Liouville, equation theorem

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... 

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,... 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

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 ... 

In the Hamiltonian formulation the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. 

The conservation of density of a mechanical system in phase space (Liouville s theorem) implies a rigorous functional relation between g(ri2) and 53(> i2,> i3,> 23)- Starting from this rigorous functional relation (an integro-differential equation ) an approximate closed equation can be obtained " by using the superposition approximation which asserts that... 

Similarly, it can be shown that Eq. [55] is a statement of the preservation of phase space volume under propagation by Hamilton s equations of motion, that is, Liouville s theorem. It is important to note that the Poincare integral invariants are also preserved under a canonical transformation of any kind and not just the propagation of Hamilton s equations. 

Consider a set of points f) in phase space with evolution associated to a differential equation z = /(z) described by the flow map f(S(0)) = > t). Liouville s theorem [ 16] states that the volume of such a set is invariant with respect to t if the divergence of / vanishes, i.e. 

To understand where Liouville s theorem comes from, recall that the variational equations of the last chapter are a system of ordinary differential equations for W t) =... 

Note that in the above proof the explicit form of the time evolution equation for is not used. Therefore the proof applies to a pure dynamical system which is described by the Liouville equation. The fluctuation dissipation theorem holds quite generedly in physiced systems near equilibrium. 

Statistical mechanics when based on Liouville s theorem yields a hierarchy of equations (BBGKY hierarchy) that makes use of the 5-particle distribution function /<)< giving the probability of finding s particles, i = 1... j, out of the N particles in the system in the positions ri r, and... 

Equation (10.24) is called the von Neumann equation after the mathematician John von Neumann, who originated the concept of the density matrix. It also is known as the Liouville equation because of its parallel to Liouville s classical statistical mechanical theorem on the density of dynamic variables in phase space. 

This results from the fact that the separation leads to a Sturm-Liouville equation for each coordinate, for which the theorem mentioned above is valid. Also in the cases where the boundary conditions do not require 4 to vanish, but only require that 4 remain finite, there are no significant changes. 

In the Hamiltonian formulation, the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. Considering an ensemble of initial conditions each representing a possible state of the system, we express the probability of a given ensemble or density distribution of system points... 

Let be a function of coordinates only, so a point in is specified by [3(A(ab - 1) - 1] coordinates and 3(A(ab - ) momenta. Identify the missing coordinate as the reaction coordinate s (so s becomes a coordinate normal to the hypersurface), and identify the momentum conjugate to s as p. Let C denote the [6(A(ab — 1) - 2]-dimensional hyperface in in which ps = 0. Assume that the % region of phase space is populated according to a Boltzmann equilibrium distribution then Liouville s theorem of classical statistical mechanics shows it will evolve into a Boltzmann equilibrium distribution at and hence also at C. Consider the one-way flux of this equilibrium ensemble of phase points through in the 5 —> P direction. This flux may be calculated quite generally, and using this calculation plus equation (2) yields... 

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. 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

This same equation has been independently derived by Caceres [90] using van Kampen s lemma [91] and the Bourret-Frisch-Pouquet theorem [92], while the theory adopted by Annunziato et al. [87] rests essentially on the Zwanzig approach of Section III, namely, a Liouville-like perspective. 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

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