# Combinatorics and Statistical Mechanics of Integer Partitions

## Abstract

**:**

## 1. Introduction

## 2. Microcanonical Ensemble of Partitions

#### 2.1. Microcanonical Table

#### 2.2. Multiplicity

#### 2.3. Microcanonical Probability

#### 2.4. Mean Distribution

## 3. Canonical Ensemble

#### 3.1. Canonical Probability

#### 3.2. Mean Canonical Distribution

## 4. A Random Walk in the Microcanonical Space: The Exchange Reaction

#### 4.1. Binary Exchange Reaction and Its Graph

- 1.
- It is bidirectional because the reverse of the exchange reaction is also a binary exchange reaction.
- 2.
- It is connected: starting from any configuration, it is always possible to reach through a series of exchange reactions a configuration with one cluster of size $M-N-1$ (giant cluster) plus $N-1$ monomers; the mass of the giant cluster can then be distributed to the other clusters to produce any other configuration of the ensemble. Therefore, any configuration can be reached from any other.
- 3.
- Every configuration is connected to $(M-N)(N-1)$ other configurations. The maximum number of units that can be transferred from a cluster with mass k is $k-1$ (cluster masses cannot be zero). The total number of units that are available for exchange within a configuration is $M-N$, and since each cluster may transfer mass any of the other $N-1$ clusters, the number of connections that depart from any configuration is $(M-N)(N-1)$.

#### 4.2. Random Walk on the Binary Exchange Reaction Graph

#### 4.3. Monte Carlo Sampling

## 5. Asymptotic Limit

#### 5.1. Microcanonical Thermodynamics

#### 5.2. Canonical Thermodynamics

## 6. Construction of the Selection Functional

## 7. Discussion

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gibbs, J.W. Elementary Principles in Statistical Mechanics; Ox Bow Press: Woodbridge, CT, USA, 1981. [Google Scholar]
- Berestycki, J. Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures. Electron. J. Probab.
**2004**, 9, 770–824. [Google Scholar] [CrossRef] - Berestycki, N.; Pitman, J. Gibbs Distributions for Random Partitions Generated by a Fragmentation Process. J. Stat. Phys.
**2007**, 127, 381–418. [Google Scholar] [CrossRef][Green Version] - Matsoukas, T. Statistical Mechanics of Discrete Multicomponent Fragmentation. Condens. Matter
**2020**, 5, 64. [Google Scholar] [CrossRef] - Durrett, R.; Granovsky, B.L.; Gueron, S. The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes. J. Theor. Probab.
**1999**, 12, 447–474. [Google Scholar] [CrossRef] - Freiman, G.A.; Granovsky, B.L. Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. Isr. J. Math.
**2002**, 130, 259–279. [Google Scholar] [CrossRef][Green Version] - Granovsky, B.L. Asymptotics of counts of small components in random structures and models of coagulation-fragmentation. arXiv
**2005**, arXiv:math/0511381. [Google Scholar] [CrossRef] - Granovsky, B.L.; Kryvoshaev, A.V. Coagulation Processes with Gibbsian Time Evolution. J. Appl. Probab.
**2012**, 49, 612–626. [Google Scholar] [CrossRef][Green Version] - Matsoukas, T. Statistical thermodynamics of clustered populations. Phys. Rev. E
**2014**, 90, 022113. [Google Scholar] [CrossRef] [PubMed][Green Version] - Matsoukas, T. Statistical Thermodynamics of Irreversible Aggregation: The Sol-Gel Transition. Sci. Rep.
**2015**, 5, 8855. [Google Scholar] [CrossRef] [PubMed][Green Version] - Matsoukas, T. The Smoluchowski Ensemble—Statistical Mechanics of Aggregation. Entropy
**2020**, 22, 1181. [Google Scholar] [CrossRef] [PubMed] - Evans, M.R.; Hanney, T. Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A Math. Gen.
**2005**, 38, R195–R240. [Google Scholar] [CrossRef][Green Version] - Erdös, P.; Lehner, J. The distribution of the number of summands in the partitions of a positive integer. Duke Math. J.
**1941**, 8, 335–345. [Google Scholar] [CrossRef][Green Version] - Vershik, A.M. Statistical mechanics of combinatorial partitions, and their limit shapes. Funct. Anal. Its Appl.
**1996**, 30, 90–105. [Google Scholar] [CrossRef] - Fatkullin, I.; Slastikov, V. Limit Shapes for Gibbs Ensembles of Partitions. J. Stat. Phys.
**2018**, 172, 1545–1563. [Google Scholar] [CrossRef][Green Version] - Erlihson, M.M.; Granovsky, B.L. Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case. Ann. L’Institut Henri Poincaré Probab. Stat.
**2008**, 44, 915–945. [Google Scholar] [CrossRef] - Adams, S.; Dickson, M. Large deviations analysis for random combinatorial partitions with counter terms. J. Phys. A Math. Theor.
**2022**, 55, 255001. [Google Scholar] [CrossRef] - Bridges, W.; Bringmann, K. Statistics for unimodal sequences. Adv. Math.
**2022**, 401, 108288. [Google Scholar] [CrossRef] - Bóna, M. A Walk Through Combinatorics—An Introduction to Enumeration and Graph Theory, 2nd ed.; World Scientific Publishing Co.: Singapore, 2006. [Google Scholar]
- Kelly, F.P. Reversibility and Stochastic Networks; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Boltzmann, L. Lecctures on Gas Theory; Dover: New York, NY, USA, 1995. [Google Scholar]
- Matsoukas, T. Thermodynamics Beyond Molecules: Statistical Thermodynamics of Probability Distributions. Entropy
**2019**, 21, 890. [Google Scholar] [CrossRef][Green Version] - Matsoukas, T. Stochastic Theory of Discrete Binary Fragmentation—Kinetics and Thermodynamics. Entropy
**2022**, 24, 229. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**Left**) Configurations undergoing an exchange reaction represent transitions that visit the space of configurations uniformly. A random walk on this space visits each configuration the same number of times. (

**Right**) The corresponding transitions between distributions visit each distribution $\mathbf{n}$ in proportion to its multinomial factor $\mathbf{n}!$.

