Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle)

Transcription

1 Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle In classical mechanics an adiabatic invariant is defined as follows[1]. Consider the Hamiltonian system with slowly varying external parameter λ: H = H(p, q; λ, λ = ɛt (1 ṗ = H q q = H p Definition: The quantity A(p, q; λ is an adiabatic invariant of the system above if for every κ > there is an ɛ > such that if < ɛ < ɛ and < t < 1/ɛ, then A(p(t, q(t; ɛt A(p(, q(; < κ. The principle of adiabatic action invariance is that the values of the actions I for an integrable system are indeed adiabatic invariants. (Note that most systems with N > 1 are not integrable, but one has to start somewhere. huh? Here s what we are saying. Consider a hamiltonian system with N position degrees of freedom (N total degrees of freedom that is a function of some fixed external parameter λ. We require that the system be integrable (have at least N independent constants of motion for any fixed value of λ in some region, say [, 1]. Since the system is integrable, its motion (trajectory is specified by I. That is, we can construct the actions I i = p dq from the i (hollow torus that the trajectory fills out. The torus, (or equivalently, the quantity I, can be considered to be a function of the starting point of the motion (p, q in phase space. It is also a function of λ. Now consider the system where λ = ɛt. I(p(, q(; is the action associated with the torus for a constant λ specified by the initial, instantaneous values of p, q, and λ. Likewise I(p(t, q(t; ɛt is the action associated with the torus (for constant λ specified by the values of p, q, and λ at the instant t. A quantity is an adiabatic invariant if we can make its maximum devation from its initial value arbitrarily small during a process in which we change an external parameter a lot (say from to 1 simply by varying the external parameter slowly enough. (Mentally translate these words into the 1

2 definition above. Our principle is that the I i are adiabatic invariants (see Arnold s quote regarding such principles. It is straightforward to get this principle of adiabatic action invariance from the averaging principle. Following Arnold: For a fixed λ we can canonically transform the system (1 into the action-angle variable system via a generating function that depends on λ. words, there is a generating function 1 F = F (I, q; λ such that via p = F q, In other φ = F I, H new = H old ( we get the new Hamiltonian system where I and φ are the action-angle variables (meaning that I i = p dq and i ( I H = φ I =, φ = ( H ω(i, λ (3 I φ Now let λ = ɛt. Now the canonical transformation generated by (the same function F is time-dependent so that the last part of ( is changed to H new H old = ( F/ t I,q = ɛ F/ λ. Thus, we have the system of equations I = ɛf(i, φ; λ (4 φ = ω(i, λ + ɛg(i, φ; λ (5 λ = ɛ (6 where g = F I λ, f = F φ λ We apply the averaging principle. The averaged system has the form (7 J i = ɛ f i i = 1,,..., N J N+1 = ɛ (8 But f i = (π N π π ( F dφ 1... dφ N φ i λ = (do the φ i integral (9 Therefore, in the averaged system the J i components corresponding to the actions I i do not change. 1 Here we use a type generating function. If we considered the actions I to be a position (Q instead of a momentum, we could use a type 1 generating function. In this case ( would be altered.

