Patent Publication Number: US-2022237346-A1

Title: Simulation method, simulation apparatus, and non-transitory computer readable medium storing program

Description:
RELATED APPLICATIONS 
     The content of Japanese Patent Application No. 2021-008846, on the basis of which priority benefits are claimed in an accompanying application data sheet, is in its entirety incorporated herein by reference. 
     BACKGROUND 
     Technical Field 
     A certain embodiment of the present invention relates to a simulation method, a simulation apparatus, and a non-transitory computer readable medium storing a program. 
     Description of Related Art 
     As a method for simulating magnetization in a magnetic body, a micromagnetic method and anatomic spin method are known in the related art. In the micromagnetic method, a magnetic body is divided into meshes of several tens of nanometers and analyzed by the finite element method. In the atomic spin method, first-principle calculation is performed in consideration of the atomic arrangement at nanometer intervals and the atomic spin. 
     SUMMARY 
     According to an embodiment of the present invention, there is provided a simulation method including: 
     coarse-graining a plurality of atoms that constitute a magnetic body to be simulated and generating a magnetic body model composed of a collection of a smaller number of particles than an original number of the atoms; 
     applying a magnetic moment to each of a plurality of the particles of the magnetic body model; 
     obtaining a magnetic field due to an interparticle exchange interaction acting between the plurality of particles of the magnetic body model, based on an interatomic exchange interaction of the magnetic body; 
     obtaining an oscillating magnetic field acting on each of the plurality of particles of the magnetic body model, based on an oscillating magnetic field originating from a thermal fluctuation acting on the atoms of the magnetic body; 
     obtaining a total magnetic field acting on each of the plurality of particles of the magnetic body model, based on the magnetic field due to the interparticle exchange interaction and the oscillating magnetic field acting on the particles of the magnetic body model; and 
     time-evolving the magnetic moment of each of the plurality of particles, based on the total magnetic field acting on each of the plurality of particles of the magnetic body model. 
     According to another embodiment of the present invention, there is provided a simulation apparatus including: 
     an input device to which simulation conditions including coarse-grained conditions are input; and 
     a processing device that obtains a distribution of a magnetic moment of a magnetic body to be simulated, based on the simulation conditions input to the input device. 
     The processing device
         coarse-grains a plurality of atoms that constitute the magnetic body, based on the input coarse-grained conditions, and generates a magnetic body model composed of a collection of a smaller number of particles than an original number of the atoms,   applies the magnetic moment to each of a plurality of the particles of the magnetic body model,   obtains a magnetic field due to an interparticle exchange interaction acting between the plurality of particles of the magnetic body model, based on an interatomic exchange interaction of the magnetic body,   obtains an oscillating magnetic field acting on each of the plurality of particles of the magnetic body model, based on an oscillating magnetic field originating from a thermal fluctuation acting on the atoms of the magnetic body,   obtains a total magnetic field acting on each of the plurality of particles of the magnetic body model, based on the magnetic field due to the interparticle exchange interaction and the oscillating magnetic field acting on the particles of the magnetic body model, and       

     time-evolves the magnetic moment of each of the plurality of particles of the magnetic body model, based on the total magnetic field. 
     According to still embodiment of the present invention, there is provided a non-transitory computer readable medium storing a program that causes a computer to execute a process including: 
     coarse-graining a plurality of atoms that constitute a magnetic body to be simulated and generating a magnetic body model composed of a collection of a smaller number of particles than an original number of the atoms; 
     applying a magnetic moment to each of a plurality of the particles of the magnetic body model; 
     obtaining a magnetic field due to an interparticle exchange interaction acting between the plurality of particles of the magnetic body model, based on an interatomic exchange interaction of the magnetic body; 
     obtaining an oscillating magnetic field acting on each of the plurality of particles of the magnetic body model, based on an oscillating magnetic field originating from a thermal fluctuation acting on the atoms of the magnetic body; 
     obtaining a total magnetic field acting on each of the plurality of particles of the magnetic body model, based on the magnetic field due to the interparticle exchange interaction and the oscillating magnetic field acting on the particles of the magnetic body model; and 
     time-evolving the magnetic moment of each of the plurality of particles of the magnetic body model, based on the total magnetic field. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a diagram schematically illustrating a plurality of atoms constituting a magnetic body to be simulated, and  FIG. 1B  is a diagram schematically illustrating a magnetic body model generated by coarse-graining a plurality of atoms constituting the magnetic body illustrated in  FIG. 1A . 
         FIG. 2  is a schematic diagram of two particles for explaining parameters V, W, and S. 
         FIG. 3  is a block diagram of a simulation apparatus according to an embodiment. 
         FIG. 4  is a flowchart of a simulation method according to the embodiment. 
         FIGS. 5A to 5D, 5F, and 5G  are diagrams illustrating the distribution of directions of magnetic moments obtained by simulation in shades, and  FIG. 5E  is a diagram schematically illustrating the directions of the magnetic moments illustrated in  FIGS. 5A to 5D . 
         FIGS. 6A and 6B  are diagrams illustrating the results of simulation with the radii r of the particles being 1 nm and 100 nm, respectively. 
         FIG. 7  is a graph showing a relationship between the normalized magnetization calculated from the simulation results and the temperature. 
     
