Universality of anomalous one-dimensional heat conductivity

Abstract

In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long-time correlation of the corresponding currents. The effective asymptotic behaviour is addressed with reference to the problem of heat transport in 1d crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with the system size L as κ ∝ L α . However, the exponent α deviates systematically from the theoretical prediction α = 1/3 proposed in a recent paper [O. Narayan, S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002) Strong spatial constraints can significantly alter transport properties. The ultimate reason is that the response to external forces depends on statistical fluctuations which, in turn, crucially depend on the system dimensionality d. A relevant example is the anomalous behaviour of heat conductivity for d ≤ 2. After the publication of the first convincing numerical evidence of a diverging thermal conductivity in anharmonic chains [1], this issue attracted a renovated interest within the theoretical community. A fairly complete overview is given in Ref. [2], where the effects of lattice dimensionality on the breakdown of Fourier's law are discussed as well. Anomalous behaviour means both a nonintegrable algebraic decay of equilibrium correlations of the heat current J(t) (the Green-Kubo integrand) at large times t → ∞ and a divergence of the finite-size conductivity κ(L) in the L → ∞ limit. This is very much reminiscent of the problem of long-time tails in fluids [3] where, in low spatial dimension, transport coefficients may not exist at all, thus implying a breakdown of the phenomenological constitutive laws of hydrodynamics. In 1d one findswhere α > 0, −1 < δ < 0, and is the equilibrium average. For small applied gradients, linear-response theory allows establishing a connection between the two exponents. By assuming that κ(L) can be estimated by cutting-off the integral in the Green-Kubo formula at the "transit time" L/v (v being some propagation velocity of excitations), one obtains κ ∝ L −δ i.e. α = −δ. Determining the asymptotic dependence of heat conductivity is not only important for assessing the universality of this phenomenon, but may be also relevant for predicting transport properties of real materials. For instance, recent molecular dynamics results obtained with phenomenological carbon potentials indicate an unusually high conductivity of single-walled nanotubes [4]: a power-law divergence with the tube length has been observed with an exponent very close to the one obtained in simple 1d models [5].The analysis of several models [2] clarified that anomalous conductivity should occur generically whenever momentum is conserved. For lattice models, this amounts to requiring that at least one acoustic phonon branch is present in the harmonic limit. The only known exception is the coupled-rotor model, where normal transport [6] is believed to arise as a co...

Anomalous Size Dependence of the Thermal Conductivity of Graphene Ribbons

“…Theoretical predictions that the intrinsic thermal conductivity K -limited by the crystal anharmonicy alone -can diverge with the crystal size L in 2D and 1D systems, continue to ignite debates [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Theoretical studies of the lattice thermal transport in 2D anharmonic Fermi-Pasta-Ulam (FPU) lattices [2][3], 2D…”

“…anharmonic Lennard-Jones lattices [3], 2D harmonic lattices with disorder [4] and 1D FPU chains [2][3]5] suggested that the lattice, i.e. phonon, thermal conductivity K diverges as log(L) for 2D lattices and as L  for 1D chains (<1), where the length L is proportional to the number of the lattice points N along the heat propagation direction [2][3][4][5].…”

“…phonon, thermal conductivity K diverges as log(L) for 2D lattices and as L  for 1D chains (<1), where the length L is proportional to the number of the lattice points N along the heat propagation direction [2][3][4][5].…”

Аналитический Подход К (Ан)гармоническим Кристаллическим Цепочкам С Самосогласованными Стохастическими Резервуарами

“…В частности, даже в одномерном контексте нет строгого вывода закона теплопроводности Фурье, согласно которому поток тепла пропорционален градиенту температуры [1]. Многие работы, почти все с использованием компьютерных вычислений [2] и с результатами, противоречащи-ми другу другу, были посвящены этой теме со времени первых строгих исследо-ваний по гармоническим цепочкам осцилляторов с термостатами на границах [3] -модели, которая не подчиняется закону Фурье. Недавно к гармонической цепочке осцилляторов вернулись в рамках аналитических исследований, но при условии, что каждый узел взаимодействует со стохастическим резервуаром.…”

Dynamics of Oscillator Chains

“…They predicted that the heat flux was proportional to the temperature difference, rather than the temperature gradient, thus showing that homogeneous harmonic chains do not exhibit normal transport properties. Although there are many aspects of linear chains, such as the inclusion of disorder or varying masses [101] that we have not considered above, our main concern here is with nonlinear chains. Numerical studies of heat conductivity in the FPU chain were reconsidered at the end of the 1960s.…”