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

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.


Introduction
Temperature and heat are notions that strictly pertain to the realm of Thermodynamics, i.e. to macroscopic objects whose microscopic states allow Local Thermodynamic Equilibrium (LTE). 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. LTE implies that macroscopic observables are not affected by microscopic fluctuations: this is part of the very definition of thermodynamic quantities.
Distinguishing features of thermodynamic laws are their applicability to space and time scales of our daily life, which are well separated from the microscopic scales as well as from the cosmological scales. Thermodynamic laws enjoy wide universality and robustness with respect to parameter variations (including variations of boundary conditions and of interaction potentials), and they make no reference to ensembles of objects: they hold for single systems. One direct consequence of LTE are the hydrodynamic equations, i.e. balance equations for locally conserved quantities, such as mass, momentum and energy [1,2,3]. In the presence of small to moderate driving, one observes the linear regime of non-equilibrium thermodynamics, a situation characterized by the linear transport coefficients.
Fluctuations are of course present in systems made of particles, see e.g. [4,5]; they may be observed and they even play a major role on certain scales and under certain conditions. This motivates a considerable fraction of current research in statistical physics, e.g. [6,7], devoted to phenomena occurring at scales much smaller than the macroscopic ones, or occurring in low dimensional (1D and 2D) systems [8,9,10,11,12]. In these phenomena, the linear transport coefficients do not always seem to exist and the robustness and universality of the thermodynamic laws appear to be violated. In particular, this is the case of phenomena at the nanometric scale, whose laws, unlike the thermodynamic laws, appear to be strongly affected by boundary conditions and by all parameters of interest [13,14,15,16,17,18,19]. This is quite natural in e.g. highly confined transport, since fluid particles interact more with the walls of their containers than with each other. Similarly, contact resistances appear to strongly depend on the microscopic details of the interactions at the interface between two materials, because such interfaces are bottlenecks for the quantities to be transported.
One interpretation of these facts is that LTE is violated in some situations, hence that thermodynamic concepts may be inappropriate [14,15]. Another interpretation is that thermodynamic notions should be generalized to properly treat a wider than thermodynamic variety of systems, including small systems. It is indeed interesting to investigate the validity and universality of the microscopic counterparts of thermodynamic quantities, in situations in which LTE is not expected to hold. This has led to the discovery of a host of (thermodynamically) anomalous behaviors, i.e. of violations of the standard thermodynamic laws.
Within the realm of 1D chains of oscillators, we investigate whether such anomalies are intrinsic to the phenomena under investigation, or are peculiar to certain definitions. For temperature, in particular, there are numerous approaches meant to extend the equilibrium definition to nonequilibrium situations, see e.g. [20,21,22,23].
In particular, keeping in mind that the fluid and solid properties are not properly established in 1D systems, we observe that chains of oscillators look more like some kind of (non-standard) fluids than like solids. Such a point of view has been expressed in the literature, see e.g. Refs. [24,25,26], in which a fluid-like (possibly fluctuating) continuum description is adopted. The fluid-like behaviour is usually attributed to the loss of crystalline structure caused by large position fluctuations that affect 1D systems [27].
In the present paper we consider chains of N oscillators and we find that: • contact with one thermostat at a given temperature may induce order O(N ), i.e. "macroscopic", fluctuations [the incoherent O( √ N ) vibrations typical of 3D systems in equilibrium states are thus replaced by collective (convective) macroscopic motions, even though the chain is bounded by still walls]; • the interaction with thermostats at different temperatures induces O(N ) distortions of the equilibrium lattice, resulting in highly inhomogeneous chains.
