+1}-x_i)({\dot{x}}_{i+1}+{\dot{x}}_i)\\ & +(x_{i+2}-x_i)F_2(x_{i+2}-x_i)({\dot{x}}_{i+2}+{\dot{x}}_i)+{\dot{x}}_i{h}_i,\end{array}$$

$$\delta =\left|\frac{{\max }_i⟨J_i⟩-{\min }_i⟨J_i⟩}{\bar{J}}\right|,\text{mbox}where\bar{J}=\frac{1}{N}\sum _i⟨J_i⟩,$$

$$J_i^n=\frac{J_i}{⟨x_{i+1}-x_i⟩},i=2,...,N-2.$$

Full text

††preprint: APS/123-QED

I Introduction

In a seminal paper, Rieder, Lebowitz and Lieb investigated the properties of chains of $N$ harmonic oscillators, interacting at their ends with stochastic heat baths Lebow . These authors proved that while energy flows from hot to cold baths, the kinetic temperature profile decreases exponentially in the direction of the hotter bath, rather than increasing, and in the bulk its slope vanishes as $N$ grows. Thus, in case the kinetic temperature equals the thermodynamic temperature, heat flows against the direction of energy, in the bulk of such 1D systems. Moreover, no steady state is reached, because at boundaries, heat flows in the opposite direction. Taken as a paradox without explanation in Ref.Lebow , this fact reveals that, in harmonic chains of oscillators, the kinetic temperature does not correspond to the thermodynamic temperature, or the energy flux does not represent a heat flux, or both. Unexpected phenomena that seem to contradict the hydrodynanmic laws of transport, e.g. currents going against the density gradient, a phenomenon called “uphill diffusion”, can be observed in several experimental settings; see e.g. Ref.ColGiaGibVer for further references and a nonequilibrium model with phase transition exhibiting uphill diffusion. However, the thermodynamic relevance of such models is still under investigation.

As a matter of fact, temperature and heat pertain to macroscopic objects with microscopic states corresponding to Local Thermodynamic Equilibrium (LTE); they cannot be directly identified with mechanical quantities such as kinetic energy and energy flux, Ref.LandauV §9, Chibbaro Chapters 3, 4 and 5. LTE is the essence of Thermodynamics: it can be viewed at once as the precondition for the existence of the thermodynamic fields, such as temperature and heat, and as the natural state of objects obeying the thermodynamic laws. The microscopic conditions under which LTE is expected to hold are extensively discussed in the literature, e.g. Prigogine Section 15.1, Spohn Section 2.3, Bellissard Section 3.3, Kreuzer Chapter 1. In short, LTE requires the existence of three well separated time and space scales, so that: 1. a macroscopic object can be subdivided in mesoscopic cells that look like a point to macroscopic observers, while containing a large number of molecules; 2. boundary effects are negligible compared to bulk effects, so that the contributions of neighboring cells to the mass and energy of a given cell are inappreciable within a cell; 3. particle interactions allow the cells to thermalize (positions and velocities become respectively uniformly and Maxwell-Boltzmann distributed) within times that are mere instants on the macroscopic scale.

That macroscopic observables are not affected by microscopic fluctuations, despite the exceedingly disordered and energetic microscopic motions, is essential for mesoscopic quantities to be sufficiently stable that thermodynamic laws apply, e.g. LandauV , §1 and §2. This is the case for a quantity that is spatially weakly inhomogeneous, when the number $N$ of particles in a cell is large, and the molecular interactions randomize positions and momenta so that, for instance, the fluctuations of a quantity $\phi$ of size $O(N)$ are order $O(\sqrt{N})$. The bulk of the cell then dominates in- and out-fluxes, and variations of $\phi$ are sufficiently slow on the mesoscopic scale. Quantitatively, the space and time scales for which this description holds depend on the properties of the microscopic components of the systems of interest, Prigogine ; Kreuzer ; Spohn ; Bellissard ; Pigo1 ; Pigo2 .

