The Theory of Partitions

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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. Holthaus, E. Kalinowski and K. Kirsten, " Condensate fluctations in trapped Bose gases: Canonical vs.

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.

Numerical estimation of the asymptotic behaviour of solid partitions of an integer - IOPscience

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. Holthaus, " From number theory to statistical mechanics: Bose—Einstein condensation in isolated traps ", Chaos, Solitons and Fractals 10 No.

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.

Weiss and M. Holthaus, " Asymptotics of the number partitioning distribution ", Europhys. Weiss, M. Block, M.

The theory of partitions

Holthaus and G. Schmieder, "Cumulants of partitions" , J. Holthaus, K. Kapale, V. Kocharovsky and M. Scully, " Master equation vs. 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.

Bhatia, M. Prasad and D.

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

Arora, " Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals ", J. Using enumeration techniques, we obtain upper and lower bounds for the number of multidimensional partitions both restricted and unrestricted. Ferreira and J.

Fontanari, "Probabilistic analysis of the number partitioning problem" , J. Furthermore, we calculate analytically the fraction of metastable states, i. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. Fontanari, "Instance space of the number partitioning problem" , J. A 33 [abstract:] "Within the replica framework we study analytically the instance space of the number partitioning problem.

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. Mertens, "Phase transition in the number partitioning problem" , Phys.

Partitions - Numberphile

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. Mertens, "The easiest hard problem: number partitioning" , A. Percus, G. Istrate and C. Moore, eds. 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.

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Mertens, "A physicist's approach to number partitioning" , Theor. Science 79— [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.

Bulletin (New Series) of the American Mathematical Society

Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search. Latapy, "Generalized integer partitions, tilings of zonotopes and lattices" , Formal Power Series and Algebraic Combinatorics: 12th International Conference, FPSAC'00, Moscow, Russia, June , Proceedings Springer, — [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.

Freiman, A. Vershik, and Yu. Yakubovich, " A local limit theorem for random strict partitions ", Theory Probab. Vershik, Funct. The geometrical language we use allows us to reformulate the problem in terms of random step functions Young diagrams.

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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. Borgs, J. Chayes and B. Within the window, i. Tran, M. Murthy, R. Bhaduri, "On the quantum density of states and partitioning an integer" , Ann. The origin of these oscillations from the quantum point of view is discussed.

Tran and R. 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.