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 φ. 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.

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. φ 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(φ), where φ 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(φ) restricts the simulation to a range of φ. The (circles in Fig. 1).
biased sampling generates a distribution ˜
P (φ) which How do we know when there is a spinodal?
is related to the distribution P (φ) of the unbiased • A spinodal is the limit of stability, and is defined by
system. the divergence of χ. χ will diverge at an inflection
We obtain a series of estimates of P (φ) by using point of the free energy.
different bias potentials Vk in different windows: Fig. 2: ∆F vs. φ for 1D long-range Ising model • Because a spinodal is a mean-field effect, a system
˜
Pk (φ) ∝ exp(βVk )Pk (φ). at three values of h. Theory: hs = 1.27 . Correct with short-range interactions cannot have a spinodal.
probability: hs ≈ 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 (φ) from Pk (φ). The free energy F (φ) ∝ P (φ) .
• 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
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