Oscillation function mean-square displacement

For the special case of the self-correlation function (n=m) B n,n,t) reveals the mean-square displacement of a polymer segment. For large p the cos in Eq. 3.16 is a rapidly oscillating function which may be replaced by the mean-value 1/2. With this approximation we can convert the sum into an integral and obtain ... 

Fig. 2 The mean square displacement of a single harmonic oscillator as a function of temperature. Units are arbitrary because (uj) and T values depend on the frequency of the oscillator and the mass of the particle. The plot shows that at low temperature the displacement is almost constant, whereas at high temperature it varies linearly with T. The change of regime occurs approximately at 0e/2...

This remarkably straightforward expression is the physical basis for the interpretation of INS spectra by lattice dynamic and molecular dynamic approaches. The Mean Square Displacement (MSD) of the scattering atom, U, is a function of the energy, co, and mass, p, of the oscillator ... 

Figure 2. (Left) Mean-squared displacement versus the time for the flow [Eq. (22)] with D = 0, and to = 1.1. Lengths and times are shown in units of L and /,2/ [i0, respectively. The best-fit (dashed) line corresponds to 2v(2) — 1.3. (Right) The diffusion coefficient DEU as a function of D for the frequency of the roll oscillation to = 1.1. The diffusivities are reported in units of (i0. The best-fit (dashed) line has the slope —p = -0.18.

Fitting procedures give information on wave functions via mean-square displacements (ufj for each vibration and effective oscillator masses. It transpires that proton dynamics for bending modes correspond very closely to isolated harmonic oscillators with a mass of 1 amu [Ikeda 2002], They are largely de-... 

Figure 2. Root-mean-square displacement of lithium ions as a function of time. Diffusion is much faster with the additional oscillating electric field (amplitude 5 kcal mol , frequency 25 MHz).

This equation is referred to as the differential equation of harmonic oscillations. By analyzing this equation, we can arrive at an important conclusion when solving a problem and arriving at an equation like that presented above, it means that the problem can be reduced to harmonic oscillations and the coefficient before the displacement function is the square of its cyclic frequency. 

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