Energy evolution in time-dependent harmonic oscillator
| dc.creator | Robnik, Marko | |
| dc.creator | Romanovski, Valery G. | |
| dc.date | 2006-08-02 | |
| dc.date.accessioned | 2026-07-07T07:22:29Z | |
| dc.date.available | 2026-07-07T07:22:29Z | |
| dc.description | The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q} + ω^2(t) q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance $μ^2$) determined by the first moment $\bar{E_1}$. For example, $μ^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (ω(T)/ω(0))^2]/2$, and all higher even moments are powers of $μ^2$, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function $ω(t)$ and is in this sense universal. In ideal adiabaticity $\bar{E_1} = ω(T) E_0/ω(0)$, and the variance $μ^2$ is zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $μ^2$ for the general case using exact WKB-theory to all orders. We prove that if $ω(t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order $m$ are continuous) $μ\propto T^{-(m+1)}$, whilst for class ${\cal C}^{\infty}$ it is known to be exponential $μ\propto \exp (-αT)$. | |
| dc.description | 26 pages, 5 figures | |
| dc.identifier | https://arxiv.org/abs/nlin/0608004 | |
| dc.identifier | http://arxiv.org/abs/nlin/0608004 | |
| dc.identifier | Open Systems and Information Dynamics, Vol. 13, No. 2, p. 197-222 (2006) | |
| dc.identifier | doi:10.1007/s11080-006-8222-0 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115676 | |
| dc.subject | Chaotic Dynamics | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.title | Energy evolution in time-dependent harmonic oscillator | |
| dc.type | text |