    
    
     DETAILED DESCRIPTION 
     In the micromagnetic method, it is difficult to perform an analysis in consideration of the interaction occurring in the microscopic region at the atomic level. The atomic spin method can reproduce microscopic physical phenomena, but the size of the calculation area that can be analyzed is small, and it is more difficult to analyze the magnetization of magnetic bodies such as magnetic heads and motor parts, due to limitation such as calculation time and memory capacity. In the atomic spin method described in the related art, a plurality of atoms are coarse-grained to reduce the number of particles to be calculated, thereby relaxing the limitation of the calculation area due to the calculation time, memory capacity, and the like. However, coarse-graining makes it impossible to reproduce the exchange interaction between atoms and the oscillating magnetic field originating from thermal fluctuation. 
     It is desirable to provide a simulation method, a simulation apparatus, and a non-transitory computer readable medium storing a program, capable of reducing the amount of calculation by coarse-graining a plurality of atoms constituting a magnetic body and reproducing the exchange interaction and the oscillating magnetic field to analyze the distribution of magnetization. 
     A simulation method and a simulation apparatus according to an embodiment will be described with reference to  FIGS. 1A to 7 . 
       FIG. 1A  is a diagram schematically illustrating a plurality of atoms  11  constituting a magnetic body  10  to be simulated. Actually, the plurality of atoms  11  are three-dimensionally distributed in the magnetic body  10 , but  FIG. 1A  illustrates an example in which the plurality of atoms  11  are two-dimensionally distributed. In  FIG. 1A , a plurality of atoms  11  located on one virtual plane in the magnetic body  10  are considered. 
     Each of the plurality of atoms  11  has an atomic spin s. The Hamiltonian H 1   exch  of the interatomic exchange interaction acting on the i-th atom  11  is defined by the following expression. 
     
       
         
           
             
               
                 
                   
                     ℋ 
                     i 
                     exch 
                   
                   = 
                   
                     
                       - 
                       J 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         z 
                       
                       ⁢ 
                       
                         
                           s 
                           i 
                         
                         · 
                         
                           s 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Here, J is an exchange interaction intensity coefficient representing the intensity of the exchange interaction between atoms, s i  and s j  are atomic spins of the i-th and j-th atoms, respectively, and sigma means the sum of all the atoms  11  adjacent to the i-th atom  11 . z is the number of atoms  11  adjacent to the i-th atom  11 . Vectors are illustrated in bold in the drawings and in expressions herein. 
     The magnetic field h i   exch  due to the interatomic exchange interaction acting on the i-th atom  11  is expressed by the following expression. 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                     exch 
                   
                   = 
                   
                     - 
                     
                       
                         ∂ 
                         
                           ℋ 
                           i 
                           exch 
                         
                       
                       
                         ∂ 
                         
                           μ 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Here, s i  in Expression (1) and μ i  in Expression (2) have the following relationship. 
       μ i   =−gμ   B   s   i   (3)
 
     Here, g is a g-factor, and usually the g-factor is about 2. μ B  is a Bohr magneton. μ i  represents the magnetic moment of one atom. 
     The magnetic field h i   exch  due to the interatomic exchange interaction acting on the i-th atom  11  is described by the following expression, by using atomic spin. 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                     exch 
                   
                   = 
                   
                     
                       - 
                       
                         J 
                         
                           g 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             B 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         z 
                       
                       ⁢ 
                       
                         s 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The temporal change of the magnetic moments μ of the plurality of atoms  11  can be expressed by the following Landau-Lifshits-Gilbert equation (LLG equation). 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       μ 
                     
                     dt 
                   
                   = 
                   
                     
                       
