Liouvilles equation

Liouville derived a remarkable equation that succinctly captures the dynamics of physical systems, describing the evolution of the probability density in time, both at and away from equilibrium. 

Consider an ensemble Af of points in phase space, each representing a system of N particles. As time evolves each microscopic state moves along a phase trajectory according to the equations of motion. The 

Let V be an arbitrary 67/-dimensional closed domain in F (Fig. 4.7). Consider S as the (6N - 1)-dimensional surface of this domain. Define v(p, ( ) to be the vector locally normal to S at = (p, q). Thinking of spac points as fluid particles, the phase space density is now a function of both position and time, p = 0- 

The continuity principle holds in this case, as there is no creation or destruction of systems. The implication is that, when considering a closed phase space domain V, the net number of points leaving S per unit time equals the rate of decrease of points in volume V. This continuity principle can be expressed as follows  

According to the divergence theorem, the left-hand side in Eq. 4.85 

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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... 

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... 

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. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

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 ... 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

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. 

The evolution of an isolated system is then given by the classical and quantum Liouville equations for the fine-grained distribution functions (i.e., the evolution is entropy-preserving) ... 

Brown, D. W., Lindenberg, K. and West, B. J. Energy transfer in condensed media. II. Comparison of stochastic Liouville equations, J.Chem.Phys., 83 (1985), 4136-4143... 

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. 

Of course, as was shown in Section V-A, this latter expression may also be derived starting from the hydrodynamical equations for the pair distribution and the Poisson equation it is also the final result of the theories developed independently by Falken-hagen and Ebeling,9 and by Friedman 12-13 in these two approaches, the starting point is a Liouville equation for the system of ions with an ad hoc stochastic term describing the interactions with the solvent. 

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 ... 

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