Under the LTE condition, matter can be considered a continuum, obeying hydrodynamic laws, i.e. balance equations for locally conserved quantities, such as mass, momentum and energy DeGroot ; CR98 ; RC02 . For small to moderate driving, they take a linear form, in which the local material properties are expressed by the linear transport coefficients. Locality implies that such coefficients do not depend on the conditions of the system far away from the considered region. The thermal conductivity of an iron bar at a given temperature at a given point in space does not depend on the conditions of the bar far from that region; cutting the bar in two, or joining it to another bar, without changing the local state, leaves unchanged its local properties.

Fluctuations remain of course present in systems made of particles; they are larger for larger systems, they may be observed AURIGA1 ; AURIGA2 , and they play a major role in many circumstances, see e.g. Refs.Mish ; HickMish . This motivates a considerable fraction of research in statistical physics, e.g. StochTherm ; CasatiRev , concerning scales much smaller than the macroscopic ones, or occurring in low dimensional (1D and 2D) systems Li1 ; Dhar ; Li2 ; LepriEd ; CasatiJ . In these phenomena, the linear transport coefficients do not always seem to exist LepriEd , the robustness and universality of the thermodynamic laws appear to be violated, and the behaviours appear to be strongly affected by boundary conditions and by all parameters that characterize a given object ExpAnomFourier ; JeppsR ; GibRon ; Zhao1 ; Zhao2 ; onorato ; Saito . It is also well known that chains of oscillators behave more like some kind of (non-standard) fluids than like solids, because of the loss of crystalline structure, caused by cumulative position fluctuations Peierls . Consequently, a fluid-like (possibly fluctuating) description has been adopted in a number of papers, cf. Refs.NaRam02 ; MaNa06 .

In driven systems, the situation is problematic also because equipartition may be violated Breaking ; Sengers ; PSV17 , the statistic describing the state of the system is model dependent, and the ergodic properties of the particles dynamics are only partially understood GGnoneq ; EWSR2016 . Hence, there is no universally accepted microscopic notion of nonequilibrium temperature MR ; Jou1 ; Jou2 ; JouBook ; Criado ; He ; PSV17 . Further, a microscopic definition of heat flux requires a clear distinction between energy transport due to macroscopic motions (convection), and transport without macroscopic motions (conduction), cf. Chapter 4 of Ref.Zema , and Section III.2 and Chapter XI of Ref.DeGroot . In 1D systems, this may not be always possible inpreparat .

One possible interpretation of these facts is that LTE is violated in some situations, hence that thermodynamic concepts may be inappropriate JeppsR ; GibRon . Another interpretation is that thermodynamic notions should be modified to treat small and strongly nonequilibrium systems, see e.g. MR ; Jou1 ; Jou2 ; He . It is therefore interesting to investigate the validity and universality of the mechanical counterparts of thermodynamic quantities, in situations in which LTE is not expected to hold, and “anomalous” phenomena have been reported.

We address such questions considering chains of $N$ Lennard-Jones oscillators interacting with deterministic baths at their ends, and without on-site potentials. Our central findings are that:

• •

thermostats at different temperatures induce $O(N)$ distortions of the equilibrium lattice, resulting in highly in-homogeneous chains;

• •

thermostats induce collective order $O(N)$ fluctuations, i.e. “macroscopic” motions. Negligible incoherent $O(\sqrt{N})$ vibrations typical of 3D equilibrium systems are thus replaced by kind of convective motions, even in chains bounded by still walls.

