Classical mechanics ergodic system

Molecular dynamics uses classical mechanics to study the evolution of a system in time. At each point in time the classical equations of motion are solved for a system of particles (atoms), interacting via a set of predefined potential functions (force field), after which the solution obtained is applied to predict positions and velocities of the particles for a (short) step in time. This step-by-step process moves the system along a trajectory in phase space. Assuming that the trajectory has sampled a sufficiently large part of phase space and the ergodicity principle is obeyed, all properties of interest can then be computed by averaging along the trajectory. In contrast to the Monte Carlo method (see below), the MD method allows one to calculate both the structural and time-dependent characteristics of the system. An interested reader can find a comprehensive description of the MD method in the books by Allen and Tildesley or Frenkel and Smit. ... 

Ergodic Classical mechanical system in which a trajectory uniformly covers a specific surface in phase space. The physics literature utilizes this term to imply uniform coverage of the surface in phase space defined by fixed total energy. 

Mixing Classical mechanical system that is ergodic and possesses additional properties associated with relaxation. 

In classical mechanics, an alternate definition of ergodicity is to say that (in the Koopman formalism, see in particular Arnold-Avez) a system is ergodic if the projector... 

The equivalence of these two averages is the essence of a very important system property called ergodicity, which deserves a separate subsection further in this chapter. pA hat is used here to emphasize that, in quantum mechanics, the Hamiltonian is an operator acting on the h function and not just a function of particle coordinates and momenta, as in classical mechanics. 

Such a system is then as ergodic as, for example, a classical gas, only the mechanisms of ergodicity are different (in one case elastic collisions, in the other decay of excited states through induced and spontaneous emission). 

The spontaneous emission in atomic problems and the decay of unstable particles are irreversible processes which manifest the ergodicity of these systems. It is therefore interesting to compare the mechanism of irreversibility which is involved to that in the usual many-body systems such as a classical gas. 

A RRKM unimolecular system obeys the ergodic principle of statistical mechanics [337]. A quantity of more utility than N t), for analyzing the classical dynamics of a micro-canonical ensemble, is the lifetime distribution Pc t), which is defined by... 

The clearest results have been obtained for classical relaxation in bound systems where the full machinery of classical ergodic theory may be utilized. These concepts have been carried over empirically to molecular scattering and decay, where the phase space is not compact and hence the ergodic theory is not directly applicable. This classical approach is the subject of Section II. Less complete information is available on the classical-quantum correspondence, which underlies step 4. This is discussed in Section III where we introduce the Liouville approach to correspondence, which, we believe, provides a unified basis for future studies in this area. Finally, the quantum picture is beginning to emerge, and Section IV summarizes a number of recent approaches relevant for a quantum-mechanical understanding of relaxation phenomena and statistical behavior in bound systems and scattering. 

As mentioned previously, the definition of an empirical potential establishes its physical accuracy those most commonly used in chemistry embody a classical treatment of pairwise particle-particle and n-body bonded interactions that can reproduce structural and conformational changes. Potentials are useful for studying the molecular mechanics (MM), e.g., structure optimization, or dynamics (MD) of systems whereby, from the ergodic hypothesis from statistical mechanics, the statistical ensemble averages (or expectation values) are taken to be equal to time averages of the system being integrated via (7). 

Ergodicity is very important for the statistical mechanical and thermodynamic descriptions to be valid for particular systems it has to be checked for each particular case. It holds for systems where the characteristic molecular relaxation times are small in comparison to the observation time scale. Polymeric and other glasses represent a classical example of nonergodic systems. They can be trapped, or configuration-ally arrested, within small regions of their phase space over very long times. 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

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