Fermi Statistics Method Applied to Model Macroscopic Demographic Data

Giuseppe Alberti

doi:doi:10.11648/j.sjams.20251304.12

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Fermi Statistics Method Applied to Model Macroscopic Demographic Data

Giuseppe Alberti^*

Published in Science Journal of Applied Mathematics and Statistics (Volume 13, Issue 4)

Received: 25 October 2025 Accepted: 12 November 2025 Published: 11 December 2025

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Abstract

The study begins by considering an abstract object (cellular automaton) able of moving -by arbitrary decision- between two given fixed positions. That is, at each clock step, it can change position or remain stationary in its current position. This object, which we call an Arbitrary Oscillator (ArbO), cannot evolve indefinitely since it may encounter ‘end-of-life’ events, which are also random. If we place quantitative limits on the number of arbitrary events and impose that the life cycle of ArbO must end in any case, we can use Fermi statistics to find the most probable distribution of fatal events along the possible sequences of choices. This distribution is represented by a recursive function that can be calculated for each total number of possible ‘life/death’ choices, which we will call Total Cases (TC). By means of a time-scale adjustment, we have associated the distribution curves of ArbO ‘fatal’ events with the demographic mortality curves (dx and qx data) of populations in the case of Italy. To better study the properties of the statistical function thus found, we attempted a continuous transposition of the recursive equation, seeking solutions to the differential equation linkable with it. With a continuous analytical expression, the characteristics of this statistical distribution can be studied more effectively. Similarities and differences with demographic mortality curves have been highlighted, attempting to explain the latter as overlaps of curves with different TC parameters. Implications with life span and more general life cycle concepts are outlined. A correlation with a more recent study using a multi-omics approach is also pointed out.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 13, Issue 4)
DOI	10.11648/j.sjams.20251304.12
Page(s)	76-91
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Cellular Automata, Fermi Statistics, Logistic Distribution, Demographic Mortality, Lifespan

References

[1]	L. A. Gavrilov and N. S. Gavrilova, “The Biology of Life Span: a Quantitative Approach” Center on Aging NORC and The University of Chicago, Illinois, USA, slide presentation available at: https://slidetodoc.com/
[2]	D. Makowiec, D. Stauffer and M. Zieliński, “Gompertz law in simple computer model of ageing of biological population”, http://arxiv.org/abs/cond-mat/0107107v1
[3]	A. Racco, M. Argollo de Menezes and T. J. Penna “Search for an unitary mortality law through a theoretical model for biological ageing”, http://arxiv.org/abs/adap-org/9709002v1
[4]	M. D. Pascariu, A. Lenart & V. Canudas-Romo (2019) “The maximum entropy mortality model: forecasting mortality using statistical moments”, Scandinavian Actuarial Journal, 2019: 8, 661-685, https://doi.org/10.1080/03461238.2019.1596974
[5]	A. Boulougari, K. Lundengård, M. Rančić, S.Silvestrov, S. Suleiman & B. Strass (2019) “Application of a power-exponential function-based model to mortality rates forecasting”, Communications in Statistics: Case Studies, Data Analysis and Applications, 5: 1, 3-10, https://doi.org/10.1080/23737484.2019.1578705
[6]	S. J. Clark “A General Age-Specific Mortality Model With an Example Indexed by Child Mortality or Both Child and Adult Mortality”, Demography. 2019 June; 56(3): 1131-1159. https://doi.org/10.1007/s13524-019-00785-3
[7]	ISTAT Web site, http://dati.istat.it
[8]	SQU Systems, Web site managed by the author and devoted to diophantine S equation systems. https://squ-systems.eu
[9]	E. Fermi, “Molecules, Crystals and Quantum Statistics”, W. A. Benjamin, 1966, pag. 264 and subseq.
[10]	P. Richmond and B. M. Roehner “Predictive implications of Gompertz’s law.”, http://arxiv.org/abs/1509.07271v1
[11]	L. A. Gavrilov and N. S. Gavrilova, “Mortality Measurement at Advanced Ages: a Study of the Social Security Administration Death Master File”, North American Actuarial Journal. 15(3): 432-447. https://doi.org/10.1080/10920277.2011.10597629
[12]	Nicolas Bacaer, Histoires des mathematiques et de populations ed. Cassini, Paris, 2008.
[13]	S. Méléard, “Modèles aléatoires en Ecologie et Evolution”, Springer.
[14]	K., Henderson\| M. Loreau “An ecological theory of changing human population dynamics”, People and Nature. 2019; 1: 31-43. https://doi.org/10.1002/pan3.8
[15]	M. Ausloos, “Gompertz and Verhulst frameworks for growth AND decay description”, arXiv: 1109.1269v2.
[16]	G. Alberti “More on the mortality conjecture: the components of demographic mortality” https://doi.org/10.13140/rg.2.2.14304.46086
[17]	Xiaotao Shen, Chuchu Wang, Xin Zhou, Wenyu Zhou, Daniel Hornburg, Si Wu & Michael P. Snyder “Nonlinear dynamics of multi-omics profiles during human aging”, Nature Aging, http://dx.doi.org/10.1038/s43587-024-00692-2