These observations combined with the results of Ref.Lebow and further literature, e.g. Refs.MR ; Jou1 ; Jou2 ; JouBook ; He ; Breaking , suggest that microscopic definitions appropriate for 3-dimensional equilibrium thermodynamic quantities, need extra scrutiny in 1D. In particular, we illustrate how $O(N)$ fluctuations and lattice distortions affect the collective behaviour of 1D systems, considering the notion of heat flux, $J$ say, given by Eq.(23) of Ref.LLP-PhyRep . This way, we confirm from a different standpoint the conclusions reached in previous studies on the inapplicability of standard hydrodynamics Hurtado ; Politi . We find that:

• •

$J$ is not spatially uniform in steady states. Variations of $J$ decrease if the baths temperature difference is reduced at constant $N$, but they do not if the mean temperature gradient is reduced increasing $N$ at constant baths temperatures.

• •

Dividing $J$ by the local mass density partially balances the lattice inhomogeneity and yields an approximately uniform quantity.

These observations should be combined with those of Refs.Politi ; inpreparat , according to which collective and molecular motions are correlated, making hard to disentangle convection from conduction, because single particles push their neighbors, producing kind of convective cascades. That difficulties do not ease when $N$ grows, indicates that LTE, hence thermodynamic quantities, cannot be established in our 1D systems. We relate this fact with the $O(N)$ growth of fluctuations and lattice distortions.

II Chains of Lennard-Jones oscillators

Consider a 1D chain of $N$ identical moving particles of equal mass $m$, and positions $x_{i}$, $i=1,...,N$. Add two particles with fixed positions, $x_{0}=0$ and $x_{N+1}=(N+1)a$, where $a$ is the lattice spacing. Let nearest neighbors interact via the Lennard-Jones potential (LJ):

[TABLE]

where $r$ is the distance between nearest neighbors: $r=|x_{i}-x_{i-1}|$ and $\epsilon>0$ is the depth of the potential well. Thus, $x_{i}=ai$, with $\,i=0,\ldots,N+1$, is a configuration of stable mechanical equilibrium for the system. We also consider interactions involving first and second nearest neighbors, with second potential given by 1st2ndpaper :

[TABLE]

where $s=|x_{i}-x_{i-2}|$. Further, we add two particles with fixed positions $x_{-1}=-a$ and $x_{N+2}=(N+2)a$. With potential $V=V_{1}+V_{2}$, the system has the usual stable mechanical equilibrium configuration $x_{i}=ai$, $\,i=-1,\ldots,N+2$. The first and last moving particles are in contact with two Nosé-Hoover thermostats, at kinetic temperatures $T_{L}$ (on the left) and $T_{R}$ (on the right) and with relaxation times $\theta_{L}$ and $\theta_{R}$. Introducing the forces

[TABLE]

the equations of motion are given by:

[TABLE]

with

[TABLE]

in the case of nearest neighbors interaction. For first and second neighbors interactions, we have:

[TABLE]

with

[TABLE]

The hard-core nature of the LJ potentials preserves the order of particles: $0<x_{1}<x_{2}<\cdots<x_{N}<(N+1)a$ holds at all times, if it does at the initial time InSome .

For such systems, a form of single particle virial relation is often found to hold Virial . That fact is usually mentioned to identify the average kinetic energy of a given particle with the temperature $T_{i}$ in position $x_{i}$ LLP-PhyRep :

[TABLE]

Here, $p_{i}$ is the momentum of particle $i$, the angular brackets ${\langle\cdot\rangle}$ denote time average, and $T_{i}$ is called single particle kinetic temperature. In the case in which $T_{L}\neq T_{R}$, the single particle kinetic temperature profile may take rather peculiar forms, compared to the linear thermodynamic temperature profiles in homogeneous solids when Fourier law holds. This is illustrated in great detail in the specialized literature, cf. LepriEd ; GibRon ; MejiaPoliti ; LLP-PhyRep ; LLP ; DharReview ; MejiaPoliti just to cite a few. Also, numerically simulated profiles of various kinds of 1D systems, appear to be sensitive to parameters such as the relaxation constants of the thermostats, the interaction parameters, the form of the boundaries etc. cf. e.g. Ref.GibRon . This is not surprising, since many correlations persist in space and time in low dimensional systems, hindering the realization of LTE JeppsR ; Hurtado ; Politi ; RJepps ; BiancaJR ; Salari ; inpreparat , and leading to anomalous behaviors.

