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Classical Magnetism and an Integral Formula Involving Modified Bessel Functions

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Published/Copyright: March 31, 2018
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 19 Issue 3-4

Abstract

We study an integral expression that is encountered in some classical spin models of magnetism. The idea is to calculate the key integral that represents the building block for the expression of the partition function of these models. The general calculation allows one to have a better look at the internal structure of the quantity of interest which, in turn, may lead to potentially new useful insights. We find out that application of two different approaches to solve the problem in a general-case scenario leads to an interesting integral formula involving modified Bessel functions of the first kind which appears to be new. We performed Monte Carlo simulations to verify the correctness of the integral formula obtained. Additional numerical integration tests lead to the same result as well. The approach under consideration, when generalized, leads to a linear integral equation that might be of interest to numerical studies of classical spin models of magnetism that rely on the well-established transfer-matrix formalism.

Keywords: General Physics; Theory and Modeling; Function Theory and Analysis
PACS: numbers; 01.55.+b; 73.43.Cd; 02.30.-f

Funding statement: This research was supported in part by National Science Foundation (NSF) Grants No. DMR-1410350 and DMR-1705084.

Acknowledgements:

The author thanks the anonymous referee for suggesting an analytical proof of the result as outlined in the Appendix.

Appendix

A Calculation Of F(a,θ)

The relation in eq. (25) can be written as:

(27)F(a,θ)=∫−1+1dyeacosθyI0asinθ1−y2=2πaI12(a) ,

based on the general definition of il(x) from eq. (5). We expand the modified Bessel function of the first kind that appears inside the integral sign by using eq. (10.25.2) of Ref. 25 that reads:

(28)Iν(z)=z2ν∑k=0∞z24kk!Γ(ν+k+1) ,

where Γ represents the gamma function. For ν=0 one has:

(29)I0(z)=∑k=0∞z2k22k(k!)2 .

We substitute the result from eq. (29) into eq. (27) and interchange summation and integration:

(30)F(a,θ)=∑k=0∞a2ksin2kθ22k(k!)2∫−1+1dyeacosθy(1−y2)kdy .

We now rely on eq. (10.32.2) of Ref. 25 which reads:

(31)Iν(z)=z2νπΓ(ν+12)∫−1+1(1−t2)ν−1/2e±ztdt .

For ν=k+1/2 such a formula leads to:

(32)∫−1+1(1−t2)ke±ztdt=πk!z2k+12Ik+12(z) .

With help from eq. (32), one transforms eq. (30) into:

(33)F(a,θ)=2πacosθ∑k=0∞1k![asin2θ2cosθ]kIk+12(acosθ) .

We now use eq. 5.7.6.(1) of Ref. 22 (pg.660) that gives:

(34)∑k=0∞tkk!Jk+ν(z)=zν/2(z−2t)−ν/2Jν(z2−2tz) ,

where Jk+ν(z) is a Bessel function of the first kind. This formula suggests that for ν=1/2 we should have:

(35)∑k=0∞tkk!Jk+12(z)=(zz−2t)1/4J12(z(z−2t)) .

The formula in eq. (10.27.6) of Ref. 25 explains how a Bessel function of the first kind with imaginary argument is related to a modified Bessel function of the first kind:

(36)Iν(z)=e−νπ2iJνzeπ2i .

This implies that, for our specific case, we have:

(37)Jν(iz)=eνπ2iIν(z) .

The next step is to write z=iξ, t=−iτ and use eq. (37) to transform eq. (35) in terms of modified Bessel functions of the first kind. After some careful algebraic manipulations of eq. (35) (note that a factor eπ4i appears in both sides of the equation and, thus, cancels out), one obtains:

(38)∑k=0∞τkk!Ik+12(ξ)=(ξξ+2τ)1/4I12(ξ(ξ+2τ)) .

The formula in eq. (38) can be immediately applied to eq. (33) with the understanding that:

(39)τ=asin2θ2cosθ   ;   ξ=acosθ .

As a result, one obtains F(a,θ)=2πaI12(a) as in eq. (27).

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Received: 2017-08-31
Accepted: 2015-03-15
Published Online: 2018-03-31
Published in Print: 2018-06-26

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2017-0193
Keywords for this article
General Physics; Theory and Modeling; Function Theory and Analysis

Articles in the same Issue

  1. Frontmatter
  3. A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay
  4. Numerical Methods for the Derivative Nonlinear Schrödinger Equation
  5. Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy
  6. Modeling of Supersonic/Hypersonic Boundary Layer Transition Using a Single-Point Approach
  7. A Novel Macromodel based on Krylov Subspace Projection Method for Micromixers with Serpentine Channels
  8. Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities
  9. Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions
  10. Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems
  11. Dynamics of Almost Periodic Solution for a Delayed Facultative Mutualism Model Involving Negative Feedback Terms
  12. Controllability of Fractional Evolution Inclusions with Noninstantaneous Impulses
  13. Analysis of a Delayed Predator–Prey System with Harvesting
  14. Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness
  15. Numerical Method for a Class of Nonlinear Singularly Perturbed Delay Differential Equations Using Parametric Cubic Spline
  16. Numerical Simulation for Shale Gas Flow in Complex Fracture System of Fractured Horizontal Well
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  21. Lie Symmetry Analysis of Boundary Layer Stagnation-Point Flow and Heat Transfer of Non-Newtonian Power-Law Fluids Over a Nonlinearly Shrinking/Stretching Sheet with Thermal Radiation
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