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The Microscopic Equations

As in the case of magnetism, the effective average electric field to which a particular atom or molecule inside a dielectric is subjected, when an external field is applied, it is not the [Pg.365]

For an isolated atom, the atomic electric dipole moment induced by the external electric field is proportional to the field  [Pg.366]

From Eq. 8.52, it is seen that the applied electric field, E, and the polarization, P, are related through a second-rank tensor called the dielectric susceptibility, Xe- Three equations, each containing three terms on the right-hand side, are needed to describe [Pg.366]

the nine components of the susceptibility can be written in a square array as  [Pg.367]

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. [Pg.367]


It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. [Pg.268]

In order to write down the microscopic equations of motion more formally, we consider a size N x N 4-neighbor lattice with periodic boundary conditions. At each site (i, j) there are four cells, each of which is associated with one of the four neighbors of site (i,j). Each cell at time t can be in one of two states defined by a Boolean variable where d = 1,..., 4 labels, respectively, the east, north,... [Pg.489]

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... [Pg.114]

The two drawbacks mentioned above can be overcome, however, by considering the ensemble-averaged version of the microscopic equation of motion, Eq. (3) ... [Pg.77]

Another result that is not evident in Fig. 33 concerns the computational times required for gathering the statistical steady-state values of various quantities (such as the slip velocity shown in Fig. 33) at comparable grid resolutions, the computational time required to solve the filtered equations is much smaller than that for the microscopic equations. This can be attributed to the fact that the structures obtained in the solution of the filtered equations are comparatively coarser than those for the microscopic TFM equations. [Pg.141]

Since Fp is a real quantity and thus iatV is imaginary, no contrast would be visible without the transfer function of the microscope (Equation 6). The lens aberrations result in a transfer of some imaginary part information at the exit plane into the real part at the image plane, which can be imaged by HRTEM. [Pg.376]

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. [Pg.73]

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. [Pg.75]

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. [Pg.76]

Thus, the fractional equilibrium state (99) can be considered as a consequence of anomalous transport of phase points in the phase space resulting in the anomalous continuity equation (104). Note that the usual form of the evolution (93) is a direct consequence of the canonical Hamiltonian form of the microscopic equations of motion. Thus, the evolution of (105) implies that the microscopic equations of motion are not canonical. The actual form of these equations has not yet been investigated. However, there are strong indications that dissipative effects on the microscopic level become important. [Pg.77]

From one point of view, (109) can be interpreted as a manifestation of the noncanonical nature of the microscopic equation of motion and supports the idea of dissipative effects on the microscopic level (for time scale t < t/). From another point of view (109) can be related to the coarse graining of the phase volume minimum cells. The concept of fractional evolution is due to the action of the averaging operator [45]. Each application of the averaging operator is equivalent to a loss of information regarding the short time mobility and is closely associated with the renormalization approach ideas [239]. [Pg.78]

The fluctuation-dissipation theorem, of which this is one example, is referred to in the Appendix. Unlike (/), is finite for / < 0. In a non-rotating system in the absence of magnetic fields, the growth of thermal fluctuations, because of the perfect reversibility of the microscopic equations of motion, is the reverse of their decay. Thus... [Pg.228]

Ip this section we will apply the actual ensemble averaging process to the microscopic equations (II.l to II.3) in succession. For Eqs. II.l and II.2 this method yields simply, when Eqs. 1.7 and 1.11 are used, two of the Maxwell equations... [Pg.320]

This is written semi-intensively in terms of the fluid density p, but total mass depends on system size via the integration limits which encompass the entire control volume. The final form of the microscopic equation of continuity is intensive because one divides by system volume and simultaneously takes the limit as each coordinate dimension approaches zero. This limiting procedure is not performed explicitly below, but the general methodology can be interpreted in that manner. The rate of accumulation of overall fluid mass within V is expressed in terms of a total time derivative, as follows ... [Pg.222]

The internal energy per unit volume of fiuid is pu, where u is the specific internal energy. Following the methodology in Section 25-2, the left side of the microscopic equation of change for internal energy, with units of energy per volume per time, is... [Pg.692]

The final form of the microscopic equation of change for internal energy is... [Pg.693]

