---

statistical mechanics
and number theory

One of the earliest, and perhaps most significant, examples of number theory influencing the development of physics was the application of Pólya's work on the Riemann zeta function to the theory of phase transitions by Lee and Yang in the early 1950's.

In 1951-2, Lee and Yang were developing this theory, and Mark Kac became aware of their conjecture which was later to become the "Lee-Yang circle theorem". It brought to his mind a theorem from Pólya's paper "Bemerkung uber die integraldarstellung der Riemannschen zeta-funktion" ("Remarks on the integral representation of Riemann's zeta function"). He realised that a slight modification of Pólya's proof could be used to prove a special case of Lee and Yang's conjecture, and brought this to their attention.

Lee and Yang were then able to adapt the reasoning and, within a couple of weeks, produce a proof of their general theorem. Kac says, "I recall Professor Yang telling me at the time that Hilfsatz II of Pólya...was one essential ingredient in their proof.".

Lee-Yang theorems provide restrictions on the locations of zeros of the analytic continuation of (real) partition functions associated with systems in statistical mechanics. Such complex zeros are associated with phase transitions. The circle theorem mentioned above states that, for certain systems, all such zeros must lie on a unit circle, but there are other cases where the zeros are restricted to a line. Recall that the Riemann Hypothesis seeks to restrict the location of zeros of the Riemann zeta function to a line in the complex plane. This parallel has turned out to be quite significant, as we shall go on to see.

In equilibrium statistical mechanics, the fundamental object of study for a system is its partition function. In the theory of the distribution of primes, the fundamental object is the Riemann zeta function. In the following paper, Bernard Julia introduced an abstract "numerical gas", called the free Riemann gas, whose constituent particles are prime numbers, and whose partition function is identical to the zeta function. This is a remarkably simple and natural construction, and leads to a number of suggestive parallels between various aspects of statistical mechanics and analytic number theory.
 

B.L. Julia, "Statistical theory of numbers", from Number Theory and Physics, M. Waldschmidt, et. al. (eds.), Springer Proceedings in Physics 47 (Springer, 1989) [outline]

B.L. Julia, "Theories statistique et thermodynamique des nombres", in: Conference de Strasbourg en l'honneur de P. Cartier, Proc. IRMA-RCP25, Vol. 44 (1993). 

"We shall in fact bring a large chunk of theoretical physics technology to bear on important mathematical problems and conversely one hopes to learn from a century of analytic number theory to understand better several issues of modern physics like the quark-gluon plasma transition and the Hagedorn critical temperature."

B.L. Julia, "Thermodynamic limit in number theory: Riemann-Beurling gases", Physica A 203 425-436.

"We study the grand canonical version of a solved statistical model, the Riemann gas: a collection of bosonic oscillators with energies the logarithms of the prime numbers. The introduction of a chemical potential \mu amounts to multiply each prime by $e^{\mu}$, the corresponding gases could be called Beurling gases because they are defined by the choice of appropriate generalized primes when considered as canonical ensembles; one finds generalized Hagedorn singularities in the temperature. The discrete spectrum can be treated as continuous in its high energy region; this approximation allows us to study the high energy level density and is applied to Beurling gases. It is expected to be accurate for the high temperature behaviour. One model (the logarithmic gases) will be studied in more detail, it corresponds to the choice of all the integers strictly larger than one as Beurling primes; we give an explicit formula for its grand canonical thermodynamic potential $F - \mu N$ in terms of a hypergeometric function and check the approximation on the Hagedorn phenomenon. Related physical situations include string theories and quark deconfinement where one needs a better understanding of the nature of the Hagedorn transitions."

"Theorie analytique des nombres et mecanique statistique quantique" - a brief discussion (in French) of Julia's work and related issues.
 

Unknown to Julia at the time of his 1989 article, George Mackey had arrived at the same idea some years earlier in the following book. Julia points out that whereas Mackey treats only the bosonic case, his own work also considers the fermionic case. Mackey's book draws a number of additional parallels between statistical mechanics and number theory.

G.W. Mackey, Unitary Group Representation in Physics, Probability, and Number Theory (Benjamin, 1978). 

"...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function -zeta'/zeta whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function." (p.300) 
 

Donald Spector, also unaware of Mackey's work, made a number of closely-related discoveries at the same time as Julia.

D. Spector, "Supersymmetry and the Möbius inversion function", Communications in Mathematical Physics 127 (1990) 239.

"We show that the Möbius inversion function of number theory can be interpreted as the operator (-1)^F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)^F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that mu(n) = 0 when n is not squarefree being equivalent to the Pauli exclusion principle...One of the results we obtain is equivalent to the prime number theorem, one of the central achievements of number theory, in which the asymptotic density of prime numbers is computed."
 

