Exact Analysis of Adiabatic Invariants in Time Dependent Harmonic Oscillator

Abstract

AbstractThe 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 linear (harmonic) oscillator, whose Newton equation \(\ddot q + \omega ^2 \left( t \right)q = 0\) 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 (microcanonical ensemble) and calculate rigorously the distribution of energy E1 after time t = T, which is fully (all moments, including the variance μ2) determined by the first moment E¯1. For example, \({{\mu ^2 = E_0^2 \left[ {\left( {{{\bar E_1 }/{E_0 }}} \right)^2 - \left( {{{\omega \left( T \right)}/{\omega \left( 0 \right)}}} \right)^2 } \right]}/{2,}}\), and all higher even moments are powers of μ2, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function É(t) and is in this sense universal, it is a normalized distribution function given by \(P(x) = \pi^{-1} (2\mu^2 - x^2)^{-\frac{1}{2}}\), where \(x = E_1 - \bar{E}_1.\ \bar{E}_1\) and μ2 can be calculated exactly in some cases. In ideal adiabaticity \(\bar{E}_1 = \omega(T)E_0/\omega(0)\), and the variance μ2 is zero, whilst for finite T we calculate \(\bar{E}_1\) , and μ2 for the general case using exact WKB-theory to all orders. We prove that if ω(t) is of class C m (all derivatives up to and including the order m are continuous) μ2∞T −2(m+1), whilst for class C∞ it is known to be exponential μ2 exp(-α T). Due to the positivity of μ2 we also see that the adiabatic invariant \(I = \bar{E}_1/\omega(T)\) at the average energy \(\bar{E}_1\) never decreases.Key wordsNonlinear dynamicsNonautonomous Hamiltonian systemsAdiabatic invariantsEnergy evolutionStatistical mechanicsMicrocanonical ensemble