Phase Space Volumes

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

When the integrator used is reversible and symplectic (preserves the phase space volume) the acceptance probability will exactly satisfy detailed balance and the walk will sample the equilibrium distribution... 

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]) ... 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

The cissertion that Hamiltonian flows preserve phase-space volumes is known as Louiville s Theorem, and is easily verified from equation 4.3 by using the Hamiltonian equations 4.5 ... 

Since V(t) = V(0) for all times t in conservative systems, Ap = 0. The presence of attractors in dissipative systems, on the other hand, implies that the available phase space volume is contracting, and thus that Ap < 0. Since chaotic motion (either in conservative or dissipative systems) yields Ai > 0, this therefore also means that, in dissipative systems, the phase space volume is both expanding along certain directions and contracting along others. 

The principles of statistical mechanics can be applied to a dynamical systeni provided that it obeys Louiville s Theorem (that is, it preserves volumes in phase space) and that its energy remains constant. The first requirement is easy since all reversible rules 4>r define bijective mappings of the phase space volume... 

If only one type of particle is present, mx = m2 however, the expressions relating the velocities before and after collision do not simplify to any great extent. If several types of particles are present, then there results one Boltzmann equation for the distribution function for each type of particle in each equation, integrals will appear for collisions with each type of particle. That is, if there are P types of particles, numbered i = 1,2,- , P, there are P distribution functions, ft /(r,vt, ), describing the system ftdrdvt is the number of particles of type i in the differential phase space volume around (r,v(). The set of Boltzmann equations for the system would then be ... 

This collision dynamics clearly satisfies the conservation laws and preserves phase space volumes. 

Hybrid MPC-MD schemes may be constructed where the mesoscopic dynamics of the bath is coupled to the molecular dynamics of solute species without introducing explicit solute-bath intermolecular forces. In such a hybrid scheme, between multiparticle collision events at times x, solute particles propagate by Newton s equations of motion in the absence of solvent forces. In order to couple solute and bath particles, the solute particles are included in the multiparticle collision step [40]. The above equations describe the dynamics provided the interaction potential is replaced by Vj(rJVs) and interactions between solute and bath particles are neglected. This type of hybrid MD-MPC dynamics also satisfies the conservation laws and preserves phase space volumes. Since bath particles can penetrate solute particles, specific structural solute-bath effects cannot be treated by this rule. However, simulations may be more efficient since the solute-solvent forces do not have to be computed. 

Although this collision rule conserves momentum and energy, in contrast to the original version of MPC dynamics, phase space volumes are not preserved. This feature arises from the fact that the collision probability depends on AV so that different system states are mapped onto the same state. Consequently, it is important to check the consistency of the results in numerical simulations to ensure that this does not lead to artifacts. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Phase space volumes, multiparticle collision dynamics, 94-95... 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Furthermore, the phase-space volumes are preserved during the Hamiltonian time evolution, according to Liouville s theorem. We denote by... 

The difference between both entropies per unit time is minus the sum of all the Lyapunov exponents which is the rate of contraction of the phase-space volumes under the effects of the nonHamiltonian forces ... 

Figure 21.6. Schematic representation of the relative phase-space volumes available to reactant, transition state, and product. A plane located at the most constricted place has the highest prohahility of being crossed only once by a molecular trajectory, which is the location of the transition state.

J. Troe Concerning the lack of dependence of the Nan911 lifetime on cluster size n discussed by Prof. Gerber, is it not possible that the excitation leads into repulsive excited electronic states from which the fragmentation is direct, that is, not related to phase-space volumes and densities of states ... 

Single-particle dipolar ACF and its spectrum Element of phase-space volume Charge of electron... 

Libration amplitude, cone angle Phase-space volume Phase of a.c. held... 

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