Numerical implementation of some reweighted path integral methods

The reweighted random series techniques provide finite-dimensional approximations to the quantum density matrix of a physical system that have fast asymptotic convergence. We study two special reweighted techniques that are based upon the Levy-Ciesielski and Wiener-Fourier series, respectively. In agreement with the theoretical predictions, we demonstrate by numerical examples that the asymptotic convergence of the two reweighted methods is cubic for smooth enough potentials. For each reweighted technique, we propose some minimalist quadrature techniques for the computation of the path averages. These quadrature techniques are designed to preserve the asymptotic convergence of the original methods.Comment: 15 pages, 10 figures, submitted to JC

Energy estimators for random series path-integral methods

We perform a thorough analysis on the choice of estimators for random series path integral methods. In particular, we show that both the thermodynamic (T-method) and the direct (H-method) energy estimators have finite variances and are straightforward to implement. It is demonstrated that the agreement between the T-method and the H-method estimators provides an important consistency check on the quality of the path integral simulations. We illustrate the behavior of the various estimators by computing the total, kinetic, and potential energies of a molecular hydrogen cluster using three different path integral techniques. Statistical tests are employed to validate the sampling strategy adopted as well as to measure the performance of the parallel random number generator utilized in the Monte Carlo simulation. Some issues raised by previous simulations of the hydrogen cluster are clarified.Comment: 15 pages, 1 figure, 3 table

Heat capacity estimators for random series path-integral methods by finite-difference schemes

Previous heat capacity estimators used in path integral simulations either have large variances that grow to infinity with the number of path variables or require the evaluation of first and second order derivatives of the potential. In the present paper, we show that the evaluation of the total energy by the T-method estimator and of the heat capacity by the TT-method estimator can be implemented by a finite difference scheme in a stable fashion. As such, the variances of the resulting estimators are finite and the evaluation of the estimators requires the potential function only. By comparison with the task of computing the partition function, the evaluation of the estimators requires k + 1 times more calls to the potential, where k is the order of the difference scheme employed. Quantum Monte Carlo simulations for the Ne_13 cluster demonstrate that a second order central-difference scheme should suffice for most applications.Comment: 11 pages, 4 figure

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.

Phase changes in selected Lennard-Jones X_{13-n}Y_n clusters

Detailed studies of the thermodynamic properties of selected binary Lennard-Jones clusters of the type X_{13-n}Y_n (where n=1,2,3) are presented. The total energy, heat capacity and first derivative of the heat capacity as a function of temperature are calculated by using the classical and path integral Monte Carlo methods combined with the parallel tempering technique. A modification in the phase change phenomena from the presence of impurity atoms and quantum effects is investigated.Comment: 14 pages, 13 figures. submitted to J. Chem. Phy

The Vascularization Pattern of the Colon and Surgical Decision in Esophageal Reconstruction with Colon. A Selective SMA and IMA Arteriographic Study

