Functional integral formulation

To proceed further, it is convenient to represent the grand canonical partition function for the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29] over Grassmann fermionic and complex hosonic fields as 

Apart from the above symmetry considerations the SB functional-integral formalism reveals additional global and local gauge invariances [25, 30-32]. 

The action S = dtL(t) is invariant under the following site and time-dependent phase transformations (0i(r)z.(r)), i.e., under the local symmetry group 5(7(2) 17(1), 

in the continuum limit, the radial gauge is introduced by representing the Bose fields by modulus and phase. 

It may be remarked that since the bosons are taken to be real, their kinetic terms, being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary conditions imposed on Bose fields ( / ()S) = iot(0)). Strictly speaking it follows from this property that all the Bose fields do no longer have dynamics of their own [25]. 

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In di.scussing these problems, it is, of course, important and natural to indicate the parallels in fields other than polymer chemistry and physics. Thus it is natural to discuss briefly any connections or similarities with Feynman path integrals, with descriptions of Brownian motion and other random processes, " and with the functional integral formulation of many body or quantum field theories. Only the most rudimentary... 

This method, powerful as it is, leads to a nonlinear implicit functional equation for the boundary motion, which must be solved by numerical means. In many cases a direct numerical attack on the governing equation and boundary conditions has been preferred but Kolodner s method has the advantage of being an exact integral formulation which does not require solution of the heat equation throughout all space at each step of the boundary motion. 

The mapping basis has been exploited in quantum-classical calculations based on a linearization of the path integral formulation of quantum correlation functions in the LAND-map method [50-52]. 

To express this in more formal mathematical terms, we cast a partial differential equation in variational form (or integral form or weak formulation) by multiplying by a suitable function, integrating over the domain where the equation is posed and applying Greens theorem. Thus, if we consider the reaction-diffusion equation... 

One may also use a memory function m(t) within the integral formulation of linear and nonlinear viscoelasticity this memory fimction m(t), which is the derivative of the relaxation function GKt), is not a measurable function. 

The functional above was used already by Gauss [12] to study classical trajectories (which explains our choice of the action symbol). Onsager and Machlup used path integral formulation to study stochastic trajectories [13]. The origin of their trajectories is different from what we discussed so far, which are mechanical trajectories. However, the functional they derive for the most probable trajectories, O [X (t)] is similar to the equation above ... 

To implement the linearized path integral formulation for time correlation functions the initial density operator must be Wigner transformed in the bath variables while it remains an operator in the quantum subsystem space. In the calculations presented below we assume that the system and bath do not interact initially. Consequently total probability density at t = 0 is of the form... 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on... 

A more efficient approach is to base the boundary-integral formulation on a fundamental solution (or more accurately a Green s function) that incorporates the relevant boundary conditions at one or more of the surfaces. In the case of a particle or drop moving near an infinite plane wall, this means finding a solution for a point force that exactly satisfies the no-slip and kinematic boundary conditions at the wall. If we were to consider the motion of a particle or drop in a tube, it would be useful to have the solution for a point force satisfying the same conditions on the tube walls. 

All first-order approximations (pertaining to integral transport theory) considered are equivalent, in accuracy, either to Pid[x where (j) stands for one of the three approximations, < fl> bd> or ( fd> fo the perturbed flux distribution and stands for either ( fl or ( bd-cf> is a better approximation to compared to better approximation to compared to we conclude that all first-order perturbation expressions in integral transport theory formulations considered in this work are equivalent, in accuracy, to some high-order approximation to Pji>. This higher accuracy can be computed, in integral formulations, using the flux and source-importance functions for the unperturbed reactor. 

W. T. Yang. Ab Initio approach for many-electron systems without invoking orbitals An Integral formulation of density-functional theory. Phys. Rev. Lett, 59(143 15 69-1572, Oct 1987. 

For a non-uniform density distribution such as a nucleus with a tapered edge, K is no longer a constant across the nucleus but instead becomes a function of r and y), the coordinates of points inside the nucleus. Obviously expression (10.6) does not hold and one must reformulate the problem as a function of r and y). For the case of absorption cross-sections a convenient integral formulation has been given by Williams for the case where K is a function of r only. However for realistic density distributions (as obtained from recent electron scattering measurements (see Sect. 37) these integrals have to be solved numerically. 

The free propagator plays an important role in path integral formulation of quantum mechanics and will be reloaded with that occasion soon bellow. Yet, it remains the problem of assigning the free Green function the stationary expression, i.e., when is not viewed as a propagator, a matter that will be as well unfolded later when describing the scattering process as a measurement tool for quantum phenomena, see the last postulate of quantum mechanics in this chapter. 

The reactive flux method is also useful in calculating rate constants in quantum systems. The path integral formulation of the reactive flux together with the use of the centroid distribution function has proved very useful for the calculation of quantum transition-state rate constants [7]. In addition new methods, such as the Meyer-Miller method [8] for semiclassical dynamics, have been used to calculate the flux-flux correlation function and the reactive flux. 

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