                         - 
                         
                           γ 
                           
                             1 
                             + 
                             
                               α 
                               2 
                             
                           
                         
                       
                       ⁢ 
                       μ 
                       × 
                       h 
                     
                     - 
                     
                       
                         
                           α 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           γ 
                         
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 α 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                              
                             μ 
                              
                           
                         
                       
                       ⁢ 
                       μ 
                       × 
                       
                         ( 
                         
                           μ 
                           × 
                           h 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Here, h is a magnetic field acting on the atoms  11 , α is an attenuation constant, and γ is a magnetic rotation ratio. 
     The magnetic moment μ(t+Δt) at time t+Δt is expressed by the following expression using the magnetic moment μ(t) at time t. 
     
       
         
           
             
               
                 
                   
                     μ 
                     ⁡ 
                     
                       ( 
                       
                         t 
                         + 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                           dt 
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The oscillating magnetic field h i   th  originating from the thermal fluctuation acting on the i-th atom  11  is expressed by the following expression. 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                     th 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           kT 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                         
                           γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             B 
                           
                           ⁢ 
                           
                             M 
                             s 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                     
                     ⁢ 
                     
                       
                         Γ 
                         i 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Here, k is the Boltzmann constant, T is the set temperature, M s  is the saturation magnetization constant, Δt is the time step width, and Γ i  (t) is a three-dimensional direction unit vector changing randomly in time. 
     Atom Coarse-Graining 
       FIG. 1B  is a diagram schematically illustrating a magnetic body model  20  generated by coarse-graining a plurality of atoms  11  constituting the magnetic body  10  illustrated in  FIG. 1A . The magnetic body model  20  is composed of a collection of coarse-grained particles  21  with a smaller number than the original number of the atoms in the magnetic body  10 . A magnetic moment p is applied to each of the plurality of particles  21 , based on the atomic spins s of the atoms  11  in the magnetic body  10 . In the calculation, the magnetic moment p of the particle  21  is, for example, a unit vector having a length of 1. 
     The magnetic field h′ i  acting on the i-th particle  21  can be obtained by the following expression. 
         h′   i   =h′   i   ext   +h′   i   dipole   +h′   i   anis   +h′   i   exch   +h′   i   th   (8)
 
     Here, h′ i   ext  is an external magnetic field, h′ i   dipole  is a magnetic field due to uniaxial crystal anisotropic interaction, and h′ i   anis  is the magnetic field due to dipole interaction, h′ i   exch  is the magnetic field due to interparticle exchange interaction, and h′ i   th  is the oscillating magnetic field. 
     The external magnetic field h′ i   ext  is generated in the entire region to be calculated and is given as a simulation condition. The magnetic field h′ i   dipole  due to uniaxial crystal anisotropic interaction, and the magnetic field h′ i   anis  due to dipole interaction can be expressed by the following expression. 
     
       
         
           
             
               
                 
                   
                     
                       h 
                       i 
                       
                         ′ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         dipole 
                       
                     
                     = 
                     
                       
                         1 
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           j 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 3 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       μ 
                                       j 
                                     
                                     · 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       ij 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   ij 
                                 
                               
                               - 
                               
                                 μ 
                                 j 
                               
                             
                             
                               r 
                               ij 
                               3 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       h 
                       i 
                       
                         ′ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         anis 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         K 
                         ⁡ 
                         
                           ( 
                           
                             
                               μ 
                               i 
                             
                             · 
                             e 
                           
                           ) 
                         
                       
                       ⁢ 
                       e 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Here, the r ij  hat is a unit vector parallel to the vector whose starting point is the position of the j-th particle  21  and the ending point is the position of the i-th particle  21 . r ij  is the distance from the j-th particle  21  to the i-th particle  21 . μ j  is the magnetic moment of the j-th particle  21 . e is a magnetization-friendly axis vector, and K is a magnetic anisotropy constant. 
     Interparticle Exchange Interaction 
     In the present embodiment, it is assumed that interparticle exchange interaction equivalent to an interatomic exchange interaction acts between two adjacent particles  21 . 
     The Hamiltonian of the interparticle exchange interaction between particles  21  of the magnetic body model  20  ( FIG. 1B ) is defined as follows. 
     
