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Mar 2005

Volume 15, Issue 1, Articles (01xxxx)

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Introduction: The Fermi–Pasta–Ulam problem—The first fifty years

David K. Campbell, Phillip Rosenau, and George M. Zaslavsky

Chaos 15, 015101 (2005); http://dx.doi.org/10.1063/1.1889345 (4 pages) | Cited 32 times

Online Publication Date: 28 March 2005

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Abstract Unavailable
Show PACS
05.45.Yv Solitons
05.45.Ac Low-dimensional chaos
05.45.Jn High-dimensional chaos
05.45.Pq Numerical simulations of chaotic systems
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations

Fermi–Pasta–Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics, and visiometrics

Norman J. Zabusky

Chaos 15, 015102 (2005); http://dx.doi.org/10.1063/1.1861554 (16 pages) | Cited 15 times

Online Publication Date: 28 March 2005

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This paper is mostly a history of the early years of nonlinear and computational physics and mathematics. I trace how the counterintuitive result of near-recurrence to an initial condition in the first scientific digital computer simulation led to the discovery of the soliton in a later computer simulation. The 1955 report by Fermi, Pasta, and Ulam (FPU) described their simulation of a one-dimensional nonlinear lattice which did not show energy equipartition. The 1965 paper by Zabusky and Kruskalshowed that the Korteweg–de Vries (KdV) nonlinear partial differential equation, a long wavelength model of the α-lattice (or cubic nonlinearity), derived by Kruskal, gave quantitatively the same results obtained by FPU. In 1967, Zabusky and Deem showed that a localized short wavelength initial excitation (then called an “optical” and now a “zone-boundary mode” excitation ) of the α-lattice revealed “n-curve” coherent states. If the initial amplitude was sufficiently large energy equipartition followed in a short time. The work of Kruskal and Miura (KM), Gardner and Greene (GG), and myself led to the appreciation of the infinity of denumerable invariants (conservation laws) for Hamiltonian systems and to a procedure by GGKM in 1967 for solving KdV exactly. The nonlinear science field exponentiated in diversity of linkages (as described in Appendix A). Included were pure and applied mathematics and all branches of basic and applied physics, including the first nonhydrodynamic application to optical solitons, as described in a brief essay (Appendix B) by Hasegawa. The growth was also manifest in the number of meetings held and institutes founded, as described briefly in Appendix D. Physicists and mathematicians in Japan, USA, and USSR (in the latter two, people associated with plasma physics) contributed to the diversification of the nonlinear paradigm which continues worldwide to the present. The last part of the paper (and Appendix C) discuss visiometrics: the visualization and quantification of simulation data, e.g., projection to lower dimensions, to facilitate understanding of nonlinear phenomena for modeling and prediction (or design). Finally, I present some recent developments that are linked to my early work by: Dritschel (vortex dynamics via contour dynamics/surgery in two and three dimensions); Friedland (pattern formation by synchronization in Hamiltonian nonlinear wave, vortex, plasma, systems, etc.); and the author (“n-curve” states and energy equipartition in a FPU lattice).
Show PACS
05.45.Yv Solitons
05.45.Xt Synchronization; coupled oscillators
02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems
42.65.Tg Optical solitons; nonlinear guided waves

Long way from the FPU-problem to chaos

G. M. Zaslavsky

Chaos 15, 015103 (2005); http://dx.doi.org/10.1063/1.1858115 (10 pages) | Cited 5 times

Online Publication Date: 28 March 2005

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This paper provides some historical comments on the study of the Fermi, Pasta, and Ulam (FPU) paper and its influence on the development of the theory of chaos. We also discuss some problems raised in the FPU paper and the links of these problems to such contemporary notions in chaos theory as ergodicity, mixing, recurrences, pseudochaos, kinetics, intermittency, etc.
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05.45.-a Nonlinear dynamics and chaos

The Fermi–Pasta–Ulam problem: Fifty years of progress

G. P. Berman and F. M. Izrailev

Chaos 15, 015104 (2005); http://dx.doi.org/10.1063/1.1855036 (18 pages) | Cited 43 times

