Heat Flow in a System of Coupled Harmonic Oscillators

瀧澤英一(Éi Iti Takizawa) ； 小林圭子(Keiko Kobayasi)

《Chinese Journal of Physics》 1卷2期 (1963/10) Pp. 59-73

並列摘要

In the present paper, the phenomenon of heat flow in the classical system of a chain consisting of 2N coupled harmonic oscillators is studied by means of the Schrödinger coordinates. Starting from the initial ensemble which corresponds to such a macroscopic state that half of the system is at temperature T and the other half is at temperature zero, the correlation functions of particles are calculated explicitly as functions of time. From the present authors' results, the average values of kinetic and potential energies are calculated. They are essentially the same as those obtained by means of the solutions in trigonometric eigenfunctions. For a large system, however, the correlation functions between different positions and the correlation functions between velocities of different particles vanish after a sufficiently long time; while the average values of the kinetic and the potential energies of each particle approach the same stationary value kT/4 after a sufficiently long time. Thus the equipartition of the energies is proved to establish in the system.On the other hand, the correlation functions between the velocity of a particle and the position of another particle do not vanish and still remain finite, even after a sufficiently long time. In other words, at the final state the instantaneous flow of energy still exists at any point throughout the system. Accordingly, in such a system of long chain, if we take good quantities such as kinetic and potential energies, they approach a stationary value; while if we take bud quantities such as correlation between velocity and displacement, they do not vanish and still remains finite. This shows that the final state thus obtained is by no means the state of thermodynamic equilibrium.