Invariant distribution

An important statistical characterization of long-term motion is provided by the invariant measure (also called the invariant distribution, or simply the probability distribution) of a mapping M x f ), Pm - satisfies two conditions ... 

The invariant distribution, p x), is a fixed point of this equation ... 

In electrochemical systems, a steady state during current flow implies that a time-invariant distribution of the concentrations of ions and neutral species, of potential, and of other parameters is maintained in any section of the cell. The distribution may be nonequilibrium, and it may be a function of current, but at a given current it is time invariant. 

S. Grossman and S. Thomae, Invariant Distributions and Stationary Correlation Functions, 32a (1977) 1353. 

An even stronger property than the ergodic property is the concept of a mixing system. For a mixing system, the finite time density p(z, t) converges, in the weak sense, to the invariant distribution Poo(z), as f 00. That is, we have, for all test functions (p in some chosen space... 

As a simple illustration, consider the harmonic oscillator with Hamiltonian H(q,p) = p /2 - - (ill and the invariant distributions obtained using several numerical methods. Applying Forward Euler to the harmonic oscillator, we have... 

An invariant distribution is a left eigenvector of II, having only non-negative elements, and corresponding to the eigenvalue 1. That is, an invariant distribution f satisfies... 

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... 

We have seen that invariant distributions correspond in general to solutions of... 

If we start with an initial distribution po which is different than peq, what sort of behavior do averages with respect to the evolving distribution p(X, t) exhibit If they converge to a stationary average, at what rate Does it matter if we start from a distribution near to (in a suitable sense) or far away from an invariant distribution ... 

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. 

Under what conditions (and in what sense) can we say that the distribution obtained in this way converges to a known invariant distribution ... 

We shall similarly consider the invariant (or native) distribution p. We write this invariant distribution as a perturbation of the (target) canonical distribution... 

The invariant distribution can, itself, be calculated by a formal expansion in powers of h, although in general this requires the solution of partial differential equations for the terms/. In some limiting cases it is possible to write down the solutions explicitly and to obtain, therefore, an understanding of the error induced in an average computed using a numerical method. 

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... 

Under suitable ergodicity assumptions (such that in the long-time limit, all distributions converge to the invariant distribution), we have... 

If we return to our scheme OYZYO, which has a simple inhomogeneity, then we can relate its invariant distribution to invariant distributions of schemes with a central OU update by writing its evolution operator as... 

It is important to note that this superconvergence property is only present in the limit of large time, i.e. this is an attribute of the invariant distribution of the discretization schemes p. As such, there are no bonuses to the weak or strong order of the superconvergent schemes, though equally there is no added cost to using the methods. 

We then ask how to analyze such schemes using the algebraic machinery developed throughout this chapter. As we solve each vector field on the constraint manifold, it is natural to consider the associated Lie derivatives on these manifolds and solve the corresponding Fokker-Planck equation for the discretization to find an invariant distribution, just as we did in the unconstrained case. This is a complicated programme and we do not develop this in detail here. 

Consider the GLA schemes of the form OXYX], making the ansatz that the invariant distribution is... 

Of course this is only a demonstration that the dynamics is compatible with p there may be other invariant distributions. 

We compare the trajectories and invariant distributions of the non-Newtonian line sampling thermostat (8.33)-(8.35) with three Newtonian-based thermostats Langevin dynamics (7.4), Nos6-Hoover (8.2)-(8.4) and Nos6-Hoover-Langevin (8.27)-(8.29) dynamics. We seek to canonically sample a one-dimensional system with double-well potential energy function given as... 

Each interacting pair thus preserves the invariant distribution. ... 

This means that the Markov chain preserves the density p as an invariant distribution. 

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