A note on an invariant for one-dimensional heat conduction

Physical Review E, 2003

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 behavior is addressed with reference to the problem of heat transport in one-dimensional crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with system size L as ϰL ␣ . However, the exponent ␣ deviates systematically from the theoretical prediction ␣ϭ1/3 proposed in a recent paper ͓O.

intechopen.com

In this book, we look at using the integral heat equation as the general method of solving the heat equation subject to given boundary conditions. We begin first by looking at x- directional heat conduction and look at the case of the insulated metal rod first. It is known from literature that the Fourier series yield a solution to this problem for given boundary conditions. But on analyzing the solution got, we notice that it is made up of an infinite number of terms and what this means is that we shall only have an approximate solution since we can’t in practice add up all the terms to infinity. To solve this problem, we solved the heat equation by first transforming it into an integral equation and then find an exact solution as shall be shown in the text later. In solving the heat equation, the temperature profiles that satisfy the boundary and initial conditions are exponential temperature profiles and hyperbolic temperature profiles as derived in literature for heat conduction in fins. For this case of insulate metal rod, we invoke L’hopital’s rule to get the steady state temperature profile. We then extend this integral transform approach to the case where there is lateral convection along the metal rod and get also both the transient and steady state solution which agrees with theory for steady state heat conduction. After that, we look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the analogous exponential and hyperbolic functions as derived in literature of conduction in fins. We use the same technique of transforming the PDE into an integral equation. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders should reduce to those of semi-infinite hollow cylinders as was the case for x-directional heat flow. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction.

Journal of Non-Equilibrium Thermodynamics, 2000

Heat conduction close-to-Fourier means that we look for a minimal extension of heat conduction theory using the usual Fourier expression of the heat flux density and modifying that of the internal energy as minimally as possible by choosing the minimal state space. Applying Liu's procedure results in the class of materials and a di¤erential equation both belonging to the close-to-Fourier case of heat conduction.

Physical Review Letters, 2001

We present the computer simulation results of a chain of hard point particles with alternating masses interacting on its extremes with two thermal baths at different temperatures. We found that the system obeys Fourier's law at the thermodynamic limit. This result is against the actual belief that one dimensional systems with momentum conservative dynamics and nonzero pressure have infinite thermal conductivity. It seems that thermal resistivity occurs in our system due to a cooperative behavior in which light particles tend to absorb much more energy than the heavier ones.

Journal of Integral Equations and Applications, 1988

Periodica Polytechnica Mechanical Engineering, 2000

This paper gives a detailed system theoretical treatment of the heat flux theory in the linear heat conduction based on the Laplace transformation method. By restricting the investigations to the simplest geometrical structures occurring in the practice, the authors prove the criteria guaranteeing the existence of the convolutional representations of the heat flux depending on the known temperature.

2016

The Mori's projection method, known as the memory function method, is an important theoretical formalism to study various transport coefficients. In the present work, we calculate the dynamical thermal conductivity in the case of metals using the memory function formalism. We introduce thermal memory functions for the first time and discuss the behavior of thermal conductivity in both the zero frequency limit and in the case of nonzero frequencies. We compare our results for the zero frequency case with the results obtained by the Bloch-Boltzmann kinetic approach and find that both approaches agree with each other. Motivated by some recent experimental advancements, we obtain several new results for the ac or the dynamical thermal conductivity.

Physical Review Letters, 2004

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 2 ͑t͒ =2Dt ␣ ͑0 Ͻ ␣ ഛ 2͒, then the thermal conductivity can be expressed in terms of the system size L as = cL ␤ with ␤ =2−2/␣. This result predicts that a normal diffusion ͑␣ =1͒ implies a normal heat conduction obeying the Fourier law ͑␤ =0͒, a superdiffusion ͑␣ Ͼ 1͒ implies an anomalous heat conduction with a divergent thermal conductivity ͑␤ Ͼ 0͒, and more interestingly, a subdiffusion ͑␣ Ͻ 1͒ implies an anomalous heat conduction with a convergent thermal conductivity ͑␤ Ͻ 0͒, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.

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