Time dependent mean field

Time-dependent mean-field Hamiltonian directly follows from (2)-(3) ... 

The result, Eq. (2.40) is known as the time-dependent mean field or time-dependent Hartree approximation. In this approximation each system is moving in the average field of the other system. 

The SCF, or mean-field, approximation does not include the effect of energy transfer processes between the modes. The Cl approach incorporates such effects in a time-independent framework, but as was noted in the previous section this method loses much of the simplicity and insight provided by the SCF model. The most natural extension of the SCF approximation that also describes energy transfer among the coupled modes in the system, and treats this effect by a mean-field approach, is the time-dependent self-consistent-field (TDSCF), or time-dependent mean-field, approximation. 

The T =0 time-dependent mean-field theory currently provides the best description of nuclear dynamics at low energies [5,6]. We consider two single-particle operators, Q, P interpreted as a collective coordinate and a collective momentum. Their nature depends on the kind of motion that we want to focus upon. We require that Q Q and P-r — P under time reversal and that IP, Q] 0. We then form a constrained Hartree-Fock (CHF) calculation on the many-body Hamiltonian H by minimizing the functional... 

Yabana K, Tazawa T, Abe Y, Bo zek P (1998) Time-dependent mean-field description for multiple electron transfer in slow ion-cluster collisions. Phys Rev A 57 R3165. doi 10.1103/PhysRevA. 57.R3165... 

A more complicated behavior of the system (3) is manifested if the time-dependent driving field and damping are taken into account. Let us assume that the driving amplitude has the form /1 (x) =/o(l + sin (Hr)), meaning that the external pump amplitude is modulated with the frequency around /0. Moreover,/) = 0 and Ai = 2 = 0. It is obvious that if we now examine Eq. (3), the situation in the phase space changes sharply. In our system there are two competitive oscillations. The first belongs to the multiperiodic evolution mentioned in Section n.D, and the second is generated by the modulated external pump field. Consequently, we observe a rich variety of nonlinear oscillations in the SHG process. 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

These fluctuations are illustrated in Fig. 9.95 in two different ways the time-dependent electric field E t) and its mean fluctuations of the amplitude 0 and phase (p are shown in an E t) diagram and in a polar phase diagram with the axes E and 2- In the latter, amplitude fluctuations cause an uncertainty of the radius r = o, whereas phase fluctuations cause an uncertainty of the phase angle (p (Fig. 9.95b). Because of Heisenberg s uncertainty relation it is not possible that both uncertainties of amplitude and phase become simultaneously zero. 

Dielectric relaxation means the adjustment of dielectric displacement (D) or polarization ( ) to the time-dependent electrical field (E). Relative permittivity (e) characterizes the capacitance ratio of a condenser filled with an insulating material and with vacuum. If the field is sinusoidal, the permittivity becomes a complex number ... 

Fig. S. The diffusion dependence of the passage time for mean-field approach System size V=100 and 200...

Turbnlent flows contain a spectrum of eddies of different size, intensity, and lifetime. However, each eddy has an element of simple shear or extension, and creates forces that lead to drop deformation. The drop sees a time-dependent deformation field even if the Reynolds-averaged velocity field does not vary in space. This is illnstrated in Figure 12-2 and is explained more fnlly in Chapter 2. In reality, forces in turbulent stirred vessels arise from both spatial and temporal velocity flnctuations. These arise fi om mean velocity gradients, interacting turbnlent eddies and impingement of jetlike flows on walls, baffles, and impeller blades. 

According to the first Maxwell equation the curl of the electrical field is zero in the absence of time dependent magnetic fields. This means that, in this case, no turbulent electrical fields occur. It is then and only then that we can define a scalar electrical potential, which we require. This is visible from the definition of curl E... 

In the LS analysis, an assembly of drops is considered. Growth proceeds by evaporation from drops withi < R and condensation onto drops R > R. The supersaturation e changes in time, so that e (x) becomes a sort of mean field due to all the other droplets and also implies a time-dependent critical radius. R (x) = a/[/"(l)e(x)]. One of the starting equations in the LS analysis is equation (A3.3.87) withi (x). 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

In the case of binary variables a G 0,1, we can let density p t) at time t represent the probability of finding value a = 1 at any site at time t. A simple counting of all possible configurations in an arbitrary neighborhood Af gives us the general mean-field expression for the time-dependence of p ... 

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