content_image/404021.jpg content_image/404022.jpg content_image/404020.jpg
# Reformulation of the Stochastic Potential Switching Algorithm and a Generalized Fourtuin-Kasteleyn Representation 

Department of Applied Physics, Tohoku University, Sendai, 980-8579 

Munetaka Sasaki 

(Dated: November 16, 2018) 

A new formulation of the stochastic potential switching algorithm is presented. This reformulation naturally leads us to a generalized Fourtuin-Kasteleyn representation of the partition function Z . A formula for internal energy E and that of heat capacity C are derived from derivatives of the partition function. We also derive a formula for the exchange probability in the replica exchange Monte Carlo method. By combining the formulae with the Stochastic Cutoff method, we can greatly reduce the computational time to perform internal energy and heat capacity measurements and the replica exchange Monte Carlo method in long-range interacting systems. Numerical simulations in three dimensional magnetic dipolar systems show the validity of the method. 

One of the most challenging subjects in the field of computational physics is to develop efficient methods for long-range interacting systems. The difficulty of long- range interacting systems comes from the fact that the number of interactions rapidly increases with increasing the number of elements N of the system. For exam- ple, in the case of systems with pairwise interactions, the number of interactions is proportional to N $^{2}$. Therefore, if one uses a naive simulation method in such systems, the computational time per one step rapidly increases in proportion to N $^{2}$, which is quite contrast to the case of short-range interacting systems in which the compu- tational time increases in proportion to N . In order to overcome the difficulty, many simulation methods have been proposed until now [1–10]. 

Recently, the author and Matsubara have developed an efficient Monte Carlo (MC) method called Stochas- tic CutOff (SCO) method for long-range interacting sys- tems [11]. The basic idea of the method is to switch long-range interactions V$_{ij}$ stochastically to either zero or a pseudointeraction ¯ V$_{ij}$ with the detailed balance con- dition satisfied. Interactions are switched by using the Stochastic Potential Switching (SPS) algorithm [12, 13]. Because most of the distant and weak interactions are eliminated by being switched to zero, the SCO method greatly reduces the number of interactions and computa- tional time in long-range interacting systems [14]. Fur- thermore, this method does not involve any approxima- tion because the detailed balance condition is satisfied strictly. Fukui and Todo have recently developed an ef- ficient MC method based on similar strategy by use of different pseudointeractions and different way of switch- ing interactions [15]. 

In the present work, we reformulate the SPS algorithm which is used in the SCO method. This reformulation gives us a generalized version of the Fourtuin-Kasteleyn representation of the partition function in the Ising fer- romagnetic model [16, 17]. This representation of the partition function is used to derive new formulae for in- 

PACS numbers: 02.70.Tt, 02.70.Rr, 05.10.Ln, 75.10.Hk 

## I. INTRODUCTION 

ternal energy and heat capacity measurements. Since these formulae consist of only terms which survive as ¯ V$_{ij}$ in the above-mentioned potential switching process, the computational time for the measurements are reduced greatly. We also derive an formula which reduces the computational time to estimate exchange probability in the replica exchange MC method [18]. In order to ver- ify the formulae, we perform some MC simulations in three dimensional magnetic dipolar systems. The results clearly show the validity of the formulae. 

The organization of the paper is as follows. In § II, we reformulate the stochastic potential switching algorithm. A generalized Fourtuin-Kasteleyn representation of the partition function is presented in this section. In § III and § IV, we derive formulae for internal energy and heat capacity measurements and a formula for the replica ex- change MC method, respectively. The validity of these formulae is confirmed numerically in § VI. Section VII is devoted to a summary and discussions. 

Before reformulating the SPS algorithm, we briefly ex- plain the SPS algorithm [12, 13]. We hereafter consider a system with pairwise long-range interactions described by the Hamiltonian 

where S$_{i}$ is a variable associated with the i -th element of the system. In this algorithm, V$_{ij}$ is stochastically switched to either ˜ V$_{ij}$ or ¯ V$_{ij}$ with a probability of P$_{ij}$ or 1 − P$_{ij}$ , respectively. The probability P$_{ij}$ is 

<img src='content_image/404021.jpg'>

where β is the inverse temperature, 

## II. REFORMULATION OF THE SPS ALGORITHM 

<img src='content_image/404022.jpg'>

<img src='content_image/404020.jpg'>