The fourth step of the general strategy in Section 25-1 is to add the microscopic equations of change given by (25-19) and (25-24). The kinetic energy and internal energy equations are repeated here, for comparison ... [Pg.693]

This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... [Pg.694]

The most important aspect of this development of the microscopic equations of change for kinetic, internal, and total energies is the identification of conductive energy flux, which allows one to construct an equation of change for specific... [Pg.696]

In computational chemistry, molecular dynamics (MD) is the most widely used methodology to study the kinetic and thermodynamic properties of atomic and molecular systems [1-3]. These properties are obtained by solving the microscopic equations of motion for the system under consideration. Due to the short time step that is needed to keep numerical stability, the time scales that can be reached in these simulations are not long, typically on the order of nanosecond to microsecond depending on the complexity of the system. For many systems, like biomolecules, this simulation time is not enough to sample conformational space or to study rare but important events. [Pg.15]

One particular difficulty in analyzing the results of the tube theory is that often papers in this area do not start by spetdlying the microscopic equations of motion at all, but rather write down the results for particular physical observables and give... [Pg.155]

In statistical mechanics, the condition of microscopic time reversal (microscopic reversibility) [16,18] expresses the invariance of the microscopic equations of evolution with respect to changing the sign of the time variable. For systems without solenoidal fields, the application of time reversal leads to the condition of detailed balance, which states that for each direct process there is a reverse process, and at equilibrimn the rate of each direct process equals the rate of the reverse process. [Pg.187]

Our next task is to write the above results, particularly Eq. (30), in terms of the variables appropriate to a continuous system. Considering first an isolated system, we suppose the microscopic equations of motion to admit the conservation laws... [Pg.272]

We emphasize that the conservation laws (31) are identities if the fluxes and densities are expressed as functions of the q, p, and the time dependence of the latter determined by the microscopic equations of motion, then the left-hand sides of the conservation laws reduce identically to zero. Stated differently, if the... [Pg.273]

The microscopic equation of motion for the number of molecules of species a is given... [Pg.300]

The procedure indeed appears analogous to that leading to transport coefficients and to specific reaction rates per unit volume. In this case it is a microscopic scale study, or experiment that permits the constitutive relation to be determined. The calculations are complex but the results obtained lead to the expression of the diffusion coefficient, to give but one example. The simplification in comparison with the microscopic equations is considerable. [Pg.598]

In order for the resulting equation and therefore Eq.(7.19) to hold we must require microreversibility, i.e. the inversion t - t does not affect the result of the ensemble averaging. But here we rely on the time reversibility of the microscopic equations of motion (at least for short times). This also is the reason for the requirements expressed in Eq. (7.20). In order for the time reversibility to hold, the Lorentz force, which is proportional to df/dt x B, must be invariant as well as the Coriolis force, proportional to dr/dt x cD, in the case of a rotating system. Therefore the magnetic field, B, and the angular velocity, u>, appear with reversed signs on the right side of Eq.(7.20). [Pg.245]

In this appendix, we give a derivation of the macroscopic equations for the fluid phase. It is obtained from the microscopic equations using a filter or smoothing kernel. The approach is similar to the classical derivation of Anderson and Jackson (1967), but some of the results are new. In the derivation, there are some tricky parts. Important in the derivation is that explicit use is made of a separation of length scales. The scale on which macroscopic quantities vary is much larger than the filtered size, which in turn is much larger than the particle size. For aU quantities (except the velocity), we win define the smoothed quantity of a microscopic property as. [Pg.182]

Above, our focus has been on the finite difference method, which is easy to implement in domains of rather simple geometry. In complex domains, it is difficult to place a grid and keep track of neighbors when the grid points are required to he along the coordinate axes. Here, we discuss another method that is not subject to this condition. We again consider the 2-D Poisson equation but now instead of the microscopic equation... [Pg.297]


See other pages where The Microscopic Equations is mentioned: [Pg.115]    [Pg.144]    [Pg.355]    [Pg.65]    [Pg.75]    [Pg.451]    [Pg.365]    [Pg.81]    [Pg.76]    [Pg.59]    [Pg.135]    [Pg.266]   


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