Phase space density, time evolution

The approach is to determine the force(s) acting on our system, write down the equations of motion, and solve Eq. (8.22). Specifically, we look at the time evolution of the phase space density, (p p,r,t)), and determine if it is increasing, decreasing, or stajung the same ... 

Structure, the position and the values of the peaks are maintained. Typically, fluctuations lower slightly (by 5-10%) the value of the principal peaks and enhance the width around these peaks. Since Q, = 500 in the simulations, this is compatible with the order of magnitude of corrections predicted on the basis of the classical inverse power law scaling of the fluctuations [11, 15]. This correction suffices, on the other hand, to erase the fine-scale characteristics of the system s evolution in phase space. A useful visualization of this smoothing action of the fluctuations is provided by Figure 17, in which the phase space density is computed by time averaging the phase... 

The state of the entire system at time t is described by the /V-particle phase space probability density function, P(x/V, t). In MPC dynamics the time evolution of this function is given by the Markov chain,... 

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... 

The basic issue is at a higher level of generality than that of the particular mechanical assumptions (Newtonian, quantum-theoretical, etc.) concerning the system. For simplicity of exposition, we deal with the classical model of N similar molecules in a closed vessel "K, intermolecular forces being conservative, and container forces having a force-function usually involving the time. Such a system is Hamiltonian, and we assume that the potentials are such that its Hamiltonian function is bounded below. The statistics of the system are conveyed by a probability density function 3F defined over the phase space QN of our Hamiltonian system. Its time evolution is completely determined by Liouville s equation... 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

There exists a second reason why Latora and Baranger have been forced to depart from the adoption of a density equation, thereby rather adopting the supposedly equivalent time evolution of a bunch of trajectories. This is due to the fact that the Lyapunov coefficients are local and might change with moving from one point of the phase space to another. It is important to stress this second reason because it is closely related to the directions which need to be followed to reveal by means of experiments the breakdown of the density perspective, and with it of quantum mechanics, in spite of the fact that so far the predictions of quantum mechanics have been found to fit very satisfactorily the experimental observation. 

Notice that in the Continuous-Time Random Walk (CTRW) as used in Klafter et al. [50], in the case where the waiting time distribution is exponential, i(t) = a expf at], the same evolution for the probability density p(x,t) and the phase-space distribution ct(x, t) occurs as that resulting from Eq. [57], This can... 

The time evolution of this phase space probability density is governed by the Liouville equation expressed as... 

Another important outcome of these considerations is the following. The uniqueness of solutions of the Newton equations of motion implies that phase point trajectories do not cross. If we follow the motions of phase points that started at a given volume element in phase space we will therefore see all these points evolving in time into an equivalent volume element, not necessarily of the same geometrical shape. The number of points in this new volume is the same as the original one, and Eq. (1.107) implies that also their density is the same. Therefore, the new volume (again, not necessarily the shape) is the same as the original one. If we think of this set of points as molecules of some multidimensional fluid, the nature of the time evolution implies that this fluid is totally incompressible. Equation (1.107) is the mathematical expression of this incompressibility property. 

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (r (f), p (f)) for a given initial condition (r (0), p (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1,2,2)) that describes the time evolution of the phase space probability density p f). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1-5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

The superscript H on the square bracket in (8.3.4) indicates that the Liouville expression inside the bracket needs to be transformed back into Hilbert space before multiplication with the Hilbert-space operator 1 = 1 - ily = I- and formation of the trace. Inverse Fourier transformation over k produces an image of the spin density. Given the form of the density matrix after the initial pulse, the spin density corresponds to 1+ (r). It is weighted by the phase evolution under the internal Hamiltonians Hx(r) during the space-encoding time ti,... 

Fig. 5,14. Spatio-temporal breathing patterns of the DBRT electron density evolution, phase portrait, and voltage evolution for (a) e = 7.0 periodic breathing, (b) e = 9.1 chaotic breathing (r = —35, Uo = —84.2895, K = 0). Time t and space x are measured in units of the tunneling time Ta and the diffusion length la, respectively. Typical values at 4K are Ta = 3.3 ps and la = 100 nm [47].

The procedure for constructing kinetic equations using the generalized Langevin equation is well known - one uses as variables in this description the p/ioje-space density fields. We could of course simply use the solute phase-space fields, (7.1), and follow the methods of Section V to obtain a formal kinetic equation for their time evolution. This procedure... 

A PWT of quasiclassical degrees of freedom provides a rigorous starting point for the treatment of molecular systems with many degrees of freedom. This leads to a convenient numerical procedure to calculate the quantal density operator, when combined with an approximation valid for short wavelengths in phase space. It allows for the introduction of effective potentials to guide the time evolution of quasiclassical trajectories. 

The solution operator describes how a function of the phase variables is mapped forward in time under the flow of the differential equation. An alternative ( dual ) perspective is in terms of the evolution of the measure (or density) of points in phase space. We begin by summarizing a few basic principles needed to provide a foundation for working with probability measures. 

Given a collection of phase space points distributed according to the density po q,p), we know that its evolution (under Langevin dynamics) after a time t will be described by... 

The continuous phase variables, which affect the behavior of each particle, may be collated into a finite c-dimensional vector field. We thus define a continuous phase vector Y(r, t) = [7 (r, t), 2(1, t. .., l (r, t)], which is clearly a function only of the external coordinates r and time t. The evolution of this field in space and time is governed by the laws of transport and interaction with the particles. The actual governing equations must involve the number density of particles in the particulate phase, which must first be identified. 

Let us try to derive an equation for the evolution in time of the phase space probability density p(p,r, t) for a random process generated by... 

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