Ensemble and time averaging

An alternative way of estimating the excited state lifetime is to compute the ratio of the MD timestep, 8t and the ensemble and time averaged transition probability... 

The behavior is clearly non-ergodic in the sense that ensemble and time average differ considerably. One direct reason is the fact that the conformational transition can only be measured for a limited time until the molecule is bleached [30,31]. On the other hand, the relaxation rate of 37 classes of single DNA molecules cover relaxation rates between 1 and 55s h This means that under the assumption that all TMR tagged DNA molecules are identical they can switch from one relaxation regime to another which stays constant during the survival time of photobleaching [31]. 

The MSDs are obtained by considering both ensemble and time averages. Particle coordinates having been known, the pair correlation functions are readily obtained. These pair correlation functions can be combined with the atomic scattering factors for the respective atoms and then convoluted to obtain a combined pair distribution function, which can be compared with experimental RDF. This is a vital step in the glass structure simulations. 

After choosing an adequate model for each different component of the system and integrating them into a final atomistic model that will be simulated, an important issue is the selection of a discretization scheme to implement the computer representation of the ion channel and its environment. Within the framework of a computer experiment, the adjective realistic is strictly related to the phenomena one wants to study, and to the resolution required to reproduce those phenomena. The basic idea for modeling many-body systems is to build a set of rules that apply to each component and let the system evolve dynamically. Ensemble and time averages are then computed to obtain observables that are compared with experiment to validate the model. A characteristic of ion channel systems is that the measurable quantities of direct biological interest evolve in times up to 12 orders of magnitude larger than the smallest atomic or molecular relaxation times (milliseconds versus femtoseconds). In comparison, solid state many-body systems collectively relax in a faster fashion, and the difference between the microscopic... 

Calculating ensemble or time averages over a relatively small number of points (perhaps a few million) and a limited number of particles (perhaps a few hundred), instead of something which approaches a macroscopic sample of perhaps 10 ° molecules/configurations. 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

For multiphase flows perturbed by the presence of particles to obtain a turbulence like behavior the local instantaneous velocity of the continuous phase can for example be decomposed adopting the Reynolds averaging procedure (i.e., other methods including time-, volume-, ensemble-, and Favre averaging have been used as well) and expressed as Vg = Vg- - < v >g, where v(. is the fluctuating component of the continuous phase velocity. Introducing the peculiar velocity for the dispersed phase this relation can be re-arranged as ... 

---