Thermal conduction of one-dimensional isotopically disordered harmonic lattice

Abstract

In the present communication we consider the one-dimensional (1D) isotopically disordered lattice with the harmonic potential. Our analytical method is adequate for any 1D lattice where potential energy can be presented as the quadratic form $U=\frac12 \sum_{i,j} q(i) U_{ij} q(j)$, where $q(i)$ -- coordinate or velocity of $i$-th particle. There are derived the closed system of equations for the temporal behavior of the correlation functions. The final expressions allow to calculate the kinetics and dynamics of the system -- energy, temperature profile, thermal conduction and others. There is developed the method for the calculation of the evolution of the eigenvalues (frequencies) and eigenvectors (relaxation times) to their stationary values. Exact results are obtained for times $\simeq 10^{14}$. The methods are suggested allowing to extend the range of the relaxation times upto $\simeq 10^{28}$. The spectrum of relaxation times reaches it constant value starting from the number of particles $N$ in the lattice $N \geq 300$. Thermal conductance $κ$ has the non-monotonic character: for the number of particles $N < 300$ $ κ$ increases as $κ\simeq 2.4 \lg N$, reaches the maximum value equal $\simeq 4.0$ at $N \simeq 300$ and then slowly decreases upto $N = 700$. The stationary state is unique and satisfies the Gibbs distribution. An excellent agreement between numerical simulations and analytical results is obtained where possible.