Zero-one law

Three Zero-One Laws and Nonequilibrium Phase Transitions in Multiscale Systems... 

Zero-one law for steady states of weakly ergodic reaction networks... 

Zero-one law for relaxation modes (eigenvectors) and lumping analysis... 

In multiscale asymptotic analysis of reaction network we found several very attractive zero-one laws. First of all, components eigenvectors are close to 0 or +1. This law together with two other zero-one laws are discussed in Section 6 "Three zero-one laws and nonequilibrium phase transitions in multiscale systems". 

We can understand better this asymptotics by using the Markov chain language. For nonseparated constants a particle in has nonzero probability to reach and nonzero probability to reach A, . The zero-one law in this simplest case means that the dynamics of the particle becomes deterministic with probability one it chooses to go to one of vertices A, A3 and to avoid another. Instead of branching, A2 A and A2 A3, we select only one way either A2 A] or A2 A3. Graphs without branching represent discrete dynamical systems. 

THREE ZERO-ONE LAWS AND NONEQUILIBRIUM PHASE TRANSITIONS IN MULTISCALE SYSTEMS... 

This is the zero-one law for multiscale networks for any l,i, the value of functional b (30) on basis vector d, b (e ), is either close to one or close to zero (with probability close to 1). We already mentioned this law in discussion of a simple example (31). The approximate equality (71) means that for each reagent A e there exists such an ergodic component G of that A transforms when t -> 00 preferably into elements of G even if there exist paths from A to other ergodic components of W. 

In general multiscale network, two type of obstacles can violate approximate equality t 1 /k.. Following the zero-one law for nonergodic multiscale networks (previous subsection) we can split the set of all vertices into two subsets. Hi and H2. The dominant reaction network dom mod( ) is a union of networks on sets without any link between sets. 

For reaction networks with well-separated constants coordinates of left eigenvectors Z are close to 0 or 1. We can use the left eigenvectors for coordinate change. For the new coordinates z,- = Vc (eigenmodes) the simplest equations hold Zi = 2 Z . The zero-one law for left eigenvectors means that the eigenmodes are (almost) sums of some components z, = some sets of numbers Vj. 

For each zero-one law specific sharp transitions exist if two systems in a one-parametric family have different zero-one steady states or relaxation modes, then somewhere between a point of jump exists. Of course, for given finite values of parameters this will be not a point of discontinuity, but rather a thin zone of fast change. At such a point the dominant system changes. We can call this change a... 

This definition of dependability is also called the Kolmogorov s zero-one law. It is one of the laws of large numbers, since according to the definition only two cases exist, either there are dependencies or there aren t. Since we already learned that a complete independency could rarely be achieved, ISO 26262 speaks of a sufficient independency. Failures of common causes or failure dependencies between functions, which can affect through different mechanisms, are often no longer analyz-able with the classical methods. In this case we can often only rely on experience. For functional dependencies we can systematically analyze a lot of things firom the functional chains and their derivation in the different horizontal abstraction levels. A barrier, independent if it is a functional or technical barrier, or whatever technology it is, could be only assessed for its sufficiency or effectiveness, in the specific context and for possible failure effects (Fig. 5.59). 

The zero-th law, which justifies the existence of the thermometer, says that two bodies A and B which are in thermal equilibrium with a third body are in thermal equilibrium with each other. There is no heat flow from one to the other, and they are said to be at the same temperature. If A and B are not in thermal equilibrium, A is said to be at a higher temperature if the heat flows from A to B when they are placed in thermal contact. The changes in temperature usually produce changes in physical properties like dimension, electrical resistance and so on. Such property variations can be used to measure the temperature changes. 

In the above general equation, k is called the rate constant, a and b are called reaction orders. Most reactions considered in introductory chemistry have a reaction order of zero, one, or two. The sum of all reaction orders for a reaction is called the overall reaction order. Rate laws cannot be predicted from the stoichiometry of a reaction. They must be determined by experiment or derived from knowledge of reaction mechanism. 

In contrast to energy, one can assign an absolute value for entropy if it is postulated that the entropy of a perfect (i.e., defect-free) solid goes to zero at absolute zero (third law). One of the implications of the third law is that every substance has a certain amount of 5" associated with it at any given temperature above absolute 0 K. 

Statistical copolymers are those in which the monomer sequence follows a specific statistical law (e.g., Markovian statistics of order zero, one, two). Random copolymers are a special case of statistical copolymers in which the nature of a monomeric unit is independent of the nature of the adjacent unit (Bernoullian or zero-order Markovian statistics). They exhibit the structure shown in Figure 6.1. If A and B are the two monomers forming the copolymer, the nomenclature is poly (A-stat-B) for statistical copolymers and poly (A-ran-B) for the random case. It should be noted that sometimes the terms random and statistical are used indistinctly. The commercial examples of these copolymers include SAN poly (styrene-ran-acrylonitrile) [4] and poly (styrene-ran-methyl methacrylate) (MMA) [5]. 

Other production terms are also restricted. The change d,U, for example, must always be zero, since energy is conserved. All changes of U must come from flux across the boundary. Similarly, the change in mass, d,M, must be zero. One cannot produce new mass. Equations (4) and (5) are expressing these conservation laws for energy and mass. 

The ideal gas equation is an example of a limiting law, that is, a scientific law that becomes exact only in some well-defined limit, which, in the case of Equation 5.6, is the limit when P— 0. Any real gas will show deviations from the ideal gas eqnation, but these deviations become progressively smaller as the pressure approaches zero. One way to see this behavior is to examine the compression factor (Z), defined as... 

This aspect appears to relate to what happens during the process of dissolution under the conditions which prevail as Pa and JVa emerge from zero i.e., the line is near the left bottom corner of the diagram, as in, e.g.. Fig. 50. The current opinion appears to be that all gases A (or liquid A) obey Henry s law, Pa = KNa, for small values of Na, i.e., when Na is closely approaching zero. The Henry s law constant does not need to be p°A, the constant of the Raoult s law. I imply from Lewis statement that as closely approaches zero, there is only one law, essentially only one line. The question is At what stage, and why, does the Henry s law line begin to... 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milhseconds) and highly reaclive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polvmerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equihbrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, aU possessing nonintegral power law or hyperbohc rate equations. 

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