Invariant quantities

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

From what has just been said with regard to carbon, it is evident that the atomicity of an element is, apparently at least, not a fixed and invariable quantity thus nitrogen is sometimes equivalent to five atoms of hydrogen, as in ammonic chloride, (i H Cl), sometimes to three atoms, as in ammonia (N" H,), mid sometimes to only one atom, as in nitrous oxide (N,0). But it is found that this variation in atomicity always takes place by the disappearance or development of an even number of bonds thus nitrogen is either a pentad, a triad, or a monad phosphorus and arsenic, either pentads or triads carbon and tin, either tetrads or dyads and sulphur, selenium, and tellurium, either hexads, tetrads, or dyads. 

Although the van der Wauls radius of an atom might thus seem to be a simple, invariant quantity, such is not the case. The size of an atom depends upon how much it is compressed by external forces and upon substituent effects. For example, in XeF ... 

Since the only invariant quantity associated with any given sphere, say S, is the number of material particles contained within it, such as N, then the only way to associate an invariant radial coordinate, say, r with S is to define it according to r = nf N), where ro is a fixed scale constant having units of length and the function / is restricted by the requirements f(Na) > f(Nb) whenever Na > Nb, f(N) > 0 for all N > 0, and /(0) = 0. To summarize, an invariant calibration of a radial coordinate in the model universe is given by r = rof(N) where... 

The right side of the equation does not depend on the position of the Gibbs dividing plane and thus, also, the left side is invariant. We divide this quantity by the surface area and obtain the invariant quantity... 

The presence of the above mentioned unphysical degrees of freedom translates into the fact that (i) a given component can in general be put to any value through the relevant coordinate transform and (ii) there must exist combination of the metric perturbations which remain unchanged under these coordinate transforms. The number of these quantities (called for obvious reasons gauge invariant quantities ) is precisely equal to the true number of physical degrees of freedom. 

From the four scalar quantities, it is possible to define two independent gauge invariant quantities, usually called the Bardeen potentials and 40... 

Note that there are some variations in the literature about the definition of these quantities sometimes the definitions of 4> and 4 are swapped, and their sign is also sometimes different. Here we choose the convention that and 4 are equal in the absence of anisotropic stress (see below), and that is the quantity that appears in the Laplacian term of the 00 part of the Einstein equations (the general relativistic analog of the Poisson equation), thus following the Newtonian convention to note the gravitational potential by. ) It is of course possible to define other scalar gauge invariant quantities. For example one can define... 

To identify the invariant quantity under Lorentz transformation it is noted that a light wave emitted from a point source at time t = 0 spreads to the surface of a sphere, radius r, such that... 

Hydrogen Hyperfine Structure and Related CPT Invariant Quantities... 

Although percolation theory deals with random systems, modeling and numerical calculations for percolation are usually carried out for a lattice of some definite geometry. To reach conclusions which do not depend on the details of lattice geometry but on dimensionality only, and thus are valid for the random system of interest, some invariant quantities must be constructed. 

Since the vector potential is not a gauge-invariant quantity, particular attention has to be paid to gauge transformations If F(rx, rjv, t) is a solution of the time-dependent Schrodinger equation... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

The term l7(mC) defined by (4.19) becomes zero because mC is an invariant quantity according to (4.31). The term defined by (4.20) is equal to zero because m is a constant and doesn t change in a collision. The local instantaneous pressure of the continuous phase might be decomposed as the sum of a mean pressure < p > and a pressure fluctuation p. However, the pressure covariance terms are normally neglected in gas-solid flows [6]. 

Lengths in the coarse-grained model can be identified with a real system by matching the characteristic molecular extension, Re(m) of one species with the experimental value. This quantity is conserved by the representation and is an example of an invariant quantity [41] mentioned in the introduction. In a generic model, these quantities are the only parameters that convey a specific physical information, establishing relevance with the energy, length and time scales of the real systems. 

In principle, it is the invariant quantities that can be expressed through certain material properties. Their number, required for each generic model parameterization, depends on the complexity of the chosen representation. 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

The definition of the invariant quantities, the 6-1 and the 9-1 S5nnbols may be inferred from a consideration of the orthogonal recoupling transformation between the different coupling schemes of irreducible products of the same degree ... 

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