Correlations and scaling in one-dimensional heat conduction

Ballistic heat conduction and mass disorder in one dimension

It is well-known that in the disordered harmonic chain, heat conduction is subballistic and the thermal conductivity (κ) scales asymptotically as lim(L--> ∞) κ ∝ L(0.5) where L is the chain length. However, using the nonequilibrium Green's function (NEGF) method and analytical modelling, we show that there exists a critical crossover length scale (LC) below which ballistic heat conduction (κ ∝ L) can coexist with mass disorder. This ballistic-to-subballistic heat conduction crossover is connected to the exponential attenuation of the phonon transmittance function Ξ i.e. Ξ(ω, L) = exp[-L/λ(ω)], where λ is the frequency-dependent attenuation length. The crossover length can be determined from the minimum attenuation length, which depends on the maximum transmitted frequency. We numerically determine the dependence of the transmittance on frequency and mass composition as well as derive a closed form estimate, which agrees closely with the numerical results. For the length-dependent thermal conductance, we also derive a closed form expression which agrees closely with numerical results and reproduces the ballistic to subballistic thermal conduction crossover. This allows us to characterize the crossover in terms of changes in the length, mass composition and temperature dependence, and also to determine the conditions under which heat conduction enters the ballistic regime. We describe how the mass composition can be modified to increase ballistic heat conduction.

Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain

Recent work has developed a nonlinear hydrodynamic fluctuation theory for a chain of coupled anharmonic oscillators governing the conserved fields, namely, stretch, momentum, and energy. The linear theory yields two propagating sound modes and one diffusing heat mode, all three with diffusive broadening. In contrast, the nonlinear theory predicts that, at long times, the sound mode correlations satisfy Kardar-Parisi-Zhang scaling, while the heat mode correlations have Lévy-walk scaling. In the present contribution we report on molecular dynamics simulations of Fermi-Pasta-Ulam chains to compute various spatiotemporal correlation functions and compare them with the predictions of the theory. We obtain very good agreement in many cases, but also some deviations.

Equilibrium time-correlation functions for one-dimensional hard-point systems

As recently proposed, the long-time behavior of equilibrium time-correlation functions for one-dimensional systems are expected to be captured by a nonlinear extension of fluctuating hydrodynamics. We outline the predictions from the theory aimed at the comparison with molecular dynamics. We report on numerical simulations of a fluid with a hard-shoulder potential and of a hard-point gas with alternating masses. These models have in common that the collision time is zero and their dynamics amounts to iterating collision by collision. The theory is well confirmed, with the twist that the nonuniversal coefficients are still changing at longest accessible times.

Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as functions of system size and temperature gradient, identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from the velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the Bhatnagar-Gross-Krook approximation. For equal-temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.

Linear response formula for open systems

An exact expression for the finite frequency response of open classical systems coupled to reservoirs is obtained. The result is valid for any conserved current. No assumption is made about the reservoirs apart from thermodynamic equilibrium. At nonzero frequencies, the expression involves correlation functions of boundary currents and cannot be put in the standard Green-Kubo form involving currents inside the system.

Linear-response formula for finite-frequency thermal conductance of open systems

An exact linear-response expression is obtained for the heat current in a classical Hamiltonian system coupled to heat baths with time-dependent temperatures. The expression is equally valid at zero and finite frequencies. We present numerical results on the frequency dependence of the response function for three different one-dimensional models of coupled oscillators connected to Langevin baths with oscillating temperatures. For momentum conserving systems, a low-frequency peak is seen that is higher than the zero-frequency response for large systems. For momentum nonconserving systems, there is no low-frequency peak. The momentum nonconserving system is expected to satisfy Fourier's law; however, at the single bond level, we do not see any clear agreement with the predictions of the diffusion equation even at low frequencies. We also derive an exact analytical expression for the response of a chain of harmonic oscillators to a (not necessarily small) temperature difference; the agreement with the linear-response simulation results for the same system is excellent.

