Time average theorem

The first, second and third terms in (3.218) have to be reformulated using the conventional time averaging theorems . The first theorem one makes use of relates the average of a time derivative to the time derivative of an average quantity, and is called the Leibnitz s rule for time averaging ... 

The time averaging theorems have been derived by [112] [47]. 

The results just obtained are special cases of a theorem that shows how a large class of time averages can be calculated in terms of the distribution function. Before demonstrating this theorem, it will be convenient for us to first discuss some useful properties of distribution functions. The most important of these are... 

The Theorem of Averages.—We are now in a position to discuss the fundamental theorem of averages10 mentioned earlier. This theorem states that all time averages of the form... 

Once again, it should be emphasized that the functional form of a set of random variables is important only insofar as it enables us to calculate their joint distribution function in terms of other known distribution functions. Once the joint distribution function of a group of random variables is known, no further reference to their fractional form is necessary in order to use the theorem of averages for the calculation of any time average of interest in connection with the given random variables. 

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

The Wiener-Khinchin theorem is a special case of Bocliner theorem applicable to time averages of stationary stochastic variables. Bochner s theorem enables the Wiener-Khinchin theorem to be applied to ensemble averaged time-correlation functions in quantum mechanics where it is difficult to think of properties as stochastic processes. 

In the exact solution it is obvious that the time average of the flux is still directed along the a>axis. The dissipation theorem (Zeldovich, 1937) may be written in the form... 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

The averaging theorem for the time derivative can be derived directly from the general transport theorem. Consider the phase k in Fig. 5.7 to be fixed in space while Ak varies as a function of time as a result of phase change. Applying Eq. (5.8) to the system in Fig. 5.7, with d/dt = 3/31, we have... 

Cf. V 23, Section 6. In the case of sinusoidal oscillations the time average of the potential energy is equal to that of the kinetic energy. The theorem of equipartition of kinetic energy therefore determines also the total energy content for harmonic oscillators. 

The ergodic theorem states the equivalence of the average of a molecular ensemble with the time average of a single molecule. We have, therefore, compared the values for the conformational transition of our standard molecule. 

In the original derivation of the classical virial theorem given by Clausius, an expression corresponding to eqn (5.30) is also obtained. In the classical case one argues that the time average of d(f p)/dt vanishes over a sufficiently long period of time or that the motion is periodic to obtain the equivalent of eqn (5.31). ... 

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... 

In order to treat the equation of motion in the same way, we apply the Reynolds decomposition procedure on the instantaneous velocity and pressure variables in (1.385) and average term by term. It can be shown by use of Leibnitz theorem that the operation of time averaging commutes with the operation of differentiating with respect to time when the limits of integration are constant [154, 106, 121, 15]). 

A special case of (3.224) is the theorem for the time average of a divergence ... 

By taking into account the particular forms of the Leibnitz s and Gauss theorems valid for time averaging (3.220), (3.224) and (3.225), the averaged generic equation (3.218) becomes ... 

From the quantum regression theorem [33], it is well known that the two-time averages (S+(f)S (f + x)) satisfy the same equations of motion as do the onetime averages (S (t)) which, on the other hand, satisfy the same equations of motion as do the density matrix elements py(f). 

The above expansion is known as the Kubo relation [14,15] and was developed within the framework of propagator methods. We will look at this alternative approach in section 2.4. When we insert the Kubo relation on the right hand side of the time-dependent Hellmann-Feynman theorem (19) and do the time averaging we obtain... 

Slater has shown that the virial theorem is also applicable in the case of quantum-mechanical systems if spatial averages replace the time averages of the classical sj tem [37—42). The proof is as follows The Schr5-dinger equation for a system of N particles is... 

The divergence theorem finds use in the derivation of the virial. Consider the motion of a collection of particles. To the ith particle a force F is exerted at some position r . The sum over each particle of the scalar product is the virial J2i r, F [19, p. 129]. The time-averaged virial is equal to the average kinetic energy... 

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