Time-averaged probability

Despite their complexity, the dynamics in these systems is fast with the molecules flipping and changing their positions. In fact these are ordered fluids composed of distinct compartments. There are no fixed positions of the molecules but distinct spaces with a maximum time averaged probability for each of the segregated units to be located and these dynamic compartments are arranged periodically in space. The structures are in thermodynamic equilibrium as indicated... 

Suppose that molecule a of the dimer considered in exercise 8.1 is excited at time t = 0, and that the excitation then oscillates between the two molecules. a) Calculate the frequency of the oscillation if the excitation energy of the two molecules is the same, (b) What would the oscillation frequency be if the excitation energy of molecule a is 1,0(X) cm above that of molecule bl (c) Calculate the time-averaged probability of finding the excitation on molecule a for the second case (Eba = a — Eb = 1 >000 cm ) in the absence of thermal relaxations, and compare your result with the probability of finding the excitation on this molecule at thermal equilibrium. 

The Boltzmaim weight appears implicitly in the way the states are chosen. The fomi of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the fomi of a Markov chain, specified by transition probabilities which are independent of the prior history of the system. 

If X is tlie failure rate of each component of a system, then X is the average nmiiber of failures per miit of time. The probability of x failures in tlie specified unit of time is obtained by substituting p = >Lt in Eq. (20.5.4) to obtain... 

The desired average is simply obtained by a time average of the given property. For example, one of the interesting properties of bulk solvents is the radial distribution function (rdf), which expresses the probability of finding a given atom type around a reference atom by... 

Slug-ring flow is the flow in which the probability of B is larger than that of C and the time-averaged void fraction is less than 0.8. 

Soil water flow is decidedly episodic. During dry times the water solutions in the soil are probably fairly concentrated and not very reactive. Time-averaged reaction rates should be roughly proportional to the fraction of time reacting minerals are in contact with thermodynamically imdersaturated (and reactive) water. In a study of the relationship between denudation rate and runoff for rivers draining igneous and metamorphic rock in Kenya, Dunne (1978) obtains the relationship of (denudation rate in tons/km per year) = 0.28 (runoff in mm/ year)°. ... 

The ionic atmosphere has a blurred (diffuse) structure. Because of thermal motion, one cannot attribute precise locations to its ions relative to the central ion one can only dehne a probability to find them at a certain point or define a time-average ionic concentration at that point (the charge of the ionic atmosphere is smeared out around the centraf ion). In DH theory, the interaction of the central ion with specific (discrete) neighboring ions is replaced by its interaction with the ionic atmosphere (i.e., with a continuum). 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

Liquid crystals, commonly referred to as the fourth state of matter, are materials which are intermediate in character between the solid and liquid states. Unlike normal isotropic liquids, they show some time-averaged positional orientation of the molecules, but they retain many of the properties of liquids, such as the ability to flow. In recent decades, liquid crystals have played an increasingly important part in our lives. Probably their most familiar application is in the information displays which provide the visual interface with microprocessor-controlled instrumentation. Liquid crystal displays have superseded more traditional display technology, such as light-emitting diodes and cathode ray tubes, for many appliances principally because of the advantages of visual appeal, low power consumption, and their ability to facilitate the miniaturisation of devices into which they are incorporated. They are encoun-... 

The bathroom, laundry, and kitchen exhausts would have to operate as normally installed and enough pressurization air would have to be added so that the intermittent operation of these exhausts would not overcome the time-averaged benefits of house pressurization. Combustion products are probably not an issue because there is no combustion device in the house. 

For ergodic systems, the probability of visiting the neighborhood of each point in phase space converges to a unique limiting value as T —> oo, such that the time average of / is equal to its ensemble average... 

EVOLUTION TIMES OF PROBABILITY DISTRIBUTIONS AND AVERAGES—EXACT SOLUTIONS OF THE KRAMERS PROBLEM... 

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