Equilibrium Correlation Functions

L 1967. Computer Experiments on Classical Fluids. II. Equilibrium Correlation Functions. tysical Review 165 201-204. 

This contrasts with relation (5.16), which led to a non-physical conservation law for J. Eqs. (5.28) and Eq. (5.30) make it possible to calculate in the high-temperature limit the relaxation of both rotational energy and momentum, avoiding any difficulties peculiar to EFA. In the next section we will find their equilibrium correlation functions and determine corresponding correlation times. 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top. instead of at reactants or products. Computer time is not wasted by waiting for the particle to escape from the well to the barrier. The method is based on the validity of Onsager s regression hypothesis,97 98 which assures that fluctuations about the equilibrium state decay on the average with the same rate as macroscopic deviations from equilibrium. It is sufficient to know the decay rate of equilibrium correlation functions. There isn t any need to determine the decay rate of the macroscopic population as in the previous subsection. 

Another way of obtaining the characteristic time scale and dynamical range of conformational dynamics is from the equilibrium correlation functions of the FRET efficiency ... 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

Let us note that formula (4.33) is a generalisation of the equilibrium correlation function of the normal co-ordinates of the macromolecule in a viscous liquid... 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

Generally speaking, the equilibrium FDT establishes a link between the dissipative part tBA(t. tr) of the linear response function %BA(t, 0 and the symmetrized equilibrium correlation function CBa MO = ([A(f ),B(f)]+) (or the derivative dCBA(t, t )jet with t < t the earlier time). 

Consider two quantum-mechanical observables A and B with thermal equilibrium correlation functions verifying the Kubo-Martin-Schwinger condition [35], that is,... 

Correspondingly, the correlation function Cvv(ti,t2) becomes the equilibrium correlation function... 

The velocity, which equilibrates at large times, is not an aging variable. Thus, Fourier analysis and the Wiener-Khintchine theorem can be used equivalently to obtain the equilibrium correlation function Cvv(t — t2). 

The evaluation of the equilibrium correlation function can be made without a problem. In, fact, adopting a quantum-like formalism we can write, for t2 > t, ... 

It seems that the conventional approach to the quantum mechanical master equation relies on the equilibrium correlation function. Thus the CTRW method used by the authors of Ref. 105, yielding time-convoluted forms of GME [96], can be made compatible with the GME derived from the adoption of the projection approach of Section III only when p > 2. The derivation of this form of GME, within the context of measurement processes, was discussed in Ref. 155. The authors of Ref. 155 studied the relaxation process of the measurement pointer itself, described by the 1/2-spin operator Ez. The pointer interacts with another 1/2-spin operator, called av, through the interaction Hamiltonian... 

If H (a,b) as provided by Eq. (3.79) is positive definite, we shall not find any theoretical difficulty in proposing the CFP as a calculation technique for this nonequilibrium process. Throughout this volume we shall consider stochastic variables A with vanishing mean value at equilibrium, that is, (A) = 0. The correlation function t) = A A t) / A A then shares the standard formal properties of the normalized equilibrium correlation functions ... 

Due to the way we followed to arrive at Eq. (5.59), the elTective damping at 1 = 0 [i.e., virtually that of Eq. (5.44)] is the result of a sort of coarse-grained measurement on the short time region of the corresponding equilibrium correlation function. In accordance with the linear response theory,(y(l)) as given by Eq. (5.59) should tend to coincide with the corresponding equilibrium correlation function as (v )cxc... 

This means that in the linear Markovian case, excitation of whatever intensity has no effect on the decay of the system, which is proved to exhibit the same decay as the equilibrium correlation function. This is in line with the theoretical remarks made above. 

In the nonlinear case, the function ij t does not vanish (at times intermediate between t = 0 and t = oo). The itinerant oscillator with an effective potential harder than the linear one is shown to result in ri(t) < 0 in accordance with the results of CFP calculations which show its decay after excitation to be faster than the corresponding equilibrium correlation function (see Fig. 11). 

Figure 18. Computer simulation of the effects of excitation on the decay of the angular velocity u. Curve 1 denotes the equilibrium correlation function Curves 2... 

Verlet L. Computer Experiments on classical fluids, n. Equilibrium correlation functions. Phys. Rev. 1968 165 201-214. 

The quantum mechanical analog of the equilibrium correlation function (6.6) is... 

Problem 8.2. Calculate the equilibrium correlation function (x(Z)x(O)) for a system undergoing the Orenstein-Uhlenbeck process. 

The second equality in (11.22) follows from the symmetry property (6.32). We have found that the dynamic susceptibility is again given in terms of equilibrium correlation functions, in this case time correlation functions involving one of the variables orB and the time derivative ofthe other. Note (c.f. Eq (1.99)) that if A is a dynamical variable, that is, a function of (r, p ) so is its time derivative. 

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