1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport

Transport and diffusion of heat in one dimensional (1D) nonlinear systems which {\it conserve momentum} is typically thought to proceed anomalously. Notable exceptions, however, exist of which the rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous transport in those low dimensional systems. Here, we develop the theory for both, momentum/heat diffusion and its corresponding momentum/heat transport. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. This result in turn relates, via the integrated momentum flux correlation, to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We put forward the intriguing hypothesis that a fluid-like momentum dynamics with a {\it finite viscosity} causes {\it normal} heat transport; its corollary being that superdiffusive momentum diffusion with an intrinsic {\it diverging viscosity} in turn yields {\it anomalous} heat transport. This very hypothesis is corroborated over wide extended time scales by use of precise molecular dynamics simulations. The numerical validation of the hypothesis involves three distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted rotator model), (ii) unbounded interactions acting at large distances (the Fermi-Pasta-Ulam $\beta$ model, or the rotator model amended with harmonic pair interactions) and (iii), a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard-Jones model).


Introduction
The investigation of heat conduction in low dimensional nonlinear lattices has attracted ever increasing attention [1,2,3] in the statistical physics community. Although early relevant work [4] can be traced back to 1993, increased activity spurred since the discovery of anomalous heat conduction occurring in one dimensional (1D) momentumconserving Fermi-Pasta-Ulam (FPU)-β lattices [5] in 1997. In those low dimensional study cases the thermal conductivity κ of the FPU-β lattice was found to diverge with the lattice size N as κ ∝ N α , with 0 < α < 1. This finding consequently yields no system-size independent thermal conductivity, thus breaking Fourier's law of heat conduction. Similar anomalous heat conduction behavior has also been identified for other archetype 1D momentum-conserving stylized nonlinear systems, such as the 1D diatomic Toda lattices [6], and, importantly, has been predicted to occur in momentumconserving physical materials, such as in carbon nanotubes [7], silicon nanowires [8] and polymer chains [9]. Experimentally, the breakdown of Fourier's law has presently been confirmed for 1D carbon nanotubes and baron-nitride nanotubes [10] and 2D suspended graphene [11].
On the other hand, the low spatial dimension alone is not the sole feature that determines whether the validity of Fourier's law holds up. For example, normal heat conduction obeying Fourier's law has been established beyond doubt for 1D nonlinear Frenkel-Kontorova (FK) [12] lattices and φ 4 lattices [13,14]. For those nonlinear lattice systems the total momentum is not conserved, being due to the presence of the on-site potentials. These numerical results for 1D lattices led to a conjecture that the property of momentum-conservation in low dimensional systems might be at the origin to give rise to anomalous heat conduction for 1D and 2D nonlinear lattices, e.g. see [1,2,15,16]. It then came as a surprise that contradictory results emerged for other stylized momentum-conserved nonlinear 1D lattices, exhibiting saturated thermal conductivities such as the rotator model [17,18] and a momentum-conserving variation of the ding-a-ling model [19]. Giardinà and Kurchan also provided a family of models with or without momentum-conservation which, however, all obey Fourier's law [20]. Therefore this situation gives rise to the dilemma of what physics is at the root for the occurrence of the breakdown of the Fourier behavior in 1D nonlinear lattices [21,22]. Most recently, relying on numerical simulations, Savin and Kosevich [23] showed that thermal conduction obeys Fourier's law for 1D momentum-conserving lattices with a 1D Lennard-Jones interaction, a Morse interaction, and as well a Coulomb-like interaction. Those numerical findings let them to conclude (erroneously, see below and in particular Sect. 3.3) that normal heat conduction emerges for momentum-conserving lattices whenever the pair interaction potentials are asymptotically free at large interaction distances.