M. Wolf, "Applications of statistical mechanics in prime number theory"

This is a summary of a recent preprint. It's a review article based on a lecture given in Budapest earlier this year (2000), and due to appear in Physica A. As well as covering the work of Julia, Spector, Knauf, etc. he introduces another abstract numerical gas. This is something like Julia's free Riemann gas, but instead of the energies of the particles (primes) being based on their magnitudes, they are based on the gaps between consecutive primes.
 

J. Baez's "This Weeks Finds in Mathematical Physics (Week 199)" discusses C* algebras and B. Julia's "free Riemann gas" concept, referencing my summary of the original article.
 

P. Leboeuf, A. G. Monastra and O. Bohigas, "The Riemannium", Regular and Chaotic Dynamics 6 (2001) 205-210.

[abstract:] "The properties of a fictitious, fermionic, many-body system based on the complex zeros of the Riemann zeta function are studied. The imaginary part of the zeros are interpreted as mean-field single-particle energies, and one fills them up to a Fermi energy E_F. The distribution of the total energy is shown to be non-Gaussian, asymmetric, and independent of E_F in the limit E_F -> infinity. The moments of the limit distribution are computed analytically. The autocorrelation function, the finite energy corrections, and a comparison with random matrix theory are also discussed."

P. Leboeuf and A.G. Monastra, "Quantum thermodynamic fluctuations of a chaotic Fermi-gas model"

[abstract:] "We investigate the thermodynamics of a Fermi gas whose single-particle energy levels are given by the complex zeros of the Riemann zeta function. This is a model for a gas, and in particular for an atomic nucleus, with an underlying fully chaotic classical dynamics. The probability distributions of the quantum fluctuations of the grand potential and entropy of the gas are computed as a function of temperature and compared, with good agreement, with general predictions obtained from random matrix theory and periodic orbit theory (based on prime numbers). In each case the universal and non-universal regimes are identified."
 

S. Torquato, A. Scardicchio, C.E. Zachary, "Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory" (preprint 09/2008)

[abstract:] "It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analytically provide exact generalizations of such a point process in $d$-dimensional Euclidean space for any $d$, which are special cases of determinantal processes. In particular, we obtain the $n$-particle correlation functions for any n, which completely specify the point processes. We also demonstrate that spin-polarized fermionic systems have these same $n$-particle correlation functions in each dimension. The point processes for any $d$ are shown to be hyperuniform. The latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law $r^{-(d+1)}$. We graphically display one- and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of $d$. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius $r$ in dimension $d$ behaves like a Poisson point process but in dimension $d+1$ for large r and finite $d$. We also show that as $d$ increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than $1/2^d$."
 

I. Bakas and M.J. Bowick, "Curiosities of arithmetic gases", Journal of Mathematical Physics 32 (7) (1991) 1881-1884

[abstract:] "Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson-parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail."

"In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented." [extensive survey article]

work by Andreas Knauf, et. al. on number theoretical spin chains

J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

"In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*- dynamical system, which admits the zeta function as partition function and the Galois group Gal(Q ^cycl/ Q) of the cyclotomic extension Q^cycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature beta = 1. The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe])."
 

D. Harari and E. Leichtnam "Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps global quelconques"

This generalises the result of Bost and Connes which interprets the Riemann zeta function as a partition function of a dynamical system (in the C*-algebra formalism) whereby the pole at s =1 is interpreted in terms of spontaneous symmetry breaking. The generalisation extends the result to general number fields, and is further improved in the following paper which generalises in such a way that the partition function becomes the appropriate Dedekind zeta function: 
 

P. Cohen "Dedekind zeta functions and quantum statistical mechanics"

P.B. Cohen, "A C*-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking", soumis aux Actes des Journees Arithmetiques de Limoges, 1997. Preprint de l'IRMA de l'UST de Lille.
 

