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Peter Virnau , M. M üller, B. Mognetti, L. Yelash, K. Binder

How to overcome free energy barriers in grandcanonical simulations. Peter Virnau , M. M üller, B. Mognetti, L. Yelash, K. Binder. Leipzig, 11/2007. Overview & Credits. Grandcanonical Simulations ( How to determine critical points?)

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Peter Virnau , M. M üller, B. Mognetti, L. Yelash, K. Binder

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  1. How to overcome free energy barriers in grandcanonical simulations Peter Virnau, M. Müller, B. Mognetti, L. Yelash, K. Binder Leipzig, 11/2007

  2. Overview & Credits Grandcanonical Simulations • (How to determine critical points?) • How to overcome free energy barriers? Successive Umbrella Sampling • (How to calculate interface tension?) Applications • Systematic approach to model and coarse-grain small molecules (Noble gases, Alkanes, CO2, benzene) G.M. Torrie and J.P. Valleau, J. Comput. Phys. 23, 187 (1977) B. Berg and T. Neuhaus, Phys. Rev. Letters 68, 9 (1992) F. Wang and D.Landau, Phys. Rev. Letters 86, 2050 (2001)

  3. Grandcanonical Simulations liquid P(n) gas particle number n Problem: Regions of small probability (free energy barriers) algorithms-> applications

  4. Free energy barriers P(n) particle number n (-H(n)/kT) modify Hamiltonian: H´(n) = H(n) + kT w(n) weight function: ideal:w(n)=ln P(n) good estimate for w(n) ->“flat” histogram algorithms-> applications

  5. How to generate a good w(n) ? P(n) particle number n 1. Histogram-reweighting (extrapolate data from previous simulation) -> limited range, tedious 2. Adjust w(n) during simulation i.e., Wang-Landau (-> violates detailed balance) algorithms-> applications

  6. Successive umbrella sampling P(n) particle number n Idea: - simulate windows successively - generate w(n) for the next window by extrapolation Adv: - works everywhere - efficient P. Virnau and M.Müller, J. Chem. Phys. (2004) algorithms-> applications

  7. Successive umbrella sampling Analysis:no violation of detailed balance -> errors are controlled and can be calculated: Errors are independent of window size single window: t = O(n2) window size one: t = n O(1) = O(n)??? D 0 t n t = n O(n) = O(n2) algorithms-> applications

  8. Noble gases • Potential:Lennard-Jones • Strategy: • Equate critical points of simulation and experiment • e and s How well is the rest of the diagram described? algorithms->applications

  9. Alkanes Mapping3CH2=1 bead Potential Lennard-Jones + FENE algorithms->applications

  10. Carbon dioxide Mapping: CO2= 1 LJ bead + quadrupole moment algorithms->applications

  11. Benzene Mapping: C6H6= 1 LJ bead + quadrupole moment algorithms->applications

  12. Take home messages How to overcome free energy barriers? Successive umbrella sampling Modeling Equate critical points of simulation and experiments to obtain simulation parameters  very good agreement with phase change data

  13. Critical points Binder cumulants U4 = <M4>/<M2>2 M = r-<r> algorithms-> applications

  14. Free energy DF/kT particle number n Relation btw. P(n) und F: F = -kT ln(Zcan) = -kT lnP(n) + const. Interface tension: g = DF/2L2 algorithms-> applications

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