Validating and implementing modified Filinov phase filtration in semiclassical dynamics.

The Schwinger-Keldysh diagram technique is usually involved in the calculation of real-time in-in correlation functions. In the case of a thermal state, one can analytically continue imaginary-time Matsubara correlation functions to real times. Nevertheless, not all real-time correlation functions usually can be obtained by such procedure. Moreover, numerical analytic continuation is an ill-posed problem. Thus, even in the case of a thermal state one may need for the Schwinger-Keldysh formalism. If the potential of a system admits degenerate minima, instantonic effects enter the game, so one should also integrate over the instantonic moduli space, including the one, corresponding to the imaginary time translational invariance. However, the Schwinger-Keldysh closed time contour explicitly breaks such invariance. We argue, that this invariance must be recovered, and show, how it can be done. After that, we construct an extension of the Schwinger-Keldysh diagram technique to instantonic systems and demonstrate it on the example of the first few-point correlation functions.

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time betavariant Planck's over 2pi tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where betavariant Planck's over 2pi exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the "nonlinear operator problem" of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.

Computing real time correlation functions on a hybrid classical/quantum computer

Within the framework of probability distributions on projective Hilbert space a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure or mixed states. It is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables the determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard et al. [Phys. Rev. Lett. {\bf 68}, 580 (1992)]. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.

On the calculation of correlation functions in nuclear matter

A method for the quantification of synchrony and oscillatory properties of neuronal activity

The viscoelasticity of a stretched polymer chain with its end particles exposed to oppositely acting forces is studied via collisional molecular dynamics and analytically. A simple model according to which a polymer molecule is a chain of particles linked through freely jointed elastic bonds is adopted. The analytical theory is in good agreement with the results of the computer simulation of time correlation functions in the range of large-scale motions of a polymer molecule. It is found that the decay of correlation functions K αβμν of fluctuations of the microscopic stress tensor of a chain, K zzzz, K zαzα , = K zαzα , (α = x, y; z is the axis along which the forces act), is slowed down, and their value increases relative to the respective correlation functions of a chain with fixed ends. The greater the force, the higher this difference. The correlation functions that are transverse with respect to the z axis do not differ from those for chains with fixed ends. The results show that, in the calculation of time correlation functions of strongly stretched polymer chains, different statistical ensembles are not equivalent; this must be taken into account in the dynamic theories of heavily deformed polymers.

Forward-backward semiclassical dynamics (FBSD) provides a rigorous and powerful methodology for calculating time correlation functions in condensed phase systems characterized by substantial quantum mechanical effects associated with zero-point motion, quantum dispersion, or identical particle exchange symmetries. The efficiency of these simulations arises from the use of classical trajectories to capture all dynamical information. However, full quantization of the density operator makes these calculations rather expensive compared to fully classical molecular dynamics simulations. This article discusses the convergence properties of various correlation functions and introduces an optimal Monte Carlo sampling scheme that leads to a significant reduction of statistical error. A simple and efficient procedure for normalizing the FBSD results is also discussed. Illustrative examples on model systems are presented.

We present a PC-based multi-tau software correlator suitable for processing dynamic light-scattering data. The correlator is based on a simple algorithm that was developed with the graphical programming language LabVIEW, according to which the incoming data are processed on line without any storage on the hard disk. By use of a standard photon-counting unit, a National Instruments Model 6602-PCI timer-counter, and a 550-MHz Pentium III personal computer, correlation functions can be worked out in full real-time over time scales of ~5 mus and in batch processing down to time scales of ~300 ns. The latter limit is imposed by the speed of data transfer between the counter and the PC's memory and thus is prone to be progressively reduced with future technological development. Testing of the correlator and evaluation of its performances were carried out by use of dilute solutions of calibrated polystyrene spheres. Our results indicate that the correlation functions are determined with such precision that the corresponding particle diameters can be recovered to within an accuracy of a few percent rms.

Multiscaling is shown to be a consequence of multifractality when a lower cutoff \ensuremath{\varepsilon} is introduced in calculations of correlation functions. After a suitable rescaling, the correlation function data for different values of \ensuremath{\varepsilon} seem to fall onto a single curve. In the multiscaling regime, however, we show that there is not a unique functional form at varying \ensuremath{\varepsilon}, but a spread very close to a single curve. For each \ensuremath{\varepsilon}, this curve can be computed analytically in terms of the f(\ensuremath{\alpha}) spectrum which characterizes the multifractal. Part of this spectrum can thus be obtained by computing only one moment of the weights at \ensuremath{\varepsilon}\ensuremath{\ne}0.

A theory of magnetic nuclear relaxation, providing a calculation of the correlation functions of complex motion of methyl groups is presented. The complex motion consists of jumps over the barrier (classical motion) and jumps through the barrier (tunneling). The Schrödinger equation has been applied in the calculation of the rate constant of tunneling jumps through the barrier. The equations for the spectral densities (ω), where ω = (ωI ± ωT), and ω = (mωI), where m = 1 or 2, ωI is the resonance angular frequency, and ωT is the angular frequency of the tunnel splitting, are derived. These spectral densities concern the motion of spin pairs inside methyl groups ("intra") and outside methyl groups ("inter"). The calculated spectral densities are applied to analyze the temperature dependencies of the spin−lattice relaxation rate in solids containing methyl groups. A wide regime of temperatures from 0 K up to the melting point is considered. The temperature at which the tunneling process ceases is discussed. The theory proposed explains the different temperature dependencies of T1 for CH3COOK obtained in the experiments caused by small amounts of CH3COOH or water impurities. The theoretical equations derived in this paper are compared to those known in the literature.

Depolarized Rayleigh wing scattering data are reported for liquid chloroform, methyl iodide and methylene chloride. The calculation of orientation correlation functions from these results and from infrared vibration-rotation bands is discussed. Fourier transforms of the infrared bands are corrected for shape changes due to variable refractive index, isotope splitting and hot bands. The resulting correlation functions are primarily a measure of orientational and collision-induced changes in polarizability or dipole moment. It is argued that the collision-induced contributions are important only at short times. The long-time behavior of the correlation functions is then compared with several models for molecular reorientation in liquids, and it is concluded that the motion of these molecules is probably best characterized by models involving random walks in angular momentum space.

A discussion of the various properties of pair correlation functions of a magnetic electron gas is presented. The cusp conditions obeyed by the many-electron wave functions for a pair of electrons with parallel and antiparallel spins in close proximity are expressed as derivative conditions on the corresponding parallel-and antiparallel-spin pair correlation functions. In particular, it is shown that in a fully ferromagnetic gas the parallel spin correlations determine the correlation energy whereas in the paramagnetic case, this function is of less importance in comparison to the antiparallel-spin correlation function. In the high-momentum limit, it is found that the momentum distribution of electrons is dominated by antiparallel-spin correlations in the partially magnetic case whereas in the fully ferromagnetic case, it is dominated by parallel-spin correlations but is smaller by a factor of ${q}^{\ensuremath{-}2}$. A calculation of the parallel-spin correlation function for the ferromagnetic gas in perturbation theory is presented. A model for the exchange correlation energy of the system which takes into account the exact conditions given here is proposed. Corresponding results for the two-dimensional electron gas are also stated.