Short-time Expansion Research Articles

Universality of decoherence for macroscopic quantum superpositions

We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian Hsys of the isolated system. This interaction-dominated decoherence limit is of importance for the emergence of classical behavior in the macroscopic domain, since it will always be the relevant regime for large enough separations between the superposed wave packets. The usual golden-rule treatment then does not apply, but we can employ a short-time expansion for the free motion while keeping the interaction Hint in full. We thus reveal decoherence as a universal short-time phenomenon largely independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. Simple analytical expressions for the decoherence time scales are obtained in the limit in which decoherence is even faster than any time scale emerging from the reservoir Hamiltonian Hres .

Heat trace asymptotics defined by transfer boundary conditions

We compute the first five terms in the short-time heat trace asymptotics expansion for an operator of the Laplace type with transfer boundary conditions using the functorial properties of these invariants.

Memory function approach to electronic diffusion in two-dimensional electron systems

Based on an expansion of the memory function, we propose an analytical approach to analyze the role played by temperature, electron effective mass, and background dielectric constant on the diffusion constant of a two-dimensional electron gas in a uniform perpendicular external magnetic field. Using a short-time expansion of the Kubo-Greenwood formula, we are able to calculate the two-body and three-body effects from the corresponding correlation functions which are obtained from the hypernetted-chain integral equation theory and the Kirkwood superposition approximation, respectively. When our results are compared with already published molecular dynamics and Monte Carlo simulation results, the agreement is excellent. At low temperature, the diffusion constant first increases and then decreases as the magnetic field is increased but decreases monotonically with increasing magnetic field at high temperature. Our calculations show that the Lorentz force induced by an applied uniform magnetic field is enhanced by increasing the dielectric constant and decreasing the electron effective mass.

On the multiple time scales in solvation dynamics: A mode-coupling theory approach

A microscopic statistical mechanical theory of solvation dynamics is presented. The theory is capable of reproducing the characteristic multiphasic behavior of the solvation correlation function typically observed in the time-resolved spectroscopic studies. The fast initial decay of the solvation correlation function is modeled on the basis of a short-time expansion, while the slowly varying long-time tail is computed within the mode-coupling theory formalism. Comparison with simulation for nonpolar solvation shows that the present theory provides a very substantial improvement over the commonly used time-dependent density functional and surrogate Hamiltonian treatments of the slow component of solvation dynamics. In addition, the range of applicability of the results obtained in this study is much wider compared to the other theories. The mode-coupling theory of solvation is tested for model nonpolar systems consisting of both a neat fluid and a system, where the chromophore size is significantly different from that of the solvent particles. A systematic study of the dependence of the solvation time scales on the size of the chromophore is carried out.

The velocity autocorrelation function and self-diffusion coefficient of fluids with steeply repulsive potentials

We consider the velocity autocorrelation function, vacf, or Cv(t) and self-diffusion coefficients, D, of steeply repulsive inverse power fluids (SRP) in which the particles interact with a pair potential, ø (r) = ε(σ/r)n. The Cv(t) are calculated numerically by molecular dynamics simulations. Accurate expressions for the short time expansion of Cv(t) to order O(t4) for n large are derived for this fluid. We propose novel expressions for Cv (t) that, for n large, spans the transition from the short time regime (expandable in even powers of time) and the longer time exponential-like regime characteristic of hard spheres. Inter alia we introduce relaxation times that characterize the duration of a collision and the decay of the velocity correlation within the mean-collision or Enskog-like relaxation time, TE.

Identification of determinism in noisy neuronal systems

Most neuronal ensembles are nonlinear excitable systems. Thus it is becoming common to apply principles derived from nonlinear dynamics to characterize neuronal systems. One important characterization is whether such systems contain deterministic behavior or are purely stochastic. Unfortunately, many methods used to make this distinction do not perform well when both determinism and high-amplitude noise are present which is often the case in physiological systems. Therefore, we propose two novel techniques for identifying determinism in experimental systems. The first, called short-time expansion analysis, examines the expansion rate of small groups of points in state space. The second, called state point forcing, derives from the technique of chaos control. The system state is forced onto a fixed point, and the subsequent behavior is analyzed. This technique can be used to verify the presence of fixed points (or unstable periodic orbits) and to assess stationarity. If these are present, it implies that the system contains determinism. We demonstrate the use and possible limitations of these two techniques in two systems: the Hénon map, a classic example of a chaotic system, and spontaneous epileptiform bursting in the rat hippocampal slice. Identifying the presence of determinism in a physiological system assists in the understanding of the system's dynamics and provides a mechanism for manipulating this behavior.

