Deterministic, classical mechanics

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

The statistical nature of the quantum theory has troubled several eminent scientists, including Einstein and Schrodinger. They were never able to accept that statistical predictions could be the last word, and searched for a deeper theory that would give precise deterministic predictions, rather that just probabilities. They were unsuccessful, and most physicists now believe that this was inevitable, as some predictions of the quantum theory, which have been verified experimentally, suggest that a completely deterministic theory such as classical mechanics cannot be correct. 

At this point we will, briefly, describe some of the fundamental qualitative differences between a quantum mechanical and a classical mechanical description. First of all, a trajectory R(t) is replaced by a wave packet, which implies that a deterministic description is replaced by a probabilistic description. x(R,t) 2 is a probability density, giving the probability of observing the nuclei at the position R at time t. In... 

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

In 1955, Born wrote an article entitled Is classical mechanics in fact deterministic (reprinted in Born (1969)). In this article Born tried to expose the emptiness of the concept of determinism. His argument is the following If determinism is not a useful concept to begin with, an... 

The question of predictability within the deterministic structure of classical mechanics was clearly appreciated by many eminent researchers in nonlinear systems theory and theoretical physics (see, e.g., Brillouin (I960)). Borel (1914) adds an additional twist to the predictability discussion. He argues that the displacement of a lump of matter with mass on the order of 1 g by as little as 1 cm and as far away as, e.g., the star Sirius is enough to preclude any prediction of the motion of the molecules of a volume of a classical gas for any longer than a firaction of a second, even if the initial conditions of the gas molecules are known with mathematical precision. Borel s example shows that many physical systems are not only sensitive to initial conditions, but also to miniscule changes in system parameters. The sensitivity to system parameters is a fundamental additional handicap for accurate long-time predictions. In the face of Borel s example, Brillouin (1960) points out that the prediction of the motion of gas molecules is not only very diflficult , as pointed... 

This equation is Schrodinger s wave equation, where h is Planck s constant and H is the Hamiltonian of the system to be investigated. The Schrodinger equation is a deterministic wave equation. This means that once ip t = 0) is given, ip t) can be calculated uniquely. Prom a conceptual point of view the situation is now completely analogous with classical mechanics, where chaos occurs in the deterministic equations of motion. If there is any deterministic quantum chaos, it must be found in the wave function ip. 

To date, quantum theory, despite its many peculiar non-classical microscopic aberrations, has been used effectively to explain experimental observations and to predict accurately physical effects in advance of the experiment. Interestingly, many of the founders of quantum mechanics later rejected it, primarily because it was a non-deterministic theory. The break from their classical mechanics view of the universe appears to have been too severe for them to accept. But a new generation of physicists was prepared to embrace the quantum mechanics and apply the methods to chemical structures. With a series of novel concepts and observations, physics and chemistry were changed forever. 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

Despite the similarity to the Gauss approach to classical mechanics, there is a key difference between the classical actions described above and the corresponding action of the stochastic difference equation. The classical actions are deterministic mechanical models the SDE is a nondeterministic approach that is based on stochastic modeling of the numerical errors introduced by the finite difference formula. 

Classical mechanics is a deterministic theory, in which the time evolution is uniquely determined for any given initial condition by the Newton equations (1.98). In quantum mechanics, the physical information associated with a given wave-function has an inherent probabilistic character, however the wavefiinction itself is rmiquely determined, again from any given initial wavefunction, by the Schrodinger equation (1.109). Nevertheless, many processes in nature appear to involve a random component in addition to their systematic evolution. What is the origin of this random character There are two answers to this question, both related to the way we observe physical systems ... 

We have already observed that the frill phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Sclrrbdinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underlying dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. 

The expectation that classical mechanics provides a simple, deterministic, easily predictable view of the dynamics of few-body systems is now recognized as a gross oversimplification. Research over the past 20 years has shown that such systems are capable of displaying relaxation to equilibrium and extreme sensitivity to both initial conditions as well as system parameters. These features, quantified subsequently, are essential characteristics of what is now termed chaotic behavior in a conservative Hamiltonian system. The relationship between chaotic behavior in conservative systems and the reaction dynamics of isolated molecules is the subject of this chapter. 

