^{1}, Gregory S. Ezra

^{2,a)}and Stephen Wiggins

^{1,b)}

### Abstract

We investigate the phase space structure and dynamics of a Hamiltonian isokinetic thermostat, for which ergodic thermostat trajectories at fixed (zero) energy generate a canonical distribution in configuration space. Model potentials studied consist of a single bistable mode plus transverse harmonic modes. Interpreting the bistable mode as a reaction(isomerization) coordinate, we establish connections with the theory of unimolecular reaction rates, in particular the formulation of isomerization rates in terms of gap times. In the context of molecular reaction rates, the distribution of gap times (or associated lifetimes) for a microcanonical ensemble initiated on the dividing surface is of great dynamical significance; an exponential lifetime distribution is usually taken to be an indicator of “statistical” behavior. Moreover, comparison of the magnitude of the phase space volume swept out by reactive trajectories as they pass through the reactant region with the total phase space volume (classical density of states) for the reactant region provides a necessary condition for ergodic dynamics. We compute gap times, associated lifetime distributions, mean gap times, reactive fluxes, reactive volumes, and total reactant phase space volumes for model thermostat systems with three and four degrees of freedom at three different temperatures. At all three temperatures, the necessary condition for ergodicity is approximately satisfied. At high temperatures a nonexponential lifetime distribution is found, while at low temperatures the lifetime is more nearly exponential. The degree of exponentiality of the lifetime distribution is quantified by computing the information entropy deficit with respect to pure exponential decay. The efficacy of the Hamiltonian isokinetic thermostat is examined by computing coordinate distributions averaged over single long trajectories initiated on the dividing surface.

P.C. and S.W. acknowledge the support of the Office of Naval Research (Grant No. N00014-01-1-0769). All three authors acknowledge the stimulating environment of the NSF sponsored Institute for Mathematics and its Applications (IMA) at the University of Minnesota, where the work reported in this paper was begun.

I. INTRODUCTION

II. THE HAMILTONIAN ISOKINETIC THERMOSTAT

A. Hamiltonian isokinetic thermostat

III. MODEL HAMILTONIANS

A. Double well potential

B. Model Hamiltonians

1. Three degrees of freedom

2. Four degrees of freedom

IV. MICROCANONICAL PHASE SPACE STRUCTURE: HAMILTONIAN AND NON-HAMILTONIAN ISOKINETIC THERMOSTAT

V. PHASE SPACE GEOMETRICAL STRUCTURES, UNIMOLECULAR REACTION RATES, AND THERMOSTAT DYNAMICS: GAP TIMES AND REACTIVE VOLUMES

A. Phase space dividing surfaces: Definition and properties

B. Phase space volumes and gap times

C. Gap time and reactant lifetime distributions

VI. NUMERICAL COMPUTATIONS FOR ISOKINETIC THERMOSTAT

A. Computations

1. System parameters

2. Computations

B. Numerical results: Three DOF

1. Lifetime distributions

2. Phase space volumes

3. Thermostat coordinate distributions

C. Numerical results: Four DOF

1. Lifetime distributions

2. Phase space volumes

3. Thermostat coordinate distributions

VII. SUMMARY AND CONCLUSIONS

### Key Topics

- Chemical reaction theory
- 22.0
- Electron densities of states
- 20.0
- Surface dynamics
- 20.0
- Phase space methods
- 16.0
- Eigenvalues
- 13.0

## Figures

(a) Section through the physical potential , Eq. (19), , , . (b) Section through the exponentiated potential, , .

(a) Section through the physical potential , Eq. (19), , , . (b) Section through the exponentiated potential, , .

Lifetime distributions for the three DOF Hamiltonian isokinetic thermostat. The lifetime distribution is derived from the distribution of gap times obtained by initiating trajectories on the incoming DS and propagating them until they cross the outgoing DS. (a) H121, . (b) H321, . (c) H521, .

Lifetime distributions for the three DOF Hamiltonian isokinetic thermostat. The lifetime distribution is derived from the distribution of gap times obtained by initiating trajectories on the incoming DS and propagating them until they cross the outgoing DS. (a) H121, . (b) H321, . (c) H521, .

Coordinate distributions for three DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) H121, . (b) H321, . (c) H521, .

Coordinate distributions for three DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) H121, . (b) H321, . (c) H521, .

Coordinate distributions for three DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) H121, . (b) H321, . (c) H521, .

Moments of the distribution of the coordinate obtained using the three DOF Hamiltonian isokinetic thermostat (squares) are compared with those for the Boltzmann distribution (circles). Odd moments for the Boltzmann distribution are identically zero. (a) H121, . (b) H321, . (c) H521, .

Moments of the distribution of the coordinate obtained using the three DOF Hamiltonian isokinetic thermostat (squares) are compared with those for the Boltzmann distribution (circles). Odd moments for the Boltzmann distribution are identically zero. (a) H121, . (b) H321, . (c) H521, .

Lifetime distributions for the four DOF Hamiltonian isokinetic thermostat. The lifetime distribution is derived from the distribution of gap times obtained by initiating trajectories on the incoming DS and propagating them until they cross the outgoing DS. (a) J121, . (b) J321, . (c) J521, .

Lifetime distributions for the four DOF Hamiltonian isokinetic thermostat. The lifetime distribution is derived from the distribution of gap times obtained by initiating trajectories on the incoming DS and propagating them until they cross the outgoing DS. (a) J121, . (b) J321, . (c) J521, .

Coordinate distributions for four DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) J121, . (b) J321, . (c) J521, .

Coordinate distributions for four DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) J121, . (b) J321, . (c) J521, .

Coordinate distributions for four DOF Hamiltonian isokinetic thermostat obtained by averaging over a single trajectory. Numerical distributions for the coordinate (histograms) are compared with the Boltzmann distribution (solid line). (a) J121, . (b) J321, . (c) J521, .

Moments of the distribution of the coordinate obtained using the four DOF Hamiltonian isokinetic thermostat (squares) are compared with those for the Boltzmann distribution (circles). Odd moments for the Boltzmann distribution are identically zero. (a) J121, . (b) J321, . (c) J521, .

Moments of the distribution of the coordinate obtained using the four DOF Hamiltonian isokinetic thermostat (squares) are compared with those for the Boltzmann distribution (circles). Odd moments for the Boltzmann distribution are identically zero. (a) J121, . (b) J321, . (c) J521, .

Comparison of the numerically determined for the four DOF harmonic potential with the theoretical expression (A14) over the energy range . The phase space volume was determined by random sampling of a phase space hypercube using points. All are divided by [cf. Eq. (A15)].

Comparison of the numerically determined for the four DOF harmonic potential with the theoretical expression (A14) over the energy range . The phase space volume was determined by random sampling of a phase space hypercube using points. All are divided by [cf. Eq. (A15)].

Number of points per energy bin vs energy (width , ) for the J321 four DOF Hamiltonian. The red curve is a fit to the numerical data using a fifth-order polynomial. The fit to the data yields a value .

Number of points per energy bin vs energy (width , ) for the J321 four DOF Hamiltonian. The red curve is a fit to the numerical data using a fifth-order polynomial. The fit to the data yields a value .

## Tables

Computed mean gap times, fluxes, reactive phase space volumes, energy surface volumes, and entropy deficits for lifetime distributions for three DOF and four DOF model Hamiltonians. Details of the computations are discussed in Appendix B.

Computed mean gap times, fluxes, reactive phase space volumes, energy surface volumes, and entropy deficits for lifetime distributions for three DOF and four DOF model Hamiltonians. Details of the computations are discussed in Appendix B.

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