A combination of density functional theory (DFT), kinetic Monte Carlo and mean-field rate theory is applied to analyze point defect migration and its effect on the observed growth of hexagonal close-packed (hcp) Zr under 1 MeV electron irradiation.
We use the mean-field rate equation approach to study a general dynamic model driven by Preferential Attachment (PA), Double PA (DPA), and a tunable PS (i.e., with any α>0), where c nodes (c<1) are deleted per node added to the network, and verify our predictions via large-scale simulations.
If the surface coverage is fixed, the size of the droplets will be inversely proportional to their density, which can be successfully described by mean-field rate equation theories[19 22].
That is, a number of studies of fully or sparsely connected integrate-and-fire networks have shown that under certain conditions, even though individual neurons exhibit Poisson-like statistics, the neurons fire asynchronously so that the total population activity evolves according to a mean-field rate equation [26 30].
Note that Eq. 1 are mean-field rate equations describing the evolution of the expected number of clones of size k.
Since all cells are identical except for their lentiviral marking, mean-field rate equations for n j (t) are identical for all j.
This is in tune with our self-consistent determination of the sweep rate, which absorbs the overlap exclusion between driver mutations [ i.e., the condition τ′′ − τ′ > τfix] into a reduced uniform or "mean-field" rate Vdrive given by Equations 11 and 12.
If f (x'| x) are the rates for the prior process, it was shown in [ 10] that the free energy (6) for this choice of approximating process is given by (8) Here (9) are the mean-field rates obtained by averaging the rates of the prior process under the approximate posterior for all species but the one under consideration.
From the mean field firing rate model, a spiking network model is obtained whose population's mean firing rate corresponds to that of the firing rate model.
The mean field firing rate model of excitatory network activity with synaptic depression (formally equivalent to the relaxation oscillator model of a single cell described above) consists in the following equations (see [19]).
