statistical mechanics
and number theory
the Riemann zeta function interpreted as a partition function
latticerelated number theory (involving Ising models, percolation, etc.)
integer partition problems and physics
entropy and number theory
number theory and statistical mechanics – general
probabilistic number theory
the Riemann zeta function interpreted as a partition function
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 19512, Lee and Yang were developing this theory, and
Mark Kac became aware of their conjecture
which was later to become the "LeeYang circle theorem". It brought to his mind
a theorem from Pólya's paper "Bemerkung uber die integraldarstellung der
Riemannschen zetafunktion" ("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.".
LeeYang 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. IRMARCP25, 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 quarkgluon plasma transition and the
Hagedorn critical temperature."
B.L. Julia, "Thermodynamic limit in number theory: RiemannBeurling
gases", Physica A 203 425436.
"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
closelyrelated 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) 205210.
[abstract:] "The properties of a fictitious, fermionic, manybody system based on the complex zeros of the Riemann
zeta function are studied. The imaginary part of the zeros are interpreted as meanfield singleparticle energies, and one fills
them up to a Fermi energy E_{F}. The distribution of the total energy is shown to be nonGaussian,
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 Fermigas model"
[abstract:] "We investigate the thermodynamics of a Fermi gas whose singleparticle 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 nonuniversal regimes are
identified."
G. Zhang, F. Martelli and S. Torquato, "Structure factor of the primes" (preprint 01/2018)
[abstract:] "Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudorandom numbers. In this article, we numerically study the pair statistics of the primes using statisticalmechanical methods, especially the structure factor $S(k)$ in an interval $M=p=M+L$ with $M$ large, and $L/M$ smaller than unity. We show that the structure factor of the prime configurations in such intervals exhibits welldefined Bragglike peaks along with a small "diffuse" contribution. This indicates that the primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any nonzero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as $M$ or $L$ increases. This suggests that the diffuse part vanishes in an appropriate infinitesystemsize limit."
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
spinpolarized 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 twodimensional 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 "hardcore" diameter that grows like the square root of $d$. The nearestneighbor 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) 18811884
[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 bosonparafermion 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."
A. Knauf, "Number theory, dynamical systems and statistical
mechanics" (1998 lecture notes)
"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) 411457.
"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 biinvariant 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 BostConnes 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:
M. Planat, P. Solé, S. Omar, "Riemann hypothesis and quantum mechanics" (preprint 12/2010)
[abstract:] "In their 1995 paper, JeanBeno\^{i}t Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function $\zeta(\beta)$, where $\beta$ is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low temperature KuboMartinSchwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as $$\phi_{\beta}(q)=N_{q1}^{\beta1} \psi_{\beta1}(N_q), $$ where $N_q=\prod_{k=1}^qp_k$ is the primorial number of order $q$ and $ \psi_b $ a generalized Dedekind $\psi$ function depending on one real parameter $b$ as $$ \psi_b (q)=q \prod_{p \in \mathcal{P,}p  q}\frac{11/p^b}{11/p}.$$ Fix a large inverse temperature $\beta >2.$ The Riemann hypothesis is then shown to be equivalent to the inequality $$ \phi_\beta (N_q)\zeta(\beta1) >e^\gamma \log \log N_q, $$ for $q$ large enough. Under RH, extra formulas for high temperatures KMS states ($1.5< \beta <2$) are derived."
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 Qlattices" (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 Qlattices 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 Lfunctions 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 RiemannHilbert 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, "BostConnesMarcolli systems for Shimura
varieties" (preprint 03/05)
[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous
to the BostConnesMarcolli system...in the case of Shimura varieties. Along the way, we
define a new BostConnes 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) 4586
[abstract:] "We describe analogies between number theory zeta functions, dynamical zeta functions,and statistical mechanics zeta
functions, with emphasis on multivariable zeta functions. We mainly consider twovariable 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 twovariable 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 selfadjoint 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 Nonlinear integral
equation involving Z(B) and V(x)."
