Numerical integration of the Langevin equation: Monte Carlo simulation

Ermak, D.L.; Buckholz, Helen

doi:10.1016/0021-9991(80)90084-4

Cited by 340 publications

(193 citation statements)

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“…First, we develop an expression for the collision kernel in the Z → ∞ limit, which converges to the correct continuum and free molecular expressions and, with an appropriate definition of Kn D , can correctly predict collision kernels in the transition regime. For this purpose, we make use of mean first passage time calculations (Klein 1952;Narsimhan and Ruckenstein 1985), with the assumptions that the colliding entities are dilute in concentration and that the motion of colliding entities is accurately described by Langevin dynamics (Chandrasekhar 1943;Sitarski and Seinfeld 1977;Ermak and Buckholz 1980;Madler et al 2006;Isella and Drossinos 2010). We further show that the transition regime collision kernel is best represented in a nondimensionalized form that can be predicted with Buckingham theorem (Buckingham 1914(Buckingham , 1915.…”

Section: Introductionmentioning

confidence: 99%

Determination of the Transition Regime Collision Kernel from Mean First Passage Times

Gopalakrishnan

Hogan

2011

Aerosol Science and Technology

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Particle-vapor molecule (condensation) and particle-particle (coagulation) collision kernels are critically important parameters for the description of particle size distribution evolution in aerosols. We use mean first passage time calculations to find a dimensionless form of the collision kernel that applies in the limit where the ratio of the masses of colliding entities (aerosol particles or vapor molecules) to gas molecule mass, Z, approaches infinity, and the colliding entities are dilute in concentration. In these calculations, the motion of colliding entities is monitored with Langevin dynamics, and the collision kernel is inferred from the time required for collision. The presented analysis reveals that the dimensionless collisional kernel (H) is a function solely of the diffusive Knudsen number (Kn D , the square root of the product of kT and the reduced mass divided by the product of the reduced friction factor and collision radius), and a number of commonly used expressions for the collision kernel also collapse to the functional form H(Kn D ), irrespective of whether they were derived for Z → ∞ or Z → 0 conditions. The examined collision kernel expressions are further found to agree within 10% of our calculations as well as with each other across the entire Kn D range. This result suggests that a single collision kernel can be used to describe all transition regime collision rates, i.e., both condensation and coagulation rates, with reasonable accuracy, and establishes that Langevin-equationbased mean first passage time calculations can serve as a simple approach to examine transition regime collision phenomena.

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Section: Introductionmentioning

confidence: 99%

Determination of the Transition Regime Collision Kernel from Mean First Passage Times

Gopalakrishnan

Hogan

2011

Aerosol Science and Technology

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“…For each initial condition, the trajectory is propagated by the method of Ermak and Buckholz. 72 The time step of propagation is 10 −3 . The system is regarded as captured in the well region when the total energy (the kinetic energy plus the potential) of the system becomes less than U min + 2k B T , where U min is the potential at the bottom of each well.…”

Section: Modelmentioning

confidence: 99%

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

Kawai¹,

Komatsuzaki²

2010

Phys. Chem. Chem. Phys.

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An analytical framework to scrutinize the physical origin, especially nonlinear dynamical effects, of rate constants in thermal fluctuation is developed.A framework to calculate the rate constants of condensed phase chemical reactions of manybody systems is presented without relying on the concept of transition state. The theory is based on a framework we developed recently adopting multidimensional underdamped Langevin equation in the region of rank-one saddle. The theory provides a reaction coordinate expressed as an analytical nonlinear functional of the position coordinates and velocities of the system (solute), the friction constants, and the random force of the environment (solvent). Up to moderately high temperature, the sign of the reaction coordinate can determine the final destination of the reaction in a thermally fluctuating media, irrespective of what values the other (nonreactive) coordinates may take. In this paper, it is shown that the reaction probability is analytically derived as the probability of the reaction coordinate being positive, and that the integration with the Boltzmann distribution of the initial conditions leads to the exact reaction rate constant when the local equilibrium holds and the quantum effect is negligible. Because of analytical nature of the theory taking into account all nonlinear effects and their combination with fluctuation and dissipation, the theory naturally provides us with the firm mathematical foundation of the origin of the reactivity of the reaction in a fluctuating media.

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“…The complex interatomic interactions were modeled using quantum-based model generalized pseudopential theory (MGPT), which have been shown to accurately describe the directional bonding in central d−electron transition metals at both ambient and under extremes of pressure and temperature [13][14][15]. An NVT ensemble (fixed Number of particles, Volume and Temperature) was implemented for these simulations, with temperature control provided by application of a Langevin thermostat [16,17]. We used a symplectic integration scheme (with a time step of 1.5 fs) as described by Martyna and co-workers (with a slight modification to incorporate the stochastic thermostat) [18][19][20].…”

Section: Beyond Finite Size Scaling In Solidification Simulationsmentioning

confidence: 99%