Ergodic mechanical systems

The single, undisturbed motion of the system, if pm-sued without limit in time, will finally traverse every phase point which is compatible with its given total energy. A mechanical system satisfying this condition is called by Boltzmann an ergodic system.  

Boltzmann and Maxwell infer from this definition the following corollaries  

For an ergodic system all motions with the same total energy take place on the same (7-path.M 

This means that all these motions differ only in the value of the constant CtrN, which appears as a constant additive to the time (cf. Integral 23c).95 

All of these motions yield the same value for the time average of any function p) of the phase variables.98 

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The basic difference between free fields and interacting fields would be of the same order as between a small (reversible) mechanical system and ergodic dissipative systems. But this means that physical states can no longer be associated with invariants of motion which no longer exist. This leads to deep changes in the structure of the theory. 

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. 

The simplest models of molecular dynamics simulations (hard or soft spheres) under stationary conditions have the property of ergodicity and mixing. A statistical mechanical system is ergodic if... 

A RRKM nnimolecnlar system obeys the ergodic principle of statistical mechanics[H],... 

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. 

The approach we shall now outline corresponds to an intermediate viewpoint. We take the limit N -> 00 very seriously and obtain in this way a statistical mechanical description. This immediately takes into account the ergodicity of such systems. 

This is a simple analogy of Birkhoff s ergodic theorem for dynamical systems, see A.I. Khinchin, Mathematical Foundation of Statistical Mechanics (Dover, New York 1949) L.E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, TX 1980) ch. 8. 

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... 

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. ... 

These phenomena lead us to a rather complicated situation. The first phenomenon reminds us of ergodicity, the realization of microcanonical distribution in systems with many degrees of freedom and the validity of statistical mechanics. We know that KAM tori cannot divide the phase space (or energy surface) for systems with many degrees of freedom, and the first phenomenon tells us that two neighborhoods in different parts of the phase space are connected not only topologically but also dynamically. In this sense the phenomenon can be considered as an elementary process of relaxation in systems with many degrees of freedom. 

A fundamental hypothesis of statistical mechanics is the ergodic theorem. Basically it says the system evolves so quickly in the phase space that it visits all of the possible phase points during the time considered. If the system is eigodic, the ensemble average is equivalent to the time average over the trajectory for the time period. The ergodidty of a system depends on the search procedure, force... 

The ergodic theorem of statistical mechanics (see also Section 1.4.2) states that, for realistic systems, these two kinds of averaging, Eqs (5.10) and (5.11) yield identical results. As example of an application of this theorem consider the total kinetic energy of the system. The corresponding dynamical variable is... 

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. 

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