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Register a new userSampling the density of states
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Authors
M. Guagnelli
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It is shown that the algorithm introduced in [1] and conceived to deal with continuous degrees of freedom models is well suited to compute the density of states in models with a discrete energy spectrum too. The q=10 D=2 Potts model is considered as a test case, and it is shown that using the Maxwell construction the interface free energy can be obtained, in the thermodynamic limit, with a good degree of accuracy.
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We apply Density Matrix Renormalization Group methods to study the phase diagram of the quantum ANNNI model in the region of low frustration where the ferromagnetic coupling is larger than the next-nearest-neighbor antiferromagnetic one. By Finite Size Scaling on lattices with up to 80 sites we locate precisely the transition line from the ferromagnetic phase to a paramagnetic phase without spatial modulation. We then measure and analyze the spin-spin correlation function in order to determine the disorder transition line where a modulation appears. We give strong numerical support to the conjecture that the Peschel-Emery one-dimensional line actually coincides with the disorder line. We also show that the critical exponent governing the vanishing of the modulation parameter at the disorder transition is $beta_q = 1/2$.
The Wang-Landau method is used to study the magnetic properties of the giant paramagnetic molecule Mo_72Fe_30 in which 30 Fe3+ ions are coupled via antiferromagnetic exchange. The two-dimensional density of states g(E,M) in energy and magnetization space is calculated using a self-adaptive version of the Wang-Landau method. From g(E,M) the magnetization and magnetic susceptibility can be calculated for any temperature and external field.
For a classical system of noninteracting particles we establish recursive integral equations for the density of states on the microcanonical ensemble. The recursion can be either on the number of particles or on the dimension of the system. The solution of the integral equations is particularly simple when the single-particle density of states in one dimension follows a power law. Otherwise it can be obtained using a Laplace transform method. Since the Laplace transform of the microcanonical density of states is the canonical partition function, it factorizes for a system of noninteracting particles and the solution of the problem is straightforward. The results are illustrated on several classical examples.
We revisit the question of describing critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. Critical exponents and central charge are determined by optimizing the exponents such as to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $lambdaphi^4$ with mass $mu^2$ and lattice spacing $a$, we demonstrate a double data collapse for the correlation length $ delta xi(mu,lambda,D)=tilde{xi} left((alpha-alpha_c)(delta/a)^{-1/ u}right)$ with $D$ the bond dimension, $delta$ the gap between eigenvalues of the transfer matrix, and $alpha_c=mu_R^2/lambda$ the parameter which fixes the critical quantum field theory.
The free energetics of water density fluctuations near a surface, and the rare low-density fluctuations in particular, serve as reliable indicators of surface hydrophobicity; the easier it is to displace the interfacial waters, the more hydrophobic the underlying surface. However, characterizing the free energetics of such rare fluctuations requires computationally expensive, non-Boltzmann sampling methods like umbrella sampling. This inherent computational expense associated with umbrella sampling makes it challenging to investigate the role of polarizability or electronic structure effects in influencing interfacial fluctuations. Importantly, it also limits the size of the volume, which can be used to probe interfacial fluctuations. The latter can be particularly important in characterizing the hydrophobicity of large surfaces with molecular-level heterogeneities, such as those presented by proteins. To overcome these challenges, here we present a method for the sparse sampling of water density fluctuations, which is roughly two orders of magnitude more efficient than umbrella sampling. We employ thermodynamic integration to estimate the free energy differences between biased ensembles, thereby circumventing the umbrella sampling requirement of overlap between adjacent biased distributions. Further, a judicious choice of the biasing potential allows such free energy differences to be estimated using short simulations, so that the free energetics of water density fluctuations are obtained using only a few, short simulations. Leveraging the efficiency of the method, we characterize water density fluctuations in the entire hydration shell of the protein, ubiquitin; a large volume containing an average of more than six hundred waters.
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