       
         
           
             
               
                 
                   
                     ℋ 
                     i 
                     
                       ′ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       exch 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           J 
                           ⁡ 
                           
                             ( 
                             
                               
                                 W 
                                 · 
                                 S 
                               
                               V 
                             
                             ) 
                           
                         
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∑ 
                           i 
                         
                         z 
                       
                       ⁢ 
                       
                         
                           μ 
                           i 
                         
                         · 
                         
                           μ 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     J is the same as the exchange interaction intensity coefficient J in Expression (1). The parameters V, W, and S will be described with reference to  FIG. 2 . μ i  and μ j  are magnetic moments of the i-th and j-th particles  21 , respectively. 
       FIG. 2  is a schematic diagram of two particles  21  for explaining parameters V, W, and S. The i-th particle  21   i  and the j-th particle  21   j  are adjacent to each other. The V on the right side of Expression (10) represents the volume of the particle  21 . S represents the surface area of the i-th particle  21   i  in the range of the solid angle Ω that allows the j-th particle  21   j  to be seen from the center O of the i-th particle  21   i . W is a parameter having a dimension of length. For example, as the value of W, the thickness of a single atomic layer located on the surface of the i-th particles  21   i  can be adopted. In this case, the value of W is equal to the diameter of the atom  11  of the magnetic body  10  ( FIG. 1A ). In  FIG. 2 , hatching is attached to a portion corresponding to the volume of W·S. 
     Next, the physical meaning of Expression (10) will be described. 
     In the magnetic body  10  ( FIG. 1A ), an interatomic exchange interaction acts between the atoms  11  adjacent to each other. The particles  21  of the magnetic body model  20  ( FIG. 1B ) are considered to represent a plurality of atoms  11 . When the interparticle interaction acting between two particles  21  is defined by using Expression (1), the state in which the interatomic exchange action is acting between two atoms  11  which are not adjacent to each other in the magnetic body  10  is reproduced. Therefore, it is considered that the interparticle exchange interaction acts only between portions facing each other at a short distance, among the surfaces of the particles  21  adjacent to each other. In the present embodiment, a surface within a range of a solid angle Ω that allows the j-th particle  21   j  to be seen from the center O of the i-th particle  21   i  is adopted as the “portions facing each other at a short distance”. 
     Further, considering that only the atoms of one atomic layer located on the surface contribute to the interparticle exchange interaction, the volume of the portion contributing to the interparticle exchange interaction is represented by W·S. The term (W·S/V) on the right side of Expression (10) corresponds to the ratio of the volume of the portion contributing to the interparticle exchange interaction to the volume of the particles  21  (hereinafter referred to as an effective volume ratio). In the calculation of the Hamiltonian H′ i   exch  of the interparticle exchange interaction, the magnetic moments μ i  and μ j  of the i-th particle  21   i  and the j-th particle  21   j  that exert the interparticle exchange interaction are multiplied by the effective volume ratio, and weakened magnetic moment is used. That is, in the simulation of the magnetic body model  20  ( FIGS. 1A and 1B ), the entire magnetic moments μ of the particles  21  do not contribute to the interparticle exchange interaction, but weakened magnetic moments (W·S/V) μ according to the effective volume ratio are considered to contribute to the interparticle exchange interaction. 
     The magnetic field h′ i   exch  due to the interparticle exchange interaction can be expressed by the following expression, by using the Hamiltonian of the interparticle exchange interaction defined by Expression (10). 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                     
                       ′ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       exch 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∂ 
                         
                           ℋ 
                           i 
                           
                             ′ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             exch 
                           
                         
                       
                       
                         ∂ 
                         
                           μ 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Oscillating Magnetic Field 
     Next, the oscillating magnetic field originating from thermal fluctuation will be described. 
     When the radius of the particle  21  that is coarse-grained atoms is λ times the atomic radius, the magnetic field h′ i   exch  due to the exchange interaction acting on the particle  21  can be expressed as follows by using the function f(λ) of λ. In the present specification, λ is referred to as a particle enlargement ratio. 
     