**Figure 2.**Exchange reactions in the ensemble $M=7$, $N=5$, visit the 15 configurations of the ensemble in proportion to the cluster weight ${w}_{k}={k}^{\alpha}$. Configurations 1–5 represent distribution ${\mathbf{n}}_{A}=(4,1,1)$; Configurations 6–15 represent distribution ${\mathbf{n}}_{A}=(3,2,0)$ (configurations are numbered in the order they appear in Table 1). (

**a**) $\alpha =0$: all configurations are visited with equal probability; (

**b**) $\alpha =4$: configurations of distribution ${\mathbf{n}}_{A}$ are visited more frequently; (

**c**) configurations of distribution ${\mathbf{n}}_{B}$ are visited more frequently. Lines show the theoretical probability calculated as $P\left(\mathbf{n}\right)/\mathbf{n}!$, where $\mathbf{n}$ is the distribution of the clusters in the configuration.

**Figure 3.**(

**a**) Triangular distribution; (

**b**) bimodal distribution of two Gaussian distributions. Symbols are MC simulations of the exchange reaction with ${w}_{k}={f}_{k}$, where ${f}_{k}$ is the triangular or the bimodal distribution. In both cases, the simulation agrees very well with the corresponding distribution.

**Table 1.**Microcanonical ensemble with $M=7$, $N=5$. (

**a**) Microcanonical table of configurations (${m}_{i}$ is the ith element of the configuration); (

**b**) list of distributions in the ensemble (${n}_{i}$ is the number of i-mers in the configuration). The table of distributions is a more concise representation of the ensemble of configurations with each distribution representing $\mathbf{n}!W\left(\mathbf{n}\right)$ distinct configurations.

(a) Configurations | ||||

m${}_{1}$ | m${}_{2}$ | m${}_{3}$ | m${}_{4}$ | m${}_{5}$ |

3 | 1 | 1 | 1 | 1 |

1 | 3 | 1 | 1 | 1 |

1 | 1 | 3 | 1 | 1 |

1 | 1 | 1 | 3 | 1 |

1 | 1 | 1 | 1 | 3 |

2 | 2 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 1 |

2 | 1 | 1 | 2 | 1 |

2 | 1 | 1 | 1 | 2 |

1 | 2 | 2 | 1 | 1 |

1 | 2 | 1 | 2 | 1 |

1 | 2 | 1 | 1 | 2 |

1 | 1 | 2 | 2 | 1 |

1 | 1 | 2 | 1 | 2 |

1 | 1 | 1 | 2 | 2 |

(b) Distributions | ||||

n${}_{1}$ | n${}_{2}$ | n${}_{3}$ | n${}_{4}$ | $\mathbf{n}\mathbf{!}$ |

4 | 0 | 1 | 0 | 5 |

3 | 2 | 0 | 0 | 10 |

**Table 2.**Canonical partitioning of microcanonical ensemble ${\mathcal{E}}_{M,N}$ with $M=7$, $N=5$. The shaded configurations form canonical ensemble ${\mathcal{C}}_{N-{N}^{\prime}|M,C}$ with ${N}^{\prime}=2$. The unshaded portion constitutes the complementary ensemble ${\mathcal{C}}_{N-{n}^{\prime}|M,N}$.

Canonical Configurations | ||||
---|---|---|---|---|

${\mathit{m}}_{\mathbf{1}}^{\prime}$ | ${\mathit{m}}_{\mathbf{2}}^{\prime}$ | ${\mathit{m}}_{\mathbf{1}}^{\u2033}$ | ${\mathit{m}}_{\mathbf{2}}^{\u2033}$ | ${\mathit{m}}_{\mathbf{3}}^{\u2033}$ |

1 | 1 | 3 | 1 | 1 |

1 | 1 | 1 | 3 | 1 |

1 | 1 | 1 | 1 | 3 |

1 | 1 | 2 | 2 | 1 |

1 | 1 | 2 | 1 | 2 |

1 | 1 | 1 | 2 | 2 |

2 | 1 | 2 | 1 | 1 |

2 | 1 | 1 | 2 | 1 |

2 | 1 | 1 | 1 | 2 |

1 | 2 | 2 | 1 | 1 |

1 | 2 | 1 | 2 | 1 |

1 | 2 | 1 | 1 | 2 |

2 | 2 | 1 | 1 | 1 |

3 | 1 | 1 | 1 | 1 |

1 | 3 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Matsoukas, T.
Combinatorics and Statistical Mechanics of Integer Partitions. *Entropy* **2023**, *25*, 385.
https://doi.org/10.3390/e25020385

**AMA Style**

Matsoukas T.
Combinatorics and Statistical Mechanics of Integer Partitions. *Entropy*. 2023; 25(2):385.
https://doi.org/10.3390/e25020385

**Chicago/Turabian Style**

Matsoukas, Themis.
2023. "Combinatorics and Statistical Mechanics of Integer Partitions" *Entropy* 25, no. 2: 385.
https://doi.org/10.3390/e25020385