In this work, we focus on heat transport in 1D momentum-conserving nonlinear lattices from another aspect, namely, the diffusion of heat. It is acknowledged that there exists a profound connection between heat conduction and heat diffusion within the region where Fourier's law is valid. Take normal heat conduction in 1D cases for example: Fourier's law states that j = −κ∂ x T , where j denotes the local heat flux and ∂ x T is the nonequilibrium temperature gradient. If we combine it with local energy conservation; i.e., ∂ t E + ∂ x j = 0 and, additionally, the relation between the local energy density E and the temperature T , E = cT (with c being the volumetric specific heat), then the familiar heat diffusion equation ∂ t T = D∂ 2 x T can be derived with the normal heat diffusivity D, reading D = κ/c.
Microscopically, normal heat diffusion can be characterized by the mean square displacement of the corresponding Helfand moment [24], which then connects to normal heat conductivity via the Green-Kubo formula. The efforts trying to bridge heat conduction and diffusion beyond the normal case have only been put forward in the recent decade [22,25,26,27,28]. Remarkably, it is only until recently that a general and rigorous connection between heat conduction and heat diffusion has been established from basic principles [29]. It is shown that in the linear response regime, the evolution of the mean square deviation (MSD) of a general energy diffusion process is determined by the heat flux autocorrelation function of the system -the central quantity that enters the Green-Kubo formula. The key ingredient for obtaining this MSD of the energy spread relies on the energy-energy correlation function C E (x, t; x , 0) [30], as subsequently rigorously shown with the recent work [29]. This thermal equilibrium excess energy-energy correlation indeed is the fundamental quantity that determines the behavior of nonequilibrium heat diffusion, as well as the nonequilibrium heat conduction in a regime not too far displaced from thermal equilibrium. Thus, using the energyenergy correlation function, we can conveniently identify whether the heat diffusion in a nonlinear lattice occurs normal or not.
With this present study we aim to shed more light on the conundrum that underpins anomalous heat transport in 1D nonlinear lattices. In doing so we study with molecular dynamics (MD) simulation for different nonlinear 1D momentum-conserving nonlinear lattices. The first one is the 1D coupled rotator lattice which has a bounded interaction potential; i.e., the potential is bounded in configuration space and therefore the motion of the particles are not confined. The second case study uses an unbounded harmonic interaction potential in combination with the coupled rotator interaction potential. As a third 1D nonlinear system we complement the rotator model with a Lennard-Jones 1Dinteraction potential, being unbounded at short interaction distances while being free at large interaction distances. This last model thus allows for bond dissociation at large interaction distances. The correlation functions for the local excess energy deviations as well as the local excess momentum are calculated via extensive equilibrium numerical MD-simulations for all these three different classes of potential landscapes.
Our studies corroborate the result that normal heat diffusion is found for the rotator lattice possessing a bounded potential. Here, we also demonstrate that in addition to normal heat diffusion the overall dynamics is accompanied by a normal momentum diffusion. We then elucidate that these two features imply that the system dynamics is ruled by the emergence of a finite shear viscosity. This observation therefore insinuates that the 1D rotator model physically mimics a fluid behavior. In clear contrast we then find that anomalous heat diffusion occurs for momentum-conserving nonlinear 1D lattices which contain an unbounded interaction potential, as it is the case also with nonlinear FPU-lattices and also the Lennard-Jones case. The anomalous heat diffusion and corresponding anomalous heat conductivity behavior is shown to be accompanied in all those test cases with an anomalous momentum superdiffusion. This latter feature causes a divergent effective viscosity, thus mimicking physically a solid-like behavior.
These findings suggest the following hypothesis: Anomalous heat transport and heat diffusion in low dimensional 1D systems which conserve momentum is governed by an inherent diverging effective viscosity behavior. Put differently, a solid-like behavior for the momentum diffusion in momentum-conserving 1D nonlinear lattices supports anomalous transport while a finite, fluid-like viscosity behavior for momentum diffusion supports normal heat transport. While we have no mathematical physics proof available for this claim we nevertheless conjecture this findings to hold up for other 1D momentumconserving models that possess such a fluid-versus solid-like viscose behavior. In particular, our extensive numerical results are contradictory to the observation made in [23] that it is the nature of the pair interaction potential, being bounded or possessing an asymptotic free part at large interaction distances that allows for dissociation, which in turn causes a normal heat transport behavior.