Importance sampling in rigid body diffusion Monte Carlo

We present an algorithm for rigid body diffusion Monte Carlo with importance sampling, which is based on a rigorous short-time expansion of the Green's function for rotational motion in three dimensions. We show that this short-time approximation provides correct sampling of the angular degrees of freedom, and provides a general way to incorporate importance sampling for all degrees of freedom. The full importance sampling algorithm significantly improves both calculational efficiency and accuracy of ground state properties, and allows rotational and bending excitations in molecular van der Waals clusters to be studied directly.

Effect of two-body and three-body correlations on the diffusion constant of two-dimensional Coulomb systems in a uniform magnetic field

We calculate the diffusion constant of a two-dimensional electron gas (2DEG) in a uniform perpendicular external magnetic field using the short-time expansion of the Kubo–Greenwood formula. We model the 2DEG as a one-component plasma. The two-body and three-body correlation functions are obtained from the hypernetted-chain integral equation theory and the Kirkwood superposition approximation, respectively. We compare the diffusion constant at two different temperatures. At the lower temperature, the diffusion constant increases as the magnetic field is increased but decreases with increasing magnetic field at the higher temperature. We compare our results when only two-body correlations are taken into account with those when both two-body and three-body correlations are included. These results may be interpreted as being due to the competition between the magnetic force and the force exerted on the target particle by the surrounding electrons in the 2DEG in the presence of a perpendicular magnetic field.

A Short-Time Quantum Mechanical Expansion Approach to Vibrational Relaxation

The vibrational relaxation rate of a “quantum” molecule embedded in a “quantum” host is approached from the perspective of a short-time expansion of the quantum force-force time correlation function. A general path integral Monte Carlo scheme is developed to compute the coefficients of the expansion, which involve the quantum thermal average of operators that depend on both position and momentum. A simple ansatz is used to connect the short-time and long-time behavior of the force-force correlation function. The method is tested on a simple model problem. Improvements of the approach and further applications are discussed.

On the dimensional reduction procedure

The issue related to the so-called dimensional reduction procedure is revisited within the Euclidean formalism. First, it is shown that for symmetric spaces, the local exact heat-kernel density is equal to the reduced one, once the harmonic sum has been succesfully performed. In the general case, due to the impossibility to deal with exact results, the short time heat-kernel asymptotics is considered. It is found that the exact heat-kernel and the dimensionally reduced one coincide up to two non trivial leading contributions in the short time expansion. Implications of these results with regard to dimensional-reduction anomaly are discussed.

An approximate short time Laplace transform inversion method

The “standard” numerical methods used for inverting the Laplace transform are based on a regularization of an exact inversion formula. They are very sensitive to noise in the Laplace transformed function. In this article we suggest a different strategy. The inversion formula we use is an approximate one, but it is stable with respect to noise. The new approximate expression is obtained from a short time expansion of the Bromwich inversion formula. We show that this approximate result can be significantly improved when iterated, while remaining stable with respect to noise. The iterated method is exact for the class of functions of type EmeaE. The method is applied to a harmonic model of the stilbene molecule, to a truncated exponent series, and to the flux–flux correlation function for the parabolic barrier. These examples demonstrate the utility of the method for application to problems of interest in molecular dynamics.

Nonlinear Stability, Thermoelastic Contact, and the Barber Condition

The behavior of a one-dimensional thermoelastic rod is modeled and analyzed. The rod is held fixed and at constant temperature at one end, while at the other end it is free to separate from or make contact with a rigid wall. At this free end we impose a pressure and gap-dependent thermal boundary condition. This condition, known as the Barber condition, couples the thermal and elastic problems. Such systems have previously been shown to undergo a bifurcation from a unique linearly stable steady-state solution to multiple steady-state solutions with alternating stability. Here, the system is studied using the asymptotic matching techniques of boundary layer theory to derive short-time, long-time, and uniform expansions. In this manner, the analysis is extended into the nonlinear regime and dynamic information about the history dependence and temporal evolution of the solution is obtained.

Short time expansion of the relative motion of molecularpairs. Application to water

A detailed analysis of the short time expansion of the relative mean squaredisplacement and mean square distance of pairs is performed for molecularfluids. A theoretical expression which relates the t2 coefficient of theexpansion to the local structure of the fluid is obtained. Good agreementbetween theoretical predictions and simulated data is found in the case ofsupercooled water.