J.G. Dueñas and N. F. Svaiter, "Zeros of the partition function in the randomized Riemann gas" (preprint 02/2014)
[abstract:] "An arithmetic gas is a second quantized mechanical system where the partition function is a Dirichlet series of a given arithmetic function. One example of such a system is known as the bosonic Riemann gas. We assume that the hamiltonian of the bosonic Riemann gas has a random variable with some probability distribution over an ensemble of hamiltonians. We discuss the singularity structure for the average free energy density of this arithmetic gas in the complex $\beta$ plane. First, assuming the Riemann hypothesis, there is a clustering of singular points along the imaginary axis coming from the nontrivial zeros of the Riemann zeta function on the critical line. This singularity structure associated to the zeros of the partition functions of the ensemble in the complex $\beta$ plane are the Fisher zeros. Second, there are also logarithmic singularities due to the poles of the Riemann zeta functions associated to the ensemble of hamiltonians. Finally we present the average energy density of the system."
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 firstnamed author at 26th International Colloquium on Group Theoretical Methods in Physics, New York, June 2006. See cs.OH/0609107 for followup talk delivered by secondnamed 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 fullyfledged 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
zerodimensional field theory; i.e. a quantum theory of noncommuting 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."
P. Dutta and D. Ghoshal, "Phase operator on $L_2(\mathbb{Q}_p$) and the zeroes of Fisher and Riemann" (preprint 02/2021)
[abstract:] "The distribution of the nontrivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang–Lee zeroes) of statistical mechanical models studied by physicists. The resolvent function of an operator akin to the phase operator, conjugate to the number operator in quantum mechanics, turns out to be important in this approach. The generalised Vladimirov derivative acting on the space $L_2(\mathbb{Q}_p)$ of complex valued locally constant functions on the $p$adic field is rather similar to the number operator. We show that a `phase operator' conjugate to it can be constructed on a subspace $L_2(p^{1}\mathbb{Z}_p)$ of $L_2(\mathbb{Q}_p)$. We discuss (at physicists' level of rigour) how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function. Finally, we extend these results to the family of Dirichlet $L$functions, using our recent construction of Vladimirov derivative like pseudodifferential operators associated with the Dirichlet characters."
S. Dwivedi, V. Kumar Singh and A. Roy, "Semiclassical limit of topological Rényi entropy in $3d$ Chern–Simons theory" (preprint 07/2020)
[abstract:] "We study the multiboundary entanglement structure of the state associated with the torus link complement $S^3\setminus T_{p,q}$ in the setup of threedimensional $SU(2)$_k Chern–Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of $k\to\infty$. We present a detailed analysis for several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large $k$ limiting value of the Rényi entropy of torus links of type $T_{p,pn}$ is the sum of two parts: (i) the universal part which is independent of $n$, and (ii) the nonuniversal or the linking part which explicitly depends on the linking number $n$. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of twodimensional topological Yang–Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological $2d$ Yang–Mills theory with $SU(2)$ gauge group. Further, the universal parts appearing in the large $k$ limits of the entanglement entropy and the minimum Rényi entropy for torus links $T_{p,pn}$ can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of $T_{p,pn}$ link in the double scaling limit of $k\to\infty$ and $n\to\infty$ and propose that the entropies converge in the double limit as well."
latticerelated
C. Newman, "Gaussian correlation inequalities for ferromagnets", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 33 (1975) 7593.
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) 389399
[abstract:] "Let V(t) be the even function on ($\infty,\infty)$ which is related to the Riemann xifunction by $\Xi (x/2)=4\int\sp{\infty}\sb{\infty}\exp (ixtV(t))dt.$ 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) 119
[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 RogersRamanujan type identities in solvable models is also reviewed."
NanXian Chen, Mi Li and Shaojun Liu, "Phonon
dispersions and elastic constants of Ni_{3}Al and Möbius inversion", Physics Letters A 195 (1994) 135143
[abstract:] "The Möbius inversion formulation corresponding to inequally weighted summations for solving some threedimensional 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 L1_{2} structures. The derivation is exact for radial potentials not only between identical atoms, but also between
distinct atoms. We have tested this formulation for Ni_{3}Al 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."
NanXian Chen, "Modified Möbius inverse formula and its
applications in physics", Phys. Rev. Lett. 64, 11931195 (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 treasuretrove 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) 441481
[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) 641645
[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 longrange 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 longrange 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) 208216
[abstract:] "We consider a onedimensional 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."
