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 [Pg.21]

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. [Pg.9]

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. [Pg.100]

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 [Pg.89]

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. [Pg.71]

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

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. [Pg.274]

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. [Pg.49]

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. [Pg.652]

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. [Pg.717]

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 [Pg.1267]

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. [Pg.625]

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. [Pg.25]

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. [Pg.181]

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. [Pg.404]

We have already observed that the full 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 Schrodinger) 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 underl)dng dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. [Pg.255]

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. [Pg.34]

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 [Pg.77]

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