Liouville s theorem

Liouville s theorem is a restatement of mechanics. The proof of the theorem consists of two steps. 

Finally, since obtains the result which concludes the proof of Liouville s theorem. 

Geometrically, Liouville s theorem means that if one follows the motion of a small phase volume in Y space, it may change its shape but its volume is invariant. In other words the motion of this volume in T space is like that of an incompressible fluid. Liouville s theorem, being a restatement of mechanics, is an important ingredient in the fomuilation of the theory of statistical ensembles, which is considered next. 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

The elements A and B therefore have equal areas. Liouville s theorem states that an element in phase space is conserved, which means that the element within which a system can be found is constant. Further, if the range Ae in the phase space is divided into equal elements, the system spends equal times passing through these elements, or cells. 

To show [115] that Liouville s theorem holds in any number of phase-space dimensions it is useful to restate some special features of Hamilton s equations,... 

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

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

Modern Information Theory is based on the invaricntivc double density functional f p(x) log [p(x)iq(x) dx. In classical or quantum mechanics a basic time-independent q(x) exists. In the case considered here, q(x) = 1 by Liouville s theorem. Cf S. Kullback, Information Theory and Statistics, Wiley, New York, 1959. 

According to Liouville s theorem (see Section 10.3.2) the spectral brilliance B cannot be increased further by any optical system, except at the expense of total flux. The brilliance is therefore an important quantity for the design of not only electron storage rings, but also beam lines with their attached monochromators and experimental equipment (see Sections 1.4 and 1.5). 

The quantities in the square brackets are just the ones known from equ. (4.19) with equ. (4.16). Due to the fixed value of pass it can be seen that for the evaluation of relative intensities the dispersion correction can be omitted. However, the transmission factor Tret(Ekin, pass) which describes the change of transmission caused by the retardation becomes very important in this case, see Fig. 4.16. It has to be determined experimentally, and in ideal cases it can be estimated on the basis of Liouville s theorem for optical systems (see Section 10.3.2). In the example shown in Fig. 4.16 the essential action of the retardation field is to change the brightness B in one dimension. (One has a one-dimensional problem because the lens produces focusing of the line source in one dimension only (for details see [GSa75]).) Following equ. (10.47) one gets (subscripts ( and r denote quantities before and after retardation)... 

This is Liouville s theorem. It can be stated as follows 80 The streaming of the (3-points in the r-space as given by Eq. (22) generates a continuous point transformation, which transforms each 2rAT-dimensional region into another one of the same volume.81... 

From Liouville s theorem, Eqs. (26) and (26 ), it follows immediately that the quantity a, which determines the distribution of the fine-grained density p, remains exactly constant during the mixing process. However, the function... 

Summarizing we oan say Boltzmann s definition of measure is not quite arbitrary in the sense that Liouville s theorem together... 

In case the collision takes place according to Newtonian mechanics, the relation (1.3) can be proved by means of Liouville s theorem. In quantum mechanics, Eq. (1.3) is practically one of the postulates of the theory, following directly from quantum mechanical calculations of transition probabilities from one state to another. For our present purpose, considering that this is an elementary discussion, we shall simply assume the correctness of relation (1.3). This relation is sometimes called the principle of microscopic reversibility. 

To conclude this section we discuss the baker s map (Farmer et al. (1983)) as an example for an area preserving mapping in two dimensions. Area preservation is of utmost importance for Hamiltonian systems, since Liouville s theorem (Landau and Lifechitz (1970), Goldstein (1976)) guarantees the preservation of phase-space volume in the course of the time evolution of a Hamiltonian system. The baker s map is a transformation of the unit square onto itself. It is constructed in the following four steps illustrated in Fig. 2.5. 

Thus, Hamilton s equations preserve the volume element on the phase space. In fact, this result is a statement of Liouville s theorem. Combining Eq. [16] with Eq. [20] leads to a statement that the probability of finding a member of the ensemble in a volume element dF about the point F, which is just /(F, t)dT, is a conserved quantity ... 

The classical phase space is formally defined in terms of generalized coordinates and momenta because it is in terms of these variables that Liouville s theorem holds. However, in Cartesian coordinates as used in the present section it is usually stiU true that pi = mci under the particular system conditions specified considering the kinetic theory of dilute gases, hence phase space can therefore be defined in terms of the coordinate and velocity variables in this particular case. Nevertheless, in the general case, for example in the presence of a magnetic field, the relation between pi and Cj is more complicated and the classical formulation is required [83]. 

The Lagrangian like control volume drdc in the six dimensional phase space may become distorted in shape as a result of the motion. But, in accordance with the Liouville s theorem, discussed in sect. 2.2.3, the new volume is simply related to the old one by the relation ... 

The simple result, Eq. 1.11, has also been obtained by Irving and Kirkwood by making use of Liouville s theorem. This, however, as the foregoing derivation shows, is not necessary at all, the law of conservation of probability being already sufficient. 

---