A. Connes and M. Marcolli, "From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices" (preprint 04/04)

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

M. Marcolli and A. Connes, "From physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory", from Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)
 

E. Ha and F. Paugam, "Bost-Connes-Marcolli systems for Shimura varieties" (preprint 03/05)

[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous to the Bost-Connes-Marcolli system...in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the "correct" symmetries and "correct" partition function. We give a formalism that applies to general Shimura data (G,X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature."   [additional background information]

J. Lagarias, "Number theory zeta functions and dynamical zeta functions", in Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Math. 237 (AMS, 1999) 45-86

[abstract:] "We describe analogies between number theory zeta functions, dynamical zeta functions,and statistical mechanics zeta functions, with emphasis on multi-variable zeta functions. We mainly consider two-variable zeta functions $\zeta_{f}(z,s)$ in which the variable $z$ is a "geometric variable", while the variable $s$ is an "arithmetic variable". The $s$-variable has a thermodynamic interpretation, in which $s$ parametrizes a family of energy functions $\phi_{s}$. We survey results on the analytic continuation and location of zeros and poles of two-variable zeta functions for four examples connected with number theory. These examples are (1) the beta transformation $f(x) = \beta x$ (mod 1), (2) the Gauss continued fraction map $f(x) = 1/x$ (mod 1), (3) zeta functions of varieties over finite fields, and (4) Riemann zeta function."

Although the following does not deal with statistical mechanics as such, the author (seemingly unaware of the works of Mackey, Julia, and Spector) presents an exactly analogous interpretation of the Riemann zeta function as a partition function, in the context of quantum entanglement:

Daniel Fivel, "The prime factorization property of entangled quantum states"

"Completely entangled quantum states are shown to factorize into tensor product of entangled states whose dimensions are powers of prime numbers...We consider processes in which factors are exchanged between entangled states and study canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension."

J.J. Garcia Moreta, Chebyshev Partition function: A connection between statistical physics and Riemann Hypothesis" (preprint 09/2006)

[abstract:] "In this paper we present a method to obtain a possible self-adjoint Hamiltonian operator so its energies satisfy Z(1/2+iE_n)=0, which is an statement equivalent to Riemann Hypothesis, first we use the explicit formula for the Chebyshev function Psi(x) and apply the change x=exp(u), after that we consider an Statistical partition function involving the Chebyshev function and its derivative so Z=Tr(exp(-BH), from the integral definition of the partition function Z we try to obtain the Hamiltonian operator assuming that H=P^{2}+V(x) by proposing a Non-linear integral equation involving Z(B) and V(x)."

A.I. Solomon, G.E.H. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, "Hopf algebra structure of a model quantum field theory" (Talk presented by first-named author at 26th International Colloquium on Group Theoretical Methods in Physics, New York, June 2006. See cs.OH/0609107 for follow-up talk delivered by second-named author.)

[abstract:] "Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra."

 

C. Newman, "Gaussian correlation inequalities for ferromagnets", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 33 (1975) 75-93.

D. Williams summarises here, saying Newman "explained that if it could be shown that a certain probability density function is 'ferromagnetic', then the Riemann Hypothesis would follow." He further notes that this density function "arises fairly naturally in the study of Brownian motion."

C. Newman, "The GHS inequality and the Riemann hypothesis", Constructive Approximation 7 No.3 (1991) 389-399

[abstract:] "Let V(t) be the even function on (-$\infty,\infty)$ which is related to the Riemann xi-function by $\Xi (x/2)=4\int\sp{\infty}\sb{-\infty}\exp (ixt-V(t))dt.$ In a proof of certain moment inequalities which are necessary for the validity of the Riemann hypothesis, it was previously shown that $V'(t)/t$ is increasing on (0,$\infty)$. We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely that $V'$ is convex on $[0,\infty)$. The possible relevance of the convexity of $V'$ to the Riemann hypothesis is discussed."
 

C.A. Tracy, "The emerging role of number theory in exactly solvable models in lattice statistical mechanics", Physica D: Nonlinear Phenomena 25 (1987) 1-19

[abstract:] "We review the Baxter model, the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory. It is shown, for example, that various physical quantities in the hard hexagon model are all expressible in terms of modular functions. The use of Rogers-Ramanujan type identities in solvable models is also reviewed."
 

Nan-Xian Chen, Mi Li and Shao-jun Liu, "Phonon dispersions and elastic constants of Ni₃Al and Möbius inversion", Physics Letters A 195 (1994) 135-143

[abstract:] "The Möbius inversion formulation corresponding to inequally weighted summations for solving some three-dimensional lattice problems introduced by Chen et al. [Phys. Lett. A 184 (1994) 347] has been used for the first time to obtain the pair potentials for fcc and L1₂ structures. The derivation is exact for radial potentials not only between identical atoms, but also between distinct atoms. We have tested this formulation for Ni₃Al using the empirical total energy function in the Rose model. The phonon dispersions and the elastic constants have been evaluated based on our calculated pair potentials and the results are in good agreement with the experimental data. This method shows a convenient route from electronic structure calculation or empirical formula of binding energy curve to mechanical properties of materials. It also indicates the potential application of the number theory method to condensed matter physics."