       
         
           
             
               
                 
                   
                     
                       h 
                       i 
                       
                         ′ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         exch 
                       
                     
                     = 
                     
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             λ 
                             ) 
                           
                         
                         ⁢ 
                         
                           h 
                           i 
                           exch 
                         
                       
                       = 
                       
                         
                           - 
                           
                             Jf 
                             ⁡ 
                             
                               ( 
                               λ 
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               j 
                             
                             z 
                           
                           ⁢ 
                           
                             μ 
                             j 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         λ 
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                           
                           
                             
                               ( 
                               
                                 λ 
                                 = 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   3 
                                   
                                     λ 
                                     · 
                                     z 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 λ 
                                 &gt; 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Here, z is the number of particles located in the vicinity. 
     By formulating the oscillating magnetic field h′ i   th  acting on the particles  21  in response to the magnetic field due to the exchange interaction shown in Expression (12) as follows, the temperature dependence of the magnetization can be reproduced. 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                     
                       ′ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       th 
                     
                   
                   = 
                   
                     
                       
                         
                           f 
                           ⁡ 
                           
                             ( 
                             λ 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         h 
                         i 
                         th 
                       
                     
                     = 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 λ 
                                 ) 
                               
                             
                             ⁢ 
                             kT 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           
                             γ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               μ 
                               B 
                             
                             ⁢ 
                             
                               M 
                               s 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                         
                       
