Adiabatic invariant

A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant. By this it is meant that if a system is varied between two end points, as the time for the variation between the end points is increased to infinity, the variation of an adiabatic invariant between the two end points goes to zero. In thermodynamics, an adiabatic process is a change that occurs without heat flow; it may be slow or fast. A reversible adiabatic process is an adiabatic process that occurs slowly compared to the time to reach equilibrium. In a reversible adiabatic process, the system is in equilibrium at all stages and the entropy is constant. In the 1st half of the 20th century the scientists that worked in quantum physics used the term "adiabatic" for reversible adiabatic processes and later for any gradually changing conditions which allow the system to adapt its configuration. The quantum mechanical definition is

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Analysis of adiabatic trapping for quasi-integrable area-preserving maps

Massimo Giovannozzi, Cedric Jean-François Hernalsteens

Trapping phenomena involving nonlinear resonances have been considered in the past in the framework of adiabatic theory. Several results are known for continuous-time dynamical systems generated by Hamiltonian flows in which the combined effect of nonlinear resonances and slow time variation of some system parameters is considered. The focus of this paper is on discrete-time dynamical systems generated by two-dimensional symplectic maps. The possibility of extending the results of neo-adiabatic theory to quasi-integrable area-preserving maps is discussed. Scaling laws are derived, which describe the adiabatic transport as a function of the system parameters using a probabilistic point of view. These laws can be particularly relevant for physical applications. The outcome of extensive numerical simulations showing the excellent agreement with the analytical estimates and scaling laws is presented and discussed in detail.

American Physical Society2014

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Geometric phases in discrete dynamical systems

Nicolas Piro Mastracchio

In order to study the behaviour of discrete dynamical systems under adiabatic cyclic variations of their parameters, we consider discrete versions of adiabatically-rotated rotators. Parallelling the studies in continuous systems, we generalize the concept of geometric phase to discrete dynamics and investigate its presence in these rotators. For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number of the system. For the discrete version of the rotated rotator considered by Berry, the rotated standard map, we further explore this connection as well as the role of the geometric phase at the onset of chaos. Further into the chaotic regime, we show that the geometric phase is also related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent. (C) 2016 Elsevier B.V. All rights reserved.

Elsevier Science Bv2016

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Bootstrap current destabilization of ideal MHD modes in three-dimensional reactor configurations

Wilfred Anthony Cooper

In current-free stellarators, the parallel current density is normally too weak to drive global external kink modes. However, at finite values of beta, the bootstrap current (BC) can provide sufficient free energy to trigger this class of mode in some stellarator systems. The effect of the BC in the collisionless 1/nu regime has been investigated in several different types of stellarator reactor systems all with a volume V similar to 1000 m(3). In quasiaxisymmetric and quasihelically symmetric stellarators, the BC is large at, finite beta and this can cause low order resonances to move into and emerge out of the plasma which in turn can destabilize global internal and external kink modes. In a six-field period system with poloidally closed contours of the magnetic field strength B, the BC is small and decreases the rotational transform only slightly. As a result, only intermediate to high n modes can become weakly destabilized. Furthermore, it is demonstrated in this system that the contours of the second adiabatic invariant J(parallel to) close poloidally for all trapped particles at finite beta* similar to 6%. This condition leads to the loss of a very small fraction of the collisionless alpha-particle orbits. In Sphellamak configurations with peaked toroidal currents required to generate nearly isodynamic maximum-B confining field structures, the BC accounts only for a small fraction of the total current. The loss of a-particles born within the inner quarter of the plasma volume is negligible while about 1/3 of those born at half volume escape the device within a slowing down time.

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