Dynamics, Monte Carlo

The Langevin equations are solved in the same way as those of MD, with the additional stochastic force drawn using a random number generator. 

Las Vegas, Atlantic City and Monte Carlo are notorious among upright citizens for day and night use of such random number generators as billiards, roulette or cards. Because of this, the idea and even the name of Monte Carlo has been accepted in mathematics, physics, chemistry and biology. The key concept is that a random number, when drawn successively many times, may serve to create a sequence of system snapshots. 

All this began from an idea of the mathematician from Lwow, then in Poland (now Lviv in the Ukraine) Stani aw Marcin Ulam. 

Perhaps an example will best explain the Monte Carlo method. I have chosen the methodology introduced to the protein folding problem by Andrzej Kolinski 

Stanistaw Ulam (1909-1984), first associated with the University of Lwow, then professor at the Harvard University, University of Wisconsin, University of Colorado, Los Alamos National Laboratory. In Los Alamos Ulam solved the most important bottleneck in hydrogen bomb construction by suggesting that pressure is the most important factor and that sufficient pressure could be achieved by using the atomic bomb as a detonator. Using this idea and an idea of Edward Teller about further amplification of the ignition effect by implosion of radiation, both scholars designed the hydrogen bomb. They both own the US patent for H-bomb production. 

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The probability of a complete Brownian path is then obtained as the product of such single-time-step transition probabilities. For other types of dynamics, such as Newtonian dynamics, Monte Carlo dynamics or general Langevin dynamics, other appropriate short-time-step transition probabilities need to be used [5, 8]. 

Levine, Y. K. (1993). Monte Carlo dynamics study of cis and trans unsaturated hydrocarbon chains, Mol. Phys., 78, 619-628. 

Hoffmann, D., Knapp, E.W. Polypeptide folding with off-lattice Monte Carlo dynamics the method. Eur. Biophys. J. 1996, 24, 387-403. 

Why the calculation based on the kinetic equations and the Monte-Carlo dynamic method should give identical results. What are the practical causes of differences between these methods What are the ways to reduce these differences ... 

A. Rey and J. Skolnick, Comparison of lattice Monte Carlo Dynamics and Brownian... 

Appropriate expressions for the short time transition probability p xt —> t+At) can be derived also for Langevin dynamics with arbitrary friction and for Monte Carlo dynamics [4,12]. 

A. Rey and J. Skolnick, Comparison of lattice Monte Carlo Dynamics and Brownian Dynamics Folding Pathways of -Helical Hairpins, Chem. Phys. 158,199-219 (1991). 

Hence, Eq. (1.48) is valid also for Monte Carlo dynamics and a new trajectory can be accepted if it connects region A with region B. 

MCD Monte Carlo Dynamics A dynamic with a stochastic choice of configurations... 

Milik, M., and Skolnick, J. 1993. Insertion of peptide chains into lipid membranes an off-lattice Monte Carlo dynamics model. Proteins. 15 10. 

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