physics- and probability-related approaches to
integer partitioning problems
(approximately chronological bibliography)

F.C. Auluck, S. Chowla and G. Gupta, "On the maximum value of the number of partitions of $n$ into $k$ parts", Journal of the Indian Mathematical Society 6 (1942) 105

F.C. Auluck and D.S. Kothari, "Statistical mechanics and the partitions of numbers", Proceedings of the Cambridge Philosophical Society 42 (1946) 272

[abstract:] "The properties of partitions of numbers extensively investigated by Hardy and Ramanujan have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy–Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar* for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose–Einstein, and Fermi–Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper."

*N. Bohr and F. Kalckar, "On the transmutation of atomic nuclei by impact of material particles. I. General theoretical remarks", Kgl. Danske Vid. Selskab. Math. Phys. Medd. 14 No. 10 (1937) [see also C. Van Lier and G.E. Uhlenbeck, "On the statistical calculation of the density of the energy levels of the nuclei", Physica 4 (1937) 531]

H.N.V. Temperley, "Statistical mechanics and the partition of numbers. I. The transition of liquid helium", Proceedings of the Royal Society of London A 199 (1949) 361–375

[abstract:] "The existing theory of 'Bose–Einstein condensation' is comparared with some results obtained from the theory of partition of numbers. Two models are examined, one in which the energy levels are all equally spaced, the other being the perfect gas model. It is concluded that orthodox theory can be relied upon at very high and at very low temperatures, also that the condensation phenomenon is a real one, but that it is not correctly described by orthodox theory, the position of the transition temperature and the form of the specific heat anomaly both being given wrongly."

H.N.V. Temperley, "Statistical mechanics and the partition of numbers: II. The form of crystal surfaces", Proceedings of the Cambridge Philosophical Society 48 (1952) 683–697

[abstract:] "The classical theory of partition of numbers is applied to the problem of determining the equilibrium profile of a simple cubic crystal. It is concluded that it may be thermodynamically profitable for the surface to be 'saw-toothed' rather than flat, the extra entropy associated with such an arrangement compensating for the additional surface energy. For both a two- and a three-dimensional 'saw-tooth' the extra entropy varies, to a first approxomation, in the same way as the surface energy, i.e. is proportional to $N^{1/2}$ or $N^{2/3}$ respectively, where $N$ is the number of molecules in a 'tooth'. For the simple cubic lattice, the entropy associated with the formation of a tooth containing $N$ atoms is estimated to be $3.3kN^{2/3}$. It is also possible to estimate the variation of the 'equilibrium roughness' of a crystal with temperature, if its surface energy is known."

V.S. Nanda, "Partition theory and thermodynamics of multi-dimensional oscillator assemblies", Proceedings of the Cambridge Philosophical Society 47 (1951) 591–601

[abstract:] "The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the 'Zustandsumme' of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck where the Hardy–Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

The present paper is concerned with the problems in partition theory of numbers corresponding to the thermodynamic assemblies of two and three-dimensional oscillators. Asymptotic expressions are deduced which constitute a generalization of the Hardy–Ramanujan formula for $p(n)$ which corresponds to an assembly of $\emph{linear oscillators}$. Generating functions similar to those considered here were studied earlier by MacMahon in his work on combinatory analysis. It is remarkable that the Zustandsumme of an assembly of a variable number of two-dimensional oscillators is identical with the generating function of plane partitions. The problem, thus, becomes one of establishing a relationship between the two seemingly different types of partitions. Further, it is noticed that a study of two-dimensional oscillator assembly is connected with the partitions of bi-partite numbers."

V.S. Nanda, "Bose–Einstein condensation and the partition theory of numbers", Proc. Nat. Inst. Sci. (India) 19 (1953) 681–690

In Section 2.3 of Bernard Julia's seminal 1990 paper "Statistical theory of numbers", the author turns briefly from multiplicative to additive number theory, in particular to generating functionals associated with integer partition problems. He relates these to the Veneziano open string model, the tachyon mode, and the phenomenon of "bosonization" which is discussed elsewhere in the paper.

