equipartition and coupled harmonic oscillator system
This question has to do with analysing how equipartition sets in for a system such as a coupled harmonic oscillator system.
Take for example, a system as shown in the figure: I now excite the system by giving a small displacement $ x_0$ to the first mass, and analyze the energy of the the two systems as: $\frac{1}{2} k x_1(t)^2 + \frac{1}{2} m v_1(t)^2$ and $\frac{1}{2} k x_2(t)^2 + \frac{1}{2}m v_2(t)^2$. The displacements and velocities calculated as per equations of motion.
Clearly, the total energy oscillates between the two oscillators sinusoidally in time. There will be instances when entire energy will be contained in the first spring-mass system, and there will be instances when the entire energy will be contained in the second spring-mass sytem. The period of oscillation of this energy depends on the coupling strength $\xi$. A weak coupling strength results in period being large, and a strong coupling strength ensures that the period is small. But oscillate it will.
Now, let me introduce some kind of dephasing in the system. That is, rather than the two oscillators interacting at a single frequency $\omega = \sqrt{k/m}$, they are now interacting with a spectrum of frequencies around this natural frequency $\omega$. 37
This spectrum could be in the form of a gaussian distribution around this natural frequency. This range of frequencies can be a result of - for example, broadening effect due to interacting with environment/ contact with bath.
My question is this: will this dephasing, due to interaction over range of frequencies rather than at a single frequency, be sufficient for the equipartition theorem to hold true in large time intervals ? That is, if I now plot energy of one oscillator system (by integrating over all frequency contributions) as a function of time - after a finite interval of time, should we expect the energy to be equally divided among the two oscillators , and remain steady? Of course, if the coupling is small, one can expect that it will take a longer time for the equilibration than if the coupling was stronger.
This is what I was expecting. However, simulation results suggest different. What I see is that equipartition seems to hold true only if coupling strength is large. For weak coupling even if I extend the time of simulation for a very long time, the first oscillator still retains bulk of its initialization energy. I am not able to understand why. Even if the coupling is weak, over a long interval of time shouldn't the systems interact sufficiently enough that equipartition sets in ? Does anybody have thoughts as to what I should introduce, for the weak coupling case, in order to still see equipartition?
(If anybody is concerned about the energy stored in the coupling spring, I can modify my analysis to look into energy stored in the normal modes - I still see the same behavior)
Thanks. 187