Spectral density defined

The quantity introduced above is the spectral density defined as the energy per unit volume per unit frequency range and is... 

In contrast to the diffusion processes, the fluctuating force 3F t) can no longer be derived from the Wiener process, and its spectral density defined as 2kBT 0 times the Fourier transform of the memory function C(f) is frequency limited and has no more the white noise characteristics. 

The cw-ESR spectrum is obtained by numerically evaluating the spectral density defined in Eq. 12.27, and here we adopt the standard methodology of spanning the Liouvillian over a proper basis set defined by the direct product... 

Against the background of the effective-mode transformation introduced above, we now address the system-bath perspective more systematically, by recasting the chain development in terms of spectral densities defining the system-bath interaction. 

Eq. 93 is assumed quasi-stationary with evolutionary power-spectral density defined by... 

Ci, the distribution of which is given by the spectral density defined as... 

Note that the spectral density defined in this way is independent of the subscript m for isotropic systems since it is based on the normalized autocorrelation functions given in Eq. 28. 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

With analogy to electric circuits, a transfer function of the antenna can be calculated and the response of the antenna to an incoming wave obtained. The output signal is usually expressed as antenna cross-section. It is defined as the ratio between the total energy absorbed by the antenna and the incident spectral density function of the incident wave. In the case of Nautilus antenna (2300 kg, 3 x 0.6 m) the cross-section is of the order of 10 25m2 Hz. 

As pointed out in the previous section, the Chebyshev operator can be viewed as a cosine propagator. By analogy, both the energy wave function and the spectrum can also be obtained using a spectral method. More specifically, the spectral density operator can be defined in terms of the conjugate Chebyshev order (k) and Chebyshev angle (0) 128 132... 

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

If, however, the transition is of a pure displacive nature, the fluctuation amplitude of the order parameter is critical and is by no means temperature-independent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc), defined by its frequency a>s and damping constant Tj, the spectral density is a Lorentzian centred at s and the... 

For analytic purposes, it is usefiil to define a spectral density of the bath modes coupled to the reaction coordinate in a given frequency range ... 

The results of the integrations depend on the spectral density, which is defined as the cosine Fourier transform of the dynamical friction Eq. (8) ... 

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

Another interesting average over a spectral density is its cumulative distribution D co0) defined by... 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

For binary systems, the spectral density J(v) is generally replaced by the gas density normalized spectral density, VG(v), defined by Eq. 5.60 below. Units thus differ by amagat2 or cm6. 

A(y) = A(v)/v. Whereas A describes the absorption of spectral intensity or energy, A is proportional to the probability of absorbing a photon per unit path length. We will not make great use of the quantity A, because the spectral density G defined above is more closely related to the squared dipole transition matrix elements, even at low frequency G is the preferred quantity. 

The spectral density is defined in terms of the matrix elements of the induced dipole moment p by the golden rule , Eq. 5.3,... 

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