F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.K. Hu and C.N. Chen, "Directed compact lattice animals, restricted partitions of an integer, and the infinite-state Potts model", Phys. Rev. Lett. 76 (1996) 173–176

[abstract:] "We consider a directed compact site lattice animal problem on the $d$-dimensional hypercubic lattice, and establish its equivalence with (i) the infinite-state Potts model and (ii) the enumeration of $(d-1)$-dimensional restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in $d = 2,3$ using known solutions of the enumeration problem. The maximum number of lattice animals of size $n$ grows as $\exp(cn^{(d-1)/d})$. Also, the infinite-state Potts model solution leads to a conjectured limiting form for the generating function of restricted partitions for $d>3$, the latter an unsolved problem in number theory."

F.Y. Wu, "The infinite-state Potts model and restricted multidimensional partitions of an integer", Mathematical and Computer Modelling 26 (1997) 269–274

[abstract:] "It is shown that the partition function of the $q$-state Potts model on a finite $d$-dimensional hypercubic lattice in the $q\rightarrow\infty$ limit is precisely the generating function of $(d-1)$-dimensional restricted partitions of an integer. For $d=2,3$, this equivalence leads to closed-form expressions of the $q=\infty$ Potts partition function. Our discussion also establishes symmetry and reciprocal properties for the generating function of restricted partitions in higher dimensions."

A.M. Vershik, "Statistical mechanics of combinatorial partitions, and their limit shapes", Funkts. Anal. Prilozh. 30 (1996) 19–30 [English translation: Funct. Anal. Appl. 30 (1996) 90–105]

A.M. Vershik, "Limit distribution of the energy of a quantum ideal gas from the viewpoint of the theory of partitions of natural numbers", Uspekhi Mat. Nauk 52 (1997) 139–146 [English translation: Russian Math. Surveys 52 (1997) 379–386]

S. Grossmann and M. Holthaus, "Microcanonical fluctuations of a Bose system's ground state occupation number", Phys. Rev. E 54 (1996) 3495–3498

[abstract:] "Employing asymptotic formulas from the partition theory of numbers, we derive the microcanonical probability distribution of the ground state occupation number for a one-dimensional ideal Bose gas confined at low temperatures by a harmonic potential. We compare the grand canonical analysis to the microcanonical one, and show how the fluctuation catastrophe characteristic for the grand canonical ensemble is avoided by the proper microcanonical approach."

M. Holthaus, E. Kalinowski and K. Kirsten, "Condensate fluctations in trapped Bose gases: Canonical vs. microcanonical ensemble", Annals of Physics 270 (1998) 198–230

[abstract:] "We study the fluctuation of the number of particles in ideal Bose–Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin–Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-dimensional harmonic traps we explore the microcanonical statistics in detail, with the help of the saddle-point method. Emphasizing the close connection between the partition theory of integer numbers and the statistical mechanics of ideal Bosons in one-dimensional harmonic traps, and utilizing thermodynamical arguments, we also derive an accurate formula for the fluctuations of the number of summands that occur when a large integer is partitioned."

S. Grossmann and M. Holthaus, "From number theory to statistical mechanics: Bose–Einstein condensation in isolated traps", Chaos, Solitons and Fractals 10 No. 4–5 (1999) 795–804

[abstract:] "We question the validity of the grand canonical ensemble for the description of Bose–Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number."

C. Weiss and M. Holthaus, "Asymptotics of the number partitioning distribution", Europhys. Lett. 59 (4) (2002) 486–492

[abstract:] "The number partitioning problem can be interpreted physically in terms of a thermally isolated noninteracting Bose gas trapped in a one-dimensional harmonic-oscillator potential. We exploit this analogy to characterize, by means of a detour to the Bose gas within the canonical ensemble, the probability distribution for finding a specified number of summands in a randomly chosen partition of an integer $n$. It is shown that this distribution approaches is asymptotics only for $n>10^{10}$."

C. Weiss, M. Block, M. Holthaus and G. Schmieder, "Cumulants of partitions", J. Phys. A 36 (2003) 1827–1844

[abstract:] "We utilize the formal equivalence between the number-partitioning problem and a harmonically trapped ideal Bose gas within the microcanonical ensemble for characterizing the probability distribution which governs the number of addends occurring in an unrestricted partition of a natural number $n$. By deriving accurate asymptotic formulae for its coefficients of skewness and excess, it is shown that this distribution remains non-Gaussian even when $n$ is made arbitrarily large. Both skewness and excess vary substantially before settling to their constant-limiting values for $n>10^{10}$."