In the following sections, we report our results about systems with various numbers of particles $N$. The parameters defining the Lennard-Jones potentials are $\epsilon=1$ and $a=1$, while the mass of the particles is $m=1$. The relaxation times of the thermostats $\theta_{L}$ and $\theta_{R}$ are set to

1. The numerical integrator used is the fourth-order Runge-Kutta method with step size $10^{-3}$. The time averages are typically taken over $O(10^{8})-O(10^{9})$ time steps in the stationary state.

III Large lattice deformations and fluctuations

The distinction between the different states of aggregation of matter is not strictly possible in 1D systems with short range interactions; one nevertheless realizes that our oscillators chains are more similar to (a kind of) compressible fluids than to solids Hurtado ; GibRon . In particular, Ref.Politi shows persistent correlations, $O(N)$ dependence of relaxation times, and the failure of standard hydrodynamics, in non-driven LJ systems.

Along similar lines of investigation, we find that temperature differences at the boundaries of the chains induce “macroscopic” deformations of the periodic structure of the lattice. For all $i$, we obtain $(\langle x_{i}\rangle-ia)\sim O(N)$, as shown in Fig.1, whose lower panel plots the quantity $\max_{i}(\langle x_{i}\rangle-ia)$ as a function of $N$.

Our second observation is that the presence of thermostats at different temperatures enhances the size of the vibrations of each particle $i$ about its average position $\langle x_{i}\rangle$. Such vibrations are order $O(i^{1/2})$ in chains without thermostats with origin in $i=0$ Peierls , which means that, for sufficiently large $i$, position fluctuations are incompatible with a crystal structure. In our framework, the length of chains is bounded, therefore the size of particle vibrations cannot indefinitely grow with particle index $i$: the vibrations are larger for particles in the bulk than for particles near the boundaries of the chains.

We find, however, that for every particle $i$, also the size of vibrations is “macroscopic”: $\sqrt{\langle x_{i}^{2}\rangle-\langle x_{i}\rangle^{2}}\sim O(N)$.

In the lower panel of Fig.2 and in Fig.3, square root fits and linear fits are compared for $N$ ranging from 64 to 6000. The square root fits are appropriate for small $N$, while at large $N$ the linear fit takes over. The

size of these vibrations appears even more striking observing that displacing by a large amount one of them, a whole collection of particles must be correspondingly displaced. Indeed, the repulsive part of the LJ potential does not allow particles’ order to be modified, as noted also in Ref.Politi . Therefore, the motion of particles about their average positions is not an irregular motion about fixed positions. In accord with the observations on persistent correlations, this motion looks like a kind of convection, although LTE and standard hydrodynamics do not hold GibRon ; NaRam02 ; MaNa06 ; Hurtado ; Politi . It follows that, in these cases, energy transport cannot be directly related to “heat” flows.

The situation is different for $T_{L}=T_{R}$. Figure 4 shows that the lattice deformations are much smaller than the lattice spacing $a$, and can be neglected. The computed values of $(\langle x_{i}\rangle-ia)$ practically vanish and do not depend on $N$. The standard deviation of the vibrations about the mean position is represented in upper panel of Fig.5 and it appears to be closer to $O(\sqrt{N})$ than to $O(N)$ as can be seen in lower panel of Fig.5. In this case, in which there is no net energy transport, the system also behaves more like a fluid than like a solid in sense closer to that of Peierls , although our results refers to a different situation.

IV Heat flux

In order to understand the effect of $O(N)$ fluctuations and lattice distortions on the behaviour of usual microscopic quantities, let us consider the “heat flux” given by Eq.(23) of Ref.LLP-PhyRep . For the case of first and second nearest neighbors interactions, that expression must be modified as follows:

[TABLE]

where $F_{1}$ and $F_{2}$ are defined by Eq.(3) and $h_{i}$ is the energy of the $i$-th particle.

