Gaussian velocity profiles

New ways of applying non-equilibrimn states in molecular dynamics continue to be invented. Arya et al. devised a novel NEMD method for extracting the shear viscosity. The method follows the decay of a Gaussian velocity profile in a simulation cell, u (y, 0) = uq exp(—Z>oy ). An analytic solution of the Navier-Stokes equation leads to a decay of the central or peak velocity, i/p, according to... 

Here, Cc is the concentration at x in an initially small cloud produced by unit source strength emitted at xs at time t s. Many experiments have shown that provided the energy and/or scale of the turbulent velocity components do not vary greatly (say more than a factor of 2) over the volume occupied by the cloud of contaminant released from the source in the presence of mean wind speed U, Cc can be approximated (near the source) by the basic Gaussian cloud profile Gc,0, i.e. 

The mean radial velocity profile has a Gaussian distribution as follows ... 

In order to obtain a steady state from Eqs. 38 dissipative heat must be removed from the system. This is achieved by the last (thermostatting) terms of the last two equations in Eqs. 38. In this respect it is essential to observe that accurate values for Uj and A are needed. Any deviations from the assumed streaming and angular velocity profiles (biased profiles) will exert unphysical forces and torques which in turn will affect the shear-induced translational and rotational ordering in the system [209,211,212]. The values for the multipliers and depend on the particular choice of the thermostat. A common choice, also adopted in the work of McWhirter and Patey, is a Gaussian isokinetic thermostat [209] which insures that the kinetic and rotational energies (calculated from the thermal velocities p" and thermal angular velocities ot) - A ) and therefore the temperature are conserved. Other possible choices are the Hoover-Nose or Nose-Hoover-chain thermostats [213-216]. 

The radial velocities were estimated by identifying a few lines (3 to 6, depending on the quality of the spectra) and fitting a Gaussian profile to each line to find the line core. We estimated the average radial velocity, and we present here only the stars with standard deviation of the mean less than 7 km s 1. 

The measurements of Rouse, Yih and Humphries (1952) [1] helped to generalize the temperature and velocity relationships for turbulent plumes from small sources, and established the Gaussian profile approximation as adequate descriptions for normalized vertical velocity (w) and temperature (7), e.g. 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

The choice of this Gaussian profile allows us to obtain a "closed" form similaritv solution. If the fluid velocity v is then expanded such that... 

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