The present study is organized as follows. In Section 2, we briefly review the state of the art of the theory for excess energy diffusion and then develop the theory describing the diffusion of excess momentum. In Section 3, we perform the numerical studies on an overall bounded interaction potential, namely the rotator model. This is then followed by studying a variant of this rotator model by complementing it with unbounded harmonic pair interactions. Finally we discuss the case with a Lennard-Jones pair interaction. These detailed numerical MD studies support the fact that it is not the mere presence or absence of the symmetry property of momentum conservation but rather the presence or absence of a fluid-like behavior, as characterized with a finite diffusivity, which is at the origin for the validity or the breakdown of Fourier's law behavior. For the case with nonlinear on-site potentials the momentum conservation is broken: the emergence of Fourier's Law in this latter situation is then ruled by nonlinear scattering processes which provide a finite mean free path behavior for the heat transfer [31]. Additional conclusions and remaining open issues are presented with our final section.

Diffusion of heat and momentum
Let us consider manifest momentum-conserving, homogeneous 1D nonlinear lattice systems with nearest neighbor interactions. Their Hamiltonian can be written in the general form 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport5 where p i denotes the momentum. The set q i are the displacements from the equilibrium position for the i-th atom with i = 0, ±1, ±2, ..., ±(N − 1)/2 where an odd value of N is assumed for the sake of convenience. V (q i+1 − q i ) is the interaction potential between neighboring sites i and i + 1. With H i we denote the local energy at site i. Moreover, throughout our numerical analysis we shall make use of periodic boundary conditions; i.e., we set q N +i = q i and p N +i = p i .

Heat diffusion
We start out with the description of heat diffusion in a discrete 1D lattice following Ref. [29]. In doing so, we introduce the energy-energy correlation function, reading: where ∆H i (t) ≡ H i (t) − H i (t) and · · · denotes the ensemble average over canonical thermal equilibrium at a temperature T and c is the specific heat per particle. Given this autocorrelation function of energy fluctuations, one can evaluate the time evolution of the excess energy distribution ρ E (i, t) starting out from an initial, near thermal equilibrium state, characterized by the initial excess energy perturbation ξ(i). If we consider the special case of a localized, small initial excess energy perturbation at the central site, ξ(i) = δ i,0 , we can use linear response theory for the excess energy distribution ρ E (i, t) to obtain [29]: This excess energy distribution remains normalized at all later times t, being due to the conservation of energy. The commonly used quantity which quantifies the speed of heat diffusion is the MSD ∆x 2 (t) E of the excess energy distribution. For a discrete 1D lattice with N sites one thus obtains with x(t) E = 0 This MSD has been shown to obey the salient second order differential equation [29]; i.e., where C J (t) denotes the autocorrelation function of total heat flux defined as wherein is the quantity that enters the well-known Green-Kubo expression for the thermal conductivity κ. For normal heat flow it explicitly reads, 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport6 The relation in (5) connects heat conduction with heat diffusion in a rigorous way. As a consequence, the investigation of heat conduction can equivalently be obtained from studying heat diffusion. The most important quantity is the energy fluctuation autocorrelation function C E (i, t; j = 0, 0) in Eq. (2); it encodes all the necessary information about heat diffusion and heat conduction. As one can defer from Eq. (4) and Eq. (5), the energy-energy correlation function C E (i, t; j = 0, 0) determines the dynamical behavior of the MSD of heat diffusion as well as the autocorrelation function of total heat flux C J (t) for heat conduction.
Take the FPU-β model which displays anomalous heat diffusion as example, this energy autocorrelation C E (i, t; j = 0, 0) follows a Levy walk distribution, being quite distinct from a normal Gaussian distribution in the long time limit [26,27,30]. This statistics then gives rise to a superdiffusive behavior for the energy spread, reading The corresponding, formally diverging anomalous thermal conductivity can be extracted to read Here, t s ∼ N/v s with N chosen sufficiently large presents the characteristic time-scale of heat diffusion wherein v s denotes the sound speed for inherent renormalized phonons [32].