The quantity $J_{i}$ is only apparently “local” because it quantifies a flow through the position of particle $i$, and not through a fixed position in space. Moreover, it implicitly requires small position fluctuations and small lattice deformations, because Eq.(11) is obtained through Fourier analysis for spatially homogeneous systems, in the limit of small wave vectors, LLP-PhyRep ; Dhar . For instance, denoting by $k$ the wave-vector, Eq.(23) of Ref.LLP-PhyRep follows from Eq.(21) only if $k(x_{n+1}-x_{n})$ is small. On the contrary, in our cases, this quantity strongly varies in space and time, and average lattice distortions are of order $O(N)$, cf. Section III. Therefore, one expects $J_{i}$ to fail, and it is interesting to investigate how that is realized, varying the relevant model parameters.

For chains with nearest neighbors Lennard-Jones interactions ($F_{2}\equiv 0$ in Eq.(11)), we find that while the steady state heat flow should not depend on position, the time average of $J_{i}$ substantially changes with $i$, cf. Fig.6.

To quantify this phenomenon, we introduce the relative variation of $\langle{J}_{i}\rangle$,

[TABLE]

In Tables 1 and 2, for average temperature gradients similar to those commonly found in the literature LLP ; Kabu , we observe that $\delta$ tends to grow with the temperature gradient, at fixed $N$. In general, however, reducing the average gradient by increasing the system size, does not lead to smaller $\delta$ observ .

We conclude that under our conditions the quantity $J_{i}$ represents neither a heat nor an energy flow, and that this is not a consequence of the size of temperature gradients, but of the size of fluctuations. These increase with growing $N$, thus preventing LTE and standard hydrodynamics in the large $N$ limit Hurtado ; GibRon ; Politi . In the next section we propose a modification of $J_{i}$ that, taking into account the deformations of the lattice, is more stable than $J_{i}$ along the chain.

V Concluding remarks

In this work we have presented numerical results concerning several kinds of 1D systems of nonlinear oscillators, in contact with two Nosé-Hoover thermostats. Scrutinizing the behaviour of mechanical quantities that are commonly considered in the specialized literature, we have investigated the fluctuations and lattice distortions, which are expected to prevent the establishment of “thermodynamic” regimes GibRon ; JeppsR ; JouBook ; Politi .

Thermodynamic properties emerge indeed from the collective behavior of very large assemblies of interacting particles, if correlations decay rapidly compared to observation time scales, and if boundary effects are negligible. While this is often the case of 3D mesoscopic cells containing large numbers of properly interacting particles, it is not obvious in 1D systems.

In particular, we have observed that temperature differences at the boundaries produce $O(N)$ fluctuations and deformations of the lattice, that result in strongly inhomogeneous systems. This should be taken into account when defining the heat conductivity. Furthermore, these $O(N)$ effects imply that larger $N$ is not going to make our systems closer to thermodynamic systems, when $N$ is increased, consequently standard hydrodynamics does not apply Hurtado ; GibRon ; Politi .

In the light of the above observations, the definition of energy transport (11) may be modified in order to take into account the lattice deformations. One possibility that may considered is to normalize the energy flow by the average distance between particles, introducing

[TABLE]

The result, shown in Fig.7, indicates that this avenue deserves further investigation, and will be the subject of future works.

ACKNOWLEDGMENTS

The authors are grateful to Carlos Mejia-Monasterio for extensive discussions and enlightening remarks. The authors are grateful to Antonio Politi for very useful suggestions. Antonio Politi, in particular has suggested the normalization of the energy flux. This work is partially supported by Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM). C.G. and C.V. acknowledge financial supports from “Fondo di Ateneo per la Ricerca 2016” and “Fondo di Ateneo per la Ricerca 2017”- Università  di Modena e Reggio Emilia.