Statistical path integral

In these conditions the quantum statistical path integral representation of quantum propagator, see Eq. (2.10) with Eq. (2.21), and Volume 1/ Chapter 4 of the present five-volume set, and takes the form... 

Expression (2.40) is called the statistical path integral in analogy to the quantum-mechanical Feynman path integral introduced as an alternative means of quantization, where all possible particle paths, which in contrast to classical Newtonian mechanics may vary due to quantum fluctuations, are integrated over [59,60]. 

Tuckerman M E and Hughes A 1998 Path integral molecular dynamics a computational approach to quantum statistical mechanics Classical and Quantum Dynamics In Condensed Phase Simulations ed B J Berne, G Ciccotti and D F Coker (Singapore World Scientific) pp 311-57... 

Hi) The use of quantum methods to obtain correct statistical static (but not dynamic) averages for heavy quantum particles. In this category path-integral methods were developed on the basis of Feynman s path... 

H. Kleinert, Path integrals in quantum mechanics, statistics and polymer physics , World Scientific, Singapore, 1995, chapters 18.5 and 18.6. 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

H. Kleinert. Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. Singapore World Scientific, 1990. 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

Feynman RP, Hibbs AR (1965) Quantum Mechanics and Path Integrals. McGraw-Hill New York, p xiv, 365 p. For the applications in quantum statistics, see chapters 10 and 11 Corrections to the errata in the book http //www.oberlin.edu/physics/dstyer/FeynmanHibbs/ and http //www.physik. fu-berlin.de/ kleinert/Feynman-Hibbs/... 

Kleinert H (2004) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. 3rd edition. World Scientific Singapore River Edge, NJ, p xxvi, 1468 p. For the quantum mechanical integral equation, see Section 1.9 For the variational perturbation theory, see Chapters... 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

Barker, J.A., A quantum-statistical Monte Carlo method Path integrals with boundary conditions, J. Chem. Phys. 1979, 70, 2914-2918... 

Path Integral Quantum Mechanics and Statistical Thermodynamics. 69... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

A NEW SERIES EXPANSION FOR PATH INTEGRALS IN STATISTICAL MECHANICS ... 

The use of path or functional integration techniques in physics offers many apparent simplifications especially in statistical mechanics. However, in practice it is usually impossible to make an explicit evaluation of the path integrals one meets. Here we shall give a very condensed account of a possible approximation scheme for the calculation of a particular class of path integrals. 

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