Time and Space

The names and symbols recommended here are in agreement with those recommended by IUPAP [4] and ISO [5.b,c]. 

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Haynes G R, Voth G A and Poliak E 1994 A theory for the activated barrier crossing rate constant in systems influenced by space and time dependent friction J. Chem. Phys. 101 7811... 

If the polarization of a given point in space and time (r, t) depends only on the driving electric field at the same coordmates, we may write tire polarization as P = P(E). In this case, we may develop the polarization m power series as P = = P - + P - + P - +, where the linear temi is = X] Jf/ Pyand the... 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

If attention is now limited to solutions separable in the space and time variables, we find... 

Noise. Technical differences exist between personal noise dosimeters and high accuracy sound level meters and these may alter the usual preference for personal monitors. But it is exposure to noise rather than general room noise that must be estimated for comparison with noise exposure criteria, the logarithmic expression and alternative means of summation (3 vs 5 db doubling) compHcate statistics. Exposure criteria for both dose and peak exposure must be evaluated, and space and time variabiUty of noise intensity can be immense. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

J. L. Burch and J. H. Waite, Jr., eds., Solar System Plasma in Space and Time, American Geophysical Union, Washiagton, D.C., 1994. 

Limited Data First, plant data are limited. Unfortunately, those easiest to obtain are not necessarily the most useful. In many cases, the measurements that are absolutely required for accurate model development are unavailable. For those that are available, the sensitivity of the parameter estimate, model evaluation, and/or subsequent conclusion to a particiilar measurement may be very low. Design or control engineers seldom look at model development as the primaiy reason for placing sensors. Further, because equipment is frequently not operated in the intended region, the sensitive locations in space and time have shifted. Finally, because the cost-effectiveness of measurements can be difficult to justify, many plants are underinstru-mented. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

If the stress is at the primary time step loeation and the veloeities are at the middle of the time step, then the resulting finite-difference equation is second-order accurate in space and time for uniform time steps and elements. If all quantities are at the primary time step, then a more complicated predictor-corrector procedure must be used to achieve second-order accuracy. A typical predictor-corrector scheme predicts the stresses at the middle of the time step and uses them to calculate the divergence of the stress tensor. 

The fluid is regarded as a continuum, and its behavior is described in terms of macroscopic properties such as velocity, pressure, density and temperature, and their space and time derivatives. A fluid particle or point in a fluid is die smallest possible element of fluid whose macroscopic properties are not influenced by individual molecules. Figure 10-1 shows die center of a small element located at position (x, y, z) with die six faces labelled N, S, E, W, T, and B. Consider a small element of fluid with sides 6x, 6y, and 6z. A systematic account... 

All fluid properties are functions of space and time, namely p(x, y, z, t), p(x, y, z, t), T(x, y, z, t), and u(x, y, z, t) for the density, pressure, temperature, and velocity vector, respectively. The element under consideration is so small that fluid properties at the faces can be expressed accurately by the first two terms of a Taylor series expansion. For example, the pressure at the E and W faces, which are both at a distance l/26x from the element center, is expressed as... 

Turbulence, which prevails in the great majority of fluid-flow situations, poses special problems. Due to the wide range of space and time scales in turbulence flow, its exact numerical simulation is possible only at relatively low Reynolds number (around 100 or below) and if the geometry is simple. 

Boundary conditions in space and time thermal, flow (venrilation, mechanical, and natural), sources of contaminants... 

When using LES, the time-dependent three-dimensional momentum and continuity are solved for. A subgrid turbulence model is used to mode the turbulent scales that are smaller than the cells. Instead of the traditional time averaging, the equations for using LES are filtered in space, and is a function of space and time. 

In particular we would like to treat some essential effects of fluctuations where we assume that, for example, thermal fluctuations exist and are localized in space and time. The effects on large lengths and long times are then of interest where the results are independent of local details of the model assumptions and therefore will have some universal validity. In particular, the development of a rough surface during growth from an initially smooth surface, the so-called effect of kinetic roughening, can be understood on these scales [42,44]. 

Here v is the space- and time-dependent velocity field, p is the density of the fluid, p is the local pressure, v is the kinematic viscosity, and / is some arbitrary body-force acting on each small element of the fluid (gravitation, for example). 

In cases where the potential is time-independent, we find that the wavefunction can be factorized into space- and time-dependent parts... 

It is a first-order differential equation in time, but second-order in the spatial variables. Space and time do not enter on an equal footing, as required by the special theory of relativity. 

Dirac (1930a) had the idea of working with a relativistic equation that was linear in the space and time derivatives. He wrote... 

Dirac s theory therefore leads to a Hamiltonian linear in the space and time variables, but with coefficients that do not commute. It turns out that these coefficients can be represented as 4 x 4 matrices, related in turn to the well-known Pauli spin matrices. I have focused on electrons in the discussion it can be shown... 

Then the mixture with droplets is quenched into the spinodal instability region to some T < Ta (Concentration c(r) within droplets starts to evolve towards the value C(,(T) > C(,(T ), but the evolution type depends crucially on the value Act = cj(T) — Ch(Ta). At small Act we have a usual diffusion with smooth changes of composition in space and time. But when Act is not mall (for our simulations Act O.2), evolution is realised via peculiar wave-like patterning shown in Figs. 8-10. 

State transitions are therefore local in both space and time individual cells evolve iteratively according to a fixed, and usually deterministic, function of the current state of that cell and its neighboring cells. One iteration step of the dynamical evolution is achieved after the simultaneous application of the rule (p to each cell in the lattice C. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Now consider the case where the system is perturbed randomly in space and time and F(t) represents a superposition of many avalanches (occurring simulta-neou.sly and independently). The total power spectrum is the (incoherent) sum of individual ( ontributions for single relaxation event due to single perturbations. 

Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. We call models of physics that assume Finite Nature Digital Mechanics. . ..we take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata. ... 

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