L. Campos Venuti, "The best quasifree approximation: reconstructing the spectrum from ground state energies" (preprint 01/2011)
[abstract:] "The sequence of ground state energy density at finite size, e_{L}, provides much more information than usually believed. Having at disposal $e_L$ for short lattice sizes, we show how to reconstruct an approximate quasiparticle dispersion for any interacting model. The accuracy of this method relies on the best possible quasifree approximation to the model, consistent with the observed values of the energy $e_L$. We also provide a simple criterion to assess whether such a quasifree approximation is valid. Perhaps most importantly, our method is able to assess whether the nature of the quasiparticles is fermionic or bosonic together with the effective boundary conditions of the model. The success and some limitations of this procedure are discussed on the hand of the spin1/2 Heisenberg model with or without explicit dimerization and of a spin1 chain with single ion anisotropy. A connection with the Riemann Hypothesis is also pointed out."
F. Vericat, "A lattice gas of prime numbers and the Riemann Hypothesis" (preprint 11/2012)
[abstract:] "In recent years there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis, perhaps motived by the apparent inaccessibility to their solution from a full rigorous mathematical point of view. Most of these kind of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this note, starting from a very simple model of onedimensional lattice gas and using the concept of equilibrium states as being described by Gibbs measures, we link classical statistical mechanics to the Riemann Hypothesis."
L. Bétermin and P. Zhang, "Minimization of energy per particle among Bravais lattices in $R^2$: Lennard–Jones and Thomas–Fermi cases" (preprint 02/2014)
[abstract:] "We study the two dimensional Lennard–Jones energy per particle of lattices and we prove that the minimizer among Bravais lattices with sufficiently large density is triangular and that is not the case for sufficiently small density. We give other results about the global minimizer of this energy. Moreover we study the energy per particle stemming from Thomas–Fermi model in two dimensions and we prove that the minimizer among Bravais lattices with fixed density is triangular. We use a result of Montgomery from number theory about the minimization of theta functions in the plane."
S. Boukraa and JM. Maillard, "Selected nonholonomic functions in lattice statistical mechanics and enumerative combinatorics" (preprint 10/2015)
[abstract:] "We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the nonlinear polynomial differential equation obtained by Tutte for the generating function of the $q$coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2+2\cos(j\pi;/n)$, the status of the $q = 4$ being unclear. We focus on the analysis of the $q = 4$ case, showing that the corresponding series is quite certainly nonholonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this $q = 4$ series modulo the first eight primes 2, 3,...,19, and show that this (probably nonholonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably nonholonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable nonholonomic functions can be seen as ratio of diagonals of rational functions, or even rational, or algebraic, functions of diagonals of rational functions."
I. Vardi,
"Deterministic percolation",
Communications in Mathematical Physics 207 (1999) 4366
[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) 275288
[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 $k\sqrt{\logz}$ 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
$\sqrt{2\pi\lambda_{c}\logz}$.
H.N.V. Temperley, "Further
results on selfavoiding walks", Physica A: Statistical and Theoretical Physics
206 (1994) 350358
[abstract:] "A Gaussian model of selfavoiding walks is studied. Not only is any cluster
integral exactly evaluable, but whole subseries can be evaluated exactly in terms of associated
Riemann zeta functions. The results are compared with information recently obtained on
selfavoiding 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 clustersums. This is split into subseries
all of which have the same radius of convergence, and the significance of this is discussed."
V. K. Varma, S. Pilati and V. E. Kravtsov, "Conduction in quasiperiodic and quasirandom lattices: Fibonacci, Riemann, and Anderson models" (preprint 07/2016)
[abstract:] "We study the ground state conduction properties of noninteracting electrons in aperiodic but nonrandom onedimensional models with chiral symmetry, and make comparisons against Anderson models with nondeterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's manyparticle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all singleparticle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the halffilled case; however, at various specific fillings $\rho$, including the values $\rho = 1/\rho^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at halffilling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to singleparticle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the offdiagonal Anderson model, albeit with different systemsize scaling of the bandcentre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted."
notes on the analogy between the functional equation
of the Riemann zeta function and the KramersWannier duality of
statistical mechanics
J.F. Burnol speculates on possible
relationships between zeta and the KramersWannier duality, Ising models
etc.
