Partition Function Formalism in the Problem of Multidimensional Integer Partitions
Department for Theoretical Physics,
Ivan Franko National University of Lviv,
12 Drahomanov St., Lviv, UA-79005, Ukraine
e-mail: andrij.rovenchak@gmail.com
Received:
Received: 23 March 2010; accepted: 1 July 2010; published online: 25 August 2010
DOI: 10.12921/cmst.2010.16.02.187-190
OAI: oai:lib.psnc.pl:729
Abstract:
The formalism based on the microcanonical treatment of a many-boson system is applied to the problem in the number theory known as the partitioning of an integer. An estimation is obtained for the asymptotic behavior of the number of multidimensional partitions into sums of various powers of integers. The obtained results are shown to reproduce the known ones for linear summands.
Key words:
bosonic systems, integer partitions, multimensional partition
References:
[1] G.W. Leibniz, Specimen de divulsionibus aequationum ad problemata indefinita in numeris rationalibus solvenda.
2. September 1674, in: G.W. Leibniz, Sämtliche Schriften und Briefe. Siebente Reihe: Mathematische Schriften, Bd. 1: 1672-1676. Geometrie – ahlentheorie – Algebra (1. Teil), Akademie-Verlag, Berlin, 740-755 (1990).
[2] L. Eulero, De partitione numerorum. Novi Commentarii Academiae scientiarum Petropolitanae 3, 125-169 (1753).
[3] G.E. Andrews, The Theory of Partitions. Addison-Wesley, Reading, Mass. (1976).
[4] G. Almkvist, Asymptotic formulas and generalized Dedekind sums. Experimental Mathematics 7, 343-359 (1998).
[5] S. Grossmann, M. Holthaus, Fluctuations of the particle number in a trapped Bose-Einstein condensate. Phys. Rev. Lett. 79, 3557-3560 (1997).
[6] M.N. Tran, M.V.N. Murthy, R.J. Bhaduri, On the quantum density of states and partitioning an integer. Ann. Phys. 311, 204-219 (2004).
[7] A. Comtet, P. Leboeuf, S.N. Majumdar, Level density of a Bose gas and extreme value statistics. Phys. Rev. Lett. 98, 070404 (4 p.) (2007).
[8] G.H. Hardy, S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17, 75-115 (1918).
[9] C.S. Srivatsan, M.V.N. Murthy, R.K. Bhaduri, Gentile statistics and restricted partitions, Pramana – J. Phys. 66, 485-494 (2006).
[10] A. Rovenchak, The relation between fractional statistics and finite bosonic systems in one-dimensional case, Fiz. Nizk. Temp. 35, 510-513 (2009); Low Temp. Phys. 35, 400-403 (2009).
[11] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions. Tenth printing. National Bureau of Standards 1972.
[12] D.P. Bhatia, M.A. Prasad, D. Arora, Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals. J. Phys. A: Math. Gen. 30, 2281-2285 (1997).
[13] V. Mustonen, R. Rajesh, Numerical estimation of the asymptotic behaviour of solid partitions of an integer. J. Phys. A: Math. Gen. 36, 6651-6659 (2003).
The formalism based on the microcanonical treatment of a many-boson system is applied to the problem in the number theory known as the partitioning of an integer. An estimation is obtained for the asymptotic behavior of the number of multidimensional partitions into sums of various powers of integers. The obtained results are shown to reproduce the known ones for linear summands.
Key words:
bosonic systems, integer partitions, multimensional partition
References:
[1] G.W. Leibniz, Specimen de divulsionibus aequationum ad problemata indefinita in numeris rationalibus solvenda.
2. September 1674, in: G.W. Leibniz, Sämtliche Schriften und Briefe. Siebente Reihe: Mathematische Schriften, Bd. 1: 1672-1676. Geometrie – ahlentheorie – Algebra (1. Teil), Akademie-Verlag, Berlin, 740-755 (1990).
[2] L. Eulero, De partitione numerorum. Novi Commentarii Academiae scientiarum Petropolitanae 3, 125-169 (1753).
[3] G.E. Andrews, The Theory of Partitions. Addison-Wesley, Reading, Mass. (1976).
[4] G. Almkvist, Asymptotic formulas and generalized Dedekind sums. Experimental Mathematics 7, 343-359 (1998).
[5] S. Grossmann, M. Holthaus, Fluctuations of the particle number in a trapped Bose-Einstein condensate. Phys. Rev. Lett. 79, 3557-3560 (1997).
[6] M.N. Tran, M.V.N. Murthy, R.J. Bhaduri, On the quantum density of states and partitioning an integer. Ann. Phys. 311, 204-219 (2004).
[7] A. Comtet, P. Leboeuf, S.N. Majumdar, Level density of a Bose gas and extreme value statistics. Phys. Rev. Lett. 98, 070404 (4 p.) (2007).
[8] G.H. Hardy, S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17, 75-115 (1918).
[9] C.S. Srivatsan, M.V.N. Murthy, R.K. Bhaduri, Gentile statistics and restricted partitions, Pramana – J. Phys. 66, 485-494 (2006).
[10] A. Rovenchak, The relation between fractional statistics and finite bosonic systems in one-dimensional case, Fiz. Nizk. Temp. 35, 510-513 (2009); Low Temp. Phys. 35, 400-403 (2009).
[11] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions. Tenth printing. National Bureau of Standards 1972.
[12] D.P. Bhatia, M.A. Prasad, D. Arora, Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals. J. Phys. A: Math. Gen. 30, 2281-2285 (1997).
[13] V. Mustonen, R. Rajesh, Numerical estimation of the asymptotic behaviour of solid partitions of an integer. J. Phys. A: Math. Gen. 36, 6651-6659 (2003).