Online Publication Date: 28 March 2005

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A brief review of the Fermi–Pasta–Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose–Einstein condensation and quantum systems of interacting Bose-particles are also considered.
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05.45.-a Nonlinear dynamics and chaos
01.30.Rr Surveys and tutorial papers; resource letters
05.30.Jp Boson systems

The Fermi–Pasta–Ulam problem as a challenge for the foundations of physics

A. Carati, L. Galgani, and A. Giorgilli

Chaos 15, 015105 (2005); http://dx.doi.org/10.1063/1.1861264 (8 pages) | Cited 13 times

Online Publication Date: 28 March 2005

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The Fermi–Pasta–Ulam (FPU) problem is discussed in connection with its physical relevance, and it is shown how apparently there exist only two possibilities: either the FPU problem is just a curiosity, or it has a fundamental role for the foundations of physics, casting a new light on the relations between classical and quantum mechanics. To this end, a short review is given of the main conceptual proposals that have been advanced. Particular emphasis is given to the perspective of a metaequilibrium scenario, which appears to be the only possible one for the FPU paradox to survive in the physically relevant case of infinitely many particles.
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05.45.-a Nonlinear dynamics and chaos
01.30.Rr Surveys and tutorial papers; resource letters

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Marco Pettini, Lapo Casetti, Monica Cerruti-Sola, Roberto Franzosi, and E. G. D. Cohen

Chaos 15, 015106 (2005); http://dx.doi.org/10.1063/1.1849131 (13 pages) | Cited 13 times

Online Publication Date: 28 March 2005

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We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi–Pasta–Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.
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02.50.Ey Stochastic processes
02.40.-k Geometry, differential geometry, and topology
02.40.Pc General topology
05.70.Fh Phase transitions: general studies

Korteweg–de Vries equation and energy sharing in Fermi–Pasta–Ulam

A. Ponno and D. Bambusi

Chaos 15, 015107 (2005); http://dx.doi.org/10.1063/1.1832772 (5 pages) | Cited 8 times

Online Publication Date: 28 March 2005

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We address the problem of equipartition in a long Fermi–Pasta–Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg–de Vries we use the latter equation to show that, corresponding to initial data à la Fermi, the time average of the energy on the kth mode decreases exponentially with k/N. The result persists in the thermodynamic limit.
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05.45.-a Nonlinear dynamics and chaos
02.30.Hq Ordinary differential equations

Time scale for energy equipartition in a two-dimensional FPU model

Giancarlo Benettin

Chaos 15, 015108 (2005); http://dx.doi.org/10.1063/1.1854278 (10 pages) | Cited 10 times

Online Publication Date: 28 March 2005

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The FPU problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in dimension two, more precisely in a model with triangular cell and nearest-neighbors Lennard-Jones interaction. The number n of degrees of freedom ranges from 182 to 6338. Energy is initially equidistributed among a small number n0 of low frequency modes, with n0 proportional to n. We study numerically the time evolution of the so-called spectral entropy and the related “effective number” neff of degrees of freedom involved in the dynamics; in this (rather typical) way we can estimate, for each n and each specific energy (energy per degree of freedom) ε, the time scale Tn(ε) for energy equipartition. Numerical results indicate that in the thermodynamic limit the equipartition times are short: more precisely, for large n at fixed ε we find a limit curve T∞(ε), and T∞ grows only as ε−1 for small ε. Larger equipartition times are obtained by lowering ε, at fixed n, below a crossover value εc(n). However, εc appears to vanish by increasing n (faster than 1/n), and the total energy E = nε, rather than ε, appears to be the relevant variable when n is large and ε<εc. In conclusion, it seems that in the thermodynamic limit, for this model and this kind of initial conditions, the FPU phenomenon, namely the lack of energy equipartition in physically reasonable times, practically disappears.
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05.45.Tp Time series analysis
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.70.Ce Thermodynamic functions and equations of state

