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Register a new userSemiclassical analysis of distinct square partitions
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We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the known leading-order expression [M. Tran {it et al.}, Ann. Phys. (N.Y.) {bf 311}, 204 (2004)], which yield a faster convergence to the average values of the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean triples $(m,p,q)$ of integers with $m^2+p^2=q^2$ play an important role in the oscillations of $P(n)$. Finally we analyze the oscillating part $delta P(n)=P(n)-P_{as}(n)$ in the spirit of semi-classical periodic orbit theory [M. Brack and R. K. Bhaduri: {it Semiclassical Physics} (Bolder, Westview Press, 2003)]. A semi-classical trace formula is derived which accurately reproduces the exact $delta P(n)$ for $n > sim 500$ using 10 pairs of `orbits. For $n > sim 4000$ only two pairs of orbits with the frequencies 4 and 5 -- belonging to the lowest Pythagorean triple (3,4,5) -- are relevant and create the prominent beating pattern in the oscillations. For $n > sim 100,000$ the beat fades away and the oscillations are given by just one pair of orbits with frequency 4.
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We consider a random walk on the fully-connected lattice with $N$ sites and study the time evolution of the number of distinct sites $s$ visited by the walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a record time $t$ when the walker visits a site of the subset for the first time. The record time $t$ is a partial covering time when $v<n$ and a total covering time when $v=n$. The probability distributions for the number of records $s$, the record value $v$ and the record (covering) time $t$, involving $r$-Stirling numbers, are obtained using generating function techniques. The mean values, variances and skewnesses are deduced from the generating functions. In the scaling limit the probability distributions for $s$ and $v$ lead to the same Gaussian density. The fluctuations of the record time $t$ are also Gaussian at partial covering, when $n-v={mathrm O}(n)$. They are distributed according to the type-I Gumbel extreme-value distribution at total covering, when $v=n$. A discrete sequence of generalized Gumbel distributions, indexed by $n-v$, is obtained at almost total covering, when $n-v={mathrm O}(1)$. These generalized Gumbel distributions are crossing over to the Gaussian distribution when $n-v$ increases.
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In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of $phi^4$ models with either nearest-neighbours and mean-field interactions.
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