M. Holthaus, K.T. Kapale, V.V. Kocharovsky and M.O. Scully, "Master equation vs. partition function: Canonical statistics of ideal Bose–Einstein condensates", Physica A 300 (2001) 433–467

[abstract:] "Within the canonical ensemble, a partially condensed ideal Bose gas with arbitrary single-particle energies is equivalent to a system of uncoupled harmonic oscillators. We exploit this equivalence for deriving a formula which expresses all cumulants of the canonical distribution governing the number of condensate particles in terms of the poles of a generalized Zeta function provided by the single-particle spectrum. This formula lends itself to systematitic asymptotic expansions which capture the non-Gaussian character of the condensate fluctuations with utmost precision even for relatively small, finite systems, as confirmed by comparison with exact numerical calculations. We use these results for assessing the accuracy of a recently developed master equation approach to the canonical condensate statistics; this approach turns out to be quite accurate even when the master equation is solved within a simple quasithermal approximation. As a further application of the cumulant formula we show that, and explain why, all cumulants of a homogeneous Bose–Einstein condensate "in a box" higher than the first retain a dependence on the boundary conditions in the thermodynamic limit."

D.P. Bhatia, M.A. Prasad and D. Arora, "Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals", J. Phys. A 30 (1997) 2281–2285

[abstract:] "There is an exact one-to-one correspondence between the number of $(d-1)$-dimensional partitions of an integer and the number of directed compact lattice animals in $d$ dimensions. Using enumeration techniques, we obtain upper and lower bounds for the number of multidimensional partitions (both restricted and unrestricted). We show that asymptotically the number of unrestricted $(d-1)$-dimensional partitions of an integer $n$ goes as $\exp(Cn^{(d-1)/d})$. We also show that for restricted partitions in $(d-1)$ dimensions (with $j$ dimensions finite, $0<j<d-1$), this number goes as $\exp(Cn^{(d-j-1)/(d-j)}(\prod_{k=1}^j L_k)^{1/(d-j)})$, where $L_k$ is the extent of the lattice along the $k$th axis."

F.F. Ferreira and J.F. Fontanari, "Probabilistic analysis of the number partitioning problem", J. Phys. A 31 (1998) 3417

[abstract:] "Given a sequence of $N$ positive real numbers $\{a_1,a_2,..., a_N \}$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. In the case that the $a_j$'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random anti-ferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large $N$ limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like $N^{-3/2}$."

F.F. Ferreira and J.F. Fontanari, "Statistical mechanics analysis of the continuous number partitioning problem", Physica A 269 (1999) 54–60

[abstract:] "The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of ${\cal A}$ and ${\cal B}$ and show that this difference is not self-averaging."

F.F. Ferreira and J.F. Fontanari, "Instance space of the number partitioning problem", J. Phys. A 33 (2000) 7265

[abstract:] "Within the replica framework we study analytically the instance space of the number partitioning problem. This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the instance) into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We show that there is an upper bound $\alpha_c N$ to the number of perfect partitions (i.e. partitions for which that difference is zero) and characterize the statistical properties of the instances for which those partitions exist. In particular, in the case that the two sets have the same cardinality (balanced partitions) we find $\alpha_c=1/2$. Moreover, we show that the disordered model resulting from the instance space approach can be viewed as a model of replicators where the random interactions are given by the Hebb rule."

S. Mertens, "Phase transition in the number partitioning problem", Phys. Rev. Lett. 81 (1998) 4281–4284

[abstract:] "Number partitioning is an NP-complete problem of combinatorial optimization. A statistical mechanics analysis reveals the existence of a phase transition that separates the easy from the hard to solve instances and that reflects the pseudo-polynomiality of number partitioning. The phase diagram and the value of the typical ground state energy are calculated."

S. Mertens, "The easiest hard problem: number partitioning", A.G. Percus, G. Istrate and C. Moore, eds., Computational Complexity and Statistical Physics (Oxford University Press, 2006) 125–140

[abstract:] "Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases."

S. Mertens, "A physicist's approach to number partitioning", Theor. Comp. Science 265 (2001) 79–108

[abstract:] "The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size $N$ to some extent corresponds to locating the minimum in an unsorted list of $O(2^N)$ numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search."

