Non-Hamiltonian dynamics

Given the possibility of generating the microcanonical ensemble from a dynamical trajectory, it is natural to ask whether other equilibrium ensembles can be generated in a similar manner. For many such ensembles, it cannot be 

Consider a general dynamical system described by the equations of motion 

If (Ff Fo) has no explicit time dependence, as is expected in equilibrium, then the partial derivative with respect to time vanishes, and Eq. [33] becomes 

Since didt and Fj dIdTf are now interchangeable, the general solution to Eq. [33] can be shown to be 

Arranging Eq. [38] so that functions of Fj appear on one side and functions of Fq appear on the other yields  

---

Equations [76] only allow for isotropic fluctuations in the volume. However, it is sometimes useful to allow the lengths and angles of the simulation box all to vary separately. Fully flexible cell simulations of this type, first carried out by Parrinello and Rahman, can also be formulated in rms of a non-Hamiltonian dynamical scheme. In such a scheme, the matrix h representing the cell, which contains the cell vectors in its columns, is incorporated as a dynamical variable. That is, nine extra variables are added to the phase space along with an additional nine from its corresponding momentum Pg matrix. In terms of the box matrix, the partition function A(N, P, T) is given by... 

Recall from the section on non-Hamiltonian dynamics that the phase space distribution function satisfies a generalized Liouville equation ... 

Because our reference propagator U q is time independent, we know that its analytic form is given by (see earlier section on Non-Hamiltonian Dynamics)... 

The use of non-Hamiltonian dynamical systems has a long history in mechanics [8] and they have recently been used to study a wide variety of problems in molecular dynamics (MD). In equilibrium molecular dynamics we can exploit non-Hamiltonian systems in order to generate statistical ensembles other than the standard microcanonical ensemble NVE) that is generated by traditional Hamiltonian dynamics. These ensembles, such as the canonical (NVT) and isothermal-isobaric (NPT) ensembles, are much better than the microcanonical ensemble for representing the actual conditions under which experiments are carried out. 

The Nose-Hoover chains method is expressed in terms of the non-Hamiltonian dynamical system with the following equations of motion ... 

J. VandeVondele and U. Rothlisberger (2001) Estimating Equilibrium Properties from Non-Hamiltonian Dynamics. J. Chem. Phys. 115, p. 7859... 

Later methods made adjustments to external forces to account for periodic boundary conditions and introduced suitable modifications of the Hamiltonian or the Newtonian equations of motion [75-78]. Considerable progress has been made since those early efforts, both with the original [79-83] and modified Hamiltonian approaches [84]. However, many subtle issues remain to be resolved. These issues concern the non-Hamiltonian nature of the models used in NEMD and the need to introduce a thermostat to obtain a stationary state. Recently Tuckerman et al. [25] have considered some statistical mechanical aspects of non-Hamiltonian dynamics and this work may provide a way to approach these problems. Although the field of NEMD has been extensively explored for simple atomic systems, its primary applications lie mainly in treating nonequilibrium phenomena in complex systems, such as transport in polymeric systems, colloidal suspensions, etc. We expect that there will be considerable activity and progress in these areas in the coming years [85]. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

We note that for the harmonic Hamiltonian in (5.25) the variance of the work approaches zero in the limit of an infinitely slow transformation, v — 0, r — oo, vt = const. However, as shown by Oberhofer et al. [13], this is not the case in general. As a consequence of adiabatic invariants of Hamiltonian dynamics, even infinitely slow transformations can result in a non-delta-like distribution of the work. Analytically solvable examples for that unexpected behavior are, for instance, harmonic Hamiltonians with time-dependent spring constants k = k t). 

Thermostated dynamical systems are deterministic systems with non-Hamiltonian forces modeling the dissipation of energy toward a thermostat [48]. The non-HamUtonian forces are chosen in such a way that the equations of... 

S. Bonella and D.F. Coker. A semi-classical limit for the mapping Hamiltonian approach to electronically non-adiabatic dynamics. J. Chem. Phys., 114 7778, 2001. 

It is not immediately obvious, by simply looking at a molecule s Hamiltonian and/or its PES, whether the unimolecular dynamics will be intrinsic RRKM or not and computer simulations as outlined here are required. Intrinsic non-RRKM dynamics is indicative of mode-specific decomposition, since different regions of phase space are not strongly coupled and a micro-canonical ensemble is not maintained during the fragmentation. The phase space structures, which give rise to intrinsic RRKM or non-RRKM behavior, are discussed in the next section. 

Equilibrium Statistical Mechanics, Non-Hamiltonian Molecular Dynamics, and Novel Applications from Resonance-Free Timesteps to Adiabatic Free Energy Dynamics... 

The organization of this document is as follows The basis of all of these methods, equilibrium statistical mechanics, will be reviewed in Sect. 2. Section 3 will discuss the use of non-Hamiltonian systems to generate important ensembles. Novel non-Hamiltonian method, such as variable transformation techniques and adiabatic free energy dynamics will be discussed in Sect. 4. Finally, some conclusions and remarks will be provided in Sect. 5. 

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. 

We begin our analysis of non-Hamiltonian systems by considering a general dynamical system of the form... 

---