Dynamics of oscillator chains from high frequency initial conditions: Comparison of ϕ4 and FPU-β models

Allan J. Lichtenberg, Vladimir V. Mirnov, and Christopher Day

Chaos 15, 015109 (2005); http://dx.doi.org/10.1063/1.1861532 (16 pages) | Cited 4 times

Online Publication Date: 28 March 2005

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The dynamics of oscillator chains are studied, starting from high frequency initial conditions (h.f.i.c.). In particular, the formation and evolution of chaotic breathers (CB’s) of the Klein–Gordon chain with quartic nonlinearity in the Hamiltonian (the ϕ4 model) are compared to the results of the previously studied Fermi–Pasta–Ulam (FPU-β) chain. We find an important difference for h.f.i.c. is that the quartic nonlinearity, which drives the high frequency phenomena, being a self-force on each individual oscillator in the ϕ4 model is significantly weaker than the quartic term in the FPU-β model, which acts between neighboring oscillators that are nearly out-of-phase. The addition of a self-force breaks the translational invariance and adds a parameter. We compare theoretical results, using the envelope approximation to reduce the discrete coupled equations to a partial differential equation for each chain, indicating that various scalings can be used to predict the relative energies at which the basic phenomena of parametric instability, breather formation and coalescence, and ultimately breather decay to energy equipartition, will occur. Detailed numerical results, comparing the two chains, are presented to verify the scalings.
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05.45.Pq Numerical simulations of chaotic systems
05.45.Xt Synchronization; coupled oscillators
02.30.Jr Partial differential equations
02.30.Hq Ordinary differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems

The anti-FPU problem

Thierry Dauxois, Ramaz Khomeriki, Francesco Piazza, and Stefano Ruffo

Chaos 15, 015110 (2005); http://dx.doi.org/10.1063/1.1854273 (11 pages) | Cited 21 times

Online Publication Date: 28 March 2005

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We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi–Pasta–Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.
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05.45.Jn High-dimensional chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Compact and almost compact breathers: A bridge between an anharmonic lattice and its continuum limit

Philip Rosenau and Steven Schochet

Chaos 15, 015111 (2005); http://dx.doi.org/10.1063/1.1852292 (18 pages) | Cited 13 times

Online Publication Date: 28 March 2005

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We demonstrate that certain strictly anharmonic one-dimensional FPU lattices with a suitable quartic site potential appended support almost-compact discrete breathers over a macroscopic localized domain that is essentially fixed independently of the sparseness of the lattice. Beyond that domain the discrete breather tails decay at a double-exponential rate in the lattice-cell index, becoming truly compact in the continuum limit. Furthermore, the discrete breather is stable for amplitudes below a sharp threshold that depends on the sparseness of the lattice. For the two-dimensional version of the problem, the continuum limit of a planar hexagonal lattice with a purely quartic interaction potential begets an isotropic multidimensional nonlinear wave equation. When a quartic site potential of the appropriate sign is appended, the continuum equation has a compactly supported radial breather solution.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.45.Pq Numerical simulations of chaotic systems
02.30.Jr Partial differential equations

Discrete breathers in Fermi–Pasta–Ulam lattices

S. Flach and A. Gorbach

Chaos 15, 015112 (2005); http://dx.doi.org/10.1063/1.1839151 (10 pages) | Cited 17 times

Online Publication Date: 28 March 2005

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We study the properties of spatially localized and time-periodic excitations—discrete breathers—in Fermi–Pasta–Ulam (FPU) chains. We provide a detailed analysis of their spatial profiles and stability properties. We especially demonstrate that the Page mode is linearly stable for symmetric FPU potentials. A resonant interaction between a localized and delocalized perturbations causes weak but finite strength instabilities for asymmetric FPU potentials. This interaction induces Fano resonances for plane waves scattered by the breather. Finally we analyze the interplay between energy thresholds for breathers in the presence of strongly asymmetric FPU potentials and the corresponding profiles of the low-frequency limit of breather families.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.45.-a Nonlinear dynamics and chaos

