Time-dependent microscopic density

A simple (atomic) liquid is adequately described by its time-dependent microscopic density p(r, f) = Sj j8(r — r (f)) or in the reciprocal space... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

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. 

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... 

Remark. We assumed that Y(t) is a Markov process. Usually, however, one is interested in materials in which a memory effect is present, because that provides more information about the microscopic magnetic moments and their interaction. In that case the above results are still formally correct, but the following qualification must be borne in mind. It is still true that p y0) is the distribution of Y at the time t0, at which the small field B is switched off. However, it is no longer true that this p(y0) uniquely specifies a subensemble and thereby the future of Y(t). It is now essential to know that the system has aged in the presence of B + AB, so that its density in phase space is canonical, not only with respect to Y, but also with respect to all other quantities that determine the future. Hence the formulas cannot be applied to time-dependent fields B(t) unless the variation is so slow that the system is able to maintain at all times the equilibrium distribution corresponding to the instantaneous B(t). 

Kinetic Theory. In the kinetic theory and nonequilibrium statistical mechanics, fluid properties are associated with averages of pruperlies of microscopic entities. Density, for example, is the average number of molecules per unit volume, times the mass per molecule. While much of the molecular theory in fluid dynamics aims to interpret processes already adequately described by the continuum approach, additional properties and processes are presented. The distribution of molecular velocities (i.e., how many molecules have each particular velocity), time-dependent adjustments of internal molecular motions, and momentum and energy transfer processes at boundaries are examples. 

In related work on the relationship of wood density to weathering rate. Sell and Feist (77) investigated the artificial weathering of 12 hardwoods and 6 softwoods. The surface erosion rate was measured with a light microscope at exposure intervals of 600 h. The results showed that the erosion rate per unit time depended predominantly on the wood density and thus on the wood cell wall thickness. The relationship was approximately linear within a wood density range of 0.3-1.0 g/cm. ... 

The principal advantage of the time correlation function method is that it provides a new set of microscopic functions for a fluid, the time correlation functions, which can be studied directly by experimental observations of the fluidt or by computer-simulated molecular dynamics. The time correlation functions depend even more sensitively on the microscopic properties of the fluid molecules than the transport coefficients, which are expressed as time integrals of the correlation functions. Thus, a further test of kinetic theory has been found it must not only lead to expressions for the transport coefficients for dilute and dense gases that are in agreement with experiment, but also describe the dependence of the time correlation functions on both time and the density of the gas. One of the principal successes of kinetic theory is that it provides a quantitatively correct description of the short- and long-time... 

In this subsection, we employ standard time-dependent perturbation theory with respect to the laser-matter interaction i/mt(i) in the wave-function formalism. Time-dependent perturbation theory may alternatively be formulated in the density-matrix formalism. The latter is more general in that it allows for the inclusion of finite temperature effects and a phenomenological description of relaxation phenomena. In this chapter, we are concerned with the description of the dynamics at conical intersections in a fully microscopic manner, and temperature effects play a minor role. The wave-function formalism is therefore appropriate for our purposes. Writing the time-dependent wave function as... 

The province of conventional dielectric measurements is here taken to be the determination of the relations of the polarization E and current density J. to the electric field in the macroscopic Maxwell equations. Proper theory should account for these relations in condensed phases as a function of state variables time dependence of applied fields and molecular parameters by appropriate statistical averaging over molecular displacements determined by the equations of motion in terms of molecular forces and fields. Simplifying assumptions and approximations are of course necessary. One kind often made and debated is use of an effective or mean local field at a molecule rather than the sum of microscopic... 

These equations of motion define the time evolution of both microscopic and macroscopic variables Coordinates and momenta can be written for each nucleus or, by well-known transformations, for the translation, rotation and vibration of each molecule. In addition to Hamilton s equations, the laws of continuum mechanics place constraints on the time-dependence of macroscopic conserved quantity such as number, momentum and energy density. For example, if the macroscopic number density of particles at point jr is denoted by p( ,t), one has a microscopic definition... 

To describe the functioning of the lEMs, theory from the field of charged membranes must be adapted for MCDI to describe the voltage-current relationship and the degree of transport of the colons. This implies that (in contrast to most membrane processes) the theory must be made dynamic (time dependent) because it has to include the fact that across the membrane the salt concentrations on either side of the membrane can be very different, and change in time. This means that approximate, phenomenological approaches based on (constant values for) transport (or transference) numbers or permselectivities are inappropriate, and that instead a microscopic theory must be used. An appropriate theory includes as input parameters the membrane ion diffusion coefficient and a membrane charge density X. 

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... 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

It is worth noting that the first term on the right hand side of this equation is time-dependent only through the translational molecular degree of freedom, r , since the isotropic part of the polarizability is not dependent on the molecule orientation. This term is directly connected with the microscopic numerical density ... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

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