Nan-Xian Chen, "Modified Möbius inverse formula and its applications in physics", Phys. Rev. Lett. 64, 1193-1195 (1990)

[abstract:] "A new theorem of inverse formula is introduced for a kind of infinite series. Thus some new results for important inverse problems in physics are presented in this paper. These are the inverse problems for obtaining the phonon density of states, the inverse blackbody radiation problem for remote sensing, and the solution for inverse Ewald summation. Of more importance, it shows the possibility of the application of number theory to physical problems."

There appears to have been a second letter in the same volume on the same topic (p. 3203), and further discussion from other authors:

A.J. Pindor, "Comment on 'Modified Möbius inverse formula and its applications in physics'", Phys. Rev. Lett. 66 (1991) 957

Ninham, et. al. begin their their 1992 survey paper:

"This paper was stimulated by a brief note of Chen, which attracted some interest. Chen showed how to effect the asymptotic solution of several standard inverse problems in statistical physics by invoking the Möbius inversion formula, an apparently obscure result of algebraic number theory. The cornerstone of Chen's analysis is equivalent to the assertion that, under modest hypotheses on the functions $\alpha$ and $\beta$, if

$\alpha(x) = \Sum_{n=1}^{\infty} \beta(nx)$ for all $x > 0$

then

$\beta(x) = \Sum_{n=1}^{\infty}\mu(n)\alpha(nx)$ for all $x > 0$

[where $\mu(n)$ is the Möbius function]

To number theorists this key result in Chen is utterly trivial and well known, and Chen is utterly trivial and well known, and Chen subsequently noted that the rather circuitous original derivation of equation 2 can be replaced by appeal to [a theorem of Hardy and Wright], yet for physicists not familiar with analysis buried in classics like Titchmarsh or Hardy and Wright some new magical tools seem to have been invented. Indeed the Editor of Nature suggested (volume 344 (1990) that by so calling in the treasure-trove of the old world some new insights of classical analysis might become accessible."
 

B.W. Ninham and B.D. Hughes, "Möbius, Mellin, and mathematical physics", Physica A: Statistical and Theoretical Physics 186 (1992) 441-481

[abstract:] "We examine some results and techniques of analytic number theory which have application, or potential application, in mathematical physics. We consider inversion formulae for lattice sums, various transformations of infinite series and products, functional equations and scaling relations, with selected applications in electrostatics and statistical mechanics. In the analysis, the Mellin transform and the Riemann zeta function play a key role."
 

M. Bazant, notes on applications of Möbius inversion in physics

"After more than a century confined to pure mathematics, the Möbius Inversion Formula of number theory and combinatorics is now finding applications in science. . . The juxtaposition of venerable knowledge in mathematics with recent insights from physics leads naturally to (i) a generalized 'Möbius Series Inversion Formula' containing all the previous cases and (ii) methods to overcome various limitations of the old theory for physical applications."

M. Bazant, "Lattice inversion problems with applications in solid state physics"
(involves Möbius function and inversion, Dirichlet multiplication)
 

Y. Wei, G. Yan and Q. Z. Yao, "Dirichlet inversion and lattice inversion problem", Computers and Mathematics with Applications, 41 (2001) 641-645

[abstract:] "Another application of Dirichlet multiplication is considered in this note. We show that Dirichlet inversion in number theory plays an important role in lattice inversion problem. With the help of this concept, lattice inversion problem becomes straightforward."
 

K. Iguchi, "Generalized Wigner lattices as a Riemann solid: Fractals in Hurwitz zeta function" (submitted to Modern Physics Letters B)

"We study the ground state configuration and the excitation energy gaps in the strong coupling limit of the extended Hubbard model with a long-range interaction in one dimension. As proved by Hubbard and Pokrovsky and Uimin, the ground state configuration is quasiperiodic and as proved by Bak and Bruinsma, the excitation energy has a finite gap which forms a devil's stair as a function of the density of particles in the system. We show that the quasiperiodicity and the fractal nature of the excitation energy come from the nature of the long-range interaction that is related to the fractal nature of the Hurwitz Zeta function and the Riemann Zeta function."
 

S. Ishiwata, S. Matsutani and Y. Ônishi, "Localized state of hard core chain and cyclotomic polynomial: hard core limit of diatomic Toda lattice", Phys. Lett. A 231 (1997) 208-216

[abstract:] "We consider a one-dimensional classical hard core chain with different alternating masses m and M. For a certain mass ratio M/m, there exists a localized state which consists of three adjacent particles and propagates. Then its mass ratio is given by a polynomial with integer coefficients, which turns out to be the cyclotomic polynomial. We can derived the complete series of such mass ratios."
 

