Statistical mechanics Liouville equation

The time evolution of the dynamical variables is controlled by the Liouvillian superoperator it. Let A = (A, A, ...) be a row vector gathering the dynamical variables of our system. According to classical statistical mechanics, the equation of motion is given by the Liouville equation... 

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

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

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. 

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 interest of Eq. (26) is in its striking analogy with the Liouville equation of classical statistical mechanics. The variables v play the role... 

The familiar Liouville equation of statistical mechanics is a special case, in which the flow is incompressible, i.e., the divergence dFv/duv vanishes, so that the factor Fv in (5.2) may be written in front of 0/fluv. This restriction was not made, however, by J. Liouville, J. Mathem. Pures Appl. 3, 342 (1838). 

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. 

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

An equation like Eq. (11-25) occurs in classical statistical mechanics, where it is called the Liouville equation. ... 

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. 

While this compact notation is convenient for the experienced user, to grasp the basic concepts a rough description may be preferable. Therefore, for the beginners that is not familiar with the concepts of statistical mechanics we reiterate the presentation of the abstract theory using an alternative notation equivalent to what is common in the fluid dynamic literature [40] [61]. The purpose is to provide introductory ideas about the practical implications of the Liouville equations describing the ensemble flow in P-space. 

Although Boltzmann may have supposed at one time that his derivation of the transport equation was based on purely mechanical arguments, it quickly became clear that this was not so, and that he had used in his derivation an assumption that is not in accord with mechanics. This assumption, the Stoss-zahlansatz, accounted for the monotonic approach of the gas to equilibrium predicted by the Boltzmann equation. " It could be justified only by supposing that the Boltzmann equation itself is the result of a more fundamental statistical-mechanical theory of gases, based on the Liouville equation. 

In order to answer these questions, we must turn to the basic equation of nonequilibrium statistical mechanics, the Liouville equation. Therefore, we no longer consider one particular container of a gas, but we consider an ensemble of similarly preparedt containers. We construct the 2Nd-dimensional phase space of positions and momenta of the N-particles in... 

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... 

The merger of probability theory and classical mechanics is accomplished by the so-called Liouville equation, which is considered to he the fiindamental equation of statistical mechanics. From this equation we can obtain a comprehensive description of both the equilibrium and nonequilibrium behavior of matter. In this chapter, we will derive the famous Liouville equation from a simple differential mass balance approach. In this case, the mass will represent a S3rstem of points in a multidimensional space. Each point contains all the information about the system at a particular time. The Liouville equation obtained here will be called upon in each of the subsequent chapters in our quest to describe the observed behavior and properties of any particular system. 

D. A. McQuarrie, Statistical Mechanics, Harper Row, New York, 1976. [Chapter 7 treats the Liouville equation.]... 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

The time-dependent classical statistical mechanics of systems of simple molecules is reviewed. The Liouville equation is derived the relationship between the generalized susceptibility and time-correlation function of molecular variables is obtained and a derivation of the generalized Langevin equation from the Liouville equation is given. The G.L.E. is then simplified and/or approximated by introducing physical assumptions that are appropriate to the problem of rotational motion in a dense fluid. Finally, the well-known expressions for spectral intensity of infrared and Raman vibration-rotation bands are reformulated in terms of time correlation functions. As an illustration, a brief discussion of the application of these results to the analysis of spectral data for liquid benzene is presented. 

The Liouville equation is the basic equation of non-equilibrium statistical mechanics. It gives the time dependence of the N-particle distribution function f (X> )> derived... 

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. 

The starting point of the statistical mechanics is the Liouville equation, named after French mathematician Joseph Liouville. It describes the time evolution of the phase space function as... 

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... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

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