Time-path integration closed

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

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... 

Different approaches have been or are being investigated. Path integral approaches scale favorably with the number of degrees of freedom. However, an efficient real time path integral treatment requires the introduction of a simple reference Hamiltonian. The propagator associated with the reference Hamiltonian has to be known in closed form and the reference Hamiltonian has to yield a resonable zero order approximation for the total dynamics. 

Implementation of the FBSD schemes requires knowledge of the initial density matrix in the coherent state representation. Usually, the initial density corresponds either to the ground vibrational state of a polyatomic molecule or a Boltzmann distribution. Below we describe ways of obtaining the coherent state matrix element through closed form expressions or in terms of an imaginary time path integral evaluated along the same Monte Carlo random walk which samples the trajectory initial conditions. 

The quantum mechanical Tr operation is represented as a path integral over all closed paths q(x) whose time (x) average is centered at the point q such that q = q= - bJo dxq(x). The centroid potential of mean force is thus obtain from a restricted summation over all paths whose zero-th Fourier mode in a Fourier expansion of the path integral is given by q. Deep tunneling reflects itself as a significant lowering of the barrier of the centroid potential of mean force. 

Now let us introduce one useful trick, so-called closed time-path contour of integration. First, note that the expression of the type... 

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. 

By comparing analogous terms in ( , x) and Q, we see that we can think of the partition function as a path integral over periodic orbits that recur in a complex time interval equal to i s flh/i = — ifih. There is no claim here that the closed paths used to generate Q correspond to actual quantum dynamics, but simply that there is an isomorphism. We therefore can refer to the equation above as the discretized path-integral (DPI) representation of the partition function. Using Feynman s notation, we have in the infinite-P limit... 

The focus of this article, however, is on a more specialized topic in path integration— the path centroid perspective. One of the many interesting ideas suggested by Feynman in his formulation and application of path integrals was the notion of the path centroid variable [1], denoted here by the symbol q. The centroid is the imaginary time average of a particular closed Feynman path q(j), which, in turn, is simply the zero-frequency Fourier mode of that path, that is. 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

There is also another way to think about how the vector potential, specifically its curl part, operates in QED. One can envision, on one hand, a static condition where the phase change of v[/ around a closed path, with no electromagnetic fields on the path, can be related to A. However, to establish this static condition, there is required a net time integral of E along the path (from previous times), that is, the electric impulse, to establish these new static conditions. An alter-... 

Figure 3. Effects of the periodic perturbation, (a) Integration path on the complex time plane. (b) Deformation of the potential by the periodical perturbation. In the case where Im t = Im ti 0— that is, the part of integration path indicated by the same broken line in (a)— the oscillation of complexified potential is amplified exponentially as shown by the broken lines, (c) Change of the tunneling trajectory with increase of the perturbation strength. In the bottom figure, a trajectory stating at ti in the close neighborhood of t c is drawn.

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