Internal dissipation of a polymer

The dynamics of flexible polymer molecules are often assumed to be governed by hydrodynamics of the solvent. However there is considerable evidence that internal dissipation of a polymer contributes as well. Here we investigate the dynamics of a single chain in the absence of solvent to characterize the nature of this internal friction. We model the chains as freely hinged but with localized bond angles and threefold symmetric dihedral angles. We show that the damping is close but not identical to Kelvin damping, which depends on the first temporal and second spatial derivative of monomer position. With no internal potential between monomers, the magnitude of the damping is small for long wavelengths and weakly damped oscillatory time dependent behavior is seen for a large range of spatial modes. When the size of the internal potential is increased, such oscillations persist, but the damping becomes larger. However underdamped motion is present even with quite strong dihedral barriers for long enough wavelengths.

Continuum and lattice heat currents for oscillator chains

We show that two commonly used definitions for the heat current give different results-through the Kubo formula-for the heat conductivity of oscillator chains. The difference exists for finite chains, and is expected to be important more generally for small structures. For a chain of N particles that are tethered at the ends, the ratio of the heat conductivities calculated with the two currents differs from unity by O(1/N). For a chain held at constant pressure, the difference from unity decays more slowly, and is consistent with O(1/Neta) with 1>eta>0.5.

Exact solution to ideal chain with fixed angular momentum

The statistical mechanics of a noninteracting polymer chain in the limit of a large number of monomers is considered when the total angular momentum L is fixed. The radius of gyration for a ring polymer in this situation is derived exactly in closed form by functional integration techniques. Even when L=0 the radius of gyration differs from that of a random walk by a prefactor of order unity. The dependence on L is discussed qualitatively and the large- L limit can be understood by physical arguments, which can also be extended to self-avoiding chains.

Polymers in a vacuum

In a variety of situations, isolated polymer molecules are found in a vacuum, and here we examine their properties. Angular momentum conservation is shown to significantly alter the average size of a chain and its conservation is only broken slowly by thermal radiation. For an ideal chain, the time autocorrelation for monomer position oscillates with a period proportional to chain length. The oscillations and damping are analyzed in detail. Short-range repulsive interactions suppress oscillations and speed up relaxation, but stretched chains still show damped oscillatory correlations.

Equilibration and universal heat conduction in fermi-pasta-ulam chains

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N(1/3), relevant for the much debated question of the universality of one-dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained.

Heat conduction and diffusion of hard disks in a narrow channel

Using molecular dynamics we study heat conduction and diffusion of hard disks in one-dimensional narrow channels. When collisions preserve momentum the heat conduction kappa diverges with the number of disks N as kappa approximately N alpha (alpha approximately 1/3) . Such a behavior is seen both when the ordering of disks is fixed ("pen-case" model), and when they can exchange their positions. Momentum conservation results also in sound-wave effects that enhance diffusive behavior and on an intermediate time scale (that diverges in the thermodynamic limit) normal diffusion takes place even in the "pen-case" model. When collisions do not preserve momentum, kappa remains finite and sound-wave effects are absent.

Universality of one-dimensional heat conductivity

We show analytically that the heat conductivity of oscillator chains diverges with system size N as N(1/3), which is the same as for one-dimensional fluids. For long cylinders, we use the hydrodynamic equations for a crystal in one dimension. This is appropriate for stiff systems such as nanotubes, where the eventual crossover to a fluid only sets in at unrealistically large . Despite the extra equation compared to a fluid, the scaling of the heat conductivity is unchanged. For strictly one-dimensional chains, we show that the dynamic equations are those of a fluid at all length scales even if the static order extends to very large . The discrepancy between our results and numerical simulations on Fermi-Pasta-Ulam chains is discussed.

Normal and anomalous heat transport in one-dimensional classical lattices

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.

Mode-coupling theory and molecular dynamics simulation for heat conduction in a chain with transverse motions

We study heat conduction in a 1D chain of particles with longitudinal as well as transverse motions. The particles are connected by 2D harmonic springs together with bending angle interactions. The problem is analyzed by mode-coupling theory and compared with molecular dynamics. We find very good, quantitative agreement for the damping of modes between a full mode-coupling theory and molecular dynamics result, and a simplified mode-coupling theory gives qualitative description of the damping. The theories predict generically that thermal conductance diverges as N(1/3) as the size N increases for systems terminated with heat baths at the ends. The N(2/5) dependence is also observed in molecular dynamics, which we attribute to crossover effect.