This may not be the case of models with on-site potentials, but it implies that the usual microscopic definitions of temperature appropriate for 3-dimensional homogeneous solids, may not be appropriate in general for 1-dimensional systems. We therefore consider a hierarchy of three "temperatures", [28] that treat differently the contributions of single particles. Moreover, we investigate the possibility of producing linear profiles in space, in cases in which the kinetic temperature profile is not linear as a function of particle number, cf. [29]. In particular, we find that: • despite local mesoscopic notions should be thermodynamically more relevant than single particle notions, two distinct mesoscopic averages of the kinetic energy, that depend on space, behave like the kinetic temperature, that depends on particle number and not on space; • the third mesoscopic definition of temprature, that we denote by Z, yields linear profiles in space in cases in which the kinetic temperature profile is not linear as a function of particle number; • the linear relation between kinetic temperature and inverse density of 1D systems, discovered in Ref. [15], is confirmed for different interaction potentials and for 2D systems as well, indicating a possible universality; • even interpreting the kinetic temperature as the temperature of a fluid, the linear relation of Ref. [15] is only vaguely reminiscent of the ideal gas law and, in accord with existing literature [24,25,26], it indicates that our 1D chains differ from standard fluids; • neglecting that even under LTE conditions the O(N ) collective position fluctuations frustrate the direct identification of energy transport and heat diffusion, usual definitions of heat flow reveal several inconsistencies (e.g. the steady state flow arising from the exact energy balance varies, rather than being constant, along the chain); • adopting nevertheless this identification, and taking gradients of Z as temperature gradients, the heat transport remains anomalous.

Chains of oscillators and "temperature" profiles
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 > 0 is the lattice spacing. Let each particle interact with its nearest neighbor via the Lennard-Jones potential (LJ): where r is the distance between nearest neighbors: r i = x i −x i−1 and ǫ > 0 is the depth of the potential well. Thus, x i = ai, with i = 0, . . . , N + 1, is a configuration of stable mechanical equilibrium for the system. We also consider chains in which interactions involve first and second nearest neighbors, with the second interaction potential given by [30]: 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, . . . , 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 θ L and θ R . Introducing the forces the equations of motion are given by: in the case of nearest neighbors interaction. For first and second neighbors interactions, we have: The hard-core nature of the LJ potentials implies that the order of particles is preserved, meaning that the sequence of inequalities 0 < x 1 < x 2 < · · · < x N −1 < x N < (N + 1)a holds at all times, provided it holds at the initial time [31]. For this kind of systems, a local form of virial relation is often found to hold and that is usually taken as a justification of the identification of local average kinetic energy and local temperature [32] (see e.g. Ref. [33] for a nonequilibrium mesoscopic view): where p i is the momentum of particle i, the angular brackets · denote time average and T i is called kinetic temperature. In the case in which T L = T R , the kinetic temperature profile may take rather peculiar forms, compared to the thermodynamic temperature profiles. This is illustrated in great detail in the specialized literature, cf. [11,15,29,32,34,35] just to cite a few. Certain S-shaped profiles define kinds of universality classes, that can be analytically expressed [29]. At the same time, numerically simulated profiles of various kinds of 1D systems, appear to be sensitive to all parameters, such as the relaxation constants of the thermostats, the interaction parameters etc. [15]. These observations are not surprising, since many correlations persist in space and time, hindering the realization of LTE in low dimensional systems [14,26,36,37,38]. This makes the dynamical quantities strongly dependent on boundary conditions, as well as on all the details affecting the dynamics and opens the doors to anomalous behaviors.
Special choices of parameters may give the impression that thermodynamic states are obtained, if normal mass diffusion or linear kinetic temperature profiles are established [14,39,40,41]. However, such states are not genuinely thermodynamic, since they are not robust with respect to variations of boundary conditions and other parameters of the dynamics at hand.
All that considered, it ought to be noted that the thermodynamic temperature is an intrinsically collective property, emerging in the dynamics of very large assemblies of particles, in correspondingly large, with respect to the microscopic scales, volumes (for solids, the volumes are fixed in space).
Moreover, surface effects must be negligible with respect to bulk effects, so that fluctuations due to the molecular motions in and out of the volume are negligible and the time scales for the establishment of the local equilibrium must be very fast compared to the observation time scales [5].
One question is whether the anomalies of the T i profiles are due to the fact that T i is too crude an approximation of the thermodynamic temperature (e.g. it is not a collective property of particles and it is a function of the particle number rather than of space) or they arise for other reasons.
In the following sections we numerically investigate some of the issues concerning temperature, heat flux and the fluid or solid nature of our oscillators chains. We have simulated systems with a number of particles N ranging from 64 to 2000. The parameters defining the Lennard-Jones potentials are ǫ = 1 and a = 1, while the mass of the particle is m = 1. The relaxation times of the thermostats θ L and θ R are set to 1. The numerical integrator used is the fourth-order Runghe-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.

