### Abstract

In this paper, a new approach to the development of Langevin-like equations for studying gas phase collisional energy tranfer and other dynamical problems is introduced based on the use of reference trajectories to describe memory effects and nonlinear interactions. In this development, the exact equations of motion are first expressed in terms of the deviations of the coordinates and momenta from some reference trajectory values and then linearized about those values. A partitioning between fast and slow variables is then assumed, and those members of the above mentioned linearized equations which refer to the fast variables are re-expressed as integral equations. A "local Brownian-like" approximation is then made in the memory kernel appearing in the integral equations to reduce them to algebraic equations, and upon substitution of these into the slow variable equations of motion, we obtain Langevin-like equations for the slow variables. In these equations the interaction between slow and fast variables appears as frictionlike and random forcelike terms, and in these terms, information about nonlinear interactions and correlated motions (including recurrences) is evaluated using the reference trajectory. In order to keep the deviations from the reference trajectory small during each collision, this trajectory is best chosen as the ensemble averaged trajectory, and we find that a good approximation to this for many problems is provided by a trajectory in which all initial vibrational energies are set equal to zero. Applications of this Langevin-like approach to several models of gas phase VT collisional energy transfer show that it is capable of quantitative predictions (errors typically <20%) of the first and second (classical) moments of the final translational distributions, provided that the initial translational energy is low enough to make the collision duration long compared with typical vibrational periods, and that the initial vibrational energy is low enough to make the deviations about the reference trajectory small. Often these restrictions are not particularly severe. For example, in a collinear Kr+CO_{2}(000) model, the average energy transfer is accurate to 5% for initial translational energies as high as 10 eV, while for a collinear He+H _{2} model, energy transfers accurate to 30% or better are obtained with five quanta of initial vibrational excitation in the H_{2}. In addition, accurate results are obtained even when the average energy transfer is of different sign than that of the reference, and in spite of the fact that the width of the translational distribution is a factor of 10 or more larger than its first moment. We also demonstrate that the Langevin equation works well when the average energy transfer becomes comparable to a quantum of vibrational energy (i.e., in the nonperturbative regime) provided that the necessary time scale separations for invoking the Langevin treatment exist.

Original language | English |
---|---|

Pages (from-to) | 2792-2801 |

Number of pages | 10 |

Journal | Journal of Chemical Physics |

Volume | 73 |

Issue number | 6 |

Publication status | Published - 1980 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Journal of Chemical Physics*,

*73*(6), 2792-2801.

**A reference trajectory approach to Langevin equations in gas phase collision dynamics.** / Schatz, George C; Moser, Mark D.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 73, no. 6, pp. 2792-2801.

}

TY - JOUR

T1 - A reference trajectory approach to Langevin equations in gas phase collision dynamics

AU - Schatz, George C

AU - Moser, Mark D.

PY - 1980

Y1 - 1980

N2 - In this paper, a new approach to the development of Langevin-like equations for studying gas phase collisional energy tranfer and other dynamical problems is introduced based on the use of reference trajectories to describe memory effects and nonlinear interactions. In this development, the exact equations of motion are first expressed in terms of the deviations of the coordinates and momenta from some reference trajectory values and then linearized about those values. A partitioning between fast and slow variables is then assumed, and those members of the above mentioned linearized equations which refer to the fast variables are re-expressed as integral equations. A "local Brownian-like" approximation is then made in the memory kernel appearing in the integral equations to reduce them to algebraic equations, and upon substitution of these into the slow variable equations of motion, we obtain Langevin-like equations for the slow variables. In these equations the interaction between slow and fast variables appears as frictionlike and random forcelike terms, and in these terms, information about nonlinear interactions and correlated motions (including recurrences) is evaluated using the reference trajectory. In order to keep the deviations from the reference trajectory small during each collision, this trajectory is best chosen as the ensemble averaged trajectory, and we find that a good approximation to this for many problems is provided by a trajectory in which all initial vibrational energies are set equal to zero. Applications of this Langevin-like approach to several models of gas phase VT collisional energy transfer show that it is capable of quantitative predictions (errors typically <20%) of the first and second (classical) moments of the final translational distributions, provided that the initial translational energy is low enough to make the collision duration long compared with typical vibrational periods, and that the initial vibrational energy is low enough to make the deviations about the reference trajectory small. Often these restrictions are not particularly severe. For example, in a collinear Kr+CO2(000) model, the average energy transfer is accurate to 5% for initial translational energies as high as 10 eV, while for a collinear He+H 2 model, energy transfers accurate to 30% or better are obtained with five quanta of initial vibrational excitation in the H2. In addition, accurate results are obtained even when the average energy transfer is of different sign than that of the reference, and in spite of the fact that the width of the translational distribution is a factor of 10 or more larger than its first moment. We also demonstrate that the Langevin equation works well when the average energy transfer becomes comparable to a quantum of vibrational energy (i.e., in the nonperturbative regime) provided that the necessary time scale separations for invoking the Langevin treatment exist.

AB - In this paper, a new approach to the development of Langevin-like equations for studying gas phase collisional energy tranfer and other dynamical problems is introduced based on the use of reference trajectories to describe memory effects and nonlinear interactions. In this development, the exact equations of motion are first expressed in terms of the deviations of the coordinates and momenta from some reference trajectory values and then linearized about those values. A partitioning between fast and slow variables is then assumed, and those members of the above mentioned linearized equations which refer to the fast variables are re-expressed as integral equations. A "local Brownian-like" approximation is then made in the memory kernel appearing in the integral equations to reduce them to algebraic equations, and upon substitution of these into the slow variable equations of motion, we obtain Langevin-like equations for the slow variables. In these equations the interaction between slow and fast variables appears as frictionlike and random forcelike terms, and in these terms, information about nonlinear interactions and correlated motions (including recurrences) is evaluated using the reference trajectory. In order to keep the deviations from the reference trajectory small during each collision, this trajectory is best chosen as the ensemble averaged trajectory, and we find that a good approximation to this for many problems is provided by a trajectory in which all initial vibrational energies are set equal to zero. Applications of this Langevin-like approach to several models of gas phase VT collisional energy transfer show that it is capable of quantitative predictions (errors typically <20%) of the first and second (classical) moments of the final translational distributions, provided that the initial translational energy is low enough to make the collision duration long compared with typical vibrational periods, and that the initial vibrational energy is low enough to make the deviations about the reference trajectory small. Often these restrictions are not particularly severe. For example, in a collinear Kr+CO2(000) model, the average energy transfer is accurate to 5% for initial translational energies as high as 10 eV, while for a collinear He+H 2 model, energy transfers accurate to 30% or better are obtained with five quanta of initial vibrational excitation in the H2. In addition, accurate results are obtained even when the average energy transfer is of different sign than that of the reference, and in spite of the fact that the width of the translational distribution is a factor of 10 or more larger than its first moment. We also demonstrate that the Langevin equation works well when the average energy transfer becomes comparable to a quantum of vibrational energy (i.e., in the nonperturbative regime) provided that the necessary time scale separations for invoking the Langevin treatment exist.

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M3 - Article

VL - 73

SP - 2792

EP - 2801

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 6

ER -