Equilibrium Time Correlation Functions

Geissler, P. L. Dellago, C., Equilibrium time correlation functions from irreversible transformations in trajectory space, J. Phys. Chem. B 2004,108, 6667-6672... 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

The focus of this chapter is exploration of the ability of mixed quantum classical approaches to capture the effects of interference and coherence in the approximate dynamics used in these different mixed quantum classical methods. As outlined below, the expectation values of computed observables are fundamentally non-equilibrium properties that are not expressible as equilibrium time correlation functions. Thus, the chapter explores the relationship between the approximations to the quantum dynamics made in these different approaches that attempt to capture quantum coherence. 

We conclude this section with some brief comments about microscopic dynamics at liquid interfaces. Molecular dynamic simulations of the dynamic properties of liquid interfaces have been limited to the calculation of equilibrium time correlation functions. The methodology of these calculations has been discussed earlier. One property that has received much attention is the molecular reorientation correlation function. If e(r) is a unit vector fixed in the molecular frame, the nth order time correlation function is defined by... 

The same dynamical system can also be used to generate an equilibrium time correlation function C b( ) = ( 4(0)B(t)). Since any physical observable can be shown to evolve according to... 

The ability to generate an equilibrium ensemble from a dynamical trajectory has a number of useful features. One can obtain not only ordinary static equilibrium properties from Eq. [25], but also dynamical information. In fact, dynamical information is available on two levels. On the one hand, equilibrium time correlation functions can be calculated, leading to the prediction of vibrational spectra, transport coefficients and so on. On the other, the trajectory allows access to the microscopic detailed motion of individual atoms. Therefore, one can, in a sense, visualize at an atomistic level the dynamical behavior of the system as a function of time, which can lead to valuable insights about chemical reaction mechanics, structural rearrangements, and other details of the system that can be captured only by visualization at this level of detail. 

The matrix equation for the Laplace transform fllxl (k, z) of the equilibrium time correlation functions flxl(k, t), where... 

The linearized transport equations (7), the equations for the equilibrium time correlation functions (13), and the equation for collective mode spectrum (14) form a general basis for the study of the dynamic behavior of a multicomponent fluid in the memory function formalism. 

In order to draw the main ideas of the GCM approach let us recall a general representation for the Laplace transform of an equilibrium time correlation function = (A t) A k(0)), derived by Mori [39],... 

Equilibrium time correlation function expressions for transport properties can be derived using linear response theory [3]. Linear response theory can be carried out directly on the Wigner transformed equations of motion to obtain the transport properties as correlation functions involving Wigner transformed quantities. Alternatively, we may carry out the linear response analysis in terms of abstract operators and insert the Wigner representation of operators in the final form for the correlation function. We use the latter route here. 

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

The phase space trajectory r (Z), p (Z) is uniquely determined by the initial conditions r (Z = 0) = r p (Z = 0) = p. There are therefore no probabilistic issues in the time evolution from Z = 0 to Z. The only uncertainty stems from the fact that our knowledge of the initial condition is probabilistic in nature. The phase space definition of the equilibrium time correlation function is therefore. 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

In the next four sections, we discuss the four principal types of application of molecular dynamics. Section 3 very briefly describes the problem of the approach to equilibrium. Section 4 deals with the evaluation of equilibrium thermodynamic functions through a discussion of the dynamical equation of state. In Section 5, we consider the evaluation of equilibrium time correlation functions, detailing the application of the combined Monte Carlo-molecular-dynamics method to the time correlation functions for self-diffusion. Section 6 deals with nonequilibrium molecular dynamics and in particular with a calculation for self-diffusion. 

First, recall that because of the stationarity of an equilibrium time correlation function... 

After the particle s initial momentum (p(0)) has come to thermal equilibrium, there will be spontaneous equilibrium momentum fluctuations. Such fluctuations are described by the equilibrium time correlation function... 

We can also examine spontaneous equilibrium fluctuations by working in the fiequency domain and considering the spectral density (Fourier-Laplace transform) of equilibrium time correlation functions. For the case of momentum fluctuations, we write... 

As in classical Brownian motion theoiy, the damping in Eq. (492), characterized by the damping constants Kjk, arises from spontaneous equilibrium fluctuations described by equilibrium time correlation functions. The oscillatory motion of the system is characterized by the frequencies S2y. The influence of microscopic interactions on the overall time evolution of the displacements from equihbrium is buried in the quantities 2y and [Kjk] or equivalently A/fy., A/ y., and Xjk. ... 

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