We propose a modified power method for computing the subdominant eigenvalue $λ_2$ of a matrix or continuous operator. Here we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfuction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing the $λ_2$ of various Markov chain transition matrices. We first computed ${λ_2}$ for several one and two dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as a function of temperature and applied magnetic field. Next, we computed $λ_2$ for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size $Δ$. Based on the $λ_2$ criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule-of-thumb of adjusting $Δ$ so the Metropolis acceptance rate is around 50% range is often sub-optimal. In general, as a function of temperature or $Δ$, $λ_2$ for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.
Exchange Monte Carlo for continuous-space Path Integral Monte Carlo simulation
The University of Tokyo（东京大学）；Institute for Physics of Intelligence, The University of Tokyo（东京大学物理智能研究所）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）
Thermodynamics of primitive model electrolytes in the symmetric and modified Poisson-Boltzmann theories. A comparative study with Monte Carlo simulations
用对称Poisson-Boltzmann理论和改进的Poisson-Boltzmann理论计算了不同摩尔浓度下原始模型电解质溶液的渗透系数、单个活度系数和平均活度系数。理论结果与Abbas等人[Fluid Phase Equilib.，2007，260，233; J. Phys. Chem. B，2009，113，5905]。修改后的泊松-玻尔兹曼预测与“精确”模拟结果之间的协议几乎是定量的一价盐，而半定量或更好的更高和多价盐。另一方面，对称的泊松-玻尔兹曼结果对于单价体系非常好，但在较高浓度和/或多价体系中往往会偏离。HCl溶液（单个活度和平均活度）和NaCl溶液（仅平均活度）的活度系数的一些最近的实验值也与对称和修正的Poisson-Boltzmann理论和蒙特卡罗模拟进行了比较。
