Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind. (English) Zbl 07885514
Summary: The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem – too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, \(K_\alpha(x)\).
{© 2024 The Author(s). Published by IOP Publishing Ltd}
{© 2024 The Author(s). Published by IOP Publishing Ltd}
MSC:
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 41A30 | Approximation by other special function classes |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
Keywords:
self-consistent expansion; modified Bessel functions of the second kind; asymptotic matching; perturbative expansions; partition functions; special functionsSoftware:
DLMFReferences:
| [1] | Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers I, 1999, Springer · Zbl 0938.34001 |
| [2] | Hinch, E. J., Perturbation Methods, 1991, Cambridge University Press · Zbl 0746.34001 |
| [3] | Baker, G. A.; Graves-Morris, P., Padé Approximants: Encyclopedia of Mathematics and It’s Applications, vol 59, 1996, Cambridge University Press · Zbl 0923.41001 |
| [4] | McComb, W. D., Renormalization Methods: A Guide for Beginners, 2003, Oxford University Press |
| [5] | Wiese, K. J., On the perturbation expansion of the KPZ equation, J. Stat. Phys., 93, 143-54, 1998 · Zbl 0963.82033 · doi:10.1023/B:JOSS.0000026730.76868.c4 |
| [6] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 1964, Dover · Zbl 0171.38503 |
| [7] | Lebedev, N. N.; Silverman, R. A., Special Functions and Their Applications, 1965, Prentice-Hall · Zbl 0131.07002 |
| [8] | Luke, Y. L., The Special Functions and Their Approximations, 1969, Academic · Zbl 0193.01701 |
| [9] | Olver, F. W J.; Olde Daalhuis, A. B.; Lozier, D. W.; Schneider, B. I.; Boisvert, R. F.; Clark, C. W.; Miller, B. R.; Saunders, B. V.; Cohl, H. S.; McClain, M. A., NIST Digital Library of Mathematical Functions, 2023 |
| [10] | Schwartz, M.; Edwards, S., Nonlinear deposition: a new approach, Europhys. Lett., 20, 301, 1992 · doi:10.1209/0295-5075/20/4/003 |
| [11] | Schwartz, M.; Edwards, S. F., Peierls-Boltzmann equation for ballistic deposition, Phys. Rev. E, 57, 5730-9, 1998 · doi:10.1103/PhysRevE.57.5730 |
| [12] | Katzav, E.; Schwartz, M., Self-consistent expansion for the Kardar-Parisi-Zhang equation with correlated noise, Phys. Rev. E, 60, 5677-80, 1999 · doi:10.1103/PhysRevE.60.5677 |
| [13] | Katzav, E.; Schwartz, M., Existence of the upper critical dimension of the Kardar-Parisi-Zhang equation, Physica A, 309, 69-78, 2002 · Zbl 0995.45002 · doi:10.1016/S0378-4371(02)00553-8 |
| [14] | Schwartz, M.; Edwards, S., Stretched exponential in non-linear stochastic field theories, Physica A, 312, 363-8, 2002 · Zbl 0997.60056 · doi:10.1016/S0378-4371(02)00608-8 |
| [15] | Katzav, E., Self-consistent expansion for the molecular beam epitaxy equation, Phys. Rev. E, 65, 2002 · doi:10.1103/PhysRevE.65.032103 |
| [16] | Katzav, E., Growing surfaces with anomalous diffusion: Results for the fractal Kardar-Parisi-Zhang equation, Phys. Rev. E, 68, 2003 · doi:10.1103/PhysRevE.68.031607 |
| [17] | Katzav, E., Self-consistent expansion results for the nonlocal Kardar-Parisi-Zhang equation, Phys. Rev. E, 68, 2003 · doi:10.1103/PhysRevE.68.046113 |
| [18] | Katzav, E.; Schwartz, M., Numerical evidence for stretched exponential relaxations in the Kardar-Parisi-Zhang equation, Phys. Rev. E, 69, 2004 · doi:10.1103/PhysRevE.69.052603 |
| [19] | Katzav, E.; Schwartz, M., Kardar-Parisi-Zhang equation with temporally correlated noise: A self-consistent approach, Phys. Rev. E, 70, 2004 · doi:10.1103/PhysRevE.70.011601 |