Momentum diffusion
The scheme for the excess energy heat diffusion can likewise be generalized for the problem of corresponding diffusion of excess momentum. For a nonlinear lattices with a Hamiltonian in Eq. (1), the translational invariance of the Hamiltonian necessarily indicates that the total momentum i p i is conserved; i.e., we have by observing that Using an analogous reasoning as put forward with the preceding subsection for heat diffusion we can define the autocorrelation function for the excess momentum fluctuation [30], reading: where , observing that p i (t) = 0 in thermal equilibrium. Following the reasoning of the previous subsection we next demonstrate that this momentum-momentum autocorrelation function describes within linear response theory the diffusion of momentum along the lattice.
1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport7 To elucidate this issue we consider alike a lattice in thermal equilibrium at temperature T . We apply a small kick of short duration to the j-th particle. The kick occurs with a constant impulse I, yielding a force kick at site j as Upon integrating the equation of motion from the moment immediately before the kick (denoted as t = 0 − ) to the moment immediately after the kick (denoted as t = 0 + ), we find that the sole effect of this kick is to change the momentum of the jth particle by an amount I. The momenta of all other particles, as well as the position of all particles remain unchanged. Formally, this is recast as The full time evolution of the momenta and positions is not analytically accessible for non-integrable nonlinear lattice systems. However, given that I is small, the validity regime of linear response is obeyed. The explicit response can be obtained by referring to canonical linear response theory for a closed system [33]. Specifically, we assume that the system has been prepared in the infinite past, t = −∞, with the canonical distribution where β T = 1/k B T and dΓ = dq 1 · · · dp 1 · · ·. With a time dependent force f j (t) applied to the jth particle, the total Hamiltonian reads H tot = H − f j (t)q j . With the system dynamics being closed, the evolution of the phase space distribution is governed by the Liouville equation where {· · · , · · ·} denotes the Poisson bracket. The linear response solution can be readily obtained up to the first order of f j , yielding The operator L is the usual Liouville operator for the original, unperturbed system, i.e. LA = {H, A} for any quantity A. Therefore, in presence of the kick-force the thermally averaged particle momenta read for t > 0 For t = 0 + , it reduces to p i (0 + ) response = Iδ i,j due to equipartition p i (0)p j (0) = mk B T δ i,j , which is consistent with Eq. (12). The conservation of total momentum necessarily implies that, i p i (t) response , is conserved as well. Evaluating this sum at time t = 0 yields i C P (i, t; j, 0) = 1 for 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport8 all later times t. The excess momentum distribution function ρ P (i, t) then assumes the form which remains normalized in the course of time t > 0; it is, however, not necessarily rigorously semi-positive (i.e. not present a manifest probability density) for all later times t.
With time evolving we notice that the momentum autocorrelation Eq. (10) describes the spread of the momentum distribution after the initial kick has occurred. As can be observed below, for increasing times t the quantity C P (j, t; j, 0) decreases (at least for some finite time). This implies the decrease of the momentum of the j'th particle. The lost momentum is transferred to its neighbors. This feature physically mimics a viscous behavior.
