Path integrals approach quantum partition function

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. 

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]... 

Equilibrium properties can be determined from the partition function Zq and this quantity can, in turn, be computed using Feynman s path integral approach to quantum mechanics in imaginary time [86]. In this representation of quantum mechanics, quantum particles are mapped onto closed paths r(f) in imaginary time f, 0 f )8ft. The path integral expression for the canonical partition function of a quantum particle is given by the P 00 limit of the quantum path discretized into P segments. 

However, the periodicity condition (4.518) for paths is to be maintained and properly implemented in approximating the effective-classical partition function (4.525) being, nevertheless, closely and powerfully related with the quantum beloved concept of stationary orbits defined/described by periodic quantum waves/paths. This way, the effective-classical path integral approach appears as the true quantum justification of the quantum atom and of the quantum stabilization of matter in general, providing reliable results without involving observables or operators relaying on special quantum postulates other than the variational principles - with universal (classical or quantum) value. 

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. ... 

---