A major difference between quantum and classical mechanics is that classical mechanics is deterministic while quantum mechanics is probabilistic (more correctly, quantum mechanics is also deterministic, but the interpretation is probabilistic). Deterministic means that Newton s equation can be integrated over time (forward or backward) and can predict where the particles are at a certain time. This, for example, allows prediction of where and when solar eclipses will occur many thousands of years in advance, with an accuracy of meters and seconds. Quantum mechanics, on the other hand, only allows calculation of the probability of a particle being at a certain place at a certain time. The probability function is given as the square of a wave function, P t,i) = P (r,f), where the wave function T is obtained by solving either the Schrodinger (non-relativistic) or Dirac (relativistic) equation. Although they appear to be the same in Figure 1.2, they differ considerably in the form of the operator H. 

The center of the controversy was that quantum mechanics is indeterministic, while classical mechanics is deterministic, although this indeterminism is not all that it seems. As will be shown later in this chapter, quantum mechanics is a fuUy deterministic theory in the Hilbert space (the space of aU possible wave functions of the sjrstem), its indeterminism pertains to the physical space in which we five. 

In physics treatments, the Liouvillian is often denoted iC. This may seem namral since (a) it allows a formal correspondence with the Schrbdinger equation and between the propagators of quantum mechanics and classical mechanics, and (b) for Hamiltonian dynamics, as we shall see, the Liouvillian is skew-adjoint (in a certain sense skew symmetric) and the inclusion of i explicitly calls attention to this fact. However, we feel it is more natural to omit the i in a treatment that includes study of both stochastic and deterministic models. 

One of the most important ramifications of the uncertainty principle is that it brought about a radical change in the philosophy of science. Classical mechanics was deterministic in nature that is to say that if the precise position and momentum of a particle or a collection of particles were known, Nev/ton s laws could be used (at least in principle) to determine all the future behavior of the particle(s). The uncertainty principle, however, tells us that there is an inherent limitation to how accurately we can measure the two quantities simultaneously. Any observation of an extremely small object (one whose wavelength is on the same magnitude or larger than the particle itself) necessarily effects a nonnegligible disturbance to the system, and thereby it influences the results. Einstein never liked the statistical nature of quantum mechanics, saying God does not play dice with the universe. Nonetheless, the quantum mechanical model is a statistical one. 

The constant xq is the largest magnitude that x attains and is called the amplitude of the oscillation. You can see that x and Vx are now determined for all values of the time, both positive and negative. This is a characteristic of classical equations of motion. We say that classical mechanics is deterministic, which means that the classical equations of motion determine the position and velocity of any particle for all time if the initial conditions are precisely specified. 

Newton s second law, F = ma, provides an equation of motion for a system that obeys classical mechanics. The solution of the classical equation of motion for the harmonic oscillator provides formulas for the position and velocity that correspond to uniform harmonic motion. The solution of the classical equation of motion for a flexible string prescribes the position and velocity of each point of the string as a function of time. These solutions are deterministic, which means that if the initial conditions are precisely specified, the motion is determined for all times. 

If the force on a particle is a known function of position, Eq. (E-1) is an equation of motion, which determines the particle s position and velocity for all values of the time if the position and velocity are known for a single time. Classical mechanics is thus said to be deterministic. The state of a system in classical mechanics is specified by giving the position and velocity of every particle in the system. All mechanical quantities such as kinetic energy and potential energy have values that are determined by the values of these coordinates and velocities, and are mechanical state functions. The kinetic energy of a point-mass particle is a state function that depends on its velocity ... 

Molecular dynamics simulation is basically very straightforward. It is a deterministic method in which the system follows a well defined trajectory in phase space. It is the only rehable method for examining time-dependent properties. Assuming the applicability of classical mechanics, it involves the simultaneous solution of the equations of motion for a small sample of particles interacting according to a predetermined force field and fixed conditions. 

In the 1800s the wave nature of light and the particulate nature of matter dominated physical explanations of the physical world. Newton s and Maxwell s equations were thought to capture all reality, conferring to it an inescapable, deterministic character. Then, at the end of the nineteenth century numerous experiments were conducted that were not reconciled with these prevailing classical mechanical notions. 

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... 

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