Ergodic principle

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

Providing the ergodic principle may be used, the time average of the fluctuating force is also zero and it will be true if the time scale of collision events is very small compared with the time interval of interest. 

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. ... 

In order to estimate the length scale at which the classical elasticity becomes non-valid, the MD on a polyethylene chain in a cubic simulation cell was performed under periodic boundary conditions at 50 K [23]. The corresponding length scales for the longitudinal and transverse directions were 1.85 and 3.81 nm, respectively. A recent work by Nikolov et al [54] estimated that rubbers above their Tg should have non-local length scale approximately 5 nm based on the calculation of the volume at which the ergodic principle is no longer valid and the network deformation is not affine. 

In principle, the transition from state A to B needs to be unbiased and ergodic... 

However, whether we have access to the configmrations in parallel or sequentially is irrelevant, which permits us to conclude that, for a sufficiently long and ergodic Markov chain, displacements will be accepted on average with the correct probability dictated by the principles of statistical physics (i.e., the probability density of a given statistical physical ensemble). 

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