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APA Style

Alberti, G. (2025). Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Science Journal of Applied Mathematics and Statistics, 13(4), 76-91. https://doi.org/10.11648/j.sjams.20251304.12

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ACS Style

Alberti, G. Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Sci. J. Appl. Math. Stat. 2025, 13(4), 76-91. doi: 10.11648/j.sjams.20251304.12

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AMA Style

Alberti G. Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Sci J Appl Math Stat. 2025;13(4):76-91. doi: 10.11648/j.sjams.20251304.12

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@article{10.11648/j.sjams.20251304.12,
  author = {Giuseppe Alberti},
  title = {Fermi Statistics Method Applied to Model Macroscopic Demographic Data},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {13},
  number = {4},
  pages = {76-91},
  doi = {10.11648/j.sjams.20251304.12},
  url = {https://doi.org/10.11648/j.sjams.20251304.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20251304.12},
  abstract = {The study begins by considering an abstract object (cellular automaton) able of moving -by arbitrary decision- between two given fixed positions. That is, at each clock step, it can change position or remain stationary in its current position. This object, which we call an Arbitrary Oscillator (ArbO), cannot evolve indefinitely since it may encounter ‘end-of-life’ events, which are also random. If we place quantitative limits on the number of arbitrary events and impose that the life cycle of ArbO must end in any case, we can use Fermi statistics to find the most probable distribution of fatal events along the possible sequences of choices. This distribution is represented by a recursive function that can be calculated for each total number of possible ‘life/death’ choices, which we will call Total Cases (TC). By means of a time-scale adjustment, we have associated the distribution curves of ArbO ‘fatal’ events with the demographic mortality curves (dx and qx data) of populations in the case of Italy. To better study the properties of the statistical function thus found, we attempted a continuous transposition of the recursive equation, seeking solutions to the differential equation linkable with it. With a continuous analytical expression, the characteristics of this statistical distribution can be studied more effectively. Similarities and differences with demographic mortality curves have been highlighted, attempting to explain the latter as overlaps of curves with different TC parameters. Implications with life span and more general life cycle concepts are outlined. A correlation with a more recent study using a multi-omics approach is also pointed out.},
 year = {2025}
}

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TY  - JOUR
T1  - Fermi Statistics Method Applied to Model Macroscopic Demographic Data
AU  - Giuseppe Alberti
Y1  - 2025/12/11
PY  - 2025
N1  - https://doi.org/10.11648/j.sjams.20251304.12
DO  - 10.11648/j.sjams.20251304.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 76
EP  - 91
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20251304.12
AB  - The study begins by considering an abstract object (cellular automaton) able of moving -by arbitrary decision- between two given fixed positions. That is, at each clock step, it can change position or remain stationary in its current position. This object, which we call an Arbitrary Oscillator (ArbO), cannot evolve indefinitely since it may encounter ‘end-of-life’ events, which are also random. If we place quantitative limits on the number of arbitrary events and impose that the life cycle of ArbO must end in any case, we can use Fermi statistics to find the most probable distribution of fatal events along the possible sequences of choices. This distribution is represented by a recursive function that can be calculated for each total number of possible ‘life/death’ choices, which we will call Total Cases (TC). By means of a time-scale adjustment, we have associated the distribution curves of ArbO ‘fatal’ events with the demographic mortality curves (dx and qx data) of populations in the case of Italy. To better study the properties of the statistical function thus found, we attempted a continuous transposition of the recursive equation, seeking solutions to the differential equation linkable with it. With a continuous analytical expression, the characteristics of this statistical distribution can be studied more effectively. Similarities and differences with demographic mortality curves have been highlighted, attempting to explain the latter as overlaps of curves with different TC parameters. Implications with life span and more general life cycle concepts are outlined. A correlation with a more recent study using a multi-omics approach is also pointed out.
VL  - 13
IS  - 4
ER  -

