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Umbrella Sampling in the Long-Range Ising Model

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rjchacko

Umbrella sampling is used to study nucleation in systems like the Ising model by adding a bias potential to restrict simulations to a range of the reaction coordinate ¦Õ. Multiple histograms of the biased distributions are combined to obtain the unbiased distribution and free energy. In the long-range Ising model, clusters can be mapped to a percolation model using a bond probability dependent on magnetization. The size of the largest cluster is used as the reaction coordinate. The choice of reaction coordinate affects results - underestimating cluster probabilities yields incorrect predictions while rigorous percolation mapping is consistent with theory. Vanishing of the free energy barrier does not necessarily imply a spinodal due to mean-field effects in long-range systems.

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Umbrella Sampling in the Long-Range Ising Model Ranjit Chacko Harvey Gould W. Klein Clark University Clark University Boston University How does the choice of the reaction coordinate affect Questions the results? ? Umbrella sampling has been used to study ? For largest cluster to be a good reaction coordinate the nucleation in the Ising model, Lennard-Jones and clusters must be statistically independent and large other particle systems. How does the order clusters must be rare. parameter affect the magnitude of the free energy barrier ?F ? ? Rigorous percolation mapping for clusters gives results consistent with theory (diamonds in Fig. 1). ? Does vanishing of ?F imply the existence of a Other choices give the wrong physics. spinodal? ? Underestimating p predicts hs to be less than mean- Fig. 1: ?F vs. ¦Õ for 1D long-range Ising model field value (crosses in Fig. 1). (L = 512, R = 64) at h = 1.2 and T = 4Tc /9. What is umbrella sampling? ? Overestimating p increases predicted value of hs . Changing p changes the value of?F . Instead of simulating the system of interest, a bias Hence, there appears to be a metastable state beyond potential V(¦Õ), where ¦Õ is the reaction coordinate, the actual limit of stability. Various workers have is added to the Hamiltonian of the original system. studied nucleation in the Ising model using p = 1 V(¦Õ) restricts the simulation to a range of ¦Õ. The (circles in Fig. 1). biased sampling generates a distribution ? P (¦Õ) which How do we know when there is a spinodal? is related to the distribution P (¦Õ) of the unbiased ? A spinodal is the limit of stability, and is defined by system. the divergence of ¦Ö. ¦Ö will diverge at an inflection We obtain a series of estimates of P (¦Õ) by using point of the free energy. different bias potentials Vk in different windows: Fig. 2: ?F vs. ¦Õ for 1D long-range Ising model ? Because a spinodal is a mean-field effect, a system ? Pk (¦Õ) ¡Ø exp(¦ÂVk )Pk (¦Õ). at three values of h. Theory: hs = 1.27 . Correct with short-range interactions cannot have a spinodal. probability: hs ¡Ö 1.25. p = 0.89: hs > 1.25. Spinodal effects can appear if the interaction is The multiple histogram method is used to calculate p = 0.02: hs 1.1. sufficiently long-range. P (¦Õ) from Pk (¦Õ). The free energy F (¦Õ) ¡Ø P (¦Õ) . ? The vanishing of ?F is not a sufficient condition for a spinodal, because the inflection point can occur at the maximum value of the order parameter. See Fig. 3. What are the clusters in the long-range Ising model? References The long-range Ising model can be rigorously [1] G. Torrie and M. Valleau, J. Comp. Phys. 23, 187 (1977). mapped onto a percolation model by placing bonds [2] A. Ferrenberg and R. Swendsen, Phys. Rev. Lett. 63, 1195 (1989). [3] W. Klein, Phys. Rev. Lett. 65, 1462 (1990). between parallel spins with bond probability [4] A. Pan and D. Chandler, J. Phys. Chem. B 108, 19681 (2004). p = 1 ? exp(?2¦ÂJ(1 + m)), [5] D. Frenkel, P. R. ten Wolde, and M. J. Ruiz-Montero, J. Chem. where m is the magnetization. The clusters are Fig. 3: Free energy vs. m for 2D nearest Phys. 102 9932 (1996). neighbor Ising model for different values of h [6] H. Wang, H. Gould, and W. Klein, Phys. Rev. E 76, 031604 (2007). statistically independent. We use the size of the [7] P. Bhimalapuram, S. Chakrabarty, and B. Bagchi, cond-mat largest cluster as a reaction coordinate. at T = 4Tc /9. ?F ¡ú 0 at h = 0.91. ¦Ö does 0702158. not diverge.