I. Vardi, "Deterministic percolation", Communications in Mathematical Physics 207 (1999) 43-66

[excerpt from introduction:] "...percolation theory has been of great interest in physics, as it is one of the simplest models to exhibit phase transitions. In this paper, I will examine how questions of percolation theory can be posed in a deterministic setting. Thus deterministic percolation is the study of unbounded walks on a single subset of a graph, e.g., defined by number theoretic conditions. This might be of interest in physics and probability theory as it studies percolation in a deterministic setting and in number theory where it can be interpreted as studying the disorder inherent in the natural numbers."

I. Vardi, "Prime percolation", Experimental Mathematics 7 (1998) 275-288

[abstract:] "This paper examines the question of whether there is an unbounded walk of bounded step size along Gaussian primes. Percolation theory predicts that for a low enough density of random Gaussian integers no walk exists, which suggests that no such walk exists along prime numbers, since they have arbitrarily small density over large enough regions. In analogy with the Cramer conjecture, I construct a random model of Gaussian primes and show that an unbounded walk of step size $k\sqrt{\log|z|}$ at $z$ exists with probability 1 if $k \gt \sqrt{2\pi\lambda_{c}}$ and does not exist with probability 1 if $k \lt \sqrt{2\pi\lambda_{c}}$ where $\lambda_{c}$ ~ 0.35 is a constant in continuum percolation, and so conjecture that the critical step size for Gaussian primes is also $\sqrt{2\pi\lambda_{c}\log|z|}$.
 

H.N.V. Temperley, "Further results on self-avoiding walks", Physica A: Statistical and Theoretical Physics 206 (1994) 350-358

[abstract:] "A Gaussian model of self-avoiding walks is studied. Not only is any cluster integral exactly evaluable, but whole sub-series can be evaluated exactly in terms of associated Riemann zeta functions. The results are compared with information recently obtained on self-avoiding walks on the plane square and simple cubic lattices and, as expected, are very similar. Use is made of the author's recent result that the reciprocal of the walks generating function is the generating function for irreducible cluster-sums. This is split into sub-series all of which have the same radius of convergence, and the significance of this is discussed."
 

notes on the analogy between the functional equation of the Riemann zeta function and the Kramers-Wannier duality of statistical mechanics
 

J.-F. Burnol speculates on possible relationships between zeta and the Kramers-Wannier duality, Ising models etc.

physics-based approaches to integer partition problems

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 > 10¹⁰."

M. Holthaus, K.T. Kapale, V.V. Kocharovsky and M.O. Scully, "Master equation vs. partition function: Canonical statistics of ideal Bose-Einstein condensates"

[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."

M. Holthaus and E. Kalinowski, "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 (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 the 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."

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

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 > 10¹⁰."
 

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."
 

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."
 

F.F. Ferreira and J.F. Fontanari, "Statistical mechanics analysis of the continuous number partitioning problem"

[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, "Probabilistic analysis of the number partitioning problem"

[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 $N^{-3/2}$."
 

S. Mertens, "A physicist's approach to number partitioning", to appear in J. Theor. Comp. Science

[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."

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", to appear in A.G. Percus, G. Istrate and C. Moore, eds., Computational Complexity and Statistical Physics (Oxford University Press, 2004)

[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."
 

A. Kubasiak, J. Korbicz, J. Zakrzewski, M. Lewenstein, "Fermi-Dirac statistics and the number theory" (preprint 07/05)

[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."
 

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."

 

D. Ruelle, "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the AMS 19 (1988) 259-268.

excerpts from E. Schrödinger's Statistical Thermodynamics wherein appear special values of the Riemann zeta function

C.E. Zachary, S. Torquato, "Hyperuniformity in point patterns and two-phase random heterogeneous media" (preprint 10/2009)

[abstract:] "Hyperuniform point patterns are characterized by vanishing infinite wavelength density fluctuations and encompass all crystal structures, certain quasi-periodic systems, and special disordered point patterns. This article generalizes the notion of hyperuniformity to include also two-phase random heterogeneous media. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal window volume as the window size increases. For microstructures of impenetrable and penetrable spheres, we derive an upper bound on the asymptotic coefficient governing local volume fraction fluctuations in terms of the corresponding quantity describing the variance in the local number density (i.e., number variance). Extensive calculations of the asymptotic number variance coefficients are performed for a number of disordered (e.g., quasiperiodic tilings, classical stealth disordered ground states, and certain determinantal point processes), quasicrystal, and ordered (e.g., Bravais and non-Bravais periodic systems) hyperuniform structures in various Euclidean space dimensions, and our results are consistent with a quantitative order metric characterizing the degree of hyperuniformity. We also present corresponding estimates for the asymptotic local volume fraction coefficients for several lattice families. Our results have interesting implications for a certain problem in number theory."