M. Latapy, "Generalized integer partitions, tilings of zonotopes and lattices", Formal Power Series and Algebraic Combinatorics: 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings (Springer, 2000) 256–267

[abstract:] "In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions."

G. Freiman, A. M. Vershik, and Yu. V. Yakubovich, "A local limit theorem for random strict partitions", Theory Probab. Appl. 44 (2000), 453–468

[abstract:] "We consider a set of partitions of natural number n on distinct summands with uniform distribution. We investigate the limit shape of the typical partition as $n\to\infty$, which was found in [A. M. Vershik, Funct. Anal. Appl., 30 (1996), pp. 90–105], and fluctuations of partitions near its limit shape. The geometrical language we use allows us to reformulate the problem in terms of random step functions (Young diagrams). We prove statements of local limit theorem type which imply that joint distribution of fluctuations in a number of points is locally asymptotically normal. The proof essentially uses the notion of a large canonical ensemble of partitions."

C. Borgs, J. Chayes and B. Pittel, "Phase transition and finite-size scaling for the integer partitioning problem", Random Structures and Algorithms 19 (2001) 247–288

[abstract:] "We consider the problem of partitioning n randomly chosen integers between $1$ and $2m$ into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called $\emph{perfect}$ if the optimum discrepancy is $0$ when the sum of all $n$ integers in the original set is even, or $1$ when the sum is odd. Parameterizing the random problem in terms of $\kappa=m/n$, we prove that the problem has a phase transition at $\kappa=1$, in the sense that for $\kappa < 1$, there are many perfect partitions with probability tending to $1$ as $n\rightarrow\infty$, whereas for $\kappa > 1$, there are no perfect partitions with probability tending to $1$. Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at $\kappa=1$.

We also determine the finite-size scaling window about the transition point: $\kappa_n=1-(2n)^{-1}\log_2n+\lambda_n/n$, by showing that the probability of a perfect partition tends to $1$, $0$, or some explicitly computable $p(\lambda)\in(0,1)$, depending on whether $\lambda_n$ tends to $-\infty, \infty$, or $\lambdaz\in(-\infty,\infty)$, respectively. For $\lambda_n\rightarrow -\infty$ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For $\lambda_n\rightarrow\infty$, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is $\Theta(2^{\lambda_n})$. Within the window, i.e., if $|\lambda_n|$ is bounded, we prove that the optimum discrepancy is bounded. Both for $\lambda_n\rightarrow\infty$ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the $k$ smallest discrepancies above the scaling window."

M.N. Tran, M.V.N. Murthy, R.K. Bhaduri, "On the quantum density of states and partitioning an integer", Ann. Phys. 311 204–219

[abstract:] "This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For $N$ bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic ($N\rightarrow\infty$) density of states is identical to the Hardy–Ramanujan formula for the partitions $p(n)$, of a number $n$ into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for $p^s(n)$, the latter being the number of partitions of $n$ into a sum of $s$-th powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain $d^s(n)$ for $\emph{distinct]$ partitions. We find that the distinct square partitions $d^2(n)$ show pronounced oscillations as a function of $n$ about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdös–Lehner formula for restricted partitions for the $s=1$ case by our method, we generalize it to obtain a new formula for distinct restricted partitions."

M.N. Tran and R.K. Bhaduri, "Number fluctuation and the fundamental theorem of arithmetic" Physical Review E 68 (2003) 026206

[abstract:] "We consider $N$ bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counter example, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations. The use of the standard canonical or grand canonical ensembles, on the other hand, gives substantial number fluctuation for the ground state. This difference between the microcanonical and canonical results cannot be accounted for within the framework of equilibrium statistical mechanics."

