Deterministic processes motion

Molecular dynamics is a deterministic process based on the simulation of molecular motion ly solving Newton s equations of motion for each atom and incrementing the position and velocity of each atom by use of a small time increment. If a molecular mechanics force field of adequate parameterization is available for the molecular system of interest and the phenomenon under study occurs within the time scale of simulation, this technique offers an extremely powerful tool for dissecting the molecular nature of the phenomenon and the details of the forces contributing to the behavior of the system. 

Subsequent workers, notably Terrell Hill and colleagues [6], extended the model to include back reactions for the chemical processes so that thermodynamic parameters such as efficiency can be calculated. In their analysis the motion of the myosin while attached to actin is modeled as a deterministic elastic relaxation often termed a power stroke. Unfortunately, this treatment of the mechanical motion as a deterministic process is not thermodynamically consistent and leads to incorrect predictions about the importance of the mechanical properties of the myosin. 

The hyperbolic expression can be derived from a picture in which the mechanical motion involves a deterministic power stroke. Although this power stroke model can be generalized to include thermodynamically consistent rate constants (see (3)), the description of the mechanical transitions as deterministic processes is not consistent with microscopic reversibility and is incorrect. 

Since the equation of motion, Eq. (4.16), contains a first derivative in time, the quantum state Y(r, t) at time t is solely determined by the initial state Y(r, 0), which, mathematically speaking, fixes the only integration constant. However, this fact demonstrates that the time evolution of a quantum mechanical system as described by Eq. (4.16) is a purely deterministic process, because a quantum mechanical state evolves uniquely according to the linear differential equation of motion at time t out of a state at an earlier time. The probabilistic character of quantum mechanics solely arises from the measurement process for which no satisfactory and consistent mathematical description is available (Bom s interpretation aims at coping with this problem see also section 4.3). 

Random motion is ubiquitous. At the molecular level, the thermal motions of atoms and molecules are random. Further, motions in macroscopic systems are often described by random processes. For example, the motion of stirred coffee is a turbulent flow that can be characterized by random velocity components. Randomness means that the movement of an individual portion of the medium (i.e., a molecule, a water parcel, etc.) cannot be described deterministically. However, if we analyze the average effect of many individual random motions, we often end up with a simple macroscopic law that depicts the mean motion of the random system (see Box 18.1). 

In principle any closed isolated physical system can be described as a Markov process by introducing all microscopic variables as components of Y. In fact, the microscopic motion in phase space is deterministic and therefore Markovian, compare (1.3). The physicist s question, however, is whether he can find a small set of variables whose behavior in time can be described as a multicomponent Markov process. The well-known, but still miraculous, experimental fact is that this is so for most many-body systems... 

To sum up, this chapter has endeavored to show that chemical processes in solution often proceed in a deterministic fashion over chemically significant distances and time scales. Ultrafast spectroscopy allows real-time observation of relative motions even when spectra are devoid of structure and has stimulated moleculear level descriptions of the early time dynamics in liquids. The implication of these findings for theories of solution phase chemical reactions are under active investigation. 

The approach developed in Section 2.2.1 is based on the independent treatment of the deterministic motion of a system and density fluctuations therein. The reaction description in terms of random process seems to be more consistent and logical. Equations (2.1.2) which were used above for a... 

With Avogadro s number of molecules participating in the above process, it would be a mistake to suppose all molecules progress through the above steps in a deterministic manner. With so many particles in motion, every possible combination of attachment is tried. For example, some clusters adsorb directly at a kink without significant diffusion. Other clusters detach from the surface and diffuse away in contrast to our macroscopic observations of growth. However,... 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

Several classical trajectories may result from such a collision process, as sketched in the figure. What makes the manifold of trajectories possible are the internal states i and j of the colliding molecules. To make that evident, let us first consider a situation where there are no internal states of the molecules and where the interaction potential only depends on the distance between the molecules, like for two hard spheres. Then there will only be one trajectory possible for a given b, , v, because the initial conditions for the deterministic classical equations of motion are completely specified. This will not be the case when the molecules have internal degrees of freedom, even if... 

In the absence of the external potential V, Eqs. (52) can be given a rigorous derivation from a microscopic Liouville equation (see Chapter I). We make the naive assumption that when an external potential driving the reaction coordinate is present, the two contributions (the deterministic motion resulting from the external potential and the fluctuation-dissipation process described by the standard generalized Langevin equation) can simply be added to each other. 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

Unlike diffusion, which is a stochastic process, particle motion in the inertial range is deterministic, except for the very important case of turbulent transport. The calculation of inertial deposition rates Is usually based either on a force balance on a particle or on a direct analysis of the equations of fluid motion in the case of colli Jing spheres. Few simple, exact solutions of the fundamental equations are available, and it is usually necessary to resort to dimensional analysis and/or numerical compulations. For a detailed review of earlier experimental and theoretical studies of the behavior of particles in the inertial range, the reader is referred to Fuchs (1964). 

In this article a simplified mass balance has been used to describe the net transport of sand over an accreting mud bottom. The combination of these two sedimentary processes controls the transition from sand to mud on the floor of the Sound. The distribution of sand may be described with three parameters an advection velocity of sand grains, an eddy-diffusion coefficient for mobile sand, and a rate of accumulation of marine mud. (Only the ratios of these quantities are needed if the distribution is in a steady state.) The motion of sand is thereby represented with both a deterministic part and a statistical part. The net, one-way advection of sand is the result of the superposition of an estuarine circulation on the tidal stream, and unpredictable variations in the rate of sand transport are represented as an eddy-diffusion process. Sand is immobilized when it is incorporated into the permanent deposit of marine mud. 

It had always been assumed that the changes in conformational state are inherently stochastic processes. However, new chaotic models suggest that channel proteins may organize fluctuations in structure into coherent patterns of motion, so that the switching between conformational states may be deterministic rather than stochastic. 

The term deterministic chaos is used to describe dynamic processes with random-looking, erratic data, in which random processes are not a dominant part of the system 46), Deterministic chaos is developed either in a dissipative system (the motion irreversible, as in systems with friction or systems exchanging energy with external media), such as the unsaturated zone, or in a conservative system (a Hamiltonian system that conserves energy), which is typical for the saturated zone. In order for a physical system to exhibit deterministic-chaotic behavior, the following conditions must be met ... 

Frequency Analysis. The statistical analysis of chaotic signals includes spectral analysis to confirm the absence of any spectral lines, since chaotic signals do not have any periodic deterministic component. Absence of spectral lines would indicate that the signal is either chaotic or stochastic. However, chaos is a complicated nonperiodic motion distinct from stochastic processes in that the amplitude of the high-frequency spectrum shows an exponential decline. The frequency spectrum can be evaluated using FFT-ba methods outlined the earlier sections. 

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