Invariant distribution uniqueness

The fundamental result for Markov chains is the following If a given Markov chain is both irreducible and aperiodic, then it can be shown that there is a unique invariant distribution such that... 

It is easy to see that the distribution converges to a unique invariant distribution (in this case a Gaussian with mean zero and variance keTm as t oo). Ergodicity is easily verified in the case of the Ornstein-Uhlenbeck equation, since we can find its exact distributional solution. However, we can still demonstrate ergodicity for more complex SDEs where it is extremely challenging (or impossible) to find a solution or even to give qualitative statements about the solution. 

Lemma 7.1 Consider two numerical schemes with associated operators governing the evolution of measure ST andTS, with unique associated invariant distributions PsT and pts respectively, such that... 

The radial distribution Wction in this form meets all the requirements for a 3D structure descriptor it is independent of the number of atoms, i.e., the size of a molecule, it is unique regarding the three-dimensional arrangement of the atoms, and it is invariant against translation and rotation of the entire molecule. Additionally, the RDF code can be restricted to specific atom types or distance ranges to represent specific information in a certain three-dimensional structure space, e.g., to describe steric hindrance or structure/activity properties of a molecule. 

In considering how the initial state of yo impacts yt, the distribution of yt given yo, denoted here as P (yJyo), needs to be examined, yt depends directly on yo because the intervening variables (yo, yi,..., yi-i) are not provided. P. yo) will eventually converge to a unique invariant (or stationary) distribution that is independent of yo or t as the chain gradually forgets its initial state, subject to regularity conditions. 

The electron denstiy distribution p(r) of an atom or molecule is an observable property that can be measured by a combination of X-ray and neutron diffraction experiments [22]. Also, it is easy to calculate p(r) once the MOs and the wave function of a molecule have been determined. The distribution p(r) is invariant with regard to any unitary transformation of the MOs. It has been shown by Hohenberg and Kohn that the energy of a molecule in its (nondegenerate) ground state is a unique functional of p(r) [23]. In other words, the physical and chemical properties of a molecule can be related to p(r). Thus, p(r) represents the best starting point for an analysis of chemical bonding. 

An analogous situation may be more familiar. Let pt be a nonthermal distribution of the internal states of, say, diatomic molecules diluted in an excess of monoatomic buffer gas. Owing to collisions, the distribution p, will relax to a final, unique equilibrium distribution. Let p be that equilibrium distribution A/J [ = A(Ej — j)] is the evolution matrix. It converts any initial distribution to a unique final distribution and it leaves the equilibrium distribution invariant p° = Ajjpf- Given the unique final equilibrium distribution p°, we have no way whatever of knowing which was the initial nonequilibrium distribution which relaxed to it. 

One-Dimensional Pore Structure Models and Pore Size. Experimental data invariably have been characterized in terms of an arbitrary model of pore structure. The most common method consists of a bundle of parallel capillary tubes of equal length and a distributed size. The pore size would be unique only if the pores were tubes of uniform size and cross-section or spherical bodies. As neither is the... 

Mathematical treatments have been developed to describe the action of a-amylase on amylose. The treatments are based on the unique properties of the exponential (or most-probable) distribution of molecular weights of the substrate, namely, that (a) the principal averages are invarient to chain-end attack if the product molecules are ignored, and (b) the ratio of the principal... 

The solutions of Eqs.(2.1.34), (2.1.35) with appropriate boundary conditions such as Eqs. (2.4.3), (2.4.4) will be called the steady states of the system. Various properties of the steady states, such as the invariant manifolds and a priori bounds, the existence and uniqueness of solutions, the asymptotic behavior, and the stability will be treated in the sections that follow. There is a strong similarity in the properties of the uniform open systems investigated in Sections 1.8,1.9 and the distributed systems to be studied now. In both types of systems the interplay between reaction and transport rates (or flow rates) creates the possibility of multiple steady states for certain types of reaction kinetics. Furthermore, the conditions for uniqueness and stability of the steady state have a common mathematical and physical basis. 

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