Extended phase space

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

We can therefore revert from the formally autonomous description in the extended phase space to an explicitly time-dependent dynamics in the original phase space with a time-dependent normal form Hamiltonian... 

Daidone, I., Amadei, A., and Di Nola, A. (2005). Thermodynamic and kinetic characterization of a beta-hairpin peptide in solution An extended phase space sampling by molecular dynamics simulations in explicit water. Proteins 59, 510-518. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

However, using a method proposed [60,62,95,112] for experimental analysis of the Hamiltonian flow in an extended phase space of the fluctuating system, we can exploit the analogy between the Wentzel-Freidlin and Pontryagin Hamiltonians arising in the analysis of fluctuations, and the energy-optimal control problem in a nonlinear oscillator. To see how this can be done, let us consider the fluctuational dynamics of the nonlinear oscillator (35). 

The original work of Andersen and of Parrinello and Rahman on the generation of the NPT or isothermal-isobaric ensemble using an extended phase space predates Nose s work on the NVT ensemble, as noted above. Applying the extended system method to generate the NPT ensemble involves the inclusion of the volume into the phase space as a dynamical variable along... 

In this section we present the concept of a generalized set of equations of motion that will describe a system coupled to an external field (e.g., shear flow, electric field, etc.). This general set of equations of motion will also describe coupling to the extended phase space (e.g., thermostats, barostats) and is of the form ... 

In this section we develop a general linear response theory for which the coupling of the system to the field in Eqs. [91] is small [i.e., fjt) 1]. Linear response theory should, in general, take into account the possibility of phase space compression coming from either the coupling to the external field or from the extended phase space variables. In traditional formulations of linear response theory, phase space compression has not been carefully considered. The present formulation treats the first level of compressibility exactly, for example, the compressibility due to the presence of an extended system (such as ther-... 

Additionally, non-Hamiltonian d5mamics can be used in applications/ methodologies such as Path-integral MD, replica-exchange methods, variable transformation techniques, free energy dynamics methods, and other new applications. Generating these alternative statistical ensembles from simulation requires the use of extended systems or extended phase space [9]. In these systems, the simulations do not only include the N coordinate and momentum vectors that are needed to describe a classical Ai-particle system, but they also include a set of additional control or extended variables that are used to drive the fluctuations required by the ensemble of interest. 

Let us introduce the extended phase space (see Appendix A) by defining a new variable T conjugated to time, such that the extended Hamiltonian becomes... 

The regularization is finally obtained considering the equations of motion in the extended phase space. 

We discuss the introduction of the extended phase space. If the Hamiltonian function H = H(P,Q,t) depends explicitly on the time, one can introduce a time-independent Hamiltonian, defined as... 

Second, calculations can be performed in a semiclassical regime, and the results plotted on a Poincare section in action-angle (I, 0) coordinates. Such diagrams may seem complicated (see figs. 10.16 and 10.17), but are at least in principle readily understood a near-horizontal line across the (1,0) plot corresponds to a torus in ordinary phase space. When periodically extended in the time coordinate, each line corresponds to a vortex tube embedded in the extended phase space of the periodically... 

Fig. 8.1 Trajectories of the Nose-Hoover Chain system may be trapped forever in restricted regions of phase space. Here a toms is shown in a projection of the full extended phase space for the harmonic oscillator with a length two thermostatting chain. (Equations of motion q = p p = -q-hpili = (p - 1)// 1 -hhih = 1)/M2. with/ti = 0.2, and/t2 = 1)...

Center Manifolds. The Center Manifold Theorem (see Carr (1981)) states that all branches of stationary and periodic states in a neighborhood of a bifurcation point are embedded in a sub-manifold of the extended phase space X M that is invariant with respect to the flow generated by the ODE (2.1). All trajectories starting on this so-called center manifold remain on it for all times. All trajectories starting from outside of it exponentially converge towards the center manifold. Specifically, static bifurcations are embedded in a two dimensional center manifold, whereas center manifolds for Hopf bifurcations are three dimensional. Figures 2.1 and 2.2 summarize the geometric properties of the flows inside a center manifold in the case of saddle-node and Hopf bifurcations, respectively. 

Then, in the extended phase space the direct product of the phase space and the parameter space) near the origin there exists a uniquely defined -smooth invariant surface of the form p = 0(a ), V (0) = 0, such that each its intersection with the plane p = constant consists of a set of closed orbits of the system (11.5.17), lying in a neighborhood of the origin at the given p. 

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

The basic Landau-Ginzburg model is valid only for relatively weak surfactants and in a limited region of the phase space. In order to find a more general mesoscopic description, valid also for strong surfactants and in a more extended region of the phase space, we derive in this section a mesoscopic Landau-Ginzburg model from the lattice CHS model [16]. 

The purpose of this work is to demonstrate that the techniques of quantum control, which were developed originally to study atoms and molecules, can be applied to the solid state. Previous work considered a simple example, the asymmetric double quantum well (ADQW). Results for this system showed that both the wave paeket dynamics and the THz emission can be controlled with simple, experimentally feasible laser pulses. This work extends the previous results to superlattices and chirped superlattices. These systems are considerably more complicated, because their dynamic phase space is much larger. They also have potential applications as solid-state devices, such as ultrafast switches or detectors. 

The problems with the adiabatic Yamada-Kawasaki distribution and its thermostatted versions can be avoided by developing a nonequilibrium phase space probability distribution for the present case of mechanical work that is analogous to the one developed in Section IVA for thermodynamic fluxes due to imposed thermodynamic gradients. The odd work is required. To obtain this, one extends the work path into the future by making it even about t ... 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

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