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Author Information

Giuseppe Alberti

Independent Researcher, Como, Italy

Contact Email

http://orcid.org/0000-0002-3041-8016

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APA Style

Alberti, G. (2025). Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Science Journal of Applied Mathematics and Statistics, 13(4), 76-91. https://doi.org/10.11648/j.sjams.20251304.12

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ACS Style

Alberti, G. Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Sci. J. Appl. Math. Stat. 2025, 13(4), 76-91. doi: 10.11648/j.sjams.20251304.12

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AMA Style

Alberti G. Fermi Statistics Method Applied to Model Macroscopic Demographic Data. Sci J Appl Math Stat. 2025;13(4):76-91. doi: 10.11648/j.sjams.20251304.12

Copy | Download

@article{10.11648/j.sjams.20251304.12,
  author = {Giuseppe Alberti},
  title = {Fermi Statistics Method Applied to Model Macroscopic Demographic Data},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {13},
  number = {4},
  pages = {76-91},
  doi = {10.11648/j.sjams.20251304.12},
  url = {https://doi.org/10.11648/j.sjams.20251304.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20251304.12},
  abstract = {The study begins by considering an abstract object (cellular automaton) able of moving -by arbitrary decision- between two given fixed positions. That is, at each clock step, it can change position or remain stationary in its current position. This object, which we call an Arbitrary Oscillator (ArbO), cannot evolve indefinitely since it may encounter ‘end-of-life’ events, which are also random. If we place quantitative limits on the number of arbitrary events and impose that the life cycle of ArbO must end in any case, we can use Fermi statistics to find the most probable distribution of fatal events along the possible sequences of choices. This distribution is represented by a recursive function that can be calculated for each total number of possible ‘life/death’ choices, which we will call Total Cases (TC). By means of a time-scale adjustment, we have associated the distribution curves of ArbO ‘fatal’ events with the demographic mortality curves (dx and qx data) of populations in the case of Italy. To better study the properties of the statistical function thus found, we attempted a continuous transposition of the recursive equation, seeking solutions to the differential equation linkable with it. With a continuous analytical expression, the characteristics of this statistical distribution can be studied more effectively. Similarities and differences with demographic mortality curves have been highlighted, attempting to explain the latter as overlaps of curves with different TC parameters. Implications with life span and more general life cycle concepts are outlined. A correlation with a more recent study using a multi-omics approach is also pointed out.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Fermi Statistics Method Applied to Model Macroscopic Demographic Data
AU  - Giuseppe Alberti
Y1  - 2025/12/11
PY  - 2025
N1  - https://doi.org/10.11648/j.sjams.20251304.12
DO  - 10.11648/j.sjams.20251304.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 76
EP  - 91
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20251304.12
AB  - The study begins by considering an abstract object (cellular automaton) able of moving -by arbitrary decision- between two given fixed positions. That is, at each clock step, it can change position or remain stationary in its current position. This object, which we call an Arbitrary Oscillator (ArbO), cannot evolve indefinitely since it may encounter ‘end-of-life’ events, which are also random. If we place quantitative limits on the number of arbitrary events and impose that the life cycle of ArbO must end in any case, we can use Fermi statistics to find the most probable distribution of fatal events along the possible sequences of choices. This distribution is represented by a recursive function that can be calculated for each total number of possible ‘life/death’ choices, which we will call Total Cases (TC). By means of a time-scale adjustment, we have associated the distribution curves of ArbO ‘fatal’ events with the demographic mortality curves (dx and qx data) of populations in the case of Italy. To better study the properties of the statistical function thus found, we attempted a continuous transposition of the recursive equation, seeking solutions to the differential equation linkable with it. With a continuous analytical expression, the characteristics of this statistical distribution can be studied more effectively. Similarities and differences with demographic mortality curves have been highlighted, attempting to explain the latter as overlaps of curves with different TC parameters. Implications with life span and more general life cycle concepts are outlined. A correlation with a more recent study using a multi-omics approach is also pointed out.
VL  - 13
IS  - 4
ER  -

Copy | Download