Liouville equation, time evolution

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. 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

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. 

In general, the time evolution of the molecular system is governed by the Liouville-von Neumann equation... 

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... 

Kadanoff and Swift have considered that the time evolution of the state is described by the Liouville equation. They also wrote down conservation equations for the number, momentum, and energy density similar to the ones given by Eqs. (1)—(3). The only difference was that in the treatment of Kadanoff and Swift the densities and currents are operators. The time derivative of the densities are replaced by the commutator of the respective density operator and the Liouville operator, L (as the Liouville operator governs the time evolution). The suffix op in the following equations stands for operator ... 

Zwanzig showed how a powerful but simple technique, known as the projection operator technique, can be used to derive the relaxation equations [12]. Let us consider a vector A(t), which represents an arbitrary property of the system. Since the time evolution of the system is given by the Liouville operator, the time evolution of the vector can be written as /4(q, t) = e,uA q), where A is the initial value. A projection operator P is defined such that it projects an arbitrary vector on A(q). P can be written as... 

So far we have treated only the case of wave packets constructed from pure states. Consider now the control of a molecular system in a mixed state in which the initial states are distributed at a finite temperature. The time evolution of the system density operator pit) is determined by the Liouville equation,... 

Instead of (r,v) we now take the distribution function n(r,v) as the state variable. The time evolution of n r,v) is governed by the Liouville equation corresponding to (72). Can the Liouville equation be cast into the form... 

It is easy to verify that time evolution (74) is the Liouville equation corresponding to the particle time evolution governed by (72). 

The macroscopic system under consideration is regarded as being composed of np 1023 particles. The state variables describing it are (26). Their time evolution is governed by (72) extended in an obvious way from one to np particles. If we pass from (26) to the state variable (27), the time evolution (72) transforms into the Liouville equation corresponding to (72), i.e., to Equation (74) with n replaced by (27) and the Poisson bracket that extends (75) in an obvious way to np particles (i.e.,... 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

Evolution of the system configuration in time is governed by the Liouville equation, which is, actually, a 67V-analogue of the continuity equation. This equation may be written in the following form... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

The crisis of the GME method is closely related to the crisis in the density matrix approach to wave-function collapse. We shall see that in the Poisson case the processes making the statistical density matrix become diagonal in the basis set of the measured variable and can be safely interpreted as generators of wave function collapse, thereby justifying the widely accepted conviction that quantum mechanics does not need either correction or generalization. In the non-Poisson case, this equivalence is lost, and, while the CTRW perspective yields correct results, no theoretical tool, based on density, exists yet to make the time evolution of a contracted Liouville equation, classical or quantum, reproduce them. 

Let us make a final comment, concerning the violation of the Green-Kubo relation. There is a close connection between the breakdown of this fundamental prescription of nonequilibrium statistical physics and the breakdown of the agreement between the density and trajectory approach. We have seen that the CTRW theory, which rests on trajectories undergoing abrupt and unpredictable jumps, establishes the pdf time evolution on the basis of v /(f), whereas the density approach to GME, resting on the Liouville equation, either classical or quantum, and on the convenient contraction over the irrelevant degrees of freedom, eventually establishes the pdf time evolution on the basis of a correlation function, the correlation function in the dynamical case... 

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]. 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

Photoinduced charge separation processes in the supramolecular triad systems D -A-A, D -A -A and D -A-A have been investigated using three potential energy surfaces and two reaction coordinates by the stochastic Liouville equation to describe their time evolution. A comparison has l n made between the predictions of this model and results involving charge separation obtained experimentally from bacterial photosynthetic reaction centres. Nitrite anion has been photoreduced to ammonia in aqueous media using [Ni(teta)] " and [Ru(bpy)3] adsorbed on a Nafion membrane. 

The time evolution of this phase space probability density is governed by the Liouville equation expressed as... 

Here we describe an alternative derivation of the Liouville equation (1.104) for the time evolution of the phase space distribution function f (r, p t). The derivation below is based on two observations First, a change inf reflects only the change in positions and momenta of particles in the system, that is, of motion of phase points in phase space, and second, that phase points are conserved, neither created nor destroyed. 

THE QUANTUM MECHANICAL DENSITY OPERATOR AND ITS TIME EVOLUTION QUANTUM DYNAMICS USING THE QUANTUM LIOUVILLE EQUATION... 

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (r (f), p (f)) for a given initial condition (r (0), p (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1,2,2)) that describes the time evolution of the phase space probability density p f). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1-5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. 

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