Path integrals approach

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

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. 

J. Lehmann, P. Reimann, and P. Hanggi, Surmounting oscillating barriers path-integral approach for weak noise, Phys. Rev. E 62, 6282 (2000). 

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. 

A dynamic description of the effect of relaxation on the probability of the adiabatic transition may be performed using various methods, e.g., a Feynman path integral approach similar to that presented in Section III (see also Refs. 81-84). Here we shall present the results for a simple model obtained by another method.85... 

A number of papers are devoted to the effect of dissipation on tunneling.81"83,103,104 Wolynes81 was one of the first to consider this problem using the Feynman path integral approach to calculate the correlation function of the reactive flux involved in the expression for the rate constant,... 

Roepstorff, G., Path Integral Approach to Quantum Physics, Springer Berlin, Heidelberg, New York, 1994... 

In the so-called primitive representation of the discretized path-integral approach [141], the canonical partition function for finite P has the form... 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

A similar procedure gives the corresponding result for the Wiener process the first term of which was obtained by Feynman and Hibbs4 using a path integral approach. 

A powerful tool for analyzing fluctuations in a nonequilibrium systems is based on the Hamiltonian [57] theory of fluctuations or alternatively on a path-integral approach to the problem [44,58-62]. The analysis requires the solution of two closely interrelated problems. The first is the evaluation of the probability density for a system to occupy a state far from the stable state in the phase space. In the stationary regime, the tails of this probability are determined by the probabilities of large fluctuations. 

S. Bonella, D. Montemayor, and D.F. Coker. Linearized path integral approach for calculating nonadiabatic time correlation functions. Proc. Natl. Acad. Sci., 102 6715-6719, 2005. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

Causo, M.S., Ciccotti, G., Montemayor, D., Bonella, S., Coker, D.F. An adiabatic linearized path integral approach for quantum time correlation functions electronic transport in metal-molten salt solutions. J. Phys. Chem. B 109 6855... 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

VIII. Finite-Size Scaling and Path Integral Approach for Quantum Criticality... 

In the path integral approach, the transition amplitude between two states of the system can be calculated by summing amplitudes for all possible paths between them. By inserting a sequence of sums over sets of intermediate states into the expression for the partition function, Eq. (48) becomes... 

VIII. FINITE-SIZE SCALING AND PATH INTEGRAL APPROACH FOR QUANTUM CRITICALITY... 

In the path integral approach, the analytical continuation of the probability amplitude to imaginary time t = —ix of closed trajectories, x(t) = x(f ), is formally equivalent to the quantum partition function Z((3), with the inverse temperature (3 = — i(t — t)/h. In path integral discrete time approach, the quantum partition function reads [175-177]... 

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