## Number Theory, Dynamical Systems and Statistical Mechanics (1998)

Citations: | 8 - 2 self |

### BibTeX

@MISC{Knauf98numbertheory,,

author = {Andreas Knauf},

title = {Number Theory, Dynamical Systems and Statistical Mechanics},

year = {1998}

}

### OpenURL

### Abstract

In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented. 1 Introduction Counting was the earliest mathematical activity. Number theory thus was among the first subjects of mathematics. It was shown by Euclid that every integer n 2 N has a unique factorization n = Y p2P p ff p (1) in terms of the primes P ae N , and a class of rings like Z sharing this property thus bears his name. To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people. A nonrepresentative example is the following, from a book by Sacks: He describes a dialogue between twins: "John would say a number --- a six-figure number. Michael would catch the number, nod, smile and seem to savour it. Then he, in turn, would say another six-figure number, and now it ...

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(Show Context)
Citation Context ...(1 < β < βcr) and M(β) = 0 (0 ≤ β < βcr). The energy function can be interpreted as the time delay of scattering geodesics in the modular domain [8]. There exist direct relations with the works [12], =-=[13]-=- by Mayer, and [11] by Lanford and Ruedin, and the Riemann Hypothesis can be related to a problem concerning the spectral radius of a related Markov chain [9]. More details can be found in the lecture... |

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(Show Context)
Citation Context ... βcr) (1 < β < βcr) and M(β) = 0 (0 ≤ β < βcr). The energy function can be interpreted as the time delay of scattering geodesics in the modular domain [8]. There exist direct relations with the works =-=[12]-=-, [13] by Mayer, and [11] by Lanford and Ruedin, and the Riemann Hypothesis can be related to a problem concerning the spectral radius of a related Markov chain [9]. More details can be found in the l... |

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(Show Context)
Citation Context ...Z(s) := ζ(s) ≡ ∞∑ ϕ(n)n −s (2) is the thermodynamic limit n=1 lim k→∞ Zk(s) = Z(s) (ℜ(s) > 2) (3) of partition functions Zk(s) = ∑∞ n=1 ϕk(n)n−s . That number-theoretical spin chain was introduced in =-=[5]-=-, see also Cvitanović [3]. The Gibbs measure for inverse temperature β ∈ R assigns probabilities σ ↦→ exp(−βHk(σ)) Zk(β) (σ ∈ Gk) (4) to the configurations of the spin chain. We denote the expectation... |

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(Show Context)
Citation Context ...zation per spin Mk(β) := 1 k k∑ 〈si〉 k (β) and its thermodynamic limit M(β). By analyzing a Perron-Frobenius operator with PF eigenvalue exp(−β · F (β)), the following statements were proved. Theorem =-=[1]-=-. The only phase transition of the number-theoretical spin chain occurs for inverse temperature βcr := 2. For lower temperatures i=1 F (β) = U(β) = 0 and M(β) = 1 (β > βcr), whereas for high temperatu... |

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(Show Context)
Citation Context ...e sense 0 ≤ (jk+1(0, t) − jk(t)) · 2 s < C · 2 −d (t ∈ G ∗ k \ {0}). For β > 0 the thermodynamic limit F (β) := lim k→∞ Fk(β) with Fk(β) := − 1 β · k ln (Zk(β)) (7) of the free energy per spin exists =-=[7]-=-. • The interaction is ferromagnetic, that is, jk(t) ≥ 0 (t ∈ G ∗ k \ {0}). (8) This is in accordance with earlier speculations (see Ruelle [14]). Note, however, that the system is not of Ising type, ... |

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(Show Context)
Citation Context ... r − l in the bulk: Ak(l, r) ≤ 1 s s 2 + 2−(k−r) This is just the borderline decay rate for a phase transition . (9) A proof of the ferromagnetic property can be based on abstract polymer models, see =-=[4]-=-, and this kind of combinatorics appears naturally in the context of numbertheoretical zeta functions [2]. Switching to a multiplicative representation si(σ) := (−1) σi of the ith spin, an important v... |

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9 |
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(Show Context)
Citation Context ... β · k ln (Zk(β)) (7) of the free energy per spin exists [7]. • The interaction is ferromagnetic, that is, jk(t) ≥ 0 (t ∈ G ∗ k \ {0}). (8) This is in accordance with earlier speculations (see Ruelle =-=[14]-=-). Note, however, that the system is not of Ising type, since multi-body interactions are present. 111• The effective interaction Ak(l, r) := ∑ jk(0, . . . , 0, 1, tl+1, . . . , tr−1, 1, 0, . . . , 0... |

8 | of the number-theoretic spin chain
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(Show Context)
Citation Context ...stical mechanics terminology, and Hk(σ) = − ∑ jk(t) · χσ(t). t∈G ∗ k The negative mean jk(0) of Hk has special properties. In the thermodynamic limit it is asymptotic to jk(0) ∼ −c · k for some c > 0 =-=[6]-=-, but it is the only coefficient whose value does not affect the Gibbs probability measure (4). When we write t ≡ (t1, . . . , tk) ∈ G∗ k \ {0} in the form t = (0, . . . , 0, 1, tl+1, . . . , tr−1, 1,... |

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7 |
Irregular Scattering, Number Theory, and Statistical Mechanics. In: Stochasticity and Quantum Chaos. Z. Haba et al, Eds
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(Show Context)
Citation Context ...r), whereas for high temperatures U(β) ≥ 1 4 (β − βcr) (1 < β < βcr) and M(β) = 0 (0 ≤ β < βcr). The energy function can be interpreted as the time delay of scattering geodesics in the modular domain =-=[8]-=-. There exist direct relations with the works [12], [13] by Mayer, and [11] by Lanford and Ruedin, and the Riemann Hypothesis can be related to a problem concerning the spectral radius of a related Ma... |

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4 |
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(Show Context)
Citation Context ...(β) = 0 (0 ≤ β < βcr). The energy function can be interpreted as the time delay of scattering geodesics in the modular domain [8]. There exist direct relations with the works [12], [13] by Mayer, and =-=[11]-=- by Lanford and Ruedin, and the Riemann Hypothesis can be related to a problem concerning the spectral radius of a related Markov chain [9]. More details can be found in the lecture notes [10]. Refere... |

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3 |
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(Show Context)
Citation Context ...s (2) is the thermodynamic limit n=1 lim k→∞ Zk(s) = Z(s) (ℜ(s) > 2) (3) of partition functions Zk(s) = ∑∞ n=1 ϕk(n)n−s . That number-theoretical spin chain was introduced in [5], see also Cvitanović =-=[3]-=-. The Gibbs measure for inverse temperature β ∈ R assigns probabilities σ ↦→ exp(−βHk(σ)) Zk(β) (σ ∈ Gk) (4) to the configurations of the spin chain. We denote the expectation of a random variable by ... |

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