Let us next assume that the kick is applied to the center particle; i.e. we explicitly set j = 0. Similarly to Eq. (4), we define the MSD of the excess momentum ∆x 2 (t) P for a discrete lattice as Because of the conservation of total momentum, in analogy to the energy continuity relation, we may define a "momentum flux" j P i via the local momentum continuity relation. To see this, we write down the Newtonian equation of motion for the i'th particle, reading dp By defining the momentum flux as j P i = −∂V (q i − q i−1 )/∂q i = ∂V (q i − q i−1 )/∂q i−1 , we obtain a discrete form of the momentum continuity relation: Note that the momentum flux j P i is actually the force exerted on particle i from particle (i − 1). Its ensemble average j P i yields the average internal pressure. Following the strategy used for heat diffusion, one can derive a corresponding relation for the second time derivative ∆x 2 (t) P . It explicitly reads: Here, the autocorrelation function of the momentum flux is given by The integration of C J P (t) over time yields in the thermodynamic limit an effective "viscosity" η in the spirit of the Green-Kubo expression [24]; i.e., 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport9 In case that the momentum diffusion occurs normal one can invoke the concept of a finite momentum diffusivity by writing Therefore, for the discrete lattices discussed here, this so introduced viscosity η precisely equals the momentum diffusivity times the atom mass, namely Given a situation where the momentum diffusion occurs superdiffusively the limit in Eq. (25) no longer exits. The integration in Eq. (24) formally diverges, thus leading to an infinite viscosity. Indeed, for the FPU-β lattice, the MSD of momentum ∆x 2 (t) P has been observed to follow an almost ballistic diffusion, obeying ∆x 2 (t) P ∝ t 1.98 [30]. Therefore, our "viscosity" η for the FPU-β lattice diverges in the thermodynamical limit as well, reading asymptotically In the context of this work we find that such an infinite viscosity indicates a manifest solid-like behavior. In distinct contrast, however, a result with a finite viscosity indicates an effective fluid-like behavior.

The hypothesis
The general folklore in the field of anomalous heat conduction [15,16] is that in momentum-conserving 1D nonlinear lattices one encounters an anomalous heat conductance behavior. The case with the rotator model, however, presents an eminent exception. So what is the physical mechanism which can explain such exceptions? An observation is that in all those presently known cases exhibiting anomalous 1D heat conductance the interaction potential has been of unbounded nature at large interaction distances. The known exceptions, predominantly the well studied case with the rotator model, do not possess such unbounded pair interactions at long distances. Obviously the form of the overall interaction does matter for the violation of Fourier's law. One may speculate that the emergence of anomalous behavior may originate from a momentum dynamics that behaves solid-like in the sense that the momentum diffusion does not support a finite effective viscosity. In contrast, a Fourier-like behavior may become possible if the inherent momentum dynamics is more fluid-like, consequently possessing a finite momentum diffusivity. A reasonable conjecture therefore is that it is the physics of momentum diffusion which rules whether heat transport occurs normal or anomalous. In short, we next test with different models the following hypothesis: Heat transport in nonlinear 1D momentum-conserving systems occurs normal whenever the momentum diffusion is normal. The corollary being that heat transport occurs 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport10 anomalous whenever the momentum diffusion is superdiffusive.

Coupled rotator dynamics
As a first test bed for the above hypothesis we scrutinize the normal heat transport behavior in a nonlinear, momentum-conserving 1D occurring with the coupled rotator lattice. Throughout the remaining we shall use Hamiltonian lattice models with corresponding dimensionless units [1,2]. The Hamiltonian for the coupled rotator lattice dynamics reads Notably, here the nonlinear, momentum-conserving interaction potential is bounded for all arguments via the cosine function. The local energy density is Without loss of generality, we consider the initial distribution of the excess energy or momentum to be a Kronecker-delta function in the lattice center. The autocorrelation functions C E (i, t; j = 0, 0) and C P (i, t; j = 0, 0) for energy and momentum are defined according to Eqs. (2) and (10). Thus, the temporo-spatial behavior of C E (i, t; j = 0, 0) and C P (i, t; j = 0, 0) describe the dynamics of energy and momentum diffusion starting out from the central position.
In order to obtain most precise numerical results, we employ MD simulations for a closed system evolving with the Liouvillian over large, extended time spans and used periodic boundary conditions. The method to obtain the correlation functions is adopted from Ref. [34]. The equations of motions are integrated with a fourth order symplectic algorithm [35].
