Phase Space and Trajectories

The state of a classical system can be completely described by specifying the positions and momenta of all particles. Space being three-dimensional, each particle has associated with it six coordinates - a system of N particles is thus characterized by 6N coordinates. The 6N-dimensional space defined by these coordinates is called the phase space of the system. At any instant in time, the system occupies one point in phase space 

For ease of notation, the position coordinates and momentum coordinates are defined as 

Over time, a dynamical system maps out a trajectory in phase space. The trajectory is the curve formed by the phase points the system passes through. We will return to consider this dynamic behavior in Section 3.2.2. 

---

If we carry out such a calculation for a large number of trajectories at the same fixed energy , we obtain a unique map of the global dynamics at that energy. The map is unique because each point on it uniquely specifies a single trajectory in phase space, and trajectories in phase space do not intersea at any given instant in time. If they did intersect, they would have to arise from identical initial conditions, and then claissical mechanics would no longer be a causal theory ... 

Figure 6. Comparison of trajectories of particle 27 in Lennard- ones and Len-nard-Jones plus Axilrod-Teller fluids at pa = 0.65. Both runs were started from the same point in phase space and trajectories shown are from time steps 500-2000 in each simulation. For this calculation the Axilrod-Teller strength constant V was assigned a value of three times that for argon given in Section 2. Note that the circles represent positions of the center of mass of the atom, not the atomic diameter.

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

In the first line we have used that 0(S [q(O)]) is always zero since all trajectories start on the reactant side. Inserting into Eq. (5.76) and interchanging the order of the phase space and time integrals gives... 

If the flow is steady in time, we speak about the x,y,z space as the phase space and the integral curves as trajectories, or in this case, the pathlines. If the fluid is also... 

The reason why the condition of Eq. (275) yields the breakdown of the correspondence between quantum and classical mechanics is evident. The quantum wave function can be identified with a trajectory if it is sharp enough, namely if U(t) < 2(f). When the width of the wave function becomes as large as the width of the channel, within which the wave function moves, the wave function motion starts to depend on the structure of the surrounding phase space, and the correspondence is broken. Using a geometrical model making E(t) decrease exponentially with time [122], the condition of Eq. (275) is shown to occur at time t = tB, where... 

Molecular dynamics uses classical mechanics to study the evolution of a system in time. At each point in time the classical equations of motion are solved for a system of particles (atoms), interacting via a set of predefined potential functions (force field), after which the solution obtained is applied to predict positions and velocities of the particles for a (short) step in time. This step-by-step process moves the system along a trajectory in phase space. Assuming that the trajectory has sampled a sufficiently large part of phase space and the ergodicity principle is obeyed, all properties of interest can then be computed by averaging along the trajectory. In contrast to the Monte Carlo method (see below), the MD method allows one to calculate both the structural and time-dependent characteristics of the system. An interested reader can find a comprehensive description of the MD method in the books by Allen and Tildesley or Frenkel and Smit. ... 

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

The virtue of this change of variables is that it allows us to visualize a phase space with trajectories frozen in it. Otherwise, if we allowed explicit time dependence, the vectors and the trajectories would always be wiggling—this would ruin the geometric picture we re trying to build. A more physical motivation is that the state of the forced harmonic oscillator is truly three-dimensional we need to know three numbers, x, x, and t, to predict the future, given the present. So a three-dimensional phase space is natural. 

The dynamics of such systems is described by the Kolmogorov-Arnold-Moser theory of nearly integrable conservative dynamical systems (see e.g. Ott (1993)). For e = 0 the fluid elements move along the streamlines and the trajectories in the phase space form tubes parallel to the time axis. Due to the periodicity in the temporal direction these tubes form tori that fill the whole phase space and are invariant surfaces for the motion of the fluid elements. Each torus... 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

Figure 9.19 Overview of the phase space and configuration space dynamics associated with the HCCH [JV = 22, l = 0] polyad. The top four plots are surfaces of section for four energies within the polyad. Only simple structures are found near the bottom (local bender) and top (counter-rotators) of the polyad. Chaos dominates at energies near the middle of the polyad, but two classes of stable trajectories coexist with chaos. The bottom four plots show the coordinate space motions of the left- and right-hand H-atoms that correspond to the two different periodic orbits (from Jacobson and Field, 2000b).

The main practical problem in the implementation of the mixed quantum-classical dynamics method described in Section 4.2.4 is the nonlocal nature of the force in the equation of motion for the stationary-phase trajectories (Equation 4.29). Surface hopping methods provide an approximate, intuitive, stochastic alternative approach that uses the average dynamics of swarm of trajectories over the coupled surfaces to approximate the behavior of the nonlocal stationary-phase trajectory. The siu--face hopping method of Tully and Preston and Tully describes nonadiabatic dynamics even for systems with many particles. Commonly, the nuclei are treated classically, but it is important to consider a large niunber of trajectories in order to sample the quantum probability distribution in the phase space and, if necessary, a statistical distribution over states. In each of the many independent trajectories, the system evolves from the initial configuration for the time necessary for the description of the event of interest. The integration of a trajec-... 

---