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Classical Methods of Molecular Simulations of Disordered Materials
Published in Alexander Bagaturyants, Vener Mikhail, Multiscale Modeling in Nanophotonics, 2017
Alexander Bagaturyants, Vener Mikhail
Phase trajectories of representation points of an ensemble cannot intersect or be tangent to each other. Otherwise, the mechanical system being in a certain state could move after the intersection or tangency point in different ways, which contradicts to the principle of mechanical causality (i.e., the uniqueness of the solution of the Hamiltonian equations). The motion of phase points obeys Liouville’s theorem. The essence of this theorem is in the following equivalent statements:For a given ensemble, points do not disappear and do not arise.Representation points move in the phase space as an incompressible liquid; that is, p(q,p,t) is constant along the phase trajectory.The phase volume ΔΓ containing a given set of moving systems of an ensemble is constant.
Chaos in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
In the Hamiltonian mechanics, an important theorem holds. It is the Liouville’s theorem stating that the initial phase space volume, defined by the sum of all possible initial conditions, does not change. In other words, the final volume of the phase space, Vf, including the trajectories of all the initial points, contained in Vi, is equal to the initial volume: Vf=Vi. This condition would mean that the uncertainty at the end of the transformation, Hf=−∑k=1Nfpklog2pk=log2(Vf), would be equal to Hi. However, the final shape is a fractal, full of holes, knots, and tendrils. It is a complex structure (see Figure 11.23)! For the determination of its volume, we are forced to circumscribe it within a larger, smooth bag, like in Figure 11.24.
Partial Differential Equations and Boundary Value Problems
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
He was the first to solve a boundary value problem by solving an equivalent integral equation, a method developed by Fredholm and Hilbert in the early 1900s into one of the major fields of modern analysis. His ingenious theory of fractional differentiation answered the long-standing question of what reasonable meaning can be assigned to the symbol dny/dxn when n is not a positive integer. He discovered the fundamental result in complex analysis now known as Liouville’s theorem—that a bounded entire function is necessarily a constant—and used it as the basis for his own theory of elliptic functions. There is also a well-known Liouville theorem in Hamiltonian mechanics, which states that volume integrals are time-invariant in phase space. His theory of the integrals of elementary functions was perhaps the most original of all his achievements, for in it he proved that such integrals as ∫e−x2dx,∫exxdx,∫sinxxdx,∫dxlogx′
Adaptive Markov chain Monte Carlo algorithms for Bayesian inference: recent advances and comparative study
Published in Structure and Infrastructure Engineering, 2019
Seung-Seop Jin, Heekun Ju, Hyung-Jo Jung
The two variables are independent (), and the momentum variable is introduced artificially to allow Hamiltonian dynamics to operate. Therefore, the variable of momentum can be marginalised, and hence Equation (7) recovers the posterior density. In addition, it is worth noting that this Hamiltonian dynamics is time-reversible and maintains the volume conservation by Liouville’s theorem (Neal, 2011).