Feynman-Kac

The solution of the sink equation (5.12), starting from an equilibrium distribution at time t = 0, can also be expressed as a path integral by using the Feynman-Kac theorem,... 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

Over the years there have been important progress in finding trial functions substantially more accurate then the pair product form for homogeneous systems [12,13]. Within the generalized Feynman-Kac formalism, it is possible to systematically improve a given trial function [13,14]. The first corrections to the pair product action with plane wave orbitals are a three-body correlation term which modifies the correlation part of the trial function (Jastrow) and a backfiow transformation which changes the orbitals and therefore the nodal structure (or the phase) of the trial function [14]. The new trial function has the form... 

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

This approach is closely related to the Feynman-Kac formula that expresses the solution of the pde as functional integral over the stochastic process. 

For systems ruled by continuous interaction potentials (e.g., helium) a fine justification of Eq. (44) can be found in Ref. 28, where the Feynman-Kac formula is used. Accordingly, the pair action in these cases can be written as... 

The last equation is known as the Feynman-Kac formula and provides the starting point for quantum Monte Carlo electronic structure calculations. The reader is referred to Refs. 11, 35, and 36 and to Monte Carlo Quantum Methods for Electronic Structure for reviews of such methods. 

Generally, the Feynman-Kac approach is not useful for obtaining properties of excited states. Some progress can be made by imposing appropriate orthogonality constraints however, because the nodes of excited state wavefunctions are not known with precision a priori, such calculations require a certain degree of skill. 

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