Diffusion characteristic time scales

Here the lattice positions i and j should be adjacent and the -function assures that one of the two lattice positions is occupied and the other one is free, r/j is a characteristic time scale for a diffusion jump. The time-dependence of the average si) is calculated by approximating the higher moments (siSj) [49]. In practice the analysis is rather involved, so we do not give further details here. An important result, for example, is the correction to the Wilson-Frenkel rate (33) at high temperatures ... 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

E.g. tryptophane residues of proteins excite at 290-295 mn but they emit photons somewhere between 310 and 350 mn. The missing energy is deposited in the tryptophane molecular enviromuent in the form of vibrational states. While the excitation process is complete in pico-seconds, the relaxation back to the initial state may take nano-seconds. While this period may appear very short, it is actually an extremely relevant time scale for proteins. Due to the inherent thermal energy, proteins move in their (aqueous) solution, they display both translational and rotational diffusion, and for both of these the characteristic time scale is nano-seconds for normal proteins. Thus we may excite the protein at time 0 and recollect some photons some nano seconds later. With the invention of lasers, as well as of very fast detectors, it is completely feasible to follow the protein relax back to its ground state with sub-nano second resolution. The relaxation process may be a simple exponential decay, although tryptophane of reasons we will not dwell on here display a multi-exponential decay. 

Since the length scales associated with the thermal lens are on the order of 10 to 1000 times the grating constant, their characteristic time scale interferes with polymer diffusion within the grating. Such thermal lensing has been ignored in many FRS experiments with pulsed laser excitation [27,46] and requires a rather complicated treatment. A detailed discussion of transient heating and finite size effects for the measurement of thermal diffusivities of liquids can be found in Ref. [47]. 

Figure 2. Evolution of the reservoir pressure for a hydraulic and a chemical loading. The pressure is scaled by the corresponding characteristic pressure while time is scaled according to the diffusion characteristic time Th-...

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

Fig. 1. Characteristic time scale and associated diffusion length of radiolytic species of the major stages of water radiolysis.

Equation [4.5.37] can be written in a more manageable way by recognizing that we can introduce a characteristic time scale for diffusion through the definition... 

By using these values, it is possible to estimate the characteristic rate of change for the internal coordinates, which can in turn be used to define characteristic time scales for phase-space advection for each internal coordinate = [ i/ i( i),..., m/ m( m)]-Analogously for diffusion a characteristic time scale is easily defined tdjj = jlDij. Also for the source term for point processes some characteristic time scales can be defined. For... 

The DE (3-95) is identical in form to the familiar heat equation for radial conduction of heat in a circular cylindrical geometry. Thus we see that the evolution in time of the steady Poiseuille velocity profile is completely analogous to the conduction of heat starting with an initial parabolic temperature profile -(1 - r2)/4. In our problem, the final steady velocity profile is established by diffusion of momentum from the wall of the tube so that the initial profile for w eventually evolves to the asymptotic value uT - 0 as 1 oo. The characteristic time scale for any diffusion process (whether it is molecular diffusion, heat conduction, or the present process) is (f y cli fl iisivity ), where tc is the characteristic distance over which diffusion occurs. In the present process, tc = R and the kinematic viscosity v plays the role of the diffusivity so that... 

In retrospect, it should perhaps have been evident from (3-95) that this would be the appropriate characteristic time scale.13 However, without the preceding discussion, the important observation of an analogy between the diffusion of momentum in start-up of a unidirectional flow and the conduction of heat would not have been evident. We shall discuss the nature of this process in more detail after we have solved (3-95) and (3-96) to obtain the time-dependent velocity profile. 

However, in this case, it is straightforward to nondimensionalize. The characteristic velocity scale is uc = U, and an appropriate characteristic length scale is lc = d. Because the velocity field is established by means of diffusion of momentum, the characteristic time scale is tc = d2 j v. Using these characteristic quantities, we find that the problem in dimensionless form is... 

If we examine (12 174) more closely, we see that there are several possible choices for the characteristic velocity. An apparently convenient choice is to designate uc = vo/d as this appears to reduce the number of parameters to the greatest extent. This choice of characteristic velocity is really saying that the characteristic time scale is the diffusion time d2/v0. The governing equations, (12-173) and (12 174), then become... 

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