Path integral relations

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

To answer the first question, we use the path integral relation... 

In the classical region of space, where the potential is less than the energy, the standard formula leads to an approximate relation between phase and modulus in the form of the following path integral ([237], Section 28)... 

The path-integral quantum mechanics relies on the basic relation for the evolution operator of the particle with the time-independent Hamiltonian H x, p) = -i- V(x) [Feynman and... 

The Feynman-Hibbs and QFH potentials have been used extensively in simulations examining quantum effects in atomic and molecular fluids [12,15,25]. We note here that the centroid molecular dynamics method [54, 55] is related and is intermediate between a full path integral simulation and the Feynman-Hibbs approximation ... 

We can expect to see future research directed at QM/MM and ab initio simulation methods to handle these electronic structure effects coupled with path integral or approximate quantum free energy methods to treat nuclear quantum effects. These topics are broadly reviewed in [32], Nuclear quantum effects for the proton in water have already received some attention [30, 76, 77]. Utilizing the various methods briefly described above (and other related approaches), free energy calculations have been performed for a wide range of problems involving proton motion [30, 67-69, 71, 72, 78-80]. 

The relation (A 7.5) first was derived by de Gennes, initiating the renormalization group or scaling approach to polymer solutions. Again these expressions need some explanation. Equation (A 7.6) defines G/ (r, r 7 q) as path integral , summing over all continuous paths r(s), 0 < s < Rh r(0) = r r(Rg) = r. It is properly defined as the continuous chain limit of the discrete... 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

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... 

Linearization methods start from a path integral representation of the forward and backward propagators in expressions for time correlation function, and combine them to describe the overall time evolution of the system in terms of a set of classical trajectories whose initial conditions are sampled from a quantity related to the Wigner transform of the quantum density operator. The linearized expression for a correlation function provides a powerful tool for describing systems in the condensed phase. The rapid decay of... 

In the non-periodic case the motion is analogous to that which in two dimensions is called a Lissajous-motion, the path being closed only in the event of a rational relation between the vks. We consider the path in the te-space, confined to a standard unit cell of the period lattice (see 13) by replacing every point on the actual path by the equivalent point in the standard cell. If there are no linear integral relations between the efc s, this path in the w-spacc approaches indefinitely near to each point in the standard cell (as proved in Appendix I). The representation of the 5-space in the w-space is continuous so in this case the path in the 5-space approaches indefinitely close to every point of an/-dimensional region. 

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