Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

Abstract

We examine the thermal conductivity and bulk viscosity of a onedimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPUαβ system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ω → 0 the frequency dependent bulk viscosity,ζ(ω), and the frequency dependent thermal conductivity,κ(ω), should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, P r ζ = kBζ(ω)/(mκ(ω)), where m is the particle mass and kB is Boltzmann's constant. This dimensionless ratio should approach a constant value as ω → 0. We use mode-coupling theory to predict the ω → 0 limit of P r ζ . Values of P r ζ obtained from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. This regime is characterized by an intermediate-frequency plateau in the value ofκ(ω). We find that the value ofκ(ω) in this plateau region is proportional to T −2 where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-αβ chains. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons.

Anomalous dynamical scaling in anharmonic chains and plasma models with multiparticle collisions

“…In order to test the scaling, data have been rescaled automatically by extracting the frequency ω max corresponding to the maximum of the curve and then shifting it to the origin, while the x axis has been renormalized according to Eqs. (16) and (17).…”

Cluster sizes in a classical Lennard-Jones chain

“…Transport in one-dimensional systems is often anomalous [2]. This has been seen for diffusion [3,4], momentum transport [5] and heat transport [2,6,7]. As a result, classical and quantum mechanical one-dimensional systems have been a playground for theorists attempting to understand the underlying mechanical basis of constitutive relations such as Fourier's law of heat conduction [8].…”

“…This is sometimes justified by noting that interatomic forces in real solids have short range, and so it might be expected to be a good approximation [8]. Additionally, the dominant mechanisms of transport in one-dimensional systems seem to be large scale collective motions [5,30,31] and so one might expect the microscopic details of the interparticle forces to play a small role. In any case, the choice of nearest-neighbour interactions makes the analytical theory tractable and additionally makes it possible to carry out simulations much faster.…”

Crossover between different universality classes: Scaling for thermal transport in one dimension

“…The underlying picture is suggested to be understood by phonons performing various kinds of continuous-time random walks (in most cases, be the Lévy walks but not always), probably induced by the peculiar phonon dispersions along with nonlinearity. The results and suggested mechanisms may provide insights into controlling the transport of heat in some 1D materials.Transport in one dimension has, for a long time, been realized to be anomalous in most cases [1,2], with signatures of a universal power-law scaling of transport coefficients, among which the heat transport has been extensively investigated in the recent decades, both by various theoretical techniques, such as the renormalization group [3], mode coupling [4,5] or cascade [6][7][8], nonlinear fluctuating hydrodynamics [9][10][11], and Lévy walks [12][13][14][15]; and also by computer simulations [16][17][18][19][20][21][22][23][24][25][26]. For all studied cases two main scaling exponents have been given the most focus, i.e., α describing the divergence of heat conductivity with space size L as L α and γ characterizing the space(x)-time(t) scaling of heat spreading density ρ(x, t) as t −1/γ ρ(t −1/γ x, t).…”

“…Unfortunately, however, depending on the focused system's different parameter regimes, different theoretical models have been employed, and different predictions have been suggested. Thus, the universality classes of both scaling exponents and their relationship [27,28] remain controversial: (i) for α, two classes: α = 1/3 [3,9,[14][15][16]19] and α = 1/2 [4,5,9] have been reported; however, the universality has been doubted [25,26] and a Fibonacci sequence of α values converging on α * = (3 − √ 5)/2 (≃ 0.382) [6][7][8] has been suggested; (ii) for γ, (a) E-mail: phyxiongdx@fzu.edu.cn two universality classes, γ = 5/3 [9-11, 13-15, 20-23] and γ = 3/2 [9-11, 22, 23] have been predicted recently. The discussion of the latter scaling exponent γ is currently very hot [9-15, 20-23, 29, 30] because it involves more detailed space and time information [31], thus it can present a very detailed prediction for heat transport.…”