Time evolution equations

The goal of kinetic theory is to find the time-evolution equation for f x,v,t), with the ultimate aim being to obtain the equilibrium properties of the system through the limiting form of f x, v,t) as t cx>. 

However, some problems remain unsolved. The three- and four-body results of Choh and Uhlenbeck and of Cohen, respectively, have to be compared with the corresponding expressions in the Prigogine s theory. Furthermore, for any concentration, one has to see how the systematic generalization of the Boltzmann equation derived by Cohen is related to the long-time evolution equation in Prigogine s theory. The aim of this work is to throw some light on these points. 

The time-evolution equation for the spherically integrated scalar-variance spectrum Eaa(K, t) obtained from (A.l) can be written as... 

The banded coherency spectrum has an analogous definition (Yeung and Pope 1993) where the energy spectra are replaced with the spectra integrated over a finite wavenumber band. From the spectral time-evolution equations, it is easily shown that, in the absence of mean scalar gradients, the time evolution of the coherency spectrum is governed by 1 dPap Tap 1 Taa 1 Tpp ... 

Integrating (A.2) over each wavenumber band yields the exact time-evolution equation... 

If the time evolution Equations (50) with the contact Hamiltonian (51) are restricted to the invariant manifold MN then they become... 

Summing up, we have arrived at the mesoscopic time evolution Equation (55) by extending the geometrical structure of equilibrium thermodynamics to the time evolution. A few observations are now in order ... 

There is also another way the mesoscopic time evolution Equation (55) can be introduced. We collect a list of well-established (i.e., well tested with experimental observations) time evolution equations on many different levels of description and try to identify their common features. This is indeed the way the time evolution Equation (55) has been first introduced. The Hamiltonian structure of the nondissipative part has been discovered first in the context of hydrodynamics by Clebsch (1895). Equations of the type (55) have started to appear in Dzyaloshinskii and Volovick (1980) and later in... 

The mass action law corresponding to the reaction (44) leads to the following time evolution equations ... 

The computations involved in the passage from (94) and (89) to the time evolution equations are straightforward. Here we shall only indicate the calculations. We note, as we already did in Section 3.1.3 (see the text following Equation (76)), that the equation x = L(px can also be written as A = A, required to hold for all sufficiently regular functions A(x). This formulation provides a systematic method for passing from the bracket to the time evolution equation. We shall illustrate it below on the example of the bracket (89), but the method can obviously be used for any bracket. We proceed as follows (i) we replace in (89) B by [Pg.109]

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]... 

Thus, the time evolution equation (3.26) can be replaced by a proper combi-... 

The perturbation reduction of the corresponding Markovian Fokker-Planck equation for the two-variable process (x t), ((t)) to an approximate one in x(t) has been carried out in Section V.A of Chapter II. For brevity we report only the approximate time-evolution equation for a x,t) up to order D-,... 

The time-evolution equations of the stability matrix M(t, 0) are the variational equation of Eq. (1), given by... 

Writing down the time evolution equation for the Wigner function, we find from the Master Equation (26)... 

This is the memory-less exact time evolution equation. In the limit when the Markovian property is expected to hold, we should obtain by comparison with (64) the result... 

Consider again the time evolution equations (2.1)—(2.3). If A is an operator representing a physical observable, the expectation value of this observable at time t is A t = ( h G) 1 1 1 (t))- We can express this same quantity differently. Define... 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators ... 

The time evolution equations in the interaction representation are easily derived from these definitions... 

Time evolution equations for reduced density operators ... 

It is actually simple to find a formal time evolution equation in P space. This formal simplicity stems from the fact that the fundamental equations of quantum dynamics, the time-dependent Schrodinger equation or the Liouville equation, are linear. Starting from the quantum Liouville equation (10.8) forthe overall system— system and bath. 

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