| [20] | Edwards, S. F.; Schwartz, M., Lagrangian statistical mechanics applied to non-linear stochastic field equations, Physica A, 303, 357-86, 2002 · Zbl 0979.82033 · doi:10.1016/S0378-4371(01)00479-4 |
| [21] | Katzav, E.; Adda-Bedia, M., Roughness of tensile crack fronts in heterogenous materials, Europhys. Lett., 76, 450, 2006 · doi:10.1209/epl/i2006-10273-7 |
| [22] | Katzav, E.; Adda-Bedia, M.; Ben Amar, M.; Boudaoud, A., Roughness of moving elastic lines: Crack and wetting fronts, Phys. Rev. E, 76, 2007 · doi:10.1103/PhysRevE.76.051601 |
| [23] | Katzav, E.; Adda-Bedia, M.; Derrida, B., Fracture surfaces of heterogeneous materials: A 2d solvable model, Europhys. Lett., 78, 2007 · Zbl 1244.74120 · doi:10.1209/0295-5075/78/46006 |
| [24] | Katzav, E.; Adda-Bedia, M., Stability and roughness of tensile cracks in disordered materials, Phys. Rev. E, 88, 2013 · doi:10.1103/PhysRevE.88.052402 |
| [25] | Li, M. S.; Nattermann, T.; Rieger, H.; Schwartz, M., Vortex lines in the three-dimensional XY model with random phase shifts, Phys. Rev. B, 54, 1996 · doi:10.1103/PhysRevB.54.16024 |
| [26] | Steinbock, C.; Katzav, E.; Boudaoud, A., Structure of fluctuating thin sheets under random forcing, Phys. Rev. Res., 4, 2022 · doi:10.1103/PhysRevResearch.4.033096 |
| [27] | Steinbock, C.; Katzav, E., Dynamics of fluctuating thin sheets under random forcing, Phys. Rev. E, 107, 2023 · doi:10.1103/PhysRevE.107.025002 |
| [28] | Steinbock, C.; Katzav, E., Thermally driven elastic membranes are quasi-linear across all scales, J. Phys. A, 56, 2023 · Zbl 1523.74073 · doi:10.1088/1751-8121/acce84 |
| [29] | Schwartz, M.; Katzav, E., The ideas behind the self-consistent expansion, J. Stat. Mech., 2008 · doi:10.1088/1742-5468/2008/04/P04023 |
| [30] | Remez, B.; Goldstein, M., From divergent perturbation theory to an exponentially convergent self-consistent expansion, Phys. Rev. D, 98, 2018 · doi:10.1103/PhysRevD.98.056017 |
| [31] | Cohen, A.; Bialy, S.; Schwartz, M., The self consistent expansion applied to the factorial function, Physica A, 463, 503-8, 2016 · Zbl 1400.82074 · doi:10.1016/j.physa.2016.07.030 |
| [32] | Tolstoy, L.; Garnett, C., (Project Gutenberg), Anna Karenina, 1901 |
| [33] | Segur, H.; Tanveer, S.; Levine, H., Asymptotics Beyond All Orders, 1991, Springer · Zbl 0920.00033 |
| [34] | Berry, M. V.; Howls, C. J., Hyperasymptotics, Proc. R. Soc. A, 430, 653-68, 1990 · Zbl 0745.34052 · doi:10.1098/rspa.1990.0111 |
| [35] | Berry, M. V.; Howls, C. J., Hyperasymptotics for integrals with saddles, Proc. R. Soc. A, 434, 657-75, 1991 · Zbl 0764.30031 · doi:10.1098/rspa.1991.0119 |
| [36] | Berry, M. V., Stokes’ phenomenon; smoothing a victorian discontinuity, Publ. Math. de l’IHÉS, 68, 211-21, 1988 · Zbl 0701.58012 · doi:10.1007/BF02698550 |
| [37] | Berry, M. V., Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. R. Soc. A, 422, 7-24, 1989 · Zbl 0683.33004 · doi:10.1098/rspa.1989.0018 |
| [38] | Rabemananjara, T R2021Resummation and machine learning techniquesPhD ThesisUniveristy of Milan |
| [39] | Palade, D. I.; Pomârjanschi, L. M., Approximations of the modified Bessel functions of the second kind \(K_{\text{\nu}} \). Applications in random field generation, Rom. J. Phys., 68, 108, 2023 |
| [40] | Balescu, R., Equilibrium and Non-Equilibrium Statistical Mechanics, 1975, Wiley · Zbl 0984.82500 |
| [41] | Plischke, M.; Bergersen, B., Equilibrium Statistical Physics, 2006, World Scientific · Zbl 1130.82001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
