Applications of statistical mechanics in number theory

Wolf, Marek

doi:10.1016/s0378-4371(99)00318-0

Cited by 29 publications

(40 citation statements)

References 26 publications

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“…It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic properties of materials that can be observed in every day life. It consists of mechanics plus theory of probabilities and has widespread applications, including applications to physical, chemical, biological systems, and other interdisciplinary applications such as optimisation techniques [26][27][28][29].…”

Section: Statistical Mechanicsmentioning

confidence: 99%

Particle Swarms and Nonextensive Statistics for Nonlinear Optimisation

Anastasiadis¹,

Magoulas

2008

TOCSJ

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Particle swarm methods are inspired from the dynamics of social interaction and employ information sharing to seek solutions to difficult optimisation problems. In this paper we introduce an approach that combines ideas from particle swarm optimisation (PSO) and the theory of nonextensive statistical mechanics. We develop two algorithms that adopt this approach and conduct an experimental study using benchmark functions to investigate their effectiveness in nonlinear optimisation. Results appear to be promising, as the tested algorithms outperform in most cases the standard PSO and other, recently proposed, PSO variants.

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Section: Statistical Mechanicsmentioning

confidence: 99%

Particle Swarms and Nonextensive Statistics for Nonlinear Optimisation

Anastasiadis¹,

Magoulas

2008

TOCSJ

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“…Higher-order gaps between the primes have been analyzed by Fourier decomposition to show patterns in the resulting power spectra [5]. The gaps between primes have been shown to exhibit a period six oscillation [6]. The histogram of the increments (the difference between two consecutive gaps between primes) has been shown to follow an exponential distribution with superposed periodic behavior of period three [7].…”

Section: Introductionmentioning

confidence: 99%

A Power Law Governing Prime Gaps

Matsushita¹,

Silva²

2016

OALib

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A prime gap is the difference between two successive prime numbers. Prime gaps are casually thought to occur randomly. However, the "k-tuple conjecture" suggests that prime gaps are non-random by estimating how often pairs, triples and larger groupings of primes will appear. The k-tuple conjecture is yet to be proven, but a very recent work presents a result that contributes to a confirmation of the k-tuple conjecture by finding unexpected biases in the distribution of consecutive primes. Here, we present another contribution to confirmation of the k-tuple conjecture based on statistical physics. The pattern we find comes in the form of a power law in the distribution of prime gaps. We find that prime gaps are proportional to the inverse of the chance of a number to be prime.

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“…Maybe, the most challenging open problem in mathematics is the so-called Riemann's hypothesis on the distribution of the (nontrivial) zeros of his zeta function, a problem directly related to the distribution of prime numbers and other mathematical issues. Also, the interplay between number theory, with the set of prime numbers playing a major role, and physics has been fruitful; see the short note by Wolf [1] and references therein for recent discussions.…”

mentioning

confidence: 99%

“…Due to the rather irregular distribution of prime numbers [1] we suspect the quantum dynamics generated by H V can strongly depend on the initial wave function, at least for small times; therefore, the initial condition was always concentrated on n 0 = 0. These dynamical tools were restricted to 0.5 ≤ λ, since for small λ it is not possible to neglect logarithmic corrections to the algebraic behaviours of C(t) and d 2 (t) [6].…”

mentioning

confidence: 99%

“…For example, in Fig. 3 the value λ = 1.0 was fixed and d 2 was computed by restricting the dynamics to eigenvalues in the range [−0.6, 0.0] (a subset of [−b (1), 0]) and also to eigenvalues in the range [1.6, 2.0] (corresponding to positive LE). In the former case d 2 (t) grows with time, while in the latter case it is clearly bounded; such results also support the presence of mobility edges in the spectrum of the prime Schrödinger operator.…”

mentioning

confidence: 99%

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