Deterministic evolution

It is illuminating to briefly discuss the significance and importance of each term in Eq. (31). The first three terms are trivial, as they are nothing but a description of the deterministic evolution in the reactive mode in the absence of coupling with the nonreactive mode the sixth term is also trivial and is a diffusional term corresponding to the effect of the stochastic force... 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

Finally, what can we measure in quantum mechanics The states themselves looses their causal evolution when interacting with an apparatus while achieving a sort of deterministic evolution therefore, the measurement is even not on the states but on operators that have an intrinsic role on the system structure from this point we recover the phenomenological idea that those operators that commutes with Hamiltonian of the system are observables and can be averaged on certain states to be measured (observed). Yet, other quantities may be measured on perturbed states of Ihe systems under investigation. 

Erosion-corrosion is another example of a process with a stochastic trigger with a deterministic evolution. In this case, the trigger is the impact of a particle onto the surface, and the resultant exposure of unpassivated metal (for a passive system) or the removal of seale or a change in mass transport may cause a current pulse, the time-evolution of whieh can be modeled according to the metal-solution system being examined [59-64]. Unfortunately, most of the current induced by an impact is not captured by an external eleetrode, and this makes interpretation complex. Indeed, correlation with the results of... 

Coming to the role of the intrinsic parameters, the general trend is that bimodality is enhanced for the parameter range for which the deterministic evolution displays two widely separated time scales. On the other hand -and this leads us to the role of initial conditions- to "probe such a time scale difference the system has to start from a state located sufficiently before the inflexion point of the deterministic potential (cf. Fig. 4) Otherwise it undergoes a rapid relaxation to the final state following essentially the deterministic path. 

Figure 15 depicts the time correlation function of the variable x. We obtain a fast initial decay similar to Figure 12, but observe that subsequently the oscillations around zero are smoothed compared to those of the deterministic evolution. In the corresponding power spectrum the principal peak appearing in the power spectrum of the deterministic limit has survived whereas the others are now hardly distinguishable from the background. 

Deterministic air quaUty models describe in a fundamental manner the individual processes that affect the evolution of pollutant concentrations. These models are based on solving the atmospheric diffusion —reaction equation, which is in essence the conservation-of-mass principle for each pollutant species... 

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. 

Figure 3.31 shows sample evolutions for p = 0, 1/4 and 3/4. The space-time pattern for p = 0 rapidly settles into an ordered state consisting of checkerboard-pattern domains, separated by two-site kinks once formed, the kinks remain locked in place. As p is slowly increased, these kinks begin to undergo annihilating random walks, much like the ones we saw earlier in the evolution of (the deterministic) rule R18. Their density decreases like pkink Grassberger, et.al, observed... 

Let us first of all consider the deterministic Life rule, or zero temperature limit of our more general stochastic rule. Using the density p to represent our state of knowledge of the system at time t, our problem is then to estimate the time-evolution of p for T = 0. 

Except for using the statistical measure p to characterize the temporal evolution, the evolution itself has so far been entirely deterministic. We now take explicit account of temperature, as introduced via equation 7.111 ... 

Initially, it was assumed that the HlV-1 population is infinite, evolution is deterministic, and antiretroviral resistance development is definite (Coffin 1995). However, our research amongst others has demonstrated that the effective population size, defined as the average number of HIV variants that produces infectious progeny is relatively small (Leigh Brown 1997 Leigh Brown and Richman 1997 Nijhnis et al. 1998). This can be explained because the majority of virus particles that are produced harbor deleterious mutations resulting in noninfectious virus. Also limited target cell availability and inactivation of potentially infectious viruses by the host... 

Deterministic rules, or a combination of deterministic and random rules, are of more value in science than rules that rely completely on chance. From a particular starting arrangement of cells and states, purely deterministic rules, such as those used by the Game of Life, will always result in exactly the same behavior. Although evolution in the forward direction always takes the same course, the CA is not necessarily reversible because there may be some patterns of cells that could be created by the transition rules from two different precursor patterns. 

Recent studies have indicated that fluidized beds may be deterministic chaotic systems (Daw etal.,1990 Daw and Harlow, 1991 Schouten and van den Bleek, 1991 van den Bleek and Schouten, 1993). Such systems are characterized by a limited ability to predict their evolution with time. If fluidized beds are deterministic chaotic systems, the scaling laws should reflect the restricted predictability associated with such systems. 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

The sequence of decisions obtained from the stochastic scheduler for all possible evolutions of the demand for the three periods is provided in Figure 9.7. The sequence of decisions obtained by the stochastic scheduler differs from that obtained by the deterministic one, e.g., xi(ti) = lOinstead ofxi(ti) = 6. The average objective for the stochastic scheduler after three periods is P = —17.65. 

The interaction of forced and natural convective flow between cathodes and anodes may produce unusual circulation patterns whose description via deterministic flow equations may prove to be rather unwieldy, if possible at all. The Markovian approach would approximate the true flow pattern by subdividing the flow volume into several zones, and characterize flow in terms of transition probabilities from one zone to others. Under steady operating conditions, they are independent of stage n, and the evolution pattern is determined by the initial probability distribution. In a similar fashion, the travel of solid pieces of impurity in the cell can be monitored, provided that the size, shape and density of the solids allow the pieces to be swept freely by electrolyte flow. 

More sophisticated calculations (14,20), using either stochastic Monte Carlo or deterministic methods, are able to consider not only different Irradiating particles but also reactant diffusion and variations In the concentration of dissolved solutes, giving the evolution of both transient and stable products as a function of time. The distribution of species within the tracks necessitates the use of nonhomogeneous kinetics (21,22) or of time dependent kinetics (23). The results agree quite well with experimental data. 

In parallel with the studies described above, which concern perfectly deterministic equations of evolution, it appeared necessary to complete the theory by studying the spontaneous fluctuations. Near equilibrium, any deviation is rapidly damped but near a bifurcation point, a fluctuation may may lead the system across the barrier. The fluctuation is then stabilized, or even amplified this is the origin of the phenomenon which Prigogine liked calling creation of order through fluctuations. More specifically, one witnesses in this way a step toward self-organization. 

The opposition of the two concepts of time is particularly striking in the famous fundamental problem of statistical mechanics How can we understand the emergence of an irreversible evolution at the level of macroscopic physics (in particular, of thermodynamics) from the deterministic and reversible Newtonian laws of mechanics ... 

We first consider a Hamiltonian, thus deterministic, system. Denoting by oo the set of all phase space coordinates of a point in phase space (which determines the instantaneous state of the system), the motion of this point is determined by a canonical transformation evolving in time, 7), with Tq = I. The function of time TfCO thus represents the trajectory passing through co at time zero. The evolution of the distribution function is obtained by the action on p of a unitary transformation Ut, related to 7) as follows ... 

The authors then ask the following question Do there exist deterministic dynamical systems that are, in a precise sense, equivalent to a monotonous Markov process The question can be reformulated in a more operational way as follows Does there exist a similarity transformation A which, when applied to a distribution function p, solution of the Liouville equation, transforms the latter into a function p that can also be interpreted as a distribution function (probability density) and whose evolution is governed by a monotonous Markov process An affirmative answer to this question requires the following conditions on A (MFC) ... 

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

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