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The concept was first introduced by S. Pancharatnam [1] as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 [2] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic.
Trigonal bipyramidal molecular shape ax = axial ligands (on unique axis) eq = equatorial ligand (in plane perpendicular to unique axis). The Berry mechanism, or Berry pseudorotation mechanism, is a type of vibration causing molecules of certain geometries to isomerize by exchanging the two axial ligands (see the figure) for two of the equatorial ones.
A generalization was next introduced as the Fortuin–Kasteleyn random cluster model, which has many connections with the Ising model and other Potts models. Bernoulli (bond) percolation on complete graphs is an example of a random graph. The critical probability is p = 1 / N , where N is the number of vertices (sites) of the graph.
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In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables.
Sir Michael Victor Berry (born 14 March 1941) is a British theoretical physicist. He is the Melville Wills Professor of Physics (Emeritus) at the University of Bristol . He is known for the Berry phase , a phenomenon observed in both quantum mechanics and classical optics , as well as Berry connection and curvature .
It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory. [10]