Liouville’s equation

A particularly convenient notation for trajectory bundle system can be introduced by using the classical Liouville equation which describes an ensemble of Hamiltonian trajectories by a phase space density / = f q, q, t). In textbooks of classical mechanics, e.g. [12], it is shown that Liouville s equation... 

We may start with Liouville s equation for monatomic particles (no internal degrees of freedom) ... 

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... 

If three-body collisions are neglected, which is permitted at sufficiently low densities, all the interactions take place between pairs of particles the two-particle distribution function will, therefore, satisfy Liouville s equation for two interacting particles. For /<2)(f + s) we may write Eq. (1-121) ... 

Chapman-Enskog solution, 35 coefficicent equations, 28 derivation from Liouville s equation, 41... 

Liouville s equation, derivation of Boltzmann s equation from, 41 Littlewood, J. E., 388 Lobachevskies method, 79,85 Local methods of solution of equations, 78... 

This is Liouville s equation, with the Liouville operator... 

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. 

In order for an ensemble to represent a system in equilibrium, the density of phase must remain constant in time. This means that Liouville s equation is satisfied, which requires that g is constant with respect to the coordinates q and p, or that g = g(a), where a = a(q,p) is a constant of the motion. Constants of motion are properties of an isolated system which do not change with time. Familiar examples are energy, linear momentum and angular momentum. For constants of motion H,a = 0. Hence, if g = g a) and a is a constant of motion, then... 

Equation (37) is the quantum statistical analogue of Liouville s equation. To find the quantum analogue of the classical principle of conservation of phase density the solution to (37) is written in the form... 

The time evolution of the probability density is induced by Hamiltonian dynamics so that it has its properties—in particular, the time-reversal symmetry. However, the solutions of Liouville s equation can also break this symmetry as it is the case for Newton s equations. This is the case if each trajectory (43) has a different probability weight than its time reversal (44) and that both are physically distinct (45). 

The basic issue is at a higher level of generality than that of the particular mechanical assumptions (Newtonian, quantum-theoretical, etc.) concerning the system. For simplicity of exposition, we deal with the classical model of N similar molecules in a closed vessel "K, intermolecular forces being conservative, and container forces having a force-function usually involving the time. Such a system is Hamiltonian, and we assume that the potentials are such that its Hamiltonian function is bounded below. The statistics of the system are conveyed by a probability density function 3F defined over the phase space QN of our Hamiltonian system. Its time evolution is completely determined by Liouville s equation... 

The question of determinateness presents itself as follows Let the initial (t = 0) values of all the macroscopically independent macroscopic variables be given the equations of macroscopic physics (thermal and hydro-dynamic equations, etc.) show that these variables evolve deterministically with t 0. Yet there are infinitely many different probability densities ( ")t = 0 which have the moments, etc., coinciding with the set of given initial macroscopic values. Each evolves (by Liouville s equation) differently, and hence may induce a different set of macroscopic expected values at / > 0. By what principle of natural selection is the class of probability densities so restricted as to restore macroscopic determinacy ... 

The problem of irreversibility lies in the contrast between the irreversible nature of the above equations of macroscopic evolution, with the reversibility of Liouville s equation governing the evolution of every probability density. ... 

It is an elementary exercise in substitution to show that such an cannot in general satisfy Liouville s equation. 

It might be possible to save the assumption in this context, by regarding it as an approximation. But this would introduce a major question of principle, since the motions which, without this approximate assumption are reversible, become, after it is made, irreversible One of the earliest applications Boltzmann made of his equation was to prove that the entropy integral increases, whereas Liouville s equation proves that it must remain strictly constant. In addition, the macroscopic determinacy appears to be established. This means that the assumption goes beyond the modest role of quantitative approximation, and assumes that of a new principle of physics—entering in a vague way through the back door. 

A generalization of Liouville s equation was presented in [10]. Applying modern techniques in differential geometry [13], as they are applied to dynamical systems [14-16], gives rise to the generalized Liouville equation of the form [17] ... 

Nakano and Yamaguchi255-258 are developing a method based on numerical solutions of Liouville s equation to describe the frequency-dependent response of an assembly of dipoles to an electric field. The aggregates are of a size such that retardation effects are significant. One novel result is that there is a sharp change of the polarizability, reminiscent of a phase transition, that occurs as the intensity of a near-resonant field is increased. 

The backward equation, Liouville s equation, for the particle density takes the form... 

As we decrease e, the density remains invariant under Liouville s equation. The prefactor means that we can think of a sequence of smooth densities taken with successively smaller values of e as approaching the limiting distribution defined in terms of the delta function by... 

In typical molecular dynamics applications with multiple bodies and complicated force laws, the motion is assumed to be chaotic. Ergodic coverage of would then have to arise from the global properties of a chaotic system (sensitivity to initial conditions and transitivity). Liouville s equation,... 

Technically, Liouville s equation refers to the continuity equation in the setting of a volume preserving flow. Here we use the term liouviUian to refer to the operator whose action on a density gives the right hand side of the general continuity equation. 

Finally in these preliminaries to writing formal response theory expressions for P(t). equation (6) applied to the distribution function f gives Liouville s equation... 

This is Liouville s equation. It completely specifies the evolution of the probability density for a system of a specified Hamiltonian. It is also equivalent to 6N Hamilton s equations of motion. 

Liouville s equation is the starting point for our discussion of non-equilibrium systems in Chapter 12. 

Consider a system at equilibrium with a phase space probability distribution function Dg. If an external field F t) that couples to a mechanical property A p, q) is applied at t = 0, the system will move away from its equilibrium state. The phase space probability distribution will now become a function of time, D = )(p, q,t), changing according to Liouville s equation, Eq. 12.1. 

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