Reconstruction of thermally-symmetrized quantum autocorrelation functions from imaginary-time data

In this paper, I propose a technique for recovering quantum dynamical information from imaginary-time data via the resolution of a one-dimensional Hamburger moment problem. It is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin. A general class of reconstruction algorithms is then identified, according to Theorem 3. The technique is advocated as especially effective for a certain class of quantum problems in continuum space, for which only a few moments are necessary. For such problems, it is argued that the derivatives at origin can be evaluated by Monte Carlo simulations via estimators of finite variances in the limit of an infinite number of path variables. Finally, a maximum entropy inversion algorithm for the Hamburger moment problem is utilized to compute the quantum rate of reaction for a one-dimensional symmetric Eckart barrier.Comment: 15 pages, no figures, to appear in Phys. Rev.

Superfluidity and Quantum Melting of para-Hydrogen clusters

Structural and superfluid properties of para-Hydrogen clusters of size up to N=40 molecules, are studied at low temperature (0.5 K < T < 4 K) by Path Integral Monte Carlo simulations. The superfluid fraction displays an interesting, non-monotonic behavior for 22 < N < 30. We interpret this dependence in terms of variations with N of the cluster structure. Superfluidity is observed at low T in clusters of as many as 27 molecules; in the temperature range considered here, quantum melting is observed in some clusters, which freeze at high temperature

Entropic effects in large-scale Monte Carlo simulations

The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Renyi divergence of order 1/2. We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for non-vanishing acceptance probabilities in the limit of large dimensions. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.Comment: minor corrections; the best compromise for the value of the epsilon parameter in Eq. A9 is now shown to be log(2); 13 pages, 4 figures, to appear in PR

Extrapolated High-Order Propagators for Path Integral Monte Carlo Simulations

We present a new class of high-order imaginary time propagators for path-integral Monte Carlo simulations by subtracting lower order propagators. By requiring all terms of the extrapolated propagator be sampled uniformly, the subtraction only affects the potential part of the path integral. The negligible violation of positivity of the resulting path integral at small time steps has no discernable affect on the accuracy of our method. Thus in principle arbitrarily high order algorithms can be devised for path-integral Monte Carlo simulations. We verify this claim is by showing that fourth, sixth, and eighth order convergence can indeed be achieved in solving for the ground state of strongly interacting quantum many-body systems such as bulk liquid $^4$He.Comment: 9 pages and 3 figures. Submitted to J. Chem. Phy

Structure of Si(114) determined by global optimization methods

In this article we report the results of global structural optimization of the Si(114) surface, which is a stable high-index orientation of silicon. We use two independent procedures recently developed for the determination of surface reconstructions, the parallel-tempering Monte Carlo method and the genetic algorithm. These procedures, coupled with the use of a highly-optimized interatomic potential for silicon, lead to finding a set of possible models for Si(114), whose energies are recalculated with ab-initio density functional methods. The most stable structure obtained here without experimental input coincides with the structure determined from scanning tunneling microscopy experiments and density functional calculations by Erwin, Baski and Whitman [Phys. Rev. Lett. 77, 687 (1996)].Comment: 19 pages, 5 figure

On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.

Feedback-optimized parallel tempering Monte Carlo

We introduce an algorithm to systematically improve the efficiency of parallel tempering Monte Carlo simulations by optimizing the simulated temperature set. Our approach is closely related to a recently introduced adaptive algorithm that optimizes the simulated statistical ensemble in generalized broad-histogram Monte Carlo simulations. Conventionally, a temperature set is chosen in such a way that the acceptance rates for replica swaps between adjacent temperatures are independent of the temperature and large enough to ensure frequent swaps. In this paper, we show that by choosing the temperatures with a modified version of the optimized ensemble feedback method we can minimize the round-trip times between the lowest and highest temperatures which effectively increases the efficiency of the parallel tempering algorithm. In particular, the density of temperatures in the optimized temperature set increases at the "bottlenecks'' of the simulation, such as phase transitions. In turn, the acceptance rates are now temperature dependent in the optimized temperature ensemble. We illustrate the feedback-optimized parallel tempering algorithm by studying the two-dimensional Ising ferromagnet and the two-dimensional fully-frustrated Ising model, and briefly discuss possible feedback schemes for systems that require configurational averages, such as spin glasses.Comment: 12 pages, 14 figure