3 I. ADIABATIC INVARIANCE IN A MORE GENERAL CONTEXT The word adiabatic appears in at least two contexts in physics. The most common context perhaps is in thermodynamics, and the second is related to sufficiently slow changes in dynamical systems, which is how we have used the word here. The two uses of adiabatic have a somewhat imprecise link regarding processes in which entropy does not change. In thermodynamics, an adiabatic process is a process in which there is no flow of heat into or out of the system considered. One can, however, do work on a system during an adiabatic process, by say, changing the volume. So the energy can certainly change. If the adiabatic process is also reversible (carried out near equilibrium then the process is isoentropic, the entropy does not change. An example of an adiabatic, non-reversible process is the free expansion of a gas, during which the entropy of the gas (and the universe increases. If we restrict ourselves to equilibrium thermodynamics however, adiabaticity implies that entropy is constant. In classical and quantum mechanics, adiabaticity has an analogous meaning, referring to a process which is in some way ordered or reversible. In quantum mechanics the adiabatic theorem states that a system that is in an energy eigenstate at time will evolve so that it always in an energy eigenstate (though the energy (eigenvalue and the eigenstate are functions of λ, given that the external parameter λ is varied with ɛ. If the system is varied too quickly then it ends up in a noticable superposition of the new eigenstates. The energy of the new state is uncertain. Since the result is a pure state however, the quantum entropy is still zero. Classically, if the process is too fast then I is not only non-constant, but its evolution depends on the initial phases φ(. That is, the surface of the initial torus evolves not to a single new torus (at time t, but to an ensemble of them. If the initial phase was not known and only the I could be observed, this would be like an increase in entropy. Besides invariance of the action for integrable systems, there is also a principle of ergodic adiabatic invariance [, 3] for completely chaotic systems (systems which, for fixed λ, the only constant of the motion, for any trajectory, is the hamiltonian H. The motion initially defines a N 1 dimensional energy shell in the N dimensional phase space. If λ is varied infinitely slowly, this shell evolves to a new energy shell such that the phase space volume Ω inside the shell is constant (Ω is the adiabatic invariant. If not, it evolves to an ensemble of 3

4 shells (a N dimensional region and Ω isn t well defined. Again, same entropy (or at least uncertainty idea. II. HARMONIC OSCILLATOR EXAMPLE I = (π 1 p dq (1 H = p /m + (1/mω q = E (11 p = ± me m ω q (1 I = 1 qmax π me m ω q dq q min (it is customary to integrate in sense that I is positive = 1 qmax me 1 mω π E q dq = 1 π I = E ω q min umax=1 me 1 u 1 π/ π/ u min = 1 1 u du cos y dy, sin y = u du mω /(E, u = mω E ( cos y dy, (nd term (13 Therefore adiabatic invariance of actions predicts that as a parameter (m or ω of the HO is changed by a significant amount but very slowly, the ratio of the oscillator s energy to its frequency remains approximately constant. The direct check to see whether or not this is true for the HO involves essentially some of the same averaging and time-scale related arguments as used in our proof of the averaging theorem. Here is a quick version with only one little leap of faith. Say we are changing ω from ω to ω uniformly over a long time τ = 1/ɛ ω(t = ω (1 + ɛt( sec 1 (14 4

5 Now Ė = H t = (1/mq (ω ω = ɛmω ω(t 1Hz (15 Imagine integrating Ė over the long time τ = 1/ɛ. Since Ė is always positive we are going to get a change in energy E that is O(1. So E is definitely not an adiabatic invariant. Now check the time derivative of the action I = E/ω (I ll suppress the extra Hz unit here. d E = dt( Ė ω ω E ω ω = ɛmq ω + ɛ E ω ω (16 = ɛ ω ω (mω q E = ɛ ω (U E ω U T = ɛω (17 ω Now lets integrate d(e/ω/dt over the long time τ to get the change in I I = ɛω τ U T ω dt (18 Is I of O(1 or O(ɛ? If it is O(1, like E then I is not an adiabatic invariant. If it is O(ɛ, then we can make I arbitrarily small by making the rate of change, ɛ, arbitrily small (although the total change in the external parameter is large. Look at the integrand in (18. The 1/ω changes slowly with time. But (U T rapidly oscillates between +E and E, two oscillations per period of the HO motion. So we do not expect the quantity t (U T /ω dt to be assymptotically proportional to t and thus we expect τ (U T /ω dt to not be O(1/ɛ but to be of some smaller order such as O(1 or O(ɛ. Then by (18 we see that I is indeed O(ɛ or smaller, and it is not O(1. Thus I = E/ω is indeed an adiabatic invariant as the principle predicted. [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, nd Ed., Springer-Verlag, New York, [] E. Ott, Phys. Rev. Lett. 4, 168 (1979. [3] C. Jarzynski, Phys. Rev. Lett. 71, 839 (