Rezumat Pattern-ul de vascularizaåie al colonului aei decizia chirurgicalã în reconstrucåia esofagianã cu colon -studiu arteriografic selectiv al AMS aei AMI Introducere: Indiferent de tehnica reconstructivã, conceptele de fundamentare din reconstrucåia visceralã au ca baza principalã suportul vascular necesar pentru grefonul de substituåie. Particularitãåile vasculare individuale pot înclina sau chiar obliga chirurgul la o anumitã opåiune cãtre unul sau altul dintre procedee. De aceea, vascularizaåia este, fãrã îndoialã, factorul care dominã mobilizarea colonului pentru reconstrucåia esofagianã. Material aei metodã: Studiul nostru arteriografic aei-a propus o investigaåie asupra tiparului vascular al celor douã surse principale ce participã prin vasele emergente la irigarea arterialã a colonului: a. mezentericã superioarã (AMS) respectiv a. mezentericã inferioarã (AMI). Nu am avut în vedere selectarea pacienåilor dupã un anumit criteriu dupã cum nu am realizat nici o excludere dintr-un anumit considerent. Lotul de studiu a constat din 49 de pacienåi care s-au prezentat în clinicã pentru o tehnicã reconstructivã, toåi aparåinând perioadei 2000-2010. În intervalul 1981-2012, au fost efectuate 187 de tehnici reconstructive pentru o indicaåie postcausticã. Din totalul de 49 de pacienåi, 11 bolnavi suferiserã intervenåii chirurgicale abdominale majore iar dintre aceaetia, 5 cu tentative nereuaeite de reconstrucåie. Rezultate: Din cei 49 de pacienåi la care s-a efectuat explorarea, arteriografia a evidenåiat o situaåie favorabilã reconstrucåiei la 31 dintre aceştia. La ceilalåi 18 pacienåi au fost identificate anomalii ori distribuåii atipice, 5 ale AMS respectiv 13 ale AMI. Decizia operatorie a fost ajustatã la 22 de bolnavi. Un lucru important de semnalat dpdv predictiv asupra viscerul de mobilizat: nu am avut necroze de grefon la pacienåii cu examinare arteriograficã preoperatorie. Concluzii: Dictate de necesitatea unei bune mobilizãri, ligaturile arteriale trebuie adaptate şi modificate în funcåie de particularitãåile de distribuåie vascularã, astfel încât sã se menåinã un flux sangvin suficient în arcada marginalã pânã la nivelul secåiunilor colice şi, implicit, în arterele drepte din vecinãtatea acestora. main grounds the mandatory vascular support for the graft replacement. Individual vascular particularities can influence or even oblige the surgeon to choose a certain procedure. This is why the vascularization is beyond doubt the dominant factor in mobilizing the colon for reconstruction. Material and method: Our arteriographic study entails an investigation upon the vascularization pattern of the two main sources that participate in the arterial irrigation of the colon via the emerging vessels: superior mesenteric artery (SMA) and inferior mesenteric artery (IMA). We did not consider certain patients upon a specific criterion; also, we did not exclude any patients due to various reasons. We took into account 49 patients as study group, all of them having registered into the clinic for a reconstructive technique, throughout the years from 2000 to 2010. From 1981 to 2012 there have been 187 reconstructive techniques performed due to post caustic pathology. From a total of 49 patients, 11 had suffered major abdominal surgeries, 5 of which had had unsuccessful reconstructive attempts. Results: Out of the 49 patients on whom we have performed the exploration, arteriography showed a favorable situation for reconstruction in 31 of them. In the other 18 patients anomalies or atypical distributions were identified, in 5 of the SMA and in 13 of the IMA, respectively. Operative decision was modified in 22 patients. One important thing to note from the point of view of the segment to be moved: we had no graft necrosis in patients with preoperative arteriographic examination. Conclusions: Due to the need for good mobilization, arterial ligations should be adjusted and modified depending on the particular vascular distribution, to maintain a sufficient blood flow in the marginal artery, in order to reach the colic sections and the straight arteries near them. Abbreviations: SMA -superior mesenteric artery; IMAinferior mesenteric artery; ICa -ileocolic artery; RCa -right colic artery; MCa -middle colic artery; LCa -left colic artery; LC acc.a -left accessory colic artery (or middle left colic artery); ILCa -inferior left colic artery; S trunk -sigmoidian trunk; Sa -sigmoidian artery; SRa -superior rectal arter

Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-\beta \hbar / 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.Comment: 13 pages, 2 figure

Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods

In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is ``experimental'', I only provide a ``physicist's'' proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in the Appendix, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present article is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.Comment: 19 pages, 4 figures; the discrete short-time approximations are now treated as independent from their continuous version; new examples of discrete short-time approximations of order three and four are given; a new appendix containing a short review on Brownian motion has been added; also, some additional explanations are provided here and there; this is the last version; to appear in Phys. Rev.