M. Planat, "Quantum 1/f noise in equilibrium: from Planck to Ramanujan", Physica A 318 (2003) 371

[abstract:] "We describe a new model of massless thermal bosons which predicts an hyperbolic fluctuation spectrum at low frequencies. It is found that the partition function per mode is the Euler generating function for unrestricted partitions $p(n)$. Thermodynamical quantities carry a strong arithmetical structure: they are given by series with Fourier coefficients equal to summatory functions $\sigma_k(n)$ of the power of divisors, with $k=-1$ for the free energy, $k=0$ for the number of particles and $k = 1$ for the internal energy. Low frequency contributions are calculated using Mellin transform methods. In particular the internal energy per mode diverges as $\frac{\tilde{E}}{kT}=\frac{\pi^2}{6 x}$ with $x=\frac{h \nu}{kT}$ in contrast to the Planck energy $\tilde{E}=kT$. The theory is applied to calculate corrections in black body radiation and in the Debye solid. Fractional energy fluctuations are found to show a $1/\nu$ power spectrum in the low frequency range. A satisfactory model of frequency fluctuations in a quartz crystal resonator follows. A sketch of the whole Ramanujan–Rademacher theory of partitions is reminded as well."

I. Junier and J. Kurchan, "Microscopic realizations of the trap model", J. Phys. A 37 (2004) 3945–3965

[abstract:] "Monte Carlo optimizations of Number Partitioning and of Diophantine approximations are microscopic realizations of 'Trap Model' dynamics. This offers a fresh look at the physics behind this model, and points at other situations in which it may apply. Our results strongly suggest that in any such realization of the Trap Model, the response and correlation functions of smooth observables obey the fluctuation-dissipation theorem even in the aging regime. Our discussion for the Number Partitioning problem may be relevant for the class of optimization problems whose cost function does not scale linearly with the size, and are thus awkward from the statistical mechanic point of view."

N.M. Chase, "Global structure of integer partition sequences" (preprint 04/2004, submitted to The Electronic Journal of Combinatorics)

[abstract:] "Integer partitions are deeply related to many phenomena in statistical physics. A question naturally arises which is of interest to physics both on "purely" theoretical and on practical, computational grounds. Is it possible to apprehend the global pattern underlying integer partition sequences and to express the global pattern compactly, in the form of a "matrix" giving all of the partitions of $N$ into exactly $M$ parts? This paper demonstrates that the global structure of integer partitions sequences (IPS) is that of a complex tree. By analyzing the structure of this tree, we derive a closed form expression for a map from $(N,M)$ to the set of all partitions of a positive integer $N$ into exactly $M$ positive integer summands without regard to order. The derivation is based on the use of modular arithmetic to solve an isomorphic combinatoric problem, that of describing the global organization of the sequence of all ordered placements of $N$ indistinguishable balls into $M$ distinguishable non-empty bins or boxes. This work has the potential to facilitate computations of important physics and to offer new insights into number theoretic problems."

A. Kubasiak, J. Korbicz, J. Zakrzewski, M. Lewenstein, "Fermi–Dirac statistics and the number theory", Europhysics Letters 72 (2005) 506

[abstract:] "We relate the Fermi–Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions."

V.P. Maslov, V.E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality: I", Mat. Zametki 83 (2008), 232–263 [Math. Notes 83 (2008) 211–237]

[abstract:] "We consider the mathematical problem of the allocation of indistinguishable particles to integer energy levels under the condition that the number of particles can be arbitrary and the total energy of the system is bounded above. Systems of integer as well as fractional dimension are considered. The occupation numbers can either be arbitrary nonnegative integers (the case of ``Bose particles'') or lie in a finite set $\{0,1,\dots,R\}$ (the case of so-called parastatistics; for example, $R = 1$ corresponds to the Fermi–Dirac statistics). Assuming that all allocations satisfying the given constraints are equiprobable, we study the phenomenon whereby, for large energies, most of the allocations tend to concentrate near the limit distribution corresponding to the given parastatistics."

V.P. Maslov, V. E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality: II", Mat. Zametki 83 (2008) 381–401 [Math. Notes 83 (2008) 345–363]

[abstract:] "We continue our study of the problem on the allocation of indistinguishable particles to integer energy levels under the condition that the total energy of the system is bounded above. It is shown that the $\emph{Bose condensation}$ phenomenon can occur in this model. Systems of dimension $d < 1$ (including negative dimensions) are studied."

