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Boundary 1.2

# Boundary 1.2 - Stochastic Boundary conditions m dv i dt =...

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Stochastic Boundary conditions - Langevin equation i i i i r U R v m dt dv m + β = The approach described above is used by M. Berkowitz and J.A. McCammon, Chem. Phys. Lett. 90 , 215 (1982). More complex and rigorous descriptions of Langevin particles has been discussed in literature, e.g. J.C. Tully, J. Chem. Phys. 73 , 1975 (1980). δ (t) β m kT 2 (0) (t)R R i i = R i – random white noise forces with Gaussian distribution centered at zero. The width of the distribution is defined by temperature and should obey the second fluctuation-dissipation theorem [R. Kubo, Rep. Prog. Theor. Phys. 33 , 425, 1965]: β – friction coefficient of an atom that, within the Debye model, can be determined from the relation β = 1/6 π ω D , where ω D is the Debye frequency, ω D = k B Θ D / ħ , and Θ D is the Debye T R n is taken from Gaussian random number generator. Methods for generating random numbers with Gaussian distribution from evenly distributed random numbers can be found in [M. Abramovitz, Handbook of Mathematical Functions , 9 th edition, 1970, p. 952] ( ) ( ) = 2 2 2 1 2 2 exp 2 i i i i R R R R W π 0 R n = Δ t 2kTm β R 2 n = The second fluctuation-dissipation theorem takes care of balancing the increase in energy due to the random fluctuating force and the decrease in energy due to the friction force. where <…> denotes average over an equilibrium ensemble and W(R i ) is the probability distribution of the random force. To implement in MD we have to average over a timestep Δ t: ( ) dt t R Δ t 1 R 1 n n t t n + = 0 R R 1 n n = + • thermal motion of particles is driven by random force • a friction force and a random force R i are added to the equation of motion

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