                       ⁢ 
                       
                         
                           Γ 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Expression (7) is used in the modification of Expression (13). 
     f(λ) in Expression (12) is a coefficient for converting the magnetic field due to the interatomic exchange interaction in the magnetic body  10  ( FIG. 1A ) into the interparticle exchange interaction in the magnetic body model  20  ( FIG. 1B ). In Expression (13), the square root of f(λ) is used as a coefficient for converting the oscillating magnetic field acting on the atoms of the magnetic body  10  into the oscillating magnetic field acting on the particles of the magnetic body model  20 . 
     Next, the physical meaning of Expression (13) will be described. Since the exchange interaction intensity coefficient J, which is the origin of spontaneous magnetization, changes due to coarse-graining, the amount of energy (Hamiltonian value) in the calculation system also changes. By changing the amount of energy dissipation in the system in response to the change in the amount of energy in the system due to coarse-graining, the temperature dependence in the system before coarse-graining can be maintained in the system after coarse-graining. 
     Since the amount of energy dissipation is the variance of the random field, that is, the square mean of Expression (7), the ratio of the amount of energy (Hamiltonian value) to the magnitude of the energy dissipation amount remains unchanged before and after coarse-graining, by multiplying the root on the right side of Expression (7) by the function f(λ) of Expression (12). That is, in Expression (13), the term of temperature fluctuation is converted such that the ratio of the Hamiltonian value to the magnitude of the energy dissipation amount does not change before and after coarse-graining. 
     Simulation Apparatus 
       FIG. 3  is a block diagram of a simulation apparatus according to an embodiment. The simulation apparatus according to the embodiment includes an input device  50 , a processing device  51 , an output device  52 , and an external storage device  53 . Simulation conditions or the like are input from the input device  50  to the processing device  51 . Further, various commands or the like are input from the operator to the input device  50 . The input device  50  includes, for example, a communication device, a removable media reading device, a keyboard, or the like. 
     The processing device  51  performs simulation calculation based on the input simulation conditions and commands. The processing device  51  is a computer including a central processing unit (CPU), a main storage device (main memory), and the like. The simulation program executed by the computer is stored in the external storage device  53 . For the external storage device  53 , for example, a hard disk drive (HDD), a solid state drive (SSD), or the like is used. The processing device  51  reads the program stored in the external storage device  53  into the main storage device and executes the program. 
     The processing device  51  outputs the simulation result to the output device  52 . The simulation result includes information indicating the magnetic moment applied to each of a plurality of particles representing the member to be analyzed, the temporal change of the physical quantity of the particle system composed of the plurality of particles, or the like. The output device  52  includes, for example, a communication device, a removable media writing device, a display, a printer, and the like. 
       FIG. 4  is a flowchart of a simulation method according to the embodiment. 
     First, the processing device  51  acquires the simulation conditions input to the input device  50  (step S 1 ). The simulation conditions include the physical property values of the magnetic body  10  ( FIG. 1A ) to be simulated, the shape of the magnetic body  10 , the external magnetic field, the coarse-grained conditions, the initial conditions, the time step width in the simulation calculation, and the like. 
     When acquiring the simulation conditions, the processing device  51  generates the magnetic body model  20  ( FIG. 1B ), based on the acquired simulation conditions (step S 2 ). Thus, the magnitude and position of the plurality of coarse-grained particles  21  ( FIG. 1B ) are determined. Further, a magnetic moment μ is applied to each of the plurality of particles  21  (step S 3 ). The direction of the magnetic moment p is set at random, for example. 
     After applying the magnetic moment p to each of the particles  21 , the magnetic moment of the particles  21  is time-evolved by using the magnetic field h′ i  acting on each particle  21  (step S 4 ). The magnetic field h′ i  acting on each particle is given by Expression (8). Each magnetic field on the right side of Expression (8) is given by Expressions (9), (10), (11), and (13). Expressions (5) and (6) are used for the time evolution of the magnetic moment of the particle  21 . Expressions (5) and (6) show the magnetic moment of the atom  11  that has not been coarse-grained, but the change in the magnetic moment of the particle  21  after coarse-graining can be also calculated using the same expression as Expressions (5) and (6). 
     The calculation in step S 4  is repeated until the end condition is satisfied. For example, when the magnetization state of the magnetic body model  20  becomes a steady state, the iterative process of step S 4  is completed. When the end condition is satisfied, the processing device  51  outputs the analysis result to the output device  52  (step S 5 ). As the analysis result, for example, the distribution of the directions of the magnetic moments μ may be displayed by a plurality of arrows, or the distribution of the directions of the magnetic moments μ may be displayed in shades of color or the like. 
     Next, the excellent effects of the above embodiment will be described. 
     In the above embodiment, the calculation time can be shortened by coarse-graining the plurality of atoms  11  ( FIG. 1A ) of the magnetic body  10 . By defining interparticle exchange interaction corresponding to an exchange interaction acting between atoms by Expressions (10) and (11), between a plurality of coarse-grained particles  21  ( FIG. 1B ), the interatomic exchange interaction can be reflected in the simulation result. Further, by defining the oscillating magnetic field acting on the particles  21  by Expression (13), the influence of the oscillating magnetic field originating from the thermal fluctuation can be reflected in the simulation result. For example, it becomes possible to reproduce, by simulation, the phase transition phenomenon when the temperature changes over the Curie temperature at which the phase transition occurs. 
     Simulation without Considering Oscillating Magnetic Field 
     Next, with reference to  FIGS. 5A to 5G , results of an actual simulation performed to check the excellent effect of the above embodiment will be described. The following simulation does not consider the oscillating magnetic field. 