V. P. Maslov and V. E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality, III", Mat. Zametki 83 (2008) 880–898 [Math. Notes 83 (2008) 804–820]

[abstract:] "We consider tuples $\{N_{ jk}\}, j=1,2,\dots,k=1,\dots,q_j$ , of nonnegative integers such that $$ \sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M. $$ Assuming that $q_j\sim j^{d-1}, 1 < d < 2$, we study how the probabilities of deviations of the sums $$ \sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } $$ $N_{jk}$ from the corresponding integrals of the Bose‐Einstein distribution depend on the choice of the interval $[j_1,j_2]$."

M. Nardelli, "On the physical interpretation of the Riemann zeta function, the rigid surface operators in gauge theory, the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula, p-adic strings, zeta strings and p-adic cosmology and mathematical connections with some sectors of string theory and number theory" (preprint 10/2008)

[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence and some aspects of the rigid surface operators in gauge theory. Furthermore, we describe the mathematical connections with some sectors of string theory (p-adic and adelic strings, p-adic cosmology) and number theory.

In the Section 1 we have described various mathematical aspects of the Riemann Hypothesis, matrix/gravity correspondence and master matrix for FZZT brane partition functions. In the Section 2, we have described some mathematical aspects of the rigid surface operators in gauge theory and some mathematical connections with various sectors of number theory, principally with the Ramanujan's modular equations (thence, prime numbers, prime natural numbers, Fibonacci's numbers, partitions of numbers, Euler's functions, etc...) and various numbers and equations related to the Lie Groups. In the Section 3, we have described some very recent mathematical results concerning the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula (connected with the Selberg zeta function), hence, we have obtained some new connections applying these results to the adelic strings and zeta strings. In the Section 4 we have described some equations concerning p-adic strings, p-adic and adelic zeta functions, zeta strings and p-adic cosmology (with regard the p-adic cosmology, some equations concerning a general class of cosmological models driven by a nonlocal scalar field inspired by string field theories). In conclusion, in the Section 5, we have showed various and interesting mathematical connections between some equations concerning the Section 1, 3 and 4."

M. Nardelli, "On some equations concerning fivebranes and knots, Wilson loops in Chern–Simons theory, cusp anomaly and integrability from string theory. Mathematical connections with some sectors of number theory" (preprint 09/2011)

[abstract:] "The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in Section 1, we have described some equations concerning the knot polynomials, the Chern–Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to $N = 4$ super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern–Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from string theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function.

Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan's modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan's equations concerning $\pi$ and, in conclusion, (iii) the mathematical connections with the golden ratio $\phi$ and with $1.375$ that is the mean real value for the number of partitions $p(n)$."

J. Roccia and P. Leboeuf, "Level density of a Fermi gas and integer partitions: a Gumbel-like finite-size correction", Phys. Rev. C 81 (2010) 044301

[abstract:] "We investigate the many-body level density of gas of non-interacting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics as well as differences with respect to the Bose gas."

V.P. Maslov, "Zeno-line, binodal, T-$\rho$ diagram and clusters as a new Bose-condensate bases on new global distributions in number theory" (preprint 07/2010)

[abstract:] "We present the correspondence principle between the T-$\rho$ diagram, the Zeno line and the binodal for a given interaction potential of Lennard–Jones type. We use this correspondence further to construct a distribution of the Bose–Einstein type for a classical gas with the help of the new notion of Bose condensate, making it possible to decrease fractal dimension while simultaneously preserving the number of particles. In so doing, we use new global distributions in number theory."

J.H. Bruinier and K. Ono, "An algebraic formula for the partition function" (A.I.M. preprint, 01/2011)

[abstract:] "We derive a formula for the partition function $p(n)$ as a finite sum of algebraic numbers. The summands are discriminant $-24n + 1$ singular moduli for a special weak Maass form that we describe in terms of Dedekind's eta-function and Eisenstein series."

A. Folsom, Z.A. Kent and Ken Ono, "$l$-adic properties of the partition function" (A.I.M. preprint, 01/2011)

popularly accessible blog piece on the remarkable discoveries described in the above two papers

[excerpt:] "Our work brings completely new ideas to the problems," Ono says. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions."

M. Psimopoulos, "Harmonic representation of combinations and partitions" (preprint 02/2011)

[abstract:] "In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory."

D. Prokhorov and A. Rovenchak, "Asymptotic formulas for integer partitions within the approach of microcanonical ensemble" (preprint 10/2012)

"The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results."


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