A. Le Méhauté, A. El Kaabouchi, L. Nivanen and Qiuping A. Wang, "Fractional dynamics, tiling equilibrium states and Riemann's zeta function" (preprint 07/09)

[abstract:] "It is argued that the generalisation of the mechanical principles to other variables than localisation, velocity and momentum leads to the laws of generalized dynamics under the condition of continuous and derivable space time. However, when the fractality arises, the mechanics principles may no more be extended especially because the time and space singularity appears on the boundary and creates curvature. There is no more equilibrium state, but only a horizon which might play a same role as equilibrium but does not close the problem - especially the problem of the invariance of the energy - which requires two complementary factors: a first one related to the closure in the dimensional space, and the second to scan dissymmetry stemming from the default of tiling the space time. A new discrete time arises from fractality. It leads irreversible thermodynamic properties. Space and time singularities lead to the relation between the above mentioned problematic and the Riemann zeta functions as well as its zeros."

A. LeClair, "Interacting Bose and Fermi gases in low dimensions and the Riemann hypothesis" (preprint, 11/2006)

[abstract:] "We apply the S-matrix based finite temperature formalism we recently developed to non-relativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. In the 2+1 dimensional case, the free energy is given in terms of Roger's dilogarithm in a way analagous to the relativistic 1+1 dimensional case. The 1-d fermionic case with a quasi-periodic 2-body potential provides a physical framework for understanding the Riemann hypothesis."

M.J. Bowick, "Finite temperature strings"

[abstract:] "These are lecture notes for the 1992 Erice Workshop on String Quantum Gravity and Physics at the Planck Energy Scale. In this talk a review of earlier work on finite temperature strings was presented. Several topics were covered, including the canonical and microcanonical ensemble of strings, the behavior of strings near the Hagedorn temperature as well as speculations on the possible phases of high temperature strings. The connection of the string ensemble and, more generally, statistical systems with an exponentially growing density of states with number theory was also discussed."

M.N. Tran, M.V.N. Murthy, R.K. Bhaduri, "On the quantum density of states and partitioning an integer" (preprint 09/03)

[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 -> infinity) 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 distinct partitions. We find that the distinct square partitions d²(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."

C. Weiss, S. Page, and M. Holthaus, "Factorising numbers with a Bose-Einstein condensate" (preprint 03/04)

[abstract:] "The problem to express a natural number N as a product of natural numbers without regard to order corresponds to a thermally isolated non-interacting Bose gas in a one-dimensional potential with logarithmic energy eigenvalues. This correspondence is used for characterising the probability distribution which governs the number of factors in a randomly selected factorisation of an asymptotically large N. Asymptotic upper bounds on both the skewness and the excess of this distribution, and on the total number of factorisations, are conjectured. The asymptotic formulas are checked against exact numerical data obtained with the help of recursion relations. It is also demonstrated that for large numbers which are the product of different primes the probability distribution approaches a Gaussian, while identical prime factors give rise to non-Gaussian statistics."

M. McGuigan, "Riemann Hypothesis and short distance fermionic Green's functions" (preprint 04/05)

[abstract:] "We show that the Green's function of a two dimensional fermion with a modified dispersion relation and short distance parameter a is given by the Lerch zeta function. The Green's function is defined on a cylinder of radius R and we show that the condition R = a yields the Riemann zeta function as a quantum transition amplitude for the fermion. We formulate the Riemann hypothesis physically as a nonzero condition on the transition amplitude between two special states associated with the point of origin and a point half way around the cylinder each of which are fixed points of a $Z_2$ transformation. By studying partial sums we show that that the transition amplitude formulation is analogous to neutrino mixing in a low dimensional context. We also derive the thermal partition function of the fermionic theory and the thermal divergence at temperature 1/a. In an alternative harmonic oscillator formalism we discuss the relation to the fermionic description of two dimensional string theory and matrix models. Finally we derive various representations of the Green's function using energy momentum integrals, point particle path integrals, and string propagators."

