Importance sampling Monte Carlo methods
0
Citation
115
Reference
10
Related Paper
Abstract:
In this chapter we want to introduce simple importance sampling Monte Carlo techniques as applied in statistical physics and which can be used for the study of phase transitions at finite temperature. We shall discuss details, algorithms, and potential sources of difficulty using the Ising model as a paradigm. It should be understood, however, that virtually all of the discussion of the application to the Ising model is relevant to other models as well, and a few such examples will also be discussed. Other models as well as sophisticated approaches to the Ising model will be discussed in later chapters. The Ising model is one of the simplest lattice models which one can imagine, and its behavior has been studied for a century. The simple Ising model consists of spins which are confined to the sites of a lattice and which may have only the values +1 or −1.Keywords:
Square-lattice Ising model
Lattice (music)
Onsager's results for the partition and correlation functions for the Ising model on a two-dimensional rectangular lattice are rederived using a Green's function technique. The definition of the Green's function is based on a recently published casting of the Ising model into a many-body fermion problem. The relation between this approach and other methods used for solving the Ising problem are indicated.
Square-lattice Ising model
Lattice (music)
Cite
Citations (8)
Square-lattice Ising model
Ising spin
Cite
Citations (11)
Square-lattice Ising model
Lattice (music)
Metropolis–Hastings algorithm
Cite
Citations (0)
We derive a relation between a general spin-3/2 Ising model and an Ising model with two sets of Ising spins. A spin-3/2 Ising model with up-down symmetry is expressed in terms of an Ashkin-Teller model and solved exactly in some cases without assuming extended Horiguchi's condition. It turns out that the system on the square lattice shows rich critical phenomena.
Square-lattice Ising model
Ising spin
Square lattice
Cite
Citations (17)
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
Square-lattice Ising model
Cite
Citations (0)
It is shown that the partition function of the Ising model in a magnetic field on an arbitrary lattice in arbitrary dimensions is expressed by an ensemble average of the partition functions of random-bond Ising models without field.
Square-lattice Ising model
Lattice (music)
Cite
Citations (2)
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices ($4\times 4$ and $4\times 6$) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
Square-lattice Ising model
Cite
Citations (0)
There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with $N$ lattice sites. It is a sum of $2^N$ exponential functions and holds for $D$-dimensional ($D=1,2,3,...$) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.
Cite
Citations (1)
Square-lattice Ising model
Duality (order theory)
Square lattice
Lattice (music)
Cite
Citations (3)
In this chapter we want to introduce simple importance sampling Monte Carlo techniques as applied in statistical physics and which can be used for the study of phase transitions at finite temperature. We shall discuss details, algorithms, and potential sources of difficulty using the Ising model as a paradigm. It should be understood, however, that virtually all of the discussion of the application to the Ising model is relevant to other models as well, and a few such examples will also be discussed. Other models as well as sophisticated approaches to the Ising model will be discussed in later chapters. The Ising model is one of the simplest lattice models which one can imagine, and its behavior has been studied for a century. The simple Ising model consists of spins which are confined to the sites of a lattice and which may have only the values +1 or −1.
Square-lattice Ising model
Lattice (music)
Cite
Citations (0)