Alternative temperatures
Given the validity of the local virial relation, one obvious mesoscopic counterpart of the kinetic temperature is the kinetic cell temperature defined as follows: subdivide the interval [0, (N + 1)a] in cells C k , k = 1, ..., M , that are sufficiently large to house a large number of particles. The borders of these cells are fictitious: particles can move freely back and forth through them. At a given instant of time τ , let the number of particles in cell C k be denoted by n k (τ ) and introduce the quantity: where χ k (x i (τ )) vanishes if the position of particle i at time τ , x i (τ ), does not belong to C k and equals 1 if x i (τ ) belongs to C k . The quantity T k (τ ) is the instantaneous mean kinetic energy per particle, within C k , at time τ . Averaging T k (τ ) in time, yields: where τ max is the total number of time steps [42]. Note that the time averaging operation does not commute with averaging over particles, in Eq. (12), hence the contribution of one particle to T C (k) weighs little if the particle spends a short time in C k , whether many or just a few particles have visited C k . Thus, in the event that all the particles visiting C k spend a short time in it, the time averaging of (12) results in a low cell temperature T C (k). This looks reasonable from certain points of view, especially when good mixing properties are realized. However, at least in principle, even a short time may suffice for a considerable amount of energy to flow out of C(k), as if there is a high temperature in C(k). Furthermore, one cannot easily ensure that the good conditions characterizing LTE states be realized in 1D systems such as ours.
The quantity T C , which is the time average of the mean kinetic energy per particle in C k , is one mesoscopic counterpart of the kinetic temperature T i , in principle quite different from T i , because: a) it is mesoscopic, as required by thermodynamics; b) it depends on space rather than on the particle number.
Nevertheless, the T C profiles do not substantially differ from the kinetic temperature profiles, for the chains with first and second neighbors Lennard-Jones interaction potential; cf. Fig.1 for the T C T C is a function of space, not of the particle number, but it converges to the kinetic temperature profile, that is a function of the particle number.
profiles of different numbers N of particles and different numbers M of cells. Figure 1 shows that increasing M , when N is sufficiently large, reproduces better and better the kinetic temperature profile. Note that even for M = N the cell C k does not contain a single particle, because C k is fixed in space, while the particles positions undergo large fluctuations, actually O(N ) fluctuations as discussed in Section 3. Therefore, the convergence of T C to the kinetic temperature, must be related to the fact that nearby particles collectively and coherently oscillate, as noted e.g. in Ref. [26]. Indeed, this fact implies that nearby particles, moving like one block, have similar kinetic energies, hence it does not matter which of them is in cell C k , at a given time t. Consequently, switching from kinetic temperature to the locally mesoscopically averaged kinetic temperature, which under LTE would correspond to the thermodynamic temperature, does not result in any substantial profile variation. This conclusion is rather robust. For instance, adding interactions with third nearest neighbors leads to the same conclusions.
In Refs. [40,43], another notion of local mesoscopic temperature, has been developed for systems of non-interacting particles, such as billiards, that can be modified to treat systems of interacting particles, as follows. Take a long observation time interval and let u(i, k) be the cardinality of I(i, k), the set of time steps in which particle i lies in cell C k . Compute first the time average of the kinetic energy of particle i, conditioned to the time that this particle has spent in cell C k : and then average over the particles that have visited cell C k , using as a weight the time spent by each of them in C k : The quantity K T (k) is a weighted average, over the particles that have visited C k , of their conditional time averaged kinetic energy K(i, k). Because in (14) the time average of the kinetic energy of a single particle, say particle i, is obtained by normalizing with the time that the particles have spent in the cell (and not with the total time of the simulation as in (12)), the contribution of particle i to K T (k) may be large even if the time it spent in C k is short. Alternatively, all particles may have spent only a short time in C k , or particle i is the only one that has visited C k . This suffices to conclude that K T (k) differs in principle from both T C (k) and T i . However, tests on our chains of oscillators show that profiles of K T (k) converge to those of the kinetic temperature, like T C (k) does.
