A New Formula for Partitions in a Set of Entities into Empty and Nonempty Subsets, and Its Application to Stochastic and Agent-Based Computational Models

Affiliation(s)

Department of Theoretical Physics, State University of Moldova, Chisinau, Republic of Moldova.

Department of Theoretical Physics, State University of Moldova, Chisinau, Republic of Moldova.

ABSTRACT

In combinatorics, a Stirling number of the second kind *S *(n,k) is the number of ways to partition a set of *n* objects into *k* nonempty subsets. The empty subsets are also added in the models presented in the article in order to describe properly the absence of the corresponding type i of state in the system, i.e. when its “share” *P*_{i }=0 . Accordingly, a new equation for partitions *P *(N, m) in a set of entities into both empty and nonempty subsets was derived. The indistinguishableness of particles (N identical atoms or molecules) makes only sense within a cluster (subset) with the size 0≤n_{i} ≥N. The first-order phase transition is indeed the case of transitions, for example in the simplest interpretation, from completely liquid state type*L *= {n_{1} =*N*, n_{2} = 0} to the completely crystalline state type*C*= {n_{1} =0, n_{2} = *N *}. These partitions are well distinguished from the physical point of view, so they are ‘typed’ differently in the model. Finally, the present developments in the physics of complex systems, in particular the structural relaxation of super-cooled liquids and glasses, are discussed by using such stochastic cluster-based models.

Cite this paper

G. Gubceac, R. Gutu and F. Paladi, "A New Formula for Partitions in a Set of Entities into Empty and Nonempty Subsets, and Its Application to Stochastic and Agent-Based Computational Models,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 14-21. doi: 10.4236/am.2013.410A3003.

G. Gubceac, R. Gutu and F. Paladi, "A New Formula for Partitions in a Set of Entities into Empty and Nonempty Subsets, and Its Application to Stochastic and Agent-Based Computational Models,"

References

[1] R. L. Graham, D. E. Knuth and O. Patashnik, “Concrete Mathematics. A Foundation for Computer Science,” 2nd Edition, Addison-Wesley Professional, Reading, 1994.

[2] J. Sandor and B. Crstici, “Handbook of Number Theory II,” Kluwer Academic Publishers, Dordrecht, 2004. http://dx.doi.org/10.1007/1-4020-2547-5

[3] F. Paladi, “On the Probabilistic Approach to Heterogene ous Structure Interactions in Agent-Based Computational Models,” Applied Mathematics and Computation, Vol. 219, No. 24, 2013, pp. 11430-11437. http://dx.doi.org/10.1016/j.amc.2013.05.042

[4] E. Bonabeau, “Agent-Based Modeling: Methods and Te chniques for Simulating Human Systems,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 99, No. 3, 2002, pp. 7280-7287. http://dx.doi.org/10.1073/pnas.082080899

[5] M. Richiardi and F. Paladi, “Jesus, Hillel and the Man of the Street. Moral and Social Norms in Heterogeneous Populations,” LABORatorio R. Revelli Working Paper 40, 2005.

http://eco83.econ.unito.it/terna/swarmfest2005papers/richiardi_paladi.pdf

[6] M. Richiardi, “Jesus vs Hillel. From Moral to Social Norms and Back,” European Journal of Economic and Social Systems, Vol. 19, No. 2, 2006, pp. 171-190.

[7] D. Kashchiev, “Nucleation. Basic Theory with Applica tions,” Butterworth-Heinemann, Oxford, 2000.

[8] C.W. Gardiner, “Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences,” 2nd Edi tion, Springer-Verlag, Berlin, 1985.

[9] A. N. Kolmogorov, “On Statistical Theory of Metal Crys tallisation (in Russian),” Izvestiya Academy of Sciences, USSR, Mathematics, Vol. 3, 1937, pp. 355-360.

[10] L. Feng, B. Li, B. Podobnik, T. Preis and H. E. Stanley, “Linking Agent-Based Models and Stochastic Models of Financial Markets,” Proceedings of the National Acad emy of Sciences of the United States of America, Vol. 109, No. 22, 2012, pp. 8388-8393. http://dx.doi.org/10.1073/pnas.1205013109

[1] R. L. Graham, D. E. Knuth and O. Patashnik, “Concrete Mathematics. A Foundation for Computer Science,” 2nd Edition, Addison-Wesley Professional, Reading, 1994.

[2] J. Sandor and B. Crstici, “Handbook of Number Theory II,” Kluwer Academic Publishers, Dordrecht, 2004. http://dx.doi.org/10.1007/1-4020-2547-5

[3] F. Paladi, “On the Probabilistic Approach to Heterogene ous Structure Interactions in Agent-Based Computational Models,” Applied Mathematics and Computation, Vol. 219, No. 24, 2013, pp. 11430-11437. http://dx.doi.org/10.1016/j.amc.2013.05.042

[4] E. Bonabeau, “Agent-Based Modeling: Methods and Te chniques for Simulating Human Systems,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 99, No. 3, 2002, pp. 7280-7287. http://dx.doi.org/10.1073/pnas.082080899

[5] M. Richiardi and F. Paladi, “Jesus, Hillel and the Man of the Street. Moral and Social Norms in Heterogeneous Populations,” LABORatorio R. Revelli Working Paper 40, 2005.

http://eco83.econ.unito.it/terna/swarmfest2005papers/richiardi_paladi.pdf

[6] M. Richiardi, “Jesus vs Hillel. From Moral to Social Norms and Back,” European Journal of Economic and Social Systems, Vol. 19, No. 2, 2006, pp. 171-190.

[7] D. Kashchiev, “Nucleation. Basic Theory with Applica tions,” Butterworth-Heinemann, Oxford, 2000.

[8] C.W. Gardiner, “Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences,” 2nd Edi tion, Springer-Verlag, Berlin, 1985.

[9] A. N. Kolmogorov, “On Statistical Theory of Metal Crys tallisation (in Russian),” Izvestiya Academy of Sciences, USSR, Mathematics, Vol. 3, 1937, pp. 355-360.

[10] L. Feng, B. Li, B. Podobnik, T. Preis and H. E. Stanley, “Linking Agent-Based Models and Stochastic Models of Financial Markets,” Proceedings of the National Acad emy of Sciences of the United States of America, Vol. 109, No. 22, 2012, pp. 8388-8393. http://dx.doi.org/10.1073/pnas.1205013109