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Consider the average number of particles with particle properties denoted by a particle state vector (x,r) (where x corresponds to particle properties like size, density, etc. also known as internal coordinates and, r corresponds to spatial position or external coordinates) dispersed in a continuous phase defined by a phase vector Y(r,t) (which again is a function of all such vectors which ...
A quantum master equation is a generalization of the idea of a master equation.Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements.
The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method , DMRG is an efficient algorithm that attempts to find the lowest-energy matrix product state wavefunction of a Hamiltonian.
In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the non-interacting Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.
The problem of computational expense can be alleviated through simplification schemes. [7] In the density fitting scheme, the four-index integrals used to describe the interaction between electron pairs are reduced to simpler two- or three-index integrals, by treating the charge densities they contain in a simplified way. This reduces the ...
Solving the equations in Wertheim's theory can be complicated, but simplifications can make their implementation less daunting. Briefly, a few extra steps are needed to compute given density and temperature. For example, when the number of hydrogen bonding donors is equal to the number of acceptors, the ESD equation becomes:
A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent quantum coherence which is a physical characteristic that is intrinsically quantum mechanical.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...