To obtain a quantity that differs even in practice from the kinetic temperature, we insist on the idea that particles may considerably matter even if they spend a short time in a given cell. To this end, we introduce the cumulative kinetic energy K C (k) of cell C k : in which all particles that have visited C k count the same. Being K(i, k) an average conditioned to the event x i (τ ) ∈ C k , the rate of energy contributed by particle i to K C (k) is independent of the extent of its permanence in C k and it may be large even if the time spent by i in the cell is small. As a consequence, K C grows with N , and it must be normalized with a normalizing factor that may depend on the particles interactions. Indeed, observe first that K C is approximately linear as a function of k, apart from contact resistances, that are larger at larger N ; cf. Fig.2 in which the linear fits of K C determined neglecting these boundary effects are drawn. Adopting different interaction potentials involving only first and second nearest neighbors leads, within our numerical accuracy, to the same profiles obtained for interactions among first, second and third nearest neighbors. One can estimate the rate c at which the slopes of the linear profiles grow with N , as done in Fig.3 for the case of Fig.2, in which the fitting parameter is estimated to be c ≈ 0.266. Then, the rescaled temperature yields approximately linear profiles that converge for growing N to an asymptotic profile, Fig.4. Unlike the cases of Ref. [41], in which the linear kinetic temperature profile results from special choices of parameters, Z is rather robust with respect to parameters variations and it is a function of space, rather than of the particles labels. Note that for 3D macroscopic systems, in which sufficiently rapid correlations decay allows LTE, Z is equivalent to the kinetic temperature. Indeed, if energy equipartition is verified, all degrees of freedom contribute the same amount of energy and all averages eventually lead to the thermodynamic temperature. On the other hand, correlations persist for long times or never decay in 1D systems [44] and the shape of temperature profiles may heavily depend on the details of the microscopic definition of temperature. This is indeed the case of our systems.

Fluid-like behaviour
Let us begin considering relation (5) of Ref. [15], that linearly links the kinetic temperature T with the "inverse density" of particles, i.e. the average distance between nearest neighbors: in 1D systems or the square of the average distance in 2D square lattices, where r i,j is the position of particle (i, j). With this notation, Eq.(5) of Ref. [15] can be written as: where β 1 and β 2 depend on the details of the dynamics and T is a function of the particle label, rather than of space. In order to encompass both the 1D and 2D cases, we have dropped this dependence from the previous equation. As already pointed out in Ref. [15] and confirmed by our present results, this linear relation is robust with respect to parameters variations and to modifications of the interaction potentials. Here, this fact is referred to 1D systems with first and second nearest neighbors Lennard-Jones potentials, as well as with first, second and third neighbors Lennard-Jones potentials. In Fig.5, x i − x i−1 is linearly fitted to ẋ 2 i for N = 128, 256, 512. The good accuracy of these linear fits is given in terms of R 2 in Table 1, where the robustness of (19) with respect to parameters variations is also evidenced. This fact is further strengthened by tests of Eq. (19) for 2D systems, that similarly satisfy the linear relation between kinetic temperature and inverse density, as functions of particle labels.
Provided the kinetic energy of single particles is related to temperature, the validity of Eq. (19) suggests that a kind of constant pressure ideal gas condition, T /P = const/ρ, may hold, where T is the local temperature, ρ is the local density, P is the constant pressure, and the "volume" is fixed, with minor variations, by the repulsive part of the potential of the boundary particles. This is not precisely the case, since β 2 is not negligible. Therefore, the system at hand cannot be considered a genuine ideal gas, from the perspective of kinetic temperature, cf.  [46], that holds for general mechanical systems; see also Ref. [33].  Table 1: Parameters of relation (19) for the cases of Fig.5. The coefficients β 1 and β 2 do not appear to depend on N or on the kind of interaction, which suggests a universal relation. Furthermore, β 2 is sizeable, indicating that the system at hand does not behave like standard ideal gases.
This apparently obvious consideration leads to a reflection on the interpretation of the results for 1D systems. The distinction between the different states of aggregation of matter is not strictly possible in 1D systems with short range interactions, but one nevertheless realizes that our oscillators chains are more similar to compressible fluids than to solids.
We observe indeed two facts, extending the work of Ref. [15]. The first is that temperature differences at the boundaries of the chains induce macroscopic deformations of the periodic structure of the lattice: for all i, one obtains ( x i − ia) ∼ O(N ). This fact is illustrated by Fig.6 and, in particular, by its right panel, where max i ( x i − ia) is plotted as a function of N .