B. Abdesselam and A. Chakrabarti, "A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices" (preprint 01/06)

[abstract:] "Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such N² x N² matrices for odd N are studied here. Presence of (N+3)(N-1)/2 free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are N possible states at each site. The trace of the transfer matrix is shown to depend on (N-1)/2 parameters. For order r, N eigenvalues consitute the trace and the remaining N^r -N eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the r-th roots of unity, or lower dimensional multiplets corresponding to factors of the order r when r is not a prime number. The modulus of any eigenvalue is of the form e^\mu\theta, where \mu is a linear combination of the free parameters, \theta being the spectral parameter. For r a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable S matrices are constructed starting from our braid matrices. Perspectives are discussed."

E. Canessa, "Theory of analogous force on number sets" (preprint 07/03)

[abstract:] "A general statistical thermodynamic theory that considers given sequences of [natural numbers] to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions p_x for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering p_x to describe complex phenomena, the theory leads to derive a distinct analogous force f_x on number sets proportional to $(\fract{\partial p_{x}}{\partial x})_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed."

A.C. Kumar and S. Dasgupta, "A small world network of prime numbers", Physica A 357 (2005) 436

[abstract:] "According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers : $n = p + q$. We construct a network where each node is a prime number and corresponding to every even number $n$, we put a link between the component primes $p$ and $q$. In most cases, an even number can be broken up in many ways, and then we chose {\em one} decomposition with a probability $|p - q|^{\alpha}$. Through computation of average shortest distance and clustering coefficient, we conclude that for $\alpha > -1.8$ the network is of small world type and for $\alpha < -1.8$ it is of regular type. We also present a theoretical justification for such behaviour."

[abstract:] "It has long been known that the gaps between consecutive prime numbers cluster on multiples of 6. Recently it was shown that the frequency of the gaps between the gaps is lower for multiples of 6 than for other values (P. Kumar et. al., "Information entropy and correlation in prime numbers"). This paper investigates "higher moments" of the prime number series and shows that they exhibit certain peculiarities. In order to remove doubts as to whether these peculiarities are related to the known clustering of the gaps on multiples of 6, the results are compared to a benchmark series of "simulated gaps"."

S. Ares and M. Castro, "Hidden structure in the randomness of the prime number sequence?", Physica A 360 (2006) 285

[abstract:] "We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."

K. Iguchi "Generalized Sommerfeld theory: Specific heat of a degenerate g-on gas in any dimension and the generalized Riemann zeta function", International Journal of Modern Physics B11, 3551-3580 (1997).

H.P. Baltes, E.R. Hilf and M. Pabst, "The long-time behaviour of the electric-field autocorrelation function in a finite photon gas", Applied Physics B: Lasers and Optics 3 (1974) 1432-0649

[note:] This involves the use of generalized Riemann zeta functions.

S. Nechaev and O. Vasilyev, "On metric structure of ultrametric spaces", J. Phys. A 37 (2004) 3783-3803

[abstract:] "In our work we have reconsidered the old problem of diffusion at the boundary of ultrametric tree from a "number theoretic" point of view. Namely, we use the modular functions (in particular, the Dedekind eta-function) to construct the "continuous" analog of the Cayley tree isometrically embedded in the Poincaré upper half-plane. Later we work with this continuous Cayley tree as with a standard function of a complex variable. In the frameworks of our approach the results of Ogielsky and Stein on dynamics on ultrametric spaces are reproduced semi-analytically/semi-numerically. The speculation on the new "geometrical" interpretation of replica n -> 0 limit is proposed."

H. Ono and H. Kuratsuji, "Statistical theory of 2-dimensional quantum vortex gas: non-canonical effect and generalized zeta function"

"The purpose of this paper is to present a quantum statistical theory of 2-dimensional vortex gas based on the generalized Hamiltonian dynamics recently developed...A remarkable consequence is that the partition function and related quantities are given in terms of the generalized Riemann zeta function. The topological phase transition is naturally understood as the pole structure of the zeta function."

B. Eckhardt, "Eigenvalue statistics in quantum ideal gases"

"The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer $\alpha>1$ one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power $\alpha/(\alpha-1)$. In particular, for $\alpha=3/2$, the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function."

M.V. Berry and P. Shukla, "Tuck's incompressibility function: statistics for zeta zeros and eigenvalues" (preprint 07/2008)

[abstract:] "For any function that is real for real $x$, positivity of Tuck's function $Q(x)=D'^2(x)/(D'^2(x)-D"(x) D(x))$ is a condition for the absence of the complex zeros close to the real axis. Study of the probability distribution $P(Q)$, for $D(x)$ with $N$ zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of $Q$ are very rare for the Riemann zeros. $P(Q)$ has singularities at $Q=0$, $Q=1$ and $Q=N$. The moments (averages of $Q^m$) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-$N$ limit of $P(Q)$ can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at $Q=1$, while the large-$Q$ decay is determined by the pole closest to the origin. Determining the large-$N$ limit of $P(Q)$ for the GUE seems difficult."

