A general formulation is presented to derive the equation of motion and demonstrate thermodynamic consistency for several classes of phase-field (PF) and PF crystal (PFC) models. It can be applied to models with a conserved and non-conserved phase-field variable, describing either locally uniform or periodic stable states, and containing slow as well as fast thermodynamic variables. The approach is based on an entropy functional formalism previously developed in the context of PF models for locally uniform states [P. Galenko and D. Jou, Phys. Rev. E 71 (2005) p.046125] and thus allows to extend several properties of the latter to PF models for periodic states, i.e., PFC models.Phase-field (PF) modeling has become a versatile tool for studying the dynamics of systems out of equilibrium and is used in numerous applications of materials science [1][2][3][4]. For instance, considering a material that is disordered at high temperature and has two stable phases at low temperature. Upon quenching the material from high to low temperature, grains with different stable phases will develop, grow, and compete with each other. PF modeling is able to describe the time evolution of such a process. In a PF model for this example, a continuous function of space and time (x, t) is introduced -the PF variable -that assumes a different constant value for both stable phases. Near an interface between two grains, the value of changes rapidly. The PF variable in this example can be interpreted as an order parameter to represent the relative mass fraction of both phases. One advantage of PF modeling is that the PF contains information on the location of all interfaces in the system, without the need for explicit interface tracking. Another advantage is that it focuses on general features that are common to the dynamics of classes of systems, and thus helps to identify generic features in newly investigated systems. System-specific details are incorporated by the interpretation given to the PF variable, and by the values given to the phenomenological parameters in the model equations. Systems that are similar according to appropriate criteria, are therefore described by the same PF model. Important classes of such models developed for slow and rapid phase
The betatron difference resonance, Q. -ZQ, = -6, where Q,,z are the number of betatron oscillations per turn, was studied at the Indiana University Cyclotron Facility (IUCF) cooler ring. The position of the beam was measured in both the horizontal and vertical planes of oc cillation after a pulsed kicker magnet was fired to produce coherent motion. The effect of the coupling resonance was clearly observed and it was found that the subsequent particle motion could be described by a simple Hamiltonian. The resonance strength and tune shift as a function of betatron amplitude were measured.
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