Microscopic time

In kinetic theory, the macroscopic quantities are found as averages over the motion of many molecules each molecular event is assumed to take place over a microscopic time interval, so that a measurement that is made over a macroscopic time interval involves many molecules. The kinetic-description is, therefore, a probabilistic one in that assumptions are made about the motion of one molecule and the results of this motion are averaged over all of the molecules of the gas, giving proper weight to the probability that the various molecules of the gas can have the assumed motion. 

Gratton, E., Breusegem, S., Sutin, J., Ruan, Q. and Barry, N. (2003). Fluorescence lifetime imaging for the two-photon microscope time-domain and frequency-domain methods. J. Biomed. Opt. 8, 381-90. 

Assuming internal microscopic time-reversal symmetry for the system of interest means that by reversing the velocity vectors of every particle, the whole system reverses its collective path. This assumption has the important consequence that in a system fluctuating around equilibrium, the fluctuations... 

It is often sensible to choose Mceii such that the simulation cell adjusts to the external pressure and the thermostat on microscopic time scales. 

The time t as it appears in Eqs, (23) or (24) is directly the time of observables, as in the left-hand side of Eq, (11), and not the microscopic time of probability amplitudes. Also it should be noticed that as the density matrix satisfies the linear equation (23) and is also linearly related to observables through Eq, (24), asymptotic procedures become especially simple to handle. 

Microscopic time-resolved measurements of the hydrate phase during gas hydrate formation, decomposition, and inhibition began only in the mid-1990s. These techniques include in situ synchrotron x-ray diffraction (Koh et al., 1996 Klapproth et al., 2003 Uchida et al., 2003), neutron diffraction (Henning et al., 2000 Koh et al., 2000 Halpern et al., 2001 Staykova et al., 2003), Raman spectroscopy (Subramanian and Sloan, 2002 Komai et al., 2004), and NMR spectroscopy (Moudrakovski et al., 2001 Kini et al., 2004 Gupta et al., 2007). 

Here we introduce reduced quantities the complex frequency z, related to co by a microscopic time scale r, and the normalized concentration G ... 

Obviously, the above algorithms are not suitable when transients of the finer scale model are involved (Raimondeau and Vlachos, 2000), as, for example, during startup, shut down, or at a short time after perturbations in macroscopic variables have occurred. The third coupling algorithm attempts fully dynamic, simultaneous solution of the two models where one passes information back and forth at each time step. This method is computationally more intensive, since it involves continuous calls of the microscopic code but eliminates the need for a priori development of accurate surfaces. As a result, it does not suffer from the problem of accuracy as this is taken care of on-the-fly. In dynamic simulation, one could take advantage of the fast relaxation of a finer (microscopic) model. What the separation of time scales between finer and coarser scale models implies is that in each (macroscopic) time step of the coarse model, one could solve the fine scale model for short (microscopic) time intervals only and pass the information into the coarse model. These ideas have been discussed for model systems in Gear and Kevrekidis (2003), Vanden-Eijnden (2003), and Weinan et al. (2003) but have not been implemented yet in realistic MC simulations. The term projective method was introduced for a specific implementation of this approach (Gear and Kevrekidis, 2003). 

Before ending tUs section, we would like to clarify the meaning of the time-scale separation between system and heat bath. If the correlation time of the variable Ag is given the exponential form of Eq. (2.25), the microscopic time is immediately provided by... 

BASIC DESCRIPTION OF RULES LEADING TO ADIABATIC ELIMINATION 43 In general, the microscopic time could be defined by... 

When y - A2 the equivalent of the microscopic time y is = y/ -Decoupling effects are present when Uj = F. To obtain an approximate value of F we can use the experimental data as follows. First, we evaluate the value of decay of the oscillation envelopes of the angular velocity autocorrelation function as a function of Equation (14) shows that this is, approximately, a Lorentzian, the linewidth of which provides the approximate expression for F. The agreement with the numerical decoupling effect is quantitatively good when the ratio ai/uf is assumed to be equal to 8.5. Simple Markovian models cannot account for decoupling effects. 

To get the results of the next section we have followed both ways. With fairly short microscopic times we have that the latter method leads to a faster convergence. This is because in such a physical condition the spectrum consists of 2/ + 1 very distinct linediapes, one for each continued fraction. A few steps of each continued fraction are then required to clearly identify each component lineshape. The reader who would like to use this approach should keep in mind these two different ways of evaluating spectra and decide which is the most convenient for the actud physical condition. Note that the direct... 

When dealing with a long microscopic time scale (slow motion), in the high-field approximation, the nonsecular terms are usually disregarded. So the Hamiltonian describing the spin system is... 

In the present work we study binary soft-sphere mixtures with a core-size ratio mass ratio m2/m = 2.0 and an equimolar concentration (ij = 0.5) for both 3-d and 2-d systems. Using a constant-temperature MD technique and the peiodic boundary conditions, we have carried out MD simulations for the models. The pair potential, Eq. (1), was cut off over the distance rlnumber density wm kept constant, i.e. n" = 0.8 the temperature was varied to achieve a desired Teff. The microscopic time scale was chosen to... 

Thus, when viewed with only half the time resolution, that being 2x rather than x, the increments of the Brownian particle position are still zero-centered Gaussian random. More generally, whatever the number of the microscopic time steps between observations M, one always finds that the increments in the particle position constitute a zero-centered Gaussian process with a variance that increases linearly with M. 

Before proceeding, we remark that just as in the Einstein theory of the Brownian motion of which the Debye theory is a rotational version, the characteristic microscopic time scale is a time interval t so long that the motion of the particle at time t is independent of its motion at time t t, but small compared to the observation time intervals [17]. It is also supposed that during the time t, which is the mean of the time intervals between collision events, any external nonstochastic forces which may be applied to the system do not alter. In the fractal waiting time picture, however, the concept of collision rate does not hold and the mean of the time intervals between collision events diverges. 

The importance of the amorphous glassy microstructure of soft particle dispersions is reflected by the great influence that the particle elastic modulus has on yielding and flow. The yield stresses of colloidal pastes and of emulsions scales like the shear modulus [13, 133, 134]. In Sect. 5, the flow curves of soft particle glasses will be shown to exhibit a remarkable universal behavior in terms ofa unique microscopic time scale tliat involves the shear modulus [13]. In Sect.4, the slip velocity... 

From the physical point of view, the motion correlation time tc alone carries precious informations concerning the microscopic times related to the motion as well as the activation energy, enthalpy or volume of the motion. The activation enthalpy A//a (activation volume A Fa) is a measure of the excess enthalpy (volume) needed to enable the given motion (translation, rotation, etc.). This is an important information to be correlated to the structure, and it may give important hints to distinguish different phases as well as... 

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