Adiabatic invariance

Table 2. Maximum variation in the vibrational energy Ef and the adiabatic invariant Jf using the energy conserving method (6).

S. Reich. Preservation of adiabatic invariants under symplectic discretization. i4pp. Numer. Math., to appear, 1998. 

Then, the adiabatic invariance Eq. (19) holds and the limit of the sequence 9e of QCMD solutions is 9bo ... 

Note that the conservation of total energy and the conservation of the adiabatic invariants associated to the Born-Oppenheimer limit of the QCMD model provide a simple test for the behavior of a numerical integrator. 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

Equation (4.24) indicates that the quantum number of the transverse x-vibration is an adiabatic invariant of the trajectory. At T=0 becomes the instantaneous zero-point spread of the transverse vibration (2co,) in agreement with the uncertainty principle. 

Burgers, J. M. [1917] Adiabatic Invariants of Mechanical Systems, Philosophical Magazine, 33, p. 514. 

We note that for the harmonic Hamiltonian in (5.25) the variance of the work approaches zero in the limit of an infinitely slow transformation, v — 0, r — oo, vt = const. However, as shown by Oberhofer et al. [13], this is not the case in general. As a consequence of adiabatic invariants of Hamiltonian dynamics, even infinitely slow transformations can result in a non-delta-like distribution of the work. Analytically solvable examples for that unexpected behavior are, for instance, harmonic Hamiltonians with time-dependent spring constants k = k t). 

For isolated Hamiltonian systems the width of the work distribution remains finite even in the limit of infinitely slow switching. This is a consequence of the so-called adiabatic invariants [16],... 

Perturbation Solution for Le = 1. The simplest case of Le = POjj/Pej = 1 is analyzed here. This assumption yields the adiabatic invariant, B(l-c) = 0-l which can be used to further simplify Eqns. (l)-(5) to... 

Quantum effects become important only at veiy low temperatures, i.e. for BIkT 1. The best candidates for observing these effects thus are hydrides. Quantum effects become more pronounced when A and/or B are species with open electronic shells see e.g. the differences between the associations N2 + N2 —> N4 and Q2 + Q2 —> 4 at temperatures below 10 K [10], Before leaving this section, it should be mentioned that alternative approaches such as the ACCS A treatment Irom [18], the perturbed rotational state treatment finm [19] and the semiclassical adiabatic invariance method Irom [20] all represent simplified variants of the here described SACM/CT approach and, like the... 

IV. Anomalous Behavior of Variance of Nonergodic Adiabatic Invariant... 

B. Adiabatic Invariant for a Simple Hamiltonian Chaotic System... 

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

IV. ANOMALOUS BEHAVIOR OF VARIANCE OF NONERGODIC ADIABATIC INVARIANT... 

In this section we will consider an nonergodic adiabatic invariant for a chaotic Hamiltonian system. Specifically, we are interested in the mixed phase space, in which tori and chaotic seas coexist. 

We consider the work in a quasi-static (QS) process and define an adiabatic invariant. In the quasi-static process, we assume that the probability density is given as p(x, t) = pme(x, c(tj). Then,... 

Now we consider the adiabatic invariant for a simple Hamiltonian chaotic system. The Hamiltonian is defined as... 

We consider a periodic change of the stochastic parameter K(t) as an external transformation to know how good the adiabatic invariant is. Since Eo(t) is the value of the energy at time t determined by the conservation of p(Eo(f)>f) = ll( o(0), 0). the deviation and the variance of the adiabatic invariance are simply related to those of the Hamiltonian as... 

We have numerically estimated the goodness of an adiabatic invariant. 

We find that the variance of the adiabatic invariant is a nice quantity to measure the complexity in phase space. In a strong chaotic case such as K 4, we can expect goodness of the ergodic adiabatic invariant. The correlation decays in a relatively short time. The variance decays as shown in Fig. 5. In the... 

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