The second fact is that the presence of thermostats at different temperatures enhances to macroscopic scales the size of the vibrations of each particle i about its average position x i . Such vibrations are order O(i 1/2 ) in isolated systems; thus, for sufficiently large particle index i they are larger than the equilibrium lattice distances and they destroy the periodicity required by a crystal [27]. In our framework, the length of chains is bounded, therefore the size of particle vibrations cannot grow monotonically with particle index i: the vibrations are larger for particles in the bulk than for particles near the boundaries of the chains.
We may nevertheless investigate the trend of vibrations in a given region of the chain, as N gets larger and larger. We find that the size of vibrations is macroscopic, i.e.
In the right panel of Fig.7 and in Fig.8 square root fits and linear fits are compared. The square root fits are appropriate for small N , while at large N the linear fit appears to take over. The size of these vibrations appears even more striking considering that the repulsive part of the LJ potential does not allow particles' order to be modified, hence displacing by a large amount one of them, a whole collection of particles has to be correspondingly displaced. Therefore, the motion about the average position of a given particle is not analogous to an irregular motion about some fixed position; it looks closer to convective macroscopic motions, although LTE cannot be invoked and the fluid at hand does not enjoy the standard hydrodynamic properties [15,24,25,26].
Consequently, in these cases, energy transport cannot be directly related to "heat" flows.     The situation is different at equilibrium, i.e. for T L = T R . In Fig.9, it is shown, for different values of N , that the mean displacements from the mechanical equilibrium positions of the particles are much smaller than the lattice spacing a: the computed values of ( x i − ia) practically vanish and do not depend on N . The standard deviation of the vibrations about the mean position is represented in left panel of Fig.10 and it appears to lie between O( √ N ) and O(N ). Even in this case in which there is no net energy transport, the system behaves more like a fluid than like a solid.

Heat flux
In 1D systems, the usual (microscopic) definitions of heat flux commonly lead to size dependent notions of heat conductivity k, that diverge as the number of particles N grows.
Two expressions concerning the transport of energy are usually considered. The first is the exact energy balance, Eq.(23) in Ref. [32], which in the case of first and second nearest neighbors interactions reads: where F 1 and F 2 are defined by Eq.(3) and h i is the energy of the i-th particle. This quantity is only apparently "local" because it quantifies a flow through the position of particle i, and not through a fixed position in space. The second expression is Eq. (17) in Ref. [32], which in our case reads: As explained in Ref. [32], Eq. (21) is an approximation of Eq. (20), that holds when the vibrations of particles about their mechanical equilibrium position are small, so that density fluctuations can be neglected. In principle, then, Eq.(21) describes the flow of energy in the chain less accurately than Eq. (20), especially because of the large fluctuations described in the previous section. In reality, Eq. (20) is also subjected to some approximation, since it is derived through a Fourier analysis, in the limit of small wave vectors which, in macroscopic objects for which the heat flows make sense, correspond to convection and not just to diffusion [32]. Apart from that, the lattice distortions and the particles fluctuations described in Section 3, suffice to frustrate the identification of the energy flow with a heat flow, in our 1D systems. Furthermore, in a stationary state, the heat flow should not depend on position [32]. Remarkably, we find instead that the stationary time average of J i does not depend on i, while J i does. In Figs.11 and 12, we report data concerning a chain of N = 64 particles with nearest neighbors Lennard-Jones interaction (i.e. F 2 ≡ 0 in Eq. (20) and Eq. (21)). In particular, for several temperature gradients, consider the relative variation of J i , Table 2 shows that δ increases significantly as the temperature gradient is increased. In Fig.11 the difference between J i and J i is shown. It is worth to note that the mean valueJ of the J i is closely approximated by the constant value J = J i , i = 1, . . . , N , as illustrated by Fig.12, where T L = 1 and T R = 10. The same figure shows also the dependence ofJ and J on N , that isJ = J ∼ N −0.69 . For comparison, we report in Tab.3 the relative variation δ computed in a chain with first and second nearest neighbors interactions. In order to asses the convergence, we consider two sets of runs with 2 · 10 9 and 4 · 10 9 time steps. 