Invariant measure

Recognizing that the information transmission is often both nonuniform (since the left and right fronts do not always propagate outward from the perturbation with a single well-defined velocity), and dependent on choice of initial states, one can instead define maximal and minimal average propagation speeds, and A  

In giving the response of the system to an impulse change to the initial configuration, Gij t) may thus be considered as being the Green Function for CA, If j i, 

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  

Although, in general, there may be many distinct invariant measures, we can single out one particular equilibrium measure by demanding that the spatial average over the distribution for (almost) all initial points xq be equal to the temporal average over the trajectory, [xq, x, X2,. ...  

If Pm x) is independent of xq then the system is said to be ergodic. In this case, various temporal averages over some function h x) may be conveniently rewritten as spatial averages over Pm x)  

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The long term behavior of any system (3) is described by so-called invariant measures a probability measure /r is invariant, iff fi f B)) = ft(B) for all measurable subsets B C F. The associated invariant sets are defined by the property that B = f B). Throughout the paper we will restrict our attention to so-called SBR-measures (cf [16]), which are robust with respect to stochastic perturbations. Such measures are the only ones of physical interest. In view of the above considerations about modelling in terms of probabilities, the following interpretation will be crucial given an invariant measure n and a measurable set B C F, the value /r(B) may be understood as the probability of finding the system within B. 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

Invariant measures correspond to fixed points of P which means that Pp = p iff /r e Ad is invariant. In what follows, we will advocate to discretize the operator P in such a way that its (matrix) approximation has an eigenvector... 

Vd satisfying PdVd = Vd, which means that Vd is an approximation of an invariant measure. For an invariant measure, any numerical discretization may be interpreted as a stochastic perturbation of the original problem. 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

For B = BiU B2, both measures are eigenmeasures of P associated with the eigenvalue A = 1. Hence, there is no unique invariant measure /t. In fact, any liii( ar combination... 

However, by constructing a nested sequence of successively larger discrete spaces and approximations therein we hope to end up with some approximation of a unique invariant measure, which is then implicitly defined via the constructed sequence of subspaces. An expression of this mathematical consideration is the multilevel structure of the suggested algorithm - details see below (Section 3.2). In physical terms, we hope that the perturbations introduced by discretization induce a unique and smooth invariant measure but are so weak that they do not destroy the essential physical structure of the problem. 

Inefficiency of Direct Simulation Suppose we want to compute the corresponding invariant measure p by direct simulation. Direct long term simulation by symplectic discretization of (1) yields the discrete solution ( ) a )i, ... 

Fig. 6. The density of the invariant measure of the potential Vi for total energy F = 4.5. Results of the subdivision approach (left) and a direct simulation with about 4.5 million steps for stepsize t = 1/30 (right).

The invariant measure corresponding to Aj = 1 has already been shown in Fig. 6. Next, we discuss the information provided by the eigenmeasure U2 corresponding to A2. The box coverings in the two parts of Fig. 7 approximate two sets Bi and B2, where the discrete density of 1 2 is positive resp. negative. We observe, that for 7 > 4.5 in (15) the energy E = 4.5 of the system would not be sufficient to move from Bi to B2 or vice versa. That is, in this case Bi and B2 would be invariant sets. Thus, we are exactly in the situation illustrated in our Gedankenexperiment in Section 3.1. 

Chapter 4 covers much of the same ground as chapter 3 but from a more formal dynamical systems theory approach. The discrete CA world is examined in the context of what is known about the behavior of continuous dynamical systems, and a number of important methodological tools developed by dynamical systems theory (i.e. Lyapunov exponents, invariant measures, and various measures of entropy and... 

The Kolmogorov consistency theorem [gnto88] asserts that any set of self- and mutually- consistent probability functions Pj, j = 1,2,..., jV may be extended to a unique shift-invariant measure on F,... 

Note that, since SnP,i = Pn for all finite block measures pn with n < N and, in particular, (Moo) = Pc, we can show that if the invariant measure Moo( ) of is a finite block measure of order n < A, then Poa ) is also invariant under An ). 

The 0 -order LST approximation of the invariant measure of is given simply by the fraction of neighborhoods that yield 1 under . 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

The relativistic invariance of the scalar product is also made explicit by Eq. (9-93) since as defined by Eq. (9-92) is a scalar, (k-x is an invariant and d 3k/k0 is the invariant measure element over the hyper-boloid k2 = m2). 

Invariably, measurements of decay of reactive molecules in solid glasses are found to be nonexponential, that is, first-order plots of ln[intensity] versus time are upwardly curved, as shown in Figure 10.3. 

Microbiologically based assay systems invariably measure the active antibiotic(s) or forms of the antibiotic that can be inhibitory to microorganisms. Immunological assays can measure both the active antibiotic as well as microbiologically inactive species. 

Finite-additive invariant measures on non-compact groups were studied by Birkhoff (1936) (see also the book of Hewitt and Ross, 1963, Chapter 4). The frequency-based Mises approach to probability theory foundations (von Mises, 1964), as well as logical foundations of probability by Carnap (1950) do not need cr-additivity. Non-Kolmogorov probability theories are discussed now in the context of quantum physics (Khrennikov, 2002), nonstandard analysis (Loeb, 1975) and many other problems (and we do not pretend provide here is a full review of related works). 

In the present case, it turns out that the isotropies already imposed on the system conspire to provide an automatic resolution of the problem that is consistent with the already assumed interpretation of time as a measure of ordered change in the model universe. To be specific, it turns out that the elapsed time associated with any given particle displacement is proportional, via a scalar field, to the invariant spatial measure attached to that displacement. Thus, physical time is defined directly in terms of the invariant measures of process with the model universe. 

Since the Raman techniques described are sensitive to (fl(t) - fi (0)>, we are interested in the properties of products of two elements of II, subject to the above constraints. Furthermore, it is necessarily true that any tensor elements of n that are related by reversed indices are identical, e.g., nxy = Flyx. This leaves us only two independent elements of R<3) to consider. It is conventional to express these independent elements in terms of rotationally invariant features of n One of these invariants (usually denoted a) is given by one third of the trace of FI (24). Since this invariant measures the average polarizability of the system, it is known as the isotropic component of n. The other invariant is generally denoted ft, and in the principal axis system of FI it is given by (24)... 

We note that in the case z > 2, a distribution with the form 1 /yz 1 would yield a divergent normalization factor because for z > 2 (p < 2) an invariant measure does not exist. If we set the system in an initial flat distribution, the pdf p(y, t) will keep changing forever, becoming sharper and sharper in the vicinity of y = 0, without ever reaching any equilibrium distribution. 

This equation represents the volume fraction of material possessing an orientation lying within invariant measure dh of the local state of interest h within an infinitesimal neighborhood of the material point at jc. Equation 2.11 can be described with a spectral basis using the simplest formulation as ... 

Invariant measure on classical phase space is an important concept in statistical theory. Suppose that there is an arbitrary phase space volume V t) at time t, which evolves to V l ) at time t. An invariant measure. 

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