### Abstract

This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward‐Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ω_{c}, which may be set equal to kT/h to simulate quantum‐mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward‐Euler/Langevin method is considered, an integral equation for the equilibrium phase‐space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second‐order partial differential equation for the equilibrium phase‐space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.

Original language | English (US) |
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Pages (from-to) | 599-645 |

Number of pages | 47 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 43 |

Issue number | 5 |

DOIs | |

State | Published - 1990 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*43*(5), 599-645. https://doi.org/10.1002/cpa.3160430503

**Analysis of the backward‐euler/langevin method for molecular dynamics.** / Peskin, Charles S.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 43, no. 5, pp. 599-645. https://doi.org/10.1002/cpa.3160430503

}

TY - JOUR

T1 - Analysis of the backward‐euler/langevin method for molecular dynamics

AU - Peskin, Charles S.

PY - 1990

Y1 - 1990

N2 - This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward‐Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ωc, which may be set equal to kT/h to simulate quantum‐mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward‐Euler/Langevin method is considered, an integral equation for the equilibrium phase‐space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second‐order partial differential equation for the equilibrium phase‐space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.

AB - This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward‐Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ωc, which may be set equal to kT/h to simulate quantum‐mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward‐Euler/Langevin method is considered, an integral equation for the equilibrium phase‐space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second‐order partial differential equation for the equilibrium phase‐space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.

UR - http://www.scopus.com/inward/record.url?scp=84990619660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990619660&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160430503

DO - 10.1002/cpa.3160430503

M3 - Article

AN - SCOPUS:84990619660

VL - 43

SP - 599

EP - 645

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 5

ER -