A.L. Kholodenko, "Statistical mechanics of 2+1 gravity from Riemann zeta function and Alexander polynomial: Exact results"

"In the recent publication (Journal of Geometry and Physics, 33 (2000) 23-102) we demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemannian surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic."

A.P.C. Malbouisson and J.M.C. Malbouisson, "Boundary dependence of the coupling constant and the mass in the vector N-component $(\lambda \phi^{4})_{D}$ theory", Journal of Physics A 35 (2002) 2263-2273.

[Abstract:] "Using the Matsubara formalism, we consider the massive $(\lambda \phi^{4})_{D}$ vector N component model in the large N limit, the system being confined between two infinite parallel planes. We investigate the behavior of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L-dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulas for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviors as functions of L are displayed. We are also able to obtainn in generic dimension D, an equation for the critical value of L corresponding to a second order phase transition in terms of the Riemann zeta-function. In D = 3 a renormalization is done and an explicit formula for the critical L is given."

S.A. Oprisal, "The classical gases in the Tsallis statistics using the generalized Riemann zeta functions", J. Phys. I France 7 (July 1997) 853-862.

[Abstract:] "In the last few years an increasing interest has been paid to fractal inspired statistics. Our aim is to describe some new insight obtained using Tsallis statistics. In the framework of the generalized statistics we described some properties of the Maxwell-Boltzmann gases. The behavior of the occupation numbers with respect to the temperature indicates similarities with Fermi gases. Using the Nernst theorem we also determine the fractal index of statistics."

R. Pearson, "Number theory and critical exponents", Phys. Rev. B 22 (1980) 3465-3470

[abstract:] "The consequences of assuming p-adic analyticity for thermodynamic functions are discussed. Rules are given for determining the denominator of a rational critical exponent from the asymptotic behavior of the coefficients of series expansions. The example of the Hamiltonian Q-state Potts model is used to demonstrate the ideas of the paper."

P. Kleban, "Crossing probabilities in critical 2-D percolation and modular forms", Physica A 281 (2000) 242-251

[abstract:] "Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises."

P. Kleban and D. Zagier, "Crossing probabilities and modular forms" (preprint 09/02)

[abstract:] "We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of "higher-order modular form" arises and its properties are discussed briefly."

J. Hilgert, D. Mayer and H. Movasati, "Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,Z)$" (preprint, 03/03)

[abstract:] "In this article we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL(2,Z)$. For this we study special eigenfunctions of the transfer operators with eigenvalues +1, which are also solutions of the Lewis equations for the groups $\Gamma_{0}(n)$ and which are determined by eigenfunctions of the transfer operator for the modular group $\PSL(2,Z)$. In the language of the Atkin-Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group $\Gamma_{0}(n)$ determine for any n a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group."

R. M. Ziff, G. E. Uhlenbeck, and M. Kac, "The Bose-Einstein Gas, Revisited", Physics Reports 32C (1977)169-248

[This involves thermodynamic uses of the Riemann and Epstein zeta functions - see in particular section 3 starting with page 218.]

P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)

See in particular sections 10.7 "Global Theory: Thermodynamic Averaging" and 10.12 "Farey Tree Thermodynamics"

[excerpt from 10.11, p.19] "The Farey series thermodynamics is of number theoretical interest, because the Farey series provide uniform coverings of the unit interval with rationals, and because they are closely related to the deepest problems in number theory, such as the Riemann hypothesis...

The Riemann hypothesis...would seem to have nothing to do with physicists' real mode-locking widths that we are interested in here. However, there is a real-line version of the Riemann hypothesis that lies very close to the mode-locking problem... The implications of this for the circle-map scaling theory have not been worked out, and it is not known whether some conjecture about the thermodynamics of irrational windings is equivalent to (or harder than) the Riemann hypothesis, but the danger lurks."

S. Ares and M. Castro, "Hidden structure in the randomness of the prime number sequence" (preprint 10/03)

[abstract:] "We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with two different but important topics: the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."

"The Prime Number Theorem obtained by statistical methods" - a heuristic argument from What is Mathematics? by Courant and Robbins

"By a procedure typical of...statistical mechanics we...[make] plausible the...law of the distribution of primes."
 

number theory and entropy