statistical mechanics – general
D. Ruelle, "Zeta functions and statistical mechanics", Asterisque 40 (1976), 167176.
D. Ruelle, "Is our mathematics natural? The case of equilibrium statistical mechanics",
Bulletin of the AMS 19 (1988) 259268.
excerpts from E. Schrödinger's Statistical Thermodynamics
wherein appear special values of the Riemann zeta function
Y.V. Fyodorov, G.A. Hiary and J.P. Keating, "Freezing transition, characteristic polynomials of random matrices, and the Riemann zetafunction" (preprint 02/2012)
[abstract:] "We argue that the freezing transition scenario, previously explored in the statistical mechanics of $1/f$noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large $N \times N$ random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zetafunction $\zeta(s)$ over sections of the critical line $s=1/2+it$ of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function."
B. Luque, I.G. Torre and L. Lacasa, "Phase transitions in number theory: From the birthday problem to Sidon sets" (preprint 10/2013)
[abstract:] "In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on gSidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of $k$ integers to be gSidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean field theory for the gSidon decision problem. We further improve the mean field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of qPotts models in condensed matter physics are briefly discussed."
C.E. Zachary, S. Torquato, "Hyperuniformity in
point patterns and twophase random heterogeneous media" (preprint 10/2009)
[abstract:] "Hyperuniform point patterns are characterized by vanishing infinite wavelength density
fluctuations and encompass all crystal structures, certain quasiperiodic systems, and special disordered
point patterns. This article generalizes the notion of hyperuniformity to include also twophase random
heterogeneous media. Hyperuniform random media do not possess infinitewavelength 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 nonBravais 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 Smatrix based finite temperature formalism we recently developed to nonrelativistic
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 1d fermionic case with a
quasiperiodic 2body potential provides a physical framework for understanding the Riemann hypothesis."
A. Roussou, J. Smyrnakis, M. Magiropoulos, Nikolaos K. Efremidis, G.M. Kavoulakis, P. Sandin, M. Ögren and M. Gulliksson, "Excitation spectrum of a mixture of two Bose gases confined in a ring potential with interaction asymmetry" (preprint 11/2017)
[abstract:] "We study the rotational properties of a twocomponent Bose–Einstein condensed gas of distinguishable atoms which are confined in a ring potential using both the meanfield approximation, as well as the method of diagonalization of the manybody Hamiltonian. We demonstrate that the angular momentum may be given to the system either via singleparticle, or "collective" excitation. Furthermore, despite the complexity of this problem, under rather typical conditions the dispersion relation takes a remarkably simple and regular form. Finally, we argue that under certain conditions the dispersion relation is determined via collective excitation. The corresponding manybody state, which, in addition to the interaction energy minimizes also the kinetic energy, is dictated by elementary number theory."
A.S. Mischenko, "Maslov's concept of phase transition from Bose–Einstein to Fermi–Dirac distribution: Results of interdisciplinary workshop in MSU" (preprint 11/2018)
[abstract:] "At the end of 2017, an interdisciplinary scientific seminar was organized at Moscow University, devoted to the study and development of a new scientific concept created by V.P. Maslov, allowing you to take a fresh look at the statistics of Bose–Einstein and Fermi–Dirac ideal gases. This new point of view allows us to interpret the indicated statistics as particular cases of statistical properties in number theory, on the one hand, and to indicate the limits of phase transitions from Bose to Fermi distributions."
D. Momeni, "Bose–Einstein condensation for an exponential density of states function and Lerch zeta function" (preprint 02/2019)
[abstract:] "I showed that how BoseEinstein condensation (BEC) in a non interacting bosonic system with exponential density of the states function yields to a new class of Lerch zeta functions. By looking on the critical temperature, I suggeted a possible strategy to prove the ``Riemann hypothesis'' problem. In a theorem and a lemma I suggested that the classical limit $\hbar\to 0$Ãƒâ€š of BEC can be used as a tool to find zeros of real part of the Riemann zeta function with complex argument. It reduces the Riemann hypothesis to a softer form. Furthermore I proposed a pair of creationannihilation operators for BEC phenomena. These set of creationannihilation operators is defined on a complex Hilbert space. They build a set up to interpret this type of BEC as a creationannihilation phenomena of the virtual hypothetical particle."
