Molecular Monte Carlo method

This article or section needs to be revised. Details should be given on the talk page. Please help to improve it , and then remove this flag.

Play media file

Monte Carlo simulation of liquid argon

The molecular Monte Carlo method , also called kinetic Monte Carlo method, (in soft matter physics mostly just called Monte Carlo ) is a Monte Carlo simulation and is used in statistical mechanics of physics for modeling equilibrium properties atomic to molecular level and can in individual cases achieve an acceleration of a factor of over 10 ¹⁰ compared to molecular dynamics. It can determine the potential energy as well as provide statistical information about the position of atoms and also determine state variables from Boltzmann statistics . Monte Carlo and Molecular Dynamics are the most common models of fluids at the atomic level. The empirical force fields that are used for this come from both ab initio calculations and experiments and are used for both molecular dynamics and the molecular Monte Carlo method. Some program systems use multi-atom models ("Coarse-grained model"), they do not apply the molecular Monte Carlo method to individual atoms, but to the center of gravity of groups of atoms or whole molecules.

procedure

First you choose the starting position of the atoms intelligently. Then you choose a new set of coordinates for the atoms; when using the Metropolis algorithm , which is the most common, individual atoms are shifted at a predefined distance and in a random direction. In addition to the Metropolis algorithm, there are e.g. B. the PHOBOS-Glauber-Monte-Carlo method. Since this happens in a force field , it usually leads to a change in the internal energy . In the Metropolis algorithm, a state is always assumed if the new energy state is lower. However, if the new state has a higher internal energy, this state is only accepted with a certain probability (less than 1), this depends on the internal energy difference of the two states as well as on the temperature, so that the molecular oscillation or the Brownian movements depicted. If one were only to assume states of lower energy, one would minimize the entropy and thus make a simulation at absolute zero. Since in the molecular Monte Carlo method the atoms are not shifted according to the force fields and Newton's laws , the Monte Carlo method can converge computationally faster, especially in gaseous states, since no (long-range) forces have to be calculated, which is due to the Diagonalization of large matrices can be processor-intensive .

In calculation steps this means:

1. Choose the starting coordinates of the atoms.
2. Check whether the termination criterion (number of steps, convergence criterion) has been met.
3. Choose a new coordinate set of atoms, here normally one or more atoms are shifted at a given distance in a random direction from the previous position.
4. Calculate the probability with which the state is accepted, here the Metropolis algorithm is the most common to determine the probability. The state is assumed in the Metropolis algorithm when the potential energy is low and in the event that the potential energy is larger, this is with a probability of e ^{- .beta. ∙ .DELTA V} adopted. (With the PHOBOS-Glauber-Monte-Carlo, the new state is assumed with a probability of .) ${\displaystyle 1/{\left(\exp \left(\beta \cdot {\Delta V}\right)+1\right)}}$
   • If the state is accepted, the coordinate set previously determined in point 3 is accepted and used as the starting point for the next MC steps.
   • If the state is not accepted, steps 3 and 4 are repeated until a state is found that is accepted.
5. Jump to point 2.

Kinetic Monte Carlo method

The kinetic Monte Carlo method is a hybrid Monte Carlo method and complements the molecular Monte Carlo method in that rates of state transitions can also be defined and thus time is indirectly modeled. The kinematic Monte Carlo method and the dynamic Monte Carlo method are largely identical.

The so-called master equation is used for the phase transition rate:

${\displaystyle {\frac {\mathrm {d} {\mathcal {P}}_{\alpha }(t)}{\mathrm {d} t}}=\left(\sum _{\beta \neq \alpha }W_{\alpha \beta }\cdot {\mathcal {P}}_{\beta }(t)\right)-\sum _{\beta \neq \alpha }W_{\beta \alpha }\cdot {\mathcal {P}}_{\alpha }(t)}$

Procedure for the rejectionless kinetic Monte Carlo simulation:

1. You define the starting position of the atoms at the point in time ${\displaystyle k}$ ${\displaystyle t=0}$
2. The transition rates are calculated for all possible transitions into the next state , with transitions that do not occur . ${\displaystyle N_{k}}$ ${\displaystyle i}$ ${\displaystyle r_{k,i}\geq 0}$ ${\displaystyle r_{k,i}=0}$
3. It is the partial sum of the transition rates: . The sum total of the transition rates is . ${\displaystyle R_{k,i}}$ ${\displaystyle R_{k,i}=\sum _{j=1}^{i}r_{k,j}}$ ${\displaystyle Q_{k}=R_{k,N_{k}}=\sum _{j=1}^{N_{k}}r_{k,j}}$
4. The states are assumed with a probability of . ${\displaystyle 0\leq {\frac {r_{k,i}}{Q_{k}}}\leq 1}$
   • A random number is determined and that transition is chosen for which: ${\displaystyle 0<\alpha \leq 1}$ ${\displaystyle r_{k,i}}$ ${\displaystyle R_{k,i-1}<\alpha \cdot Q_{k}\leq R_{k,i}}$
5. The time is set to set, with which is a random number between 0 to 1. ${\displaystyle t_{i}=t_{k}+\Delta t}$ ${\displaystyle \Delta t={\frac {ln(1/a)}{Q_{k}}},}$ ${\displaystyle a}$
6. Steps 2–5 are repeated until the termination criterion is met.

Quantum Monte Carlo method

Quantum Monte Carlo method algorithm

In the Quantum Monte Carlo method, a quantum many-body system is simulated. The procedure here is very similar to the procedure for atomistic modeling:

1. Initial condition
2. Suggest a move
3. Determination of the probability
4. Metropolis: acceptance / rejection
5. (If accepted :) Update of the electron position
6. Calculate the energy
7. Jump to point 2 until the termination criterion is met
8. Output of the result

Difference to molecular dynamics

→ Main article : Molecular dynamics

The molecular Monte Carlo method is preferred for gaseous states, whereas molecular dynamics is preferred for solid or liquid states. The methods can also be combined or alternated to achieve faster convergence. In the Monte Carlo method, neither the dynamics nor the time of the system are mapped directly, but only the state variables can be determined.

