The Liouville equation

Finally, there is the change of density due to changes only in the velocity of one or more particles as a consequence of acceleration. Providing there are no external forces, the acceleration is due to forces between particles (e.g. a and 6) which, for convenience, will be considered as only pairwise additive. If the interaction energy is U and the particle mass is m, the acceleration is l/m)Vrf7(ra6) when the particles are separated by rab. This force causes the velocity of both particles a and b to change, so that the 

Adding all these three components together gives 9 at n i 

This is the Liouville equation and it states that the total time rate of change of the density, pN, is zero. There are many excellent discussions of this equation and its properties. Those of Resibois and De Leener [490], Rice and Gray [513], Zwanzig [540] and Forster [453] are as comprehensive and readable as any. 

Inspection of eqn. (291) reveals the immense task involved in solving 

One way around the difficulty with the collision term of eqn. (291) is to restrict attention to hard spheres, which only interact with one another when they axe in contact. In essence, the range of the interaction has been reduced almost to zero and the time of collision is then infinitesimally short. This has another very useful consequence. The probability of two collisions occurring simultaneously is negligible. Events just before and just after the collision can be described without reference to any other collision. A collision between particles a and b alters the velocity and position of both particles before and after the collision, but all other 

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A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

Using these vectors, we can rewrite the Liouville equation for the two-level system as... 

The Liouville equation dictates how the classical statistical mechanical distribution fiinction t)... 

The Liouville equation applies to any ensemble, equilibrium or not. Equilibrium means that should be... 

The idea is now to replace the formal solution of the Liouville equation by the discretized version. The middle term gf the propagator in Eq. (51) can be further decomposed by an additional Trotter factorization to obtain... 

Quantum statistical mechanics with the concepts of mixed states, density operators and the Liouville equation. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

In this equation L>[ is the number of rotation operators in the set. Equation (15) is the MPC analogue of the Liouville equation for a system obeying Newtonian dynamics. 

In the previous section we discussed the effective Hamiltonian method a main feature of this method is that it results in the appearance of damping operator T in the Liouville equation. However, the damping operator is introduced in an ad hoc manner. In this section we shall show that the damping operator results from the interaction between the system and heat bath. 

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

To calculate x( ) we have to calculate the polarizability P(t), which is related to the reduced density matrix p(f). [Here, for convenience, p is used instead of cr(f).] The reduced density matrix satisfies the Liouville equation ... 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

After these technical preliminaries, we may consider the solution of the Liouville equation (10). However, we shall not discuss the most general situation but we shall limit ourselves to the special case where ... 

In order to make clear that this theory is not derived from the Liouville equation, we use here a notation different from the rest of the paper. 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

The distribution function p,v(x1.. . xN, t) in the 6AT-dimen-sional phase space represents the probability of finding particle i with phase x ----- rf, p4, particle / with phase x, = r, etc.,.. . at the instant t. This function obeys the Liouville equation... 

The Liouville equation can be integrated over all the phases Xf — Tj, p except those of the sub-group of particles 1,2,.. ., s. One obtains then ... 

The equation (6) for s = 1 connects fx to /2, which is itself connected to /3. The ensemble of equations (6) constitutes the hierarchy derived independently by Bogolubov, Bom, Green, Kirkwood, and Yvon. This hierarchy is equivalent to the Liouville equation and to try to solve it is equivalent to studying the trajectories of 1023 particles whose phases at the initial instant are known. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

The method developed by I. Prigogine and his collaborators (see, for example, ref. 14) is a perturbation calculation to study the iV-body problem. The point of departure is the Liouville equation (2) of which one looks for the solution in the limit of a large system ... 

On the other hand, the solution of the Liouville equation (2) is written formally ... 

Taking the logarithmic derivative of (A.36) with respect to r, multiplying on both sides by <0 S T 0>, and using finally the Liouville equation (2), we have (see A.33) ... 

Starting from the Liouville equation as the fundamental microscopic evolution equation for the dynamics of all phase-space variables, MCT uses... 

The dynamics of classical as well as of quantum systems can be described by a Liouville equation for the time-dependent density. In quantum mechanics, the Liouville equation for the density operator p(t) reads... 

The standard proposal for the Liouville equation for this QC density operator is... 

The research was greatly facilitated by two important elements. The (formal, perturbative) solution of the Liouville equation is greatly simplified by a Fourier representation (see Appendix). The latter allows one to easily identify the various types of statistical correlations between the particles. The traditional dynamics thus becomes a dynamics of correlations. The latter is completed by... 

The starting point of classical statistical mechanics is the exact equation of evolution of the distribution function p in phase space the Liouville equation, which Prigogine always wrote in the form... 

The authors then ask the following question Do there exist deterministic dynamical systems that are, in a precise sense, equivalent to a monotonous Markov process The question can be reformulated in a more operational way as follows Does there exist a similarity transformation A which, when applied to a distribution function p, solution of the Liouville equation, transforms the latter into a function p that can also be interpreted as a distribution function (probability density) and whose evolution is governed by a monotonous Markov process An affirmative answer to this question requires the following conditions on A (MFC) ... 

Upon trying to solve the equations of motion (or the Liouville equation) by a series in powers of the interaction potential, it is found that all terms contain factors or products of factors of the form [X) > where G>k J)... 

In short, the distributivity of the transformation f/t implies that retains the reducibility of the Liouville equation into a pair of Schrodinger equations. Furthermore, this transformation retains the time-reversal invariance of these equations, since the free-motion equations [Eqs. (15)] are time-reversal invariant. 

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