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
manyparticle 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 HardyRamanujan
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 powerlaw spectrum
yields the partitioning formula for p^{s}(n),
the latter being the number of partitions of n into a sum of
sth 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 d^{2}(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 ErdösLehner
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 noninteracting Bose gas in a onedimensional 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 nonGaussian 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,
Smatrices" (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 N^{2}
x N^{2} matrices for odd N are studied here. Presence of (N+3)(N1)/2
free parameters is the crucial feature of our models, setting them apart from other wellknown ones. There
are N possible states at each site. The trace of the transfer matrix is shown to depend on
(N1)/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 zerosum
multiplets formed by the rth 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."
G.G. Szpiro, "The
gaps between the gaps: some patterns in the prime number sequence", Physica A: Statistical Mechanics and its Applications
341 (2004) 607617
[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 HardyLittlewood conjecture,
the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."
K. Iguchi
"Generalized Sommerfeld theory: Specific heat of a degenerate gon
gas in any dimension and the generalized Riemann zeta function",
International Journal of Modern Physics B11, 35513580
(1997).
H.P. Baltes, E.R. Hilf and M. Pabst, "The longtime behaviour of the electricfield
autocorrelation function in a finite photon gas", Applied Physics B: Lasers and Optics 3
(1974) 14320649
[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) 37833803
[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 etafunction) to construct the "continuous" analog of the Cayley tree
isometrically embedded in the Poincaré upper halfplane. 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 semianalytically/seminumerically.
The speculation on the new "geometrical" interpretation of replica n > 0 limit is proposed."
P.J. Forrester, "Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble" (preprint 06/2013)
[abstract:] "It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these interrelationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp((\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ denotes the Riemann zeta function. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations."
H. Ono and H. Kuratsuji,
"Statistical theory of 2dimensional quantum vortex gas: noncanonical
effect and generalized zeta function"
"The purpose of this paper is to present a quantum statistical theory
of 2dimensional 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 $\alpha/(\alpha1)$. 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 (Poissondistributed) 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) 23102) 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 pseudoAnosov 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 3manifolds (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."
M. Takahashi, "Correlation function and simplified TBA equations for XXZ chain" (preprint 12/2010)
[abstract:] "The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by ${\rm log} 2$ and Riemann's zeta functions with odd integer argument $\zeta(3), \zeta(5), \zeta(7),....$. We can calculate density submatrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions.
Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with YangYang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$."
C. Pineda and T. Prosen, "Nonuniversal level statistics in a chaotic quantum spin chain", Phys. Rev. E 76 (2007) 061127
[abstract:] "We study the level statistics of an interacting multiqubit system, namely the kicked Ising spin chain, in the regime of quantum chaos. Long range quasienergy level statistics show effects analogous to the ones observed in semiclassical systems due to the presence of classical periodic orbits, while short range level statistics display perfect statistical agreement with random matrix theory. Even though our system possesses no classical limit, our result suggest existence of an important nonuniversal system specific behavior at short time scale, which clearly goes beyond finite size effects in random matrix theory."
[author comment:] "It [attempts] to calculate the dimension of a Hilbert space associated with rotationally invariant systems of $n$ spins. The dimension was given in terms of the Möbius function."
A. Klümper, D. Nawrath and J. Suzuki, "Correlation functions of the integrable isotropic spin1 chain: algebraic expressions for arbitrary temperature" (preprint 04/2013)
[abstract:] "We derive algebraic formulas for the density matrices of finite segments of the integrable $su(2)$ isotropic spin1 chain in the thermodynamic limit. We give explicit results for the 2 and 3 site cases for arbitrary temperature $T$ and zero field. In the zero temperature limit the correlation functions are given in elementary form in terms of Riemann's zeta function at even integer arguments."
A.P.C. Malbouisson and J.M.C. Malbouisson, "Boundary dependence of the coupling constant and the mass in the vector Ncomponent
$(\lambda \phi^{4})_{D}$ theory", Journal of Physics A
35 (2002) 22632273.
[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 Wickordered model in D = 3 we are able to give an exact formula to the Ldependence of the
coupling constant. For the nonWickordered model we indicate how expressions for the coupling constant and the mass
can be obtained for arbitrary dimension D in the smallL regime. Closed exact formulas for the
Ldependent 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 zetafunction. 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) 853862.
[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
MaxwellBoltzmann 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."