In Fig. 1 (a), we depict the correlation functions C E (i, t; j = 0, 0) for the energy diffusion versus evolving relative time span t. For sufficiently large times t we observe that the energy autocorrelation function C E (i, t; j = 0, 0) evolves very closely into a Gaussian distribution function (but still spatially bounded with the causal cone as determined by the speed of sound); i.e., its profile is perfectly given by with D E denoting the diffusion constant for heat diffusion. As a result, the MSD of heat diffusion ∆x 2 (t) E then depicts at for sufficiently long time t a linear dependence in time t, being the hall mark for normal diffusion. In summary, normal diffusion for heat is very precisely corroborated numerically with the findings depicted with Fig. 1 (c). Accordingly, heat diffusion theory in [29] for normal diffusion of heat ∆x 2 (t) E implies then that the heat conduction behavior is normal as well, with the heat conductivity given by κ = cD E . This Gaussian behavior for C E (i, t; j = 0, 0) with its corresponding linear timedependence of the MSD for heat diffusion ∆x 2 (t) E ∝ t has been observed previously in nonlinear 1D lattices which explicitly do break momentum conservation by including an on-site potential. For example, this is so for the case of 1D lattices with a φ 4 on-site potential [30]. In the latter case it is agreed among all practitioners that normal heat conduction occurs beyond any doubt [13,14]. The situation with momentum-conserving 1D-coupled rotator lattices, however, is far from being settled in the literature [21,22]. Here the possibility for a diverging thermal conductivity in the thermodynamic limit is still considered as an option by some practitioners. The present state of the art is nonconclusive although prior extensive numerical simulations, using either the Green-Kubo method or the Non-Equilibrium Molecular Dynamics (NEMD) method, both seem to indicate that the thermal conductivity is size-independent [17,18]. The source of the ongoing dispute is that the numerical results stemming either from the Green-Kubo method and/or the NEMD method, all performed for finite lattice sizes, may possibly not be consistent with manifest asymptotic results in the thermodynamical limit.
In contrast, as we emphasized with the previous section, the energy autocorrelation function C E (i, t; j = 0, 0) constitutes a fundamental, detailed measure yielding information well beyond the MSD of energy spread ∆x 2 (t) E [29,32]. This is so because of its equivalence with the Green-Kubo formula, which derives from the salient relation detailed with Eq. (5). Put differently, the temporal-spatial distribution of C E (i, t; j = 0, 0) yields improved, more detailed insight as compared to a method that solely evaluates via MD the Green-Kubo expression.
Next we study the diffusion of the excess momentum via the momentum autocorrelation function C P (i, t; j = 0, 0). Our findings are depicted with Fig. 1 (b). One finds that not only does the energy diffusion obey a Gaussian behavior but likewise also the momentum diffusion occurs Gaussian within our explored large regimes of correlation time spans t.
This behavior of C P (i, t; j = 0, 0) in this coupled rotator lattice possessing a bounded interaction potential is therefore radically distinct from the behavior of the C P (i, t; j = 0, 0) occurring in the momentum-conserving in FPU-β lattice [30]. Our MSD of the excess momentum ∆x 2 (t) P nicely follows a perfect linear time dependence, as can be deduced from Fig. 1 (d).
According to Eq. (24), the viscosity η for this coupled rotator 1D lattice is therefore finite. Put differently, it exhibits a fluidic-like characteristics referred to in the previous section. In distinct contrast, the viscosity η for the FPU-β lattice is diverging towards infinity in the thermodynamic limit; thus displaying the solid-like characteristics, as discussed in section 2, cf. see Eq. (27).

Amended Rotator dynamics with harmonic interactions
In testing further our hypothesis we next amend the rotator coupling by adding an additional unbounded harmonic interaction potential. This transforms the original rotator 1D lattice with bounded interaction into a momentum-conserving 1D lattice, but now with an unbounded pair interaction via the harmonic contribution. The Hamiltonian 1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport14 for this so amended rotator model reads: where K denotes the strength of the harmonic interaction. The total momentum is still conserved.
Using the same numerical procedure we numerically study the heat and momentum diffusion for this set up. In Fig. 2 (a), the energy autocorrelation function C E (i, t; j = 0, 0) at different correlation times is shown. The finite broadened side peaks exhibited by C E (i, t; j = 0, 0) imply that heat conduction no longer proceeds normal; instead an anomalous, faster-than-linear superdiffusive time dependence of the MSD of the energy spread ∆x 2 (t) E is depicted with Fig. 2 (c). This numerically confirms that heat conduction in this unbounded 1D lattice is rendered anomalous. Our numerical fit exhibits this superdiffusive heat spreading, growing as ∆x 2 (t) E ∝ t 1.40 .
