Volume in phase space

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

The exact propagator for a Hamiltonian system for any given time increment At is symplectic. As a consequence it possesses the Liouville property of preserving volume in phase space. 

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... 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

Liouville theorem and related forms The Helmholtz-Lagrange relation given in equ. (4.46) is related to many other forms which all state certain conservation laws (the Clausius theorem, Abbe s relation, the Liouville theorem). The most important one in the present context is the Liouville theorem [Lio38] which describes the invariance of the volume in phase space. The content of this theorem will be discussed and represented finally in a slightly different form which allows a new access to the luminosity introduced in equ. (4.14). 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

In using Eq. (67), we are faced with the problem of determining g(r, p), the number of states corresponding to each volume in phase space. Classical mechanics does not quantize variables, so there is no obvious way of doing... 

We note that volumes in phase space have units of kg m2/s = J s for each Cartesian coordinate. In order to have a dimensionless degeneracy, g, we divide volumes in phase space, dr dp, by a constant, h2, for which h has units of J s. In other words, g(r, p) = h independent of position and momentum ... 

In quantum mechanics, Heisenberg s uncertainty principle states that there is a limit to which we can know the product of the uncertainties in a coordinate and its corresponding momentum, AxApx. Thus, even in quantum mechanics, there is a minimum volume in phase space in which we can localize a particle. 

The Lorenz system is dissipative volumes in phase space contract under the flow. To see this, we must first ask how do volumes evolve ... 

This distinction between a < and a = exemplifies a broader theme in nonlinear dynamics. In general, if a map or flow contracts volumes in phase space, it is called dissipative. Dissipative systems commonly arise as models of physical situations involving friction, viscosity, or some other process that dissipates energy. In contrast, area-preserving maps are associated with conservative systems, particularly with the Hamiltonian systems of classical mechanics. 

According to Gleick (1987, p. 149), Henon became interested in the problem after hearing a lecture by the physicist Yves Pomeau, in which Pomeau described the numerical difficulties he had encountered in trying to resolve the tightly packed sheets of the Lorenz attractor. The difficulties stem from the rapid volume contraction in the Lorenz system after one circuit around the attractor, a volume in phase space is typically squashed by a factor of about 14,000 (Lorenz 1963). 

In particular, the evolution of a system in time also represents a canonical transformation which implies that the volume in phase space is conserved as it evolves in time. This is known as the Liouville theorem (1809-1882). 

In an engineering view the ensemble of system points moving through phase space behaves much like a fluid in a multidimensional space, and there are numerous similarities between our imagination of the ensemble and the well known notions of fluid dynamics [35]. Then, the substantial derivative in fluid dynamics corresponds to a derivative of the density as we follow the motion of a particular differential volume of the ensemble in time. The material derivative is thus similar to the Lagrangian picture in fluid d3mamics in which individual particles are followed in time. The partial derivative is defined at fixed (q,p). It can be interpreted as if we consider a particular fixed control volume in phase space and measure the time variation of the density as the ensemble of system points flows by us. The partial derivative at a fixed point in phase space thus resembles the Eulerian viewpoint in fluid dynamics. 

Ensemble of System Points Moving Through a Fixed Volume in Phase Space... 

Hence S(y) is as much a constraint on P(y) as, say, the given total strength y. One sometimes writes loosely S(y) = In V, where V is the available volume in phase space. If there are no other constraints, this volume is uniformly sampled. 

Even in the chaotic limit the effective number of states Ne will be below the available number N0. The reason is the fluctuations of the intensities about their mean values. Fluctuations imply lower entropy or, equivalently, lower Ne. To estimate the importance of fluctuation in reducing the available volume in phase space, we use (3.12) with = p°. Then, using (3.32),... 

This means that the determinant of the Jacobian of the flow in phase space is equal to one. Consequently, the volume in phase space is conserved (the Liouville theorem). 

Asymptotic stability never appears, because it is not possible for the eigenvalues A3, A4 to be both inside the unit circle. This is also a consequence of the fact that the volume in phase space is conserved. 

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