Phase space continuous

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

The Boltzman probability distribution function P may be written either in a discrete energy representation or in a continuous phase space formulation. 

Dissipative systems whether described as continuous flows or Poincare maps are characterized by the presence of some sort of internal friction that tends to contract phase space volume elements. They are roughly analogous to irreversible CA systems. Contraction in phase space allows such systems to approach a subset of the phase space, C P, called an attractor, as t — oo. Although there is no universally accepted definition of an attractor, it is intuitively reasonable to demand that it satisfy the following three properties ([ruelle71], [eckmanSl]) ... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

The probability density p(p, q, t) in phase space satisfies the continuity equation... 

A two-dimensional cartoon is helpful in understanding overlap relationships between the important phase spaces /, and /. As illustrated in Fig. 6.1, there are four possible ways that /, and / can be related (a) / can form a subset of /, (b) /, and r may almost coincide, i.e., have complete overlap (c) T( and / may have partial overlap and (d) / and r may have no overlap. For simplicity, in Fig. 6.1 we have only considered the case in which / is continuous in space but the same principle applies to more-complicated situations, in which regions of / are separated in space. 

The Jarzynski identity can be used to calculate the free energy difference between two states 0 and 1 with Hamiltonians J%(z) and -A (z). To do that we consider a Hamiltonian -AA iz, A) depending on the phase-space point z and the control parameter A. This Hamiltonian is defined in such a way that A0 corresponds to the Hamiltonian of the initial state, Af(z, A0) = Atfo (z), and Ai to the Hamiltonian of the final state, Ai) = Aif z). By changing A continuously from A0 to Ai the Hamiltonian of the initial state is transformed into that of the final state. The free energy difference ... 

Motion of the virtual ensemble in phase space may be likened to fluid flow by introducing a 2n-dimcnsional vector of phase velocity v with components qi,pi(i = 1,2,..., n). Since the systems of the virtual ensemble can neither be created nor destroyed in the flow, the equation of continuity... 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

A major interest to sequential IPN s relates to dual phase continuity, defined as a region of space where each of two phases maintain some degree of connectivity. An example is an air filter with the air flowing through it. A Maxwell demon can transverse all space within both the filter phase and the air phase, both phases being continuous. 

The corresponding structure of fast-slow time separation in phase space is not necessarily a smooth slow invariant manifold, but may be similar to a "crazy quilt" and may consist of fragments of various dimensions that do not join smoothly or even continuously. 

Detonation in gases has been mote thoroughly investigated than in condensed phases, and continues to receive the major share of attention. In a broad sense the chain of events is the same in both cases (Ref 7, p 2), but in gases is spread out in both space and time, being therefore more easily observed... 

The beauty of the prior approximations is that by assuming a mean-field influence of solvation we can continue to work in a phase space having the same dimensionality as that for the gas phase that being the case, analysis using the tools of TST is mechanically identical for the two phases. When the solvent is not fully equilibrated with the complete reaction path, however, the reacting system can no longer legitimately be described exclusively in terms of solute coordinates. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

A statistical analysis of the fluctuational trajectories is based on the measurements of the prehistory probability distribution [60] ph(q, t qy, tf) (see Section IIIC). By investigating the prehistory probability distribution experimentally, one can establish the area of phase space within which optimal paths are well defined, specifically, where the tube of fluctuational paths around an optimal path is narrow. The prehistory distribution thus provides information about both the optimal path and the probability that it will be followed. In practice the method essentially reduces to continuously following the dynamics of the system and constructing the distribution of all realizations of the fluctuational trajectories that transfer it from a state of equilibrium to a prescribed remote state. 

This is a continuity equation in phase space. When this equation is written out,... 

An alternative formulation [3] of the expression for the rate constant that combines steps (i) and (ii) above is possible, and therefore results in an expression for the rate constant that is independent of the chosen surface, as long as it does not exclude significant parts of the reactant phase space. The expression also forms a convenient basis for developing a quantum version of the theory. We begin with the reformulation of the classical expression and continue with the quantum expression in the following section. 

Fig. 2 a Representation of the 3D diffraction data from the monolayer tubulin crystals. Amplitudes (shown here) and phases vary continuously along reciprocal lattice lines that pass through each of the diffraction spots in the diffraction pattern of an untilted crystal, shown in the horizontal plane. The two planes perpendicular to this plane show amplitudes along some of the lattice lines, b Three lattice line curves. The vertical axis represents intensity in an arbitrary unit, while the horizontal axis is z, the height along the lattice line in reciprocal space. Note the odd numbered curve (17,6) although the intensities are weak, the peaks are well defined because of the large number of measurements... 

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