       FIGS. 5A to 5D, 5F, and 5G  are diagrams illustrating the distribution of the directions of magnetic moments obtained by simulation in shades.  FIG. 5E  is a diagram schematically illustrating the directions of the magnetic moments illustrated in  FIGS. 5A to 5D . The calculation area in the simulation is a two-dimensional square with a side length of 50 nm. An xy Cartesian coordinate system is defined in the calculation area. When the radii of the coarse-grained particles  21  is 1 nm and 7.5 nm, respectively, the magnetic moments are time-evolved until the magnetic moment distribution of the particles  21  reaches a steady state. The particles  21  are arranged at the positions of the lattice points of the square lattice, and as initial conditions, the distribution of the directions of the magnetic moments are the same in all of  FIGS. 5A to 5D, 5F, and 5G . 
       FIGS. 5A and 5B  illustrate the simulation results of the magnetic moments when the radius r of the coarse-grained particle  21  is 1 nm.  FIGS. 5C, 5D, 5F, and 5G  illustrate the simulation results of the magnetic moments when the radius r of the coarse-grained particle  21  is 7.5 nm. Note that  FIGS. 5F and 5G  illustrate the results of simulations performed under the condition that the interparticle exchange interaction does not act between the coarse-grained particles  21 . 
       FIGS. 5A, 5C, and 5F  illustrate the magnitudes of the y components of the magnetic moments, and  FIGS. 5B, 5D, and 5G  illustrate the magnitudes of the x components of the magnetic moments. The region where the absolute values of the x and y components of the magnetic moments are large is illustrated relatively dark. The outline of the direction of the magnetic moment of each region divided by shades in  FIGS. 5A to 5D  is illustrated by an arrow in  FIG. 5E . 
     In the simulations in which the interparticle exchange interaction is considered, the results of simulation ( FIGS. 5A and 5B ) with the particle radius r set to 1 nm and the results of simulation ( FIGS. 5C and 5D ) with the particle radius r set to 7.5 nm, a clear magnetic domain structure with aligned magnetic moment direction is checked. On the other hand, the magnetic domain structure does not appear in the results of the simulation ( FIGS. 5F and 5G ) in which the interparticle exchange interaction is not considered. From this simulation results, it can be seen that the interatomic exchange interaction of the magnetic body  10  to be simulated is appropriately reproduced in the coarse-grained magnetic body model  20 . 
     Next, the results of the simulation performed to check the degree of influence of the exchange interaction will be described with reference to  FIGS. 6A and 6B . 
       FIGS. 6A and 6B  are diagrams illustrating the results of simulation with the radii r of the particles  21  being 1 nm and 100 nm, respectively. In  FIGS. 6A and 6B , the directions of the magnetic moments when the distribution of the magnetic moment reaches a steady state are indicated by arrows. The simulation area is a two-dimensional rectangle, and 24 and 9 particles  21  are arranged in the length direction and the width direction, respectively. 
     In the simulation results illustrated in  FIG. 6A , the magnetic moments of all the particles  21  are oriented in substantially the same direction. This is because the interparticle exchange interaction acts stronger than the uniaxial crystal anisotropic interaction and the dipole interaction. On the other hand, in the simulation result illustrated in  FIG. 6B , the annular magnetic domain structure is checked. This is because the interparticle exchange interaction is relatively weakened, and the uniaxial crystal anisotropic interaction and the dipole interaction become apparent. 
     In both the simulations of  FIGS. 6A and 6B , the number of target particles  21  is the same. Therefore, the calculation times for both are almost equal. Further, in the simulation of  FIG. 6A , the rectangular region of 48 nm in width and 18 nm in length is the calculation target, whereas in the simulation of  FIG. 6B , the rectangular region of 4800 nm in width and 1800 nm in length is the calculation target. In this way, by adopting the method according to the above embodiment, it is possible to expand the calculation area while suppressing the lengthening of the calculation time. Thus, it is possible to suppress an increase in calculation cost when simulating the magnetic moments of a large magnetic body. 
     Simulation Considering Oscillating Magnetic Field 
     Next, with reference to  FIG. 7 , results of another actual simulation performed to check the excellent effect of the above embodiment will be described. In the following simulation, the magnetic field and the oscillating magnetic field due to the interparticle exchange interaction are considered. 
     The particle enlargement ratios λ are set to 1, 10, or 100, and the magnetization temperature characteristics are obtained by simulation. The crystal structure is a body-centered cubic lattice (BCC), and the number of crystal lattices is 22×22×22. The value of iron is used as the physical property value of the object to be analyzed. Calculations are performed until steady state is reached at each of the plurality of temperatures. The magnitude M of the average vector of the magnetic moments of all the particles to be analyzed in the steady state is obtained. 
       FIG. 7  is a graph showing the relationship between the normalized magnetization calculated from the simulation results and the temperature. The horizontal axis represents the temperature in the unit “K”, and represents the normalized magnetization in which the magnitude of the average vector of the magnetic moments of all the particles is normalized by the saturation magnetization M s . The circle symbol, square symbol, and triangle symbol in the graph indicate the simulation results when the particle enlargement ratios λ are 1, 10, and 100, respectively. 
     When the particle enlargement ratio λ is 1, 10, or 100, the magnetization decreases as the temperature rises, and when the temperature slightly exceeds 1000 K, the magnetization becomes almost zero. The temperature at which the magnetization becomes almost zero is almost equal to the Curie temperature of iron 1043K. 
     From the simulation shown in  FIG. 7 , it is confirmed that it is possible to apply the method according to the present embodiment to perform a simulation reflecting the interatomic exchange interaction and the oscillating magnetic field. 
     Next, a modified example of the above embodiment will be described. 
     In the above embodiment, as illustrated in Expression (10), when determining the Hamiltonian of the interparticle exchange interaction, a value obtained by weakening the magnetic moment applied to the particles  21  according to the value of (W·S/V) is used. That is, the magnetic field due to the interparticle exchange interaction is calculated by weakening the interparticle exchange interaction. The coefficient for weakening the magnetic moment applied to the particle  21  is not limited to (W·S/V), and other coefficients less than 1 may be used. By weakening the interparticle exchange interaction, it is possible to make the uniaxial crystal anisotropic interaction and the dipole interaction apparent, while considering the interparticle exchange interaction. The coefficient for weakening the magnetic moment may be set to a value larger than 0 and smaller than 1, based on the magnitude and shape of the magnetic body  10  ( FIG. 1A ) to be simulated, the physical property value of the magnetic body, and the like. 
     It should be understood that the invention is not limited to the above-described embodiment, but may be modified into various forms on the basis of the spirit of the invention. Additionally, the modifications are included in the scope of the invention.