Localized waves in nonlinear oscillator chains

Gérard Iooss and Guillaume James

Chaos 15, 015113 (2005); http://dx.doi.org/10.1063/1.1836151 (15 pages) | Cited 11 times

Online Publication Date: 28 March 2005

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This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi–Pasta–Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgässner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.
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01.30.Rr Surveys and tutorial papers; resource letters
05.45.Xt Synchronization; coupled oscillators
05.45.Yv Solitons
02.30.Hq Ordinary differential equations
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

On the problem of periodicity and hidden solitons for the KdV model

Jüri Engelbrecht and Andrus Salupere

Chaos 15, 015114 (2005); http://dx.doi.org/10.1063/1.1858781 (6 pages) | Cited 2 times

Online Publication Date: 28 March 2005

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In continuum limit, the Fermi-Pasta-Ulam lattice is modeled by a Korteweg–de Vries (KdV) equation. It is shown that the long-time behavior of a KdV soliton train emerging from a harmonic excitation has a regular periodicity of right- and left-going trajectories. In a soliton train not all the solitons are visible, the solitons with smaller amplitude are hidden and their influence is seen through the changes of phase shifts of larger solitons. In the case of an external harmonic force several resonance schemes are revealed where both visible and hidden solitons have important roles. The weak, moderate, strong, and dominating fields are distinguished and the corresponding solution types presented.
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05.45.Yv Solitons

Nonlinear lattice dynamics of Bose–Einstein condensates

Mason A. Porter, R. Carretero-González, P. G. Kevrekidis, and Boris A. Malomed

Chaos 15, 015115 (2005); http://dx.doi.org/10.1063/1.1858114 (9 pages) | Cited 16 times

Online Publication Date: 28 March 2005

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The Fermi–Pasta–Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in nonmetallic solids and develop “experimental” techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems—including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose–Einstein condensates (BECs) in optical lattices. BECs, in particular, due to their widely ranging and easily manipulated dynamical apparatuses—with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials—provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.
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05.45.Ac Low-dimensional chaos
05.45.Yv Solitons
05.30.Jp Boson systems
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
87.14.G- Nucleic acids
42.65.Jx Beam trapping, self-focusing and defocusing; self-phase modulation
01.30.Rr Surveys and tutorial papers; resource letters

Anomalous deterministic transport

Roberto Artuso and Giampaolo Cristadoro

Chaos 15, 015116 (2005); http://dx.doi.org/10.1063/1.1832811 (7 pages) | Cited 1 time

Online Publication Date: 28 March 2005

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We present a series of results on deterministic transport in chaotic system, obtained in the framework of periodic orbits theory. The emphasis is on intermittent systems, where deviations from complete chaos may induce anomalies on the asymptotic moments’ growth.
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05.45.Ac Low-dimensional chaos
05.45.−a
56.0C.d

Normal and anomalous heat transport in one-dimensional classical lattices

Tomaž Prosen and David K. Campbell

Chaos 15, 015117 (2005); http://dx.doi.org/10.1063/1.1868532 (17 pages) | Cited 11 times

Online Publication Date: 28 March 2005

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We present analytic and numerical results on several models of one-dimensional (1D) classical lattices with the goal of determining the origins of anomalous heat transport and the conditions for normal transport in these systems. Some of the recent results in the literature are reviewed and several original “toy” models are added that provide key elements to determine which dynamical properties are necessary and which are sufficient for certain types of heat transport. We demonstrate with numerical examples that chaos in the sense of positivity of Lyapunov exponents is neither necessary nor sufficient to guarantee normal transport in 1D lattices. Quite surprisingly, we find that in the absence of momentum conservation, even ergodicity of an isolated system is not necessary for the normal transport. Specifically, we demonstrate clearly the validity of the Fourier law in a pseudo-integrable particle chain.
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05.45.Ac Low-dimensional chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Studies of thermal conductivity in Fermi–Pasta–Ulam-like lattices