2.3 · 10 −5 3.9 · 10 −6 2.2 · 10 −6 4.6 · 10 −6 2.0 · 10 −5 0.0001 0.0006 0.0033 Table 2: Relative variation δ of the average fluxes J i and δ of J i , respectively defined by Eq. (20) and (21). A chain with N = 64 particles and nearest neighbors interactions is considered. Averages are computed over 4 · 10 9 time steps and, moreover, T L = 1 while T R takes eight different values.  Table 3: Relative variation δ of the fluxes J i computed for a chain of N = 64 particles and first and second nearest neighbors interactions. As in Tab.2, T L = 1 while T R takes eight different values. δ 1 is computed averaging over 2 · 10 9 time steps and δ 2 over 4 · 10 9 time steps. The relative variation for T R = 64 of J i is δ 2 = 0.004.  The dependence of J i on the particle number i may be related to the fact that the diffusion and convection contributions to the energy flow about particle i cannot be separated from each other. Thus, the "exact" flow (20) does not denote a genuine heat flow. This may be reconciled with Eq.(29) of Ref. [32], or with page 462 of Ref. [9], assuming that the approximations through which that expression has been derived, require small temperature gradients. For instance, Eq.(23) of Ref. [32] follows from Eq.(21) of the same reference only if k(x n+1 − x n ) is small, but this quatity sensibly varies in space and time, for large N and positive gradients. Also, the heat flux is a collective property that emerges at the mesoscopic scale, when LTE holds, which is not our case. The above results could thus be blamed on these facts.
We may nevertheless investigate the properties of Fourier's law where k is the thermal conductivity, J is the stationary state value of the flux computed according to either Eq. (20) or Eq. (21) and ∇T is given in terms of Z, rather than in terms of the kinetic temperature. Because of its dependence on i, J i cannot be used to compute k. However, one may indifferently useJ or J for such a calculation, since they are equal. Thus, let us introduce the quantity where m Z is the slope of the linear part of the Z profile, a is the lattice spacing and N the number of particles. This definition may in principle affect the calculation of the thermal conductivity k. For instance, using the extensive quantity K C to express ∇T , one finds that k tends to zero as N grows, cf. left panel of Fig.13. On the other hand, using Z, we obtain k ∼ N 0.43 , cf. right panel of Fig.13. This rate of growth is not far from other estimates of k based on the kinetic temperature, for chains with nearest neighbors interactions (see Wang et al. in [11]) and it indicates that modifications of the notions of heat flux and/or of temperature may lead to the intermediate, non-anomalous behavior.

Concluding remarks
In this work we have presented numerical results concerning several kinds of one dimensional systems of nonlinear oscillators, in contact with two heat baths. In particular we have scrutinized quantities that are commonly considered in the specialized literature, because low dimensional systems are bound to violate LTE, the condition under which the robustness (universality) of thermodynamic laws is guaranteed. At the same time, LTE is but a sufficient condition for the thermodynamic laws to hold and its violations do not necessarily imply a breakdown of these laws and of their robustness. Therefore, the question arises about the definitions of the quantities of physical interest and of their dependence on LTE [15,14].
In this respect, one first observes that thermodynamic properties emerge from the collective behavior of very large assemblies of interacting particles, if correlation decay rapidly compared to observation time scales and boundary effects are negligible. This is often the case of 3D mesoscopic cells containing large numbers of properly interacting particles. Considering the kinetic temperature in 1D systems, we have shown that grouping particles together does not suffice to obtain thermodynamic behaviors. We have also shown that a mesoscopic definition of temperature that yields linear profiles does not suffice to cure the divergence of the heat conductivity, computed from the usual definitions of heat flow. However, these definitions reveal various inconsistencies, in addition to the fact that O(N ) collective motions prevent the direct identification of energy transport and heat flow.
The O(N ) fluctuations suggest that the behavior of 1D systems is closer to that of some kind of fluid than to that of solids. However, these motions are only vaguely reminiscent of convective motions of fluids. Furthermore, temperature differences at the boundaries of the systems of interest produce macroscopic, O(N ), deformations of the regular lattice, that result in strongly inhomogeneous systems. This should be taken into account when defining the heat conductivity and will be the subject of a future paper.