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Umbrella Sampling in the Long-Range Ising Model

  • 1. Umbrella Sampling in the Long-Range Ising Model Ranjit Chacko Harvey Gould W. Klein Clark University Clark University Boston University How does the choice of the reaction coordinate affect Questions the results? ? Umbrella sampling has been used to study ? For largest cluster to be a good reaction coordinate the nucleation in the Ising model, Lennard-Jones and clusters must be statistically independent and large other particle systems. How does the order clusters must be rare. parameter affect the magnitude of the free energy barrier ?F ? ? Rigorous percolation mapping for clusters gives results consistent with theory (diamonds in Fig. 1). ? Does vanishing of ?F imply the existence of a Other choices give the wrong physics. spinodal? ? Underestimating p predicts hs to be less than mean- Fig. 1: ?F vs. ¦Õ for 1D long-range Ising model field value (crosses in Fig. 1). (L = 512, R = 64) at h = 1.2 and T = 4Tc /9. What is umbrella sampling? ? Overestimating p increases predicted value of hs . Changing p changes the value of?F . Instead of simulating the system of interest, a bias Hence, there appears to be a metastable state beyond potential V(¦Õ), where ¦Õ is the reaction coordinate, the actual limit of stability. Various workers have is added to the Hamiltonian of the original system. studied nucleation in the Ising model using p = 1 V(¦Õ) restricts the simulation to a range of ¦Õ. The (circles in Fig. 1). biased sampling generates a distribution ? P (¦Õ) which How do we know when there is a spinodal? is related to the distribution P (¦Õ) of the unbiased ? A spinodal is the limit of stability, and is defined by system. the divergence of ¦Ö. ¦Ö will diverge at an inflection We obtain a series of estimates of P (¦Õ) by using point of the free energy. different bias potentials Vk in different windows: Fig. 2: ?F vs. ¦Õ for 1D long-range Ising model ? Because a spinodal is a mean-field effect, a system ? Pk (¦Õ) ¡Ø exp(¦ÂVk )Pk (¦Õ). at three values of h. Theory: hs = 1.27 . Correct with short-range interactions cannot have a spinodal. probability: hs ¡Ö 1.25. p = 0.89: hs > 1.25. Spinodal effects can appear if the interaction is The multiple histogram method is used to calculate p = 0.02: hs 1.1. sufficiently long-range. P (¦Õ) from Pk (¦Õ). The free energy F (¦Õ) ¡Ø P (¦Õ) . ? The vanishing of ?F is not a sufficient condition for a spinodal, because the inflection point can occur at the maximum value of the order parameter. See Fig. 3. What are the clusters in the long-range Ising model? References The long-range Ising model can be rigorously [1] G. Torrie and M. Valleau, J. Comp. Phys. 23, 187 (1977). mapped onto a percolation model by placing bonds [2] A. Ferrenberg and R. Swendsen, Phys. Rev. Lett. 63, 1195 (1989). [3] W. Klein, Phys. Rev. Lett. 65, 1462 (1990). between parallel spins with bond probability [4] A. Pan and D. Chandler, J. Phys. Chem. B 108, 19681 (2004). p = 1 ? exp(?2¦ÂJ(1 + m)), [5] D. Frenkel, P. R. ten Wolde, and M. J. Ruiz-Montero, J. Chem. where m is the magnetization. The clusters are Fig. 3: Free energy vs. m for 2D nearest Phys. 102 9932 (1996). neighbor Ising model for different values of h [6] H. Wang, H. Gould, and W. Klein, Phys. Rev. E 76, 031604 (2007). statistically independent. We use the size of the [7] P. Bhimalapuram, S. Chakrabarty, and B. Bagchi, cond-mat largest cluster as a reaction coordinate. at T = 4Tc /9. ?F ¡ú 0 at h = 0.91. ¦Ö does 0702158. not diverge.