Phase Space and the Liouville Equation

By aiming my eyeballs in different directions -THEODOR SEUSS GEISEL, I Had Trouble Getting to SoUa SoUew 

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Phase space, the classical distribution function, and the Liouville equation... 

The quantum analogs of the phase space distribution function and the Liouville equation discussed in Section 1.2.2 are the density operator and the quantum Liouville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

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. 

We consider an ensemble of systems each containing n atoms. Thus, q = (<71, , < 3n), P = (pi, , P3n), and dpdq = Ilf" (dp dqi). We assume that all interactions are known. As time evolves, each point will trace out a trajectory that will be independent of the trajectories of the other systems, since they represent isolated systems with no coupling between them. Since the Hamilton equations of motion, Eq. (4.63), determine the trajectory of each system point in phase space, they must also determine the density p(p, q, t) at any time t if the dependence of p on p and q is known at some initial time to. This trajectory is given by the Liouville equation of motion that is derived below. 

Equation [16] is known as the Liouville equation and is, in fact, a statement of the conservation of the phase space probability density. Indeed, it can be seen that the Liouville equation takes the form of a continuity equation for a flow field on the phase space satisfying the incompressibility condition dIdT F = 0. Thus, given an initial phase space distribution function /(F, 0) and some appropriate boundary conditions on the phase space satisfied by f, Eq. [16] can be used to determine /(F, t) at any time t later. 

Equation [43] was first derived in Ref. 15, where we represented V (r) by a generic Jacobian function/(F), and it represents a correct generalization of the Liouville equation to account for the nonvanishing compressibility of phase space. Equation [43] can be derived in many ways. A general approach starts with a statement of continuity valid for a space with any metric (see Appendix 2). One can examine the transformation from one set of phase space coordinates r to another F. The metric determinant transforms according to... 

Here we describe an alternative derivation of the Liouville equation (1.104) for the time evolution of the phase space distribution function f (r, p t). The derivation below is based on two observations First, a change inf reflects only the change in positions and momenta of particles in the system, that is, of motion of phase points in phase space, and second, that phase points are conserved, neither created nor destroyed. 

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 is called the Liouville equation or continuity equation. It tells us how the distribution of phase points varies in time and space. 

In order to answer these questions, we must turn to the basic equation of nonequilibrium statistical mechanics, the Liouville equation. Therefore, we no longer consider one particular container of a gas, but we consider an ensemble of similarly preparedt containers. We construct the 2Nd-dimensional phase space of positions and momenta of the N-particles in... 

The phase space distribution function is thus seen to evolve in time according to Eqs. (9). Note the difference in the sign accompanying L in Eqs. (6) and (9). Equation (9b) is called the Liouville equation. Substitution of the explicit form of L from Eq. (4) gives various forms for the Liouville equation ... 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

The Liouville equation, Eq. (2.6), describes the behavior of the collection of phase points as they move through a multidimensional space, or phase space, representing the position and momentum coordinates of all molecules in the system. The phase points tend to be concentrated in regions of phase space where it is most likely to find the N molecules with a certain momentum and position. Thus, the density function pN can be interpreted (aside from a normalization constant) as a probability density function, i.e., pjvdr- dp is proportional to the probability of finding a phase point in a multidimensional region between (r, p ) and (r - - d r, + d p ) at any time t. 

Multiplying the Liouville equation, Eq. (2.6), written in terms of, by a (r, )/iV and integrating over all phase space gives... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

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]. 

Since we have discovered the underlying Hamiltonian structure of the QCMD model we are able to apply methods commonly used to construct suitable numerical integrators for Hamiltonian systems. Therefore we transform the QCMD equations (1) into the Liouville formalism. To this end, we introduce a new state z in the phase space, z = and define the nonlinear... 

This is the probability of finding particle 1 with coordinate rx and velocity vx (within drx and dVj), particle 2 with coordinate r2 and velocity v2 (within phase space with velocity rather than momentum for convenience since only one type of particle is being considered, this causes no difficulties in Liouville s equation.) The -particle probability distribution function ( < N) is... 

This relationship is known as Liouville s equation. It is the fundamental theorem of statistical mechanics and is valid only for the phase space of generalized coordinates and momenta, whereas it is not, for example valid for the phase space of generalized coordinates and velocities. This is the basic reason that one chooses the Hamiltonian equations of motion in statistical mechanics. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

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