Vlasov equation

MSN. 167. I. Prigogine and T. Petrosky, Semigroup representation of the Vlasov equation, J. Plasma Phys. 59, 611-618. 

The situation is similar to the one encountered in the kinetic theory of dilute plasmas. 510 To lowest order in the density+) the one-particle distribution function of the electrons obeys the Vlasov equation. The next order approximation consists of two coupled equations for the one-particle and two-particle distribution functions. On the other hand, in the kinetic theory of gases Bogolyubovft) has proposed an... 

Bagchi and co-workers [47-50] have explored the role of translational diffusion in the dynamics of solvation by employing a Smoluchowski-Vlasov equation (see also Calef and Wolyness [37] and Nichols and Calef [42]). A significant contribution to polarization relaxation is observed in certain cases. It is found that the Onsager inverted snowball model is correct only when the rotational diffusion mechanism of solvation dominates the polarization relaxation. The Onsager model significantly breaks down when there is an important translational contribution to the polarization relaxation [47-50]. In fact, translational effects can rapidly accelerate solvation near the probe. In certain cases, the predicted behavior can actually approach the uniform continuum result that rs = t,. 

From collisionality considerations in fusion edge plasmas one can conclude that a collisional transport model for the charged plasma components is adequate at least for transport along the field lines. By contrast, core plasma transport in the only relevant transport directions there, across the magnetic field, is often studied on the basis of the Vlasov-equation (see below), e.g., in the drift-wave theory of plasma transport. 

The paper is oiganized as follows. In foe following section, foe site-site Smoluchowsky -Vlasov equation for foe solvent dynamics is bridly disclosed. In section HI, a preliminary resdts for foe solvation dynamics of an ion in a variety of solvent is presented. 

We note that the second term on the rhs of (18) describes a diffusion process, which is induced by the Vlasov field Vp = -kgT f dr c( r - r )[n(r, l) - nr,]. Equation(18) with or without the random current together with its generalization to two-component system and to polar liquids have been playing important roles and now generally called a Smoluchowski-Vlasov equation. As an apphcation of the L-D equation (18) with the F-D theorem (9), we calculated the dynamic structure factor of a simple liquid... 

The nonlinear Smoluchowski-Vlasov equation is calculated to investigate nonlinear effects on solvation dynamics. While a linear response has been assumed for free energy in equilibrium solvent, the equation includes dynamical nonlinear terms. The solvent density function is expanded in terms of spherical harmonics for orientation of solvent molecules, and then only terms for =0 and 1, and m=0 are taken. The calculated results agree qualitatively with that obtained by many molecular dynamics simulations. In the long-term region, solvent relaxation for a change from a neutral solute to a charged one is slower than that obtained by the linearized equation. Further, in the model, the nonlinear terms lessen effects of acceleration by the translational diffusion on solvent relaxation. 

Figure 1. Normalized functions S(t) in solvation dynamics calculated by the nonlinear Smoluchowski-Vlasov equation for a change in a solute charge from 0 to c (z=0- 1) and from e to 0 (z= 1 - 0) (solid lines), and by the linearized equation (dashed line).

Calculation with the variation of translational diffusion coefficients showed that the nonlinear effects lessened effects of translational diffusion (Fig. 2). In the linearized equation, solvent relaxed quickly for a large translational diffusion coefficient. That is, a value of S(t) at t=(2DR) decreased with the increase in the translational diffusion coefficient for the linearized equation. For calculation with the nonlinear Smoluchowski-Vlasov equation (1), however, a value of S(t) at t=(2DR) increased slightly with the translational diffusion coefficient. Thus, the ratio of a nonlinear value to a linear value increased with the translational diffusion coefficient. 

Strictly speaking, the relative contribution of bound states within the collisionless models is in principle indeterminate, which is, eventually, the consequence of the time-reversibility of Vlasov equation. The matter is that the stationary solutions of the Vlasov equation are dependent on the way of formation of the steady state of the system. Thus, to tackle this problem, one has to employ additional considerations or principles for evaluating the number of trapped ions. 

Notice that the above equations also follow from the stationary solution of the Vlasov equation with the appropriate boundary conditions (Maxwellian distributions at the infinity and zero value of distribution functions with positive radial velocity at the grain surface). It should be noted, that in the derivation of the density for bound ions, we also start from the Maxwellian distribution, though the finite trajectories do not reach the infinity and, therefore, cannot be... 

The memory function M(k, pp" t) represents effects from the dynamics of collisional processes. Before embarking on the survey of the results of the generalized kinetic theory, let us see briefly how the basic equations of classical kinetic theories can be recovered by means of the memory-function equation (5.38). For instance, the Vlasov equation can be obtained by completely ignoring the memory term in Eq. (5.38) ... 

In the (younger) field of cluster physics, semiclassical methods have been less systematically explored, although several applications exist [2, 119, 120]. The Vlasov equation suffers from the same formal [121] and numerical [122] deficiencies as its nuclear counterpart, and this defect can be cured in a similar (expensive) way [123]. A more promising alternative is here offered by TDTF [124]. Careful studies of the phase-space structure of the electron cloud in a cluster show that the local degree of anisotropy of the Fermi sphere remains reasonably small in the dynamical situations considered here. The situation is more gratifying than in the nuclear case, because of the overwhelming dominance of the dipole response in the cluster case, a mode for which velocity anisotropy is small. 

The Vlasov-Newton equation of motion of/is the closed equation ... 

The Vlasov-Newton equation has many steady solutions describing a self-gravitating cluster. This is easy to show in the spherically symmetric case (the situation we shall restrict in this work, except for a few remarks at the end of this section). If one assumes a given r(r) in the steady state, the general steady solution of Eq. (4) is a somewhat arbitrary function of the constants of the motion of a single mass in this given external held, namely a funchon/(E, I ) where niE is the total energy of a star in a potenhal (r) such that r(r) = —(r/r) [d r)/dr] and where — (r.v) is the square of the... 

Suppose Eq. (6) has a solution with the given asymptotic conditions, which holds true in a wide range of cases [2] then one associates to a given/( ) a solution of the steady state of the Vlasov-Newton equation. There are various restrictions on possible functions/( ) It must be positive or zero and such that the total mass is finite. Of course, as we said, this is not enough to tell what function f E) is to be chosen. Moreover, knowing l>(r), it is possible in principle to find the function/(E) from Eq. (6) by writing the left-hand side as a function of <1) (instead of r). Then there remains to invert an Abel transform to get back/(E). We shall comment now on the impossibility of applying the usual methods of equilibrium statistical mechanics to the present problem (that is, the determination of f E) from a principle of maximization of entropy for instance). 

Site-Site Smoluchowsky-Vlasov (SSSV) Equation... 

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