Multi-scale models

Modeling method depending on the order of magnitude

The molecular Monte Carlo method and molecular dynamics are semi-empirical methods and are suitable for systems that can no longer be modeled with ab initio calculations in a reasonable computing time, but in which, in contrast to continuum mechanics , the individual atoms are still used and not only depicts their density.

Individual evidence

1. ↑ ^{a b} Jihan Kim, Jocelyn M. Rodgers, Manuel Athenes, Berend Smit: Molecular monte carlo simulations using graphics processing units: To waste recycle or not? In: Journal of chemical theory and computation . tape 7 , no. 10 . ACS Publications, October 2011, ISSN  1549-9618 , p. 3208-3222 , doi : 10.1021 / ct200474j , PMID 26598157 .
2. ↑ ^{a b c d e f} Bingqing Cheng : Monte Carlo is not just a sampling method. It's also a philosophy. Quora , 2004, accessed July 23, 2017 . 
3. ^ ^{A b c} Alfred B. Bortz, Malvin H. Kalos, Joel L. Lebowitz: A new algorithm for Monte Carlo simulation of Ising spin systems . In: Collection of Computational Physics . tape 17 , no. 1 . Elsevier, 1975, ISSN  0021-9991 , pp. 10-18 , doi : 10.1016 / 0021-9991 (75) 90060-1 , bibcode : 1975JCoPh..17 ... 10B .
4. ↑ ^{a b c} Johannes Schlundt: Modeling and simulation Monte Carlo simulation. (PDF) Universität Hamburg, January 7, 2013, pp. 19-21 , accessed on July 4, 2017 . 
5. ^ David P. Landau, Kurt Binder: A guide to Monte Carlo simulations in statistical physics . 3. Edition. Cambridge university press, New York, NY, USA 2009, ISBN 978-0-511-65176-2 , bibcode : 2009gmcs.book ..... L (538 pp., PDF ).
6. ↑ ^{a b} Ruslan Zinetullin: For the calibration of kinetic Monte Carlo simulations through molecular dynamics . Faculty of Physics, University of Duisburg-Essen, Duisburg November 16, 2010 (147 pages).
7. ↑ ^{a b c} David J. Earl, Michael W. Deem: Monte Carlo simulations . In: Andreas Kukol (Ed.): Molecular modeling of proteins . Humana Press, 2008, ISBN 978-1-59745-177-2 , ISSN  1064-3745 , pp. 25–36 , doi : 10.1007 / 978-1-59745-177-2_2 ( springer.com [PDF; accessed July 26, 2017]).
8. ^ ^{A b} William L. Jorgensen, Julian Tirado-Rives: Monte Carlo vs molecular dynamics for conformational sampling . In: The anthology of Physical Chemistry . tape 100 , no. 34 . ACS Publications, August 1996, pp. 14508-14513 , doi : 10.1021 / jp960880x .
9. ↑ DNA model introduction. In: oxDNA. Retrieved July 2, 2017 . 
10. ↑ Coarse-Grained Molecular Dynamics. Retrieved July 6, 2017 . 
11. ↑ Coarse-grained models. Retrieved July 6, 2017 . 
12. ↑ Kurt Binder: Monte Carlo and molecular dynamics simulations in polymer science . Oxford University Press, 1995 ( limited preview in Google Book Search).
13. ↑ ^{a b c d e f g h i j} Jan K. Labanowski: Molecular Dynamics and Monte Carlo. In: The Computational Chemistry List. Retrieved July 7, 2017 . 
14. ↑ ^{a b c} David Holec: 308,882 atomistic materials modeling . TU Vienna, Vienna November 2016 ( tuwien.ac.at [accessed on November 28, 2016] techreport).
15. ↑ ^{a b} Daan Frenkel, Berend Smit: Monte Carlo simulations and molecular dynamics . January 4, 2007.
16. ↑ Ahmad Kadoura, Amgad Salama, Shuyu Sun: Speeding up Monte Carlo molecular simulation by a non-conservative early rejection scheme . In: Mol Simul . tape 42 , no. 3 . Taylor & Francis, April 2016, ISSN  0892-7022 , pp. 229–241 , doi : 10.1080 / 08927022.2015.1025268 (English).
17. ^ ^{A b} Frank Michael Kuhn: Kinetic Monte Carlo simulations of reactions on supported nanoparticles . Ed .: O. Deutschmann, L. Kunz, S. Tischer. Institute for Technical Chemistry and Polymer Chemistry of the Faculty of Chemistry and Biosciences; Karlsruhe Institute of Technology , Karlsruhe November 8, 2011, chap. 2.2, p. 7–24 (73 pp., PDF [accessed July 4, 2017] diploma thesis).
18. ^ ^{A b c d e f g} Johannes Schlundt: Written elaboration on the lecture Monte Carlo simulation. (PDF) University of Hamburg, March 20, 2013, pp. 9–10 , accessed on July 4, 2017 . 
19. ↑ ^{a b c d e f g h} Jose Guilherme Vilhena: An introduction to Quantum Monte Carlo. (PDF) In: TDDFT.org. Martin Luther University Halle-Wittenberg - Institute for Physics, July 7, 2010, accessed on July 22, 2017 .  
20. ^ ^{A b} Monte Carlo or Molecular Dynamics. In: Condensed Matter Theory, University of Durham . Accessed July 1, 2017 .