Time correlation functions statistical errors

From the simulation data, the correlation time is found by integrating the time correlation function as shown in Eq. (6). While the correlation functions may be easily computed from the trajectories, statistical errors due to finite trajectory length limit the useful data to short delay times r [16,17]. The variance of a Gaus-... 

To estimate errors in structural or other properties, expressed not just as numbers but as functions of either distance (e.g., g(r)), or time (e.g., time correlation functions), a similar analysis should be carried out for different values of the argument. For instance, when the time correlation function is calculated, much better statistics is obtained for short times, as there are much more data available. The time correlation function values for times comparable to the length of the simulation run are obtained with larger errors, as the average is taken over only a few pairs of values. 

It is evident that this is a way of rewriting the exact solution of Eq. (47). However, it is interesting to recover the fluctuation-dissipation prediction from a perspective that might lead to a free diffusion with no upper limit if an error is made that does not take into account the statistical properties of the fluctuation E,(f). Let us evaluate the correlation function of E,(f). Using the property of Eq. (48) and moving to the asymptotic time limit reflecting the microscopic equilibrium condition, we obtain... 

Fig. 7 Decay of translational (U) and rotational (12) velocity correlations of a suspended sphere. The time-dependent velocities of the sphere are shown as solid symbols the relaxation of the corresponding velocity autocorrelation functions are shown as open symbols (with statistical error bars). A sufficiently large fluid volume was used so that the periodic boundary conditions had no effect on the numerical results for times up to r = 1,000 in lattice units (h = b = 1). The solid lines are theoretical results, obtained by an inverse Laplace transform of the frequency-dependent friction coefficients [175] of a sphere of appropriate size (a = 2.6) and mass (pj/p = 12) the kinematic viscosity of the pure fluid = 1/6...

Statistical analysis of the results was performed using the software Statistica 5.5 (Stat Soft). Maximum lipase activities and time to reach the maximum were calculated through fitting of kinetic curves. The maximum was estimated by derivation of the fits. Empirical models were built to fit maximum lipase activity in the function of incubation temperature (T), moisture of the cake (%M), and supplementation (%00). The experimental error estimated from the duplicates was considered in the parameter estimation. The choice of the best model to describe the influence of the variables on lipase activity was based on the correlation coefficient (r2) and on the x2 test. The model that best fits the experimental data is presented in Table 2. 

Figure 2.9 Electron relaxation dynamics for GaAs (100). (a) Compares the hot electron lifetimes as a function of excess energy (above the valence band) of a pristine surface prepared using MBE methods with device-grade GaAs under the same conditions. The higher surface defect density of the device-grade material increases the relaxation rate by a factor of 4 to 5. (b) The electron distribution as a function of excess energy for various time delays between the two-pulse correlation for MBE GaAs. The dotted lines indicate a statistical distribution corresponding to an elevated electronic temperature. The distribution does not correspond to a Fermi-Dirac distribution until approximately 400 fs. The deviation from a statistical distribution is shown in (c) where the size of the error bars on the effective electron temperature quantifies this deviation.

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