Vibrational motions of hydrogen bonds from the short time relative pair dynamics

The short time dynamics of pairs of water molecules, initially lying in the first coordination shell, is investigated via molecular dynamics simulation. The introduction of the generalized time-dependent pair distribution function allows to obtain a relationship between the dynamics and the local structure. The relationship explains the different short time behaviors between the hydrogen-bonded molecules and the structural defects, and the lack of the free flight time dependence of the mean-square distance. The centers of mass vibrational motion of hydrogen-bonded molecules influences the relative pair dynamics beyond the short time expansion. An approach, based on the instantaneous normal modes theory, is proposed to derive the vibrational motion of the hydrogen bonds. Its general applicability is stressed and the particular relevance for studying systems whose dynamics is determined by strong oriented interactions is suggested.

Two-point spectral correlations for star graphs

The eigenvalues of the Schrodinger operator on a graph G are related via an exact trace formula to periodic orbits on G. This connection is used to calculate two-point spectral statistics for a particular family of graphs, called star graphs, in the limit as the number of edges tends to infinity. Combinatorial techniques are used to evaluate both the diagonal (same orbit) and off-diagonal (different orbit) contributions to the sum over pairs of orbits involved. In this way, a general formula is derived for terms in the (short-time) expansion of the form factor K() in powers of , and the first few are computed explicitly. The result demonstrates that K() is neither Poissonian nor random matrix, but an intermediate between the two. Off-diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients.

On the Diffusion-Convection-Reaction Equations

The expansion of moments technique for generating short-time expansions for the moments associated with the probability density p(x, t) is used to solve the diffusion-convection-reaction equations. The obtained results are compared with those found by other theories such as Lie group theory and operator technique.

Asymptotics of the heat kernel on rank-1 locally symmetric spaces

We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.

Stochastic diagrams for critical point spectra

A new technique for calculating the time-evolution, correlations and steady state spectra for nonlinear stochastic differential equations is presented. To illustrate the method, we consider examples involving cubic nonlinearities in an N-dimensional phase-space. These serve as a useful paradigm for describing critical point phase transitions in numerous equilibrium and non-equilibrium systems. The technique presented here is not perturbative. It consists in developing the stochastic variable as a power series in time, and using this to compute the short time expansion for the correlation functions. This, in turn, is extrapolated to large times and Fourier transformed to obtain the spectrum. A stochastic diagram technique is developed to facilitate computation of the coefficients of the relevant power series expansion. Two different types of long-time extrapolation technique, involving either simple exponentials or logarithmic rational approximations, are evaluated for third-order diagrams. The analytical results thus obtained are compared with numerical simulations, together with exact results available in special cases. The agreement is excellent in the neighborhood of the critical point. Finally, we emphasize that the technique is also applicable to more general stochastic problems involving spatial variation in addition to temporal variation.

Dynamics of atomic pairs in a Lennard-Jones fluid: Mean relative displacement analysis

We have analyzed the dynamics of atomic pairs in a simple fluid in terms of the mean relative displacement of two atoms with a given initial separation. The short-time expansion of the mean relative displacement has been derived exactly beyond the constant acceleration approximation (CAA) (up to the ${t}^{4}$ order). For a Lennard-Jones fluid, the mean relative displacements calculated by either the CAA or the ${t}^{4}$-order approximation have been compared with the results obtained from a molecular-dynamics simulation. Through studying the relative motions of atomic pairs with different initial separations, we give a physical picture of the dynamics of neighboring atoms beyond the CAA.

Immediate and early renal function after living donor transplantation.

In order to assess the immediate renal function after living donor transplantation, renal function was compared in eight renal allograft recipients and their living related kidney donors during the first 24 h after transplantation. Substantial and comparable intraoperative volume loading with Ringer's acetate and mannitol was performed together with the administration of frusemide. Glomerular filtration rate (GFR) and effective renal plasma flow (ERPF) were estimated by the clearances of inulin and p-aminohippurane, respectively. Tubular reabsorptive function and injury were estimated from the clearance of lithium, the fractional excretion of sodium and the urinary excretion of N-acetyl-beta-glucosaminidase. One hour after completion of surgery, GFR (54 +/- 7 ml/min) and ERPF (294 +/- 35 ml/min) were only 30% lower in the grafts than in the remaining donor kidneys, increasing to similar levels within 3 h. Only minor tubular dysfunction and injury were revealed in the grafted kidneys, and these tended to normalize within 24 h. By the present transplantation procedure comprising short ischaemia time and substantial volume expansion combined with mannitol and frusemide administration, kidneys from living donors regain nearly normal function within a few hours after transplantation.