The question then arises whether this anomalous heat transport behavior is also reflected with a solid-like, divergent behavior for the momentum diffusion. The numerically evaluated momentum autocorrelation function C P (i, t; j = 0, 0) vs. the lattice site are depicted in Fig. 2(b) for different correlation times. With the presence of the harmonic interaction term the diffusion of momentum behaves quite similar to the FPU-β lattice, and thus are distinctly different from the strict rotator model. We identify two main side humps that are moving outwards with a velocity which essentially remains constant. The vanishing central peak indicates that the momentum of the particle being kicked is quickly lost. Its momentum is quickly transferred to its neighbors. Indeed, it can be observed in Fig. 3(b) that C P (i = 0, t; j = 0, 0) decays fast to zero. On the contrary, for the rotator model C P (i = 0, t; j = 0, 0) remains finite and decays slowly in the form of ∼ t −1/2 as shown with Fig. 3(a). Therefore, the loss of momentum correlation in the rotator model is not as efficient; this being due to a finite effective viscosity.
With the approximate constant velocity spreading for the MSD of the momentum ∆x 2 (t) P should consistently occur ballistic. Indeed this feature is corroborated numerically with ∆x 2 (t) P ∝ t δ where δ ≈ 2, as can be deduced from Fig. 2(d). As a consequence, the "viscosity" for this so amended, unbounded rotator lattice behaves solid-like, assuming an infinite viscosity value.

Testing a Lennard-Jones pair interaction
From the preceding two test case one is led to speculate that it may well be the unbounded part of the interaction potential that is at the cause for a normal heat transport behavior in nonlinear 1D momentum-conserving lattices. This reasoning is even further reinforced in view of the recent numerical studies by Savin and Kosevich [23] which numerically find that heat conductivity remains finite in 1D interaction potentials possessing a regime that allows for dissociation at asymptotic large interaction distances as it occurs with the Lennard-Jones 1D case. If so, then for our hypothesis to hold up  Fig. 1 for rotator model and in Fig. 2 for the amended rotator model.
we should find that in this case the viscous behavior should emerge to be normal as well.
Using the same numerical schemes as for the foregoing two cases we next test the hypothesis for a Lennard-Jones setup. The corresponding Hamiltonian is given by using the same parameters as in Savin and Kosevich's paper; i.e., σ = 2 −1/6 and a binding energy ε = 1/72 [23]. Here, the pair interaction potential is unbounded at short interaction distances but becomes free at large interaction distances, allowing dissociation. The autocorrelation functions C E (i, t; j = 0, 0) and C P (i, t; j = 0, 0) for energy and momentum are defined as before with Eqs. (2) and (10). In Fig. 4 (a), we depict the correlation functions C E (i, t; j = 0, 0) for the energy diffusion versus correlation time t. For sufficient large times t we observe that the energy autocorrelation function C E (i, t; j = 0, 0) evolves with two broadened side peaks, being rather distinct from a Gaussian-like energy distribution spreading. Its corresponding energy MSD is therefore also not proportional to time t. In fact it assumes at long times a power-law like behavior, being just below an overall ballistic spreading, see Fig.  4 (c).
