Partition function, path integral method

As in the standard path integral method for finite temperature systems [19], the above pseudo partition function can be regarded as a configurational integral of classical polymers. However, in the variational path integral, the classical isomorphic systems consist of open-chain polymers. Furthermore, distributions of end-point coordinates at = 0 and M are affected by the trial wavefunction and respectively. 

The centroid path integral method described above enable us to conveniently determine KIEs by directly computing the ratio of the quantum partition functions for two different isotopes through free energy perturbation (FEP) theory. The use of mass perturbation in free-particle bisection sampling scheme results in a major improvement in computation accuracy for KIE calculations such that secondary kinetic isotope effects and heavy atom isotope effects can be reliably obtained. The PI-FEP/UM method is the only practical approach to yield computed secondary KIEs sufficiently accurate to be compared with experiments. ... 

Let us now turn to the case T -> 0. First of all, somewhat suspicious is the combination of the continuous integral (2.1) with the discrete partition function Zq (2.13), usual for CLTST. This serious deficiency cannot be circumvented in the framework of this CLTST-based formalism, and a more rigorous reasoning is needed to describe the quantum situation. Introduction of adequate methods will be the objective of the next sections devoted to the path-integral formalism, so here we... 

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. 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Edwards approach (Ref. [10]) is based on field-theoretic path-integral representation of the partition function Wn(R0, RN, N) defining the probability density of the fact that end points of an JV-link chain are placed at the points R0 and Rn, respectively, and the chain turns n times around the string (the obstacle). The same problem in a slightly different way was considered by Prager and Frisch by using the combinatorial methods [11] and later by Saito and Chen by employing Fourier analysis [12]. 

Eq. (4.8) is the approximate path integral expression for the partition function used in the DPI method. The exact path integral expressions arising from Eq. (4.8) are obtained in the limit that JVf i-+ oo. Noting that... 

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