Multicanonical Monte Carlo ensemble growth algorithm

We present an ensemble Monte Carlo growth method to sample the equilibrium thermodynamic properties of random chains. The method is based on the multicanonical technique of computing the density of states in the energy space. Such a quantity is temperature independent, and therefore microcanonical and canonical thermodynamic quantities, including the free energy, entropy, and thermal averages, can be obtained by reweighting with a Boltzmann factor. The algorithm we present combines two approaches: The first is the Monte Carlo ensemble growth method, where a “population” of samples in the state space is considered, as opposed to traditional sampling by long random walks, or iterative single-chain growth. The second is the flat-histogram Monte Carlo, similar to the popular Wang-Landau sampling, or to multicanonical chain-growth sampling. We discuss the performance and relative simplicity of the proposed algorithm, and we apply it to known test cases.

Figure 1

Density of states (top) and specific heat (bottom) for ISAW chains with $N=21$ on a square lattice, and $N=28$ on a cubic lattice. Markers represent our simulations, while dots (connected by solid lines, for eye guidance) are from exact enumeration. The error bars (reported) are smaller than marker sizes. The agreement between the simulation and the exact values is remarkable.

Figure 2

Normalized density of states (and specific heat, inset) of an ISAW chain on a square lattice, for lengths $N=40,41,42$. The number of replicas is $M_0=3\times {10}^5$. The solid lines connect the exact enumeration results from Refs. [16, 17].

Figure 3

Density of states of a $N=40$ ISAW confined in a spherical three-dimensional cavity, as a function of the cavity radius $R$. The curves are shifted to zero at their maximum value.