Partition function, path integral

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

The functional (4.8) permits one to study the set of paths which actually contribute to the partition-function path integral thereby leading to the determinant (4.17). Namely, the symmetric Green function for the deviation from the instanton path x(t) is given by [Benderskii et al. 1992a]... 

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... 

Unlike the trivial solution x = 0, the instanton, as well as the solution x(t) = x, is not the minimum of the action S[x(t)], but a saddle point, because there is at least one direction in the space of functions x(t), i.e. towards the absolute minimum x(t) = 0, in which the action decreases. Hence if we were to try to use the approximation of steepest descents in the path integral (3.13), we would get divergences from these two saddle points. This is not surprising, because the partition function corresponding to the unbounded Hamiltonian does diverge. 

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

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. 

The path-integral (PI) representation of the quantum canonical partition function Qqm for a quantized particle can be written in terms of the effective centroid potential IT as a classical configuration integral ... 

In centroid path integral, the canonical QM partition function of a hybrid quantum and classical system, consisting of one quantized atom for convenience, can be written as follows ... 

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

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. 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

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]. 

Let us remind that the problem of the partition function calculation for a closed polymer chain with topological constraints is usually formulated as integration over the set fl of closed paths from a fixed topological class, or with fixed value of the topological invariant ... 

The partition function then is written as a path integral [245], for the blend [Eq. (98) ... 

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