Stochastic approximation Monte Carlo and Wang–Landau Monte Carlo applied to a continuum polymer model
Introduction
Wang–Landau Monte Carlo simulations belong to the broader class of flat-histogram Monte Carlo simulations aimed at obtaining an estimate of the density of states of a system. They were first applied to spin models [1], [2] but their use was soon extended to polymer models [3], [4], [5], [6], [7], [8], [9]. In the last ten years WLMC has been shown to be a very powerful tool for the determination of the density of states of polymer models, see, e.g., [10], [11], [12], [13], [14], [15], [16] or reviews in [17], [18], [19], comparable to multi-canonical simulation approaches [20], [21], [22]. The convergence properties of the original formulation of WLMC has been an issue of controversy [23], [24] and simulations clearly pointed to a saturation of the final error [25] irrespective of the simulation effort employed. This has led to the suggestion of a modification of the original method [26], [27] which was shown to converge in the selected applications. However, it has been pointed out [28] that the practical convergence properties also of this version of WLMC may be strongly dependent on the physical model it is applied to.
In parallel to much of this development, Stochastic Approximation Monte Carlo (SAMC) has been formulated [29], [30] in the context of stochastic optimization problems. Liang et al. [30] showed that WLMC could be seen as a version of SAMC and using the mathematical background of stochastic approximation methods they proved the convergence of SAMC. Understanding the necessary conditions for this convergence also clarified why the modified WLMC [26], [27] converged, whereas the original version [1], [2] yielded an excellent (and adjustable) approximation to the density of states, but did not strictly speaking converge.
We will discuss in the next section the theoretical background of the SAMC and WLMC ideas and discuss in which way WLMC can be seen as a version of SAMC. This discussion will also point out strong and weak points for the practical use of both methods. An abridged first comparison of the two methods has been presented in [31]. In Section 3 we will introduce the model for which both methods will then be compared with respect to their practical usefulness. Section 4 will then compare the two methods concerning quality of convergence with a given simulation time effort and discuss the tunability of the SAMC method, which has not been systematically analyzed for applications to physics problems so far, by suitable choice of its free parameters. Finally, Section 5 will present our conclusions.
Section snippets
Theoretical background
The idea of the WLMC method [1], [2] is that an unbiased random walk over micro-states (random walk in configuration space) will visit every admissible energy value of a Hamiltonian defined on this configuration space proportional to the number of micro-states, , with that energy (or the measure of the set of points in configuration space having an energy in the interval for continuous ranges of energies and states). The quantity, , is the density of states of the model. When
Model
We study a model based on the model of tangent hard sphere chains used in [11], [12]. All repeat units in the chain except bonded ones have an attractive interaction of the square-well type The hard sphere diameter, , sets the length scale, and the well depth, sets the energy (temperature) scale. The admissible energy values of this model are discrete where is the number of monomer pairs with a distance falling into the well range. The admissible
Results
We begin the discussion of our results with an analysis of the CPU time needed by the WLMC and SAMC simulations to determine the density of states of chains with lengths .
For the SAMC method we empirically find the following dependence of the program run-time, , on simulation parameters Here is the final value of , i.e., equivalent to the final value in WLMC. The term is the total number of simulation time steps, counted in units of
Conclusions
We have presented in this work a discussion and comparative study of Wang–Landau Monte Carlo (WLMC) and Stochastic Approximation Monte Carlo (SAMC). The latter can be considered a generalization and mathematical background to WLMC and provides proof of convergence for WLMC simulations employing a time dependence of the modification factor. We have assessed the relative practical merits of the simulation algorithms when applied to a continuum polymer model comprised of tangent or fused hard
Acknowledgment
The authors acknowledge funding by the German Science Foundation in SFB TRR102 projects A2 and A7.
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