How about the behavior for momentum diffusion? Does it support our hypothesis that the viscosity in this case is indeed diverging, being solid-like? Fig. 4(b) for different correlation times exhibits again no Gaussian like spreading, but as in the case of the rotator dynamics amended with unbounded harmonic interactions a spreading involving two broadened side humps. We numerically identify that the two main side humps are  The MSD of the energy ∆x 2 (t) E and momentum ∆x 2 (t) P , respectively. The time dependence does not support a normal behavior but rather indicates anomalous heat transport for both energy and momentum diffusion. The solid blue power law lines serve as a guide to the eye for the data in the large time regime. The parameters in the numerical simulations are N = 5001, σ = 2 −1/6 and ε = 1/72, which are the same as in Savin and Kosevich's paper [23]. The calculated equilibrium temperature is at T ≈ 0.002. moving outwards at practically constant velocity. The momentum excess distribution C P (i, t; j = 0, 0) for the Lennard-Jones lattice then yields for the MSD of the momentum ∆x 2 (t) P a nearly perfect ballistic-like behavior, ∆x 2 (t) P ∝ t δ with δ ≈ 2, as deduced from Fig. 4 (d). In agreement with our hypothesis we thus find a diverging viscosity for this set up. Our findings not only contradict the recent results reported with [23], predicting therein a normal behavior for heat transport, but as well make evident that it is not the shape of the interaction potential which determines whether transport proceeds normal or anomalous. The quantifier that evidently rules the anomalous versus normal transport of heat in 1D momentum-conserving lattice systems is rather the physics of the momentum dynamics; namely being fluid-like for normal heat transport and being solid-like for anomalous heat transport.

Conclusions
With this work we studied the diffusion of both, heat and momentum in nonlinear momentum-conserving systems that possess different shapes for the interaction potential. Using recent results of [29] we started out showing that for energy diffusion there exists a close relationship between the behavior of excess energy diffusion and the overall conductivity behavior for thermal heat transport. This relationship has then been generalized alike for the case of momentum diffusion in 1D nonlinear lattices, yielding a momentum diffusivity which relates to the time derivative of the asymptotic MSD for excess momentum, see in Eq. (25). The consideration of momentum diffusion offers the possibility to quantify an effective viscosity, being proportional to the momentum diffusivity. For normal momentum diffusion this effective viscosity is finite while it diverges with increasing time t if the intrinsic momentum diffusion occurs superdiffusive. Based on the physical behavior of momentum diffusion we have put forward a hypothesis for the occurrence normal versus anomalous heat transport for 1D nonlinear systems that conserve total momentum. It states that the character of the momentum dynamics, being either fluid-like (finite viscosity) or solid-like (infinite viscosity), is at the root of the overall behavior for heat transport and heat diffusion. In particular for normal heat transport to occur the corresponding momentum diffusion proceeds normal, exhibiting a fluid-like behavior.
We tested the hypothesis with extended numerical means by considering different classes of 1D nonlinear lattice dynamics; namely (i) a setup with globally bounded pair interactions of which the rotator model is the archetype representant [17,18], (ii) using this latter set up but amended with unbounded, harmonic pair interactions (doing so mimics the situation of the frequently studied FPU-β model), and (iii) a set up with one-sided bounded interactions at large distances as it is the case with a Lennard-Jones pair interaction potential. In all those cases the study of the momentum diffusion and the accompanying heat diffusion validated our hypothesis: it is the very value of this effective viscosity of momentum diffusion which rules the overall heat transport behavior, occurring normal if the effective viscosity is fluid-like, or anomalous if the effective viscosity is solid-like.
As a result, the commonly assumed expectation that in 1D momentum-conserving nonlinear systems heat transport occurs always anomalous does not hold up to a closer inspection. A known counter example in recent years has been provided with the afore mentioned coupled rotator dynamics. Interestingly enough, however, it is also not the nature of the interaction potential, being either (i) globally bounded (ii) unbounded at large interaction distances or (iii) being unbounded for short distances while approaching free motion at large interaction distances, that is able to differentiate between anomalous versus normal heat conduction/diffusion. At present we are not able to prove on mathematical physics grounds our criterion; nor can we say whether the hypothesis remains valid in the quantum regime. Moreover, for now an open issue left for future studies is whether the criterion can be extended to anomalous/normal heat flow occurring in two-dimensional momentum-conserving lattice systems. Typically, the anomalous heat conductance then tends to diverge in system size logarithmically [2]. -Last but not least, this complexity of normal versus anomalous heat transport in low dimensions might possibly be put to constructive use when designing 1D low dimensional devices for function, such as is the case for the timely topic of "phononics" [36].

Acknowledgements
The numerical calculations were carried out at Shanghai Supercomputer Center, which has been supported by the NSF