Analysis of path integrals at low temperature : Box formula, occupation time and ergodic approximation

We study the low temperature behaviour of path integrals for a simple one-dimensional model. Starting from the Feynman-Kac formula, we derive a new functional representation of the density matrix at finite temperature, in terms of the occupation times of Brownian motions constrained to stay within boxes with finite sizes. From that representation, we infer a kind of ergodic approximation, which only involves double ordinary integrals. As shown by its applications to different confining potentials, the ergodic approximation turns out to be quite efficient, especially in the low-temperature regime where other usual approximations fail

Le confessioni religiose tra libertà di vivere nella realtà dell'ordinamento statale e potere di creare norme giuridiche all'interno dello Stato. Il caso della Chiesa di Scientology.

L'A. si pone nella prospettiva di delineare il possibile significato della compresenza delle due distinte previsioni, l’una nell’art. 8, l’altra nell’art. 19 della Costituzione, a garanzia del fenomeno associativo religioso. La ricerca è stata redatta, fra l’altro, in considerazione della circostanza che, ad una prima sommaria lettura della Carta Costituzionale, il lettore potrebbe essere indotto a pensare che il fenomeno religioso, ed in particolare quello che si svolge in forma associata, goda di una maggiore preferenza, rispetto a quello che si esprime in forma individuale, per essere più volte disciplinato in Costituzione. Problema, questo, il quale non sembra abbia sinora ricevuto adeguato rilievo dalla dottrina ecclesiasticista che si è meritevolmente impegnata sul tema. Si è posto al riguardo l’interrogativo che nasce dalla presenza di due distinte previsioni costituzionali a garanzia del fenomeno associativo religioso: la prima contenuta nell’art. 8 il quale - com’è noto - regola con novità di linguaggio la posizione dei culti diversi dalla religione cattolica, garantisce alle confessioni religiose l’eguale libertà davanti alla legge, il diritto di organizzarsi secondo i propri statuti ed il potere di concludere intese con gli organi dello Stato; la seconda nell’art. 19 che riconosce la libertà di professare liberamente la propria fede religiosa in qualsiasi forma “individuale o associata”. In ordine ai numerosi problemi che la previsione costituzionale dell’art. 8 solleva, si è riflettuto soprattutto sul dubbio se la fattispecie in essa contemplata sia sostanzialmente ripetitiva e si risolva in un inutile duplicato di quanto disposto dall’art. 19 per la parte che riguarda tale aspetto, oppure se, con ciascuna di tali disposizioni, il legislatore costituzionale abbia voluto conferire uno specifico rilievo a due distinte sfere dell’esperienza giuridica con caratteristiche proprie, per struttura e sistemi di garanzie. L’analisi intrapresa ha interessato anche la ricostruzione della definizione giuridica di confessione religiosa, nonché l’accertamento del carattere confessionale della Chiesa di Scientology alla luce delle sentenze della giurisprudenza costituzionale e di legittimità

Three applications of path integrals: equilibrium and kinetic isotope effects, and the temperature dependence of the rate constant of the [1,5] sigmatropic hydrogen shift in (Z)-1,3-pentadiene

Recent experiments have confirmed the importance of nuclear quantum effects even in large biomolecules at physiological temperature. Here we describe how the path integral formalism can be used to describe rigorously the nuclear quantum effects on equilibrium and kinetic properties of molecules. Specifically, we explain how path integrals can be employed to evaluate the equilibrium (EIE) and kinetic (KIE) isotope effects, and the temperature dependence of the rate constant. The methodology is applied to the [1,5] sigmatropic hydrogen shift in pentadiene. Both the KIE and the temperature dependence of the rate constant confirm the importance of tunneling and other nuclear quantum effects as well as of the anharmonicity of the potential energy surface. Moreover, previous results on the KIE were improved by using a combination of a high level electronic structure calculation within the harmonic approximation with a path integral anharmonicity correction using a lower level method.Comment: 9 pages, 4 figure