Stefano Lepri, Roberto Livi, and Antonio Politi

Chaos 15, 015118 (2005); http://dx.doi.org/10.1063/1.1854281 (9 pages) | Cited 14 times

Online Publication Date: 28 March 2005

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The pioneering computer simulations of the energy relaxation mechanisms performed by Fermi, Pasta, and Ulam (FPU) can be considered as the first attempt of understanding energy relaxation and thus heat conduction in lattices of nonlinear oscillators. In this paper we describe the most recent achievements about the divergence of heat conductivity with the system size in one-dimensional (1D) and two-dimensional FPU-like lattices. The anomalous behavior is particularly evident at low energies, where it is enhanced by the quasiharmonic character of the lattice dynamics. Remarkably, anomalies persist also in the strongly chaotic region where long-time tails develop in the current autocorrelation function. A modal analysis of the 1D case is also presented in order to gain further insight about the role played by boundary conditions.
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05.70.Ce Thermodynamic functions and equations of state
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.45.Pq Numerical simulations of chaotic systems
05.45.Xt Synchronization; coupled oscillators
03.65.Ge Solutions of wave equations: bound states

Heat conduction in the Frenkel–Kontorova model

Bambi Hu and Lei Yang

Chaos 15, 015119 (2005); http://dx.doi.org/10.1063/1.1862552 (9 pages) | Cited 32 times

Online Publication Date: 28 March 2005

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Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a first-principle derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are far from clear. In this paper we will review recent works on heat conduction in a simple nonintegrable model called the Frenkel–Kontorova model. The thermal conductivity of this model has been found to be finite. We will study the dependence of the thermal conductivity on the temperature and other parameters of the model such as the strength and the periodicity of the external potential. We will also discuss other related problems such as phase transitions and finite-size effects. The study of heat conduction is not only of theoretical interest but also of practical interest. We will show various recent designs of thermal rectifiers and thermal diodes by coupling nonlinear chains together. The study of heat conduction in low dimensions is also important to the understanding of the thermal properties of carbon nanotubes.
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02.30.Nw Fourier analysis
05.20.-y Classical statistical mechanics
05.70.Fh Phase transitions: general studies
44.10.+i Heat conduction
47.27.T- Turbulent transport processes
05.70.Ce Thermodynamic functions and equations of state

Controlling the heat flow: Now it is possible

Giulio Casati

Chaos 15, 015120 (2005); http://dx.doi.org/10.1063/1.1869912 (9 pages) | Cited 12 times

Online Publication Date: 28 March 2005

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We discuss the problem of heat conduction in 1D nonlinear chains in relation to the dynamical properties of the system. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in linear mixing systems. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability. We then show that, acting on the parameter which controls the strength of the on site potential inside a segment of the chain, we induce a transition from conducting to insulating behavior in the whole system. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.
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05.45.Ac Low-dimensional chaos
05.60.Cd Classical transport
44.10.+i Heat conduction
02.60.Cb Numerical simulation; solution of equations

Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels

Baowen Li, Jiao Wang, Lei Wang, and Gang Zhang

Chaos 15, 015121 (2005); http://dx.doi.org/10.1063/1.1832791 (13 pages) | Cited 43 times

Online Publication Date: 28 March 2005

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We study anomalous heat conduction and anomalous diffusion in low-dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is σ2(t) = 2Dtα (0<α ⩽ 2), then the thermal conductivity can be expressed in terms of the system size L as κ = cLβ with β = 2−2/α. This result predicts that a normal diffusion (α = 1) implies a normal heat conduction obeying the Fourier law (β = 0), a superdiffusion (α>1) implies an anomalous heat conduction with a divergent thermal conductivity (β>0), and more interestingly, a subdiffusion (α<1) implies an anomalous heat conduction with a convergent thermal conductivity (β<0), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.
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44.10.+i Heat conduction
05.45.−a
05.70.Ln Nonequilibrium and irreversible thermodynamics
66.70.-f Nonelectronic thermal conduction and heat-pulse propagation in solids; thermal waves
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