P. Tempesta, "Group entropies, correlation laws and zeta functions" (preprint 05/2011)
[abstract:] "The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann–Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback–Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function."
S. Tanaka, "Distribution of the Riemann zeros represented by the Fermi gas" (preprint 10/2010)
[abstract:] "The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of onedimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the nontrivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly. "
R. Pearson, "Number theory and critical exponents",
Phys. Rev. B 22 (1980) 34653470
[abstract:] "The consequences of assuming padic 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 Qstate Potts model is used to demonstrate the ideas of the paper."
P. Kleban, "Crossing
probabilities in critical 2D percolation and modular forms", Physica A 281
(2000) 242251
[abstract:] "Crossing probabilities for critical 2D 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 "horizontalvertical" 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 2D
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 "higherorder 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 AtkinLehner 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."
D. Schumayer and D.A.W. Hutchinson, "Physics of the Riemann hypothesis", Rev. Mod. Phys. 83 (2011) 307–330
[abstract:] "Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundredyearold problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its polestructure encodes when particles can undergo Bose–Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader."
R. M. Ziff, G. E. Uhlenbeck, and M. Kac, "The Bose–Einstein Gas, Revisited", Physics Reports
32C (1977)169248
[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
modelocking widths that we are interested in here. However, there is a realline
version of the Riemann hypothesis that lies very close to the modelocking problem...
The implications of this for the circlemap 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."
N. Chair, "Trigonometrical sums connected with onedimensional lattice, the chiral Potts model and number theory using the residue operator" (preprint 06/2012)
[abstract:] "We have recently developed a method for obtaining exact twopoint resistance on the square $N$cycle, the complete graph minus $N$ edges of the opposite vertices, here, a similar method is used to obtain closed formulas of certain trigonometrical sums that arise in connection with onedimensional lattice and the chiral Potts model. By modifying slightly, the previous computations, then, we are able to get new closed formulas for different trigonometrical sums, some of which appear in number theory."
B. BasuMallick, T. Bhattacharyya and D. Sen, "Clusters of bound particles in the derivative deltafunction Bose gas" (preprint 10/2001)
[abstract:] "In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative deltafunction Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasimomenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative deltafunction Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant."
B. BasuMallick, T. Bhattacharyya and D. Sen, "Clusters of bound particles in a quantum integrable manybody system and number theory" (preprint 10/2014)
[abstract:] "We construct clusters of bound particles for a quantum integrable derivative deltafunction Bose gas in one dimension. It is found that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasimomenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. Interestingly, there exists a connection between the above mentioned special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles for the derivative deltafunction Bose gas and the determination of various properties of these clusters like their size and their stability under a variation of the coupling constant."
A. Dias Ribeiro, "Level density of a Bose gas: Beyond the saddle point approximation" (preprint 11/2014)
[abstract:] "The present article is concerned with the use of approximations in the calculation of the manybody density of states (MBDS) of a system with total energy $E$, composed by $N$ bosons. In the meanfield framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the groundstate is sufficiently populated. Alternatively, we derive a uniform formula for MBDS, which is potentially able to deal with this regime. In the case of the onedimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method."
Y. Kong, "Packing dimers on $(2p+1)\times (2q+1)$ lattices" (preprint 10/2014)
[abstract:] "We use computational method to investigate the number of ways to pack dimers on oddbyodd lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1)\times (2k+1)$ odd square lattices have some remarkable numbertheoretical properties in parallel to those of closepacked dimers on $2k\times 2k$ even square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of oddbyodd lattice strips with any width $n = 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomerdimer problem might be solvable."
M.V.N. Murthy, M. Brack and R.K. Bhaduri, "On the asymptotic distinct prime partitions of integers" (preprint 04/2019)
[abstract:] "We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddlepoint approximation. We find that our result of $Q_{as}(n)$, which includes two higherorder corrections to the leading term in its exponent and a preexponential correction factor, approximates the exact $Q(n)$ far better than its simple leadingorder exponential form given so far in the literature."
R. Dong and M. Khalkhali, "Second quantization and the spectral action" (preprint 03/2019)
[abstract:] "We show that by incorporating chemical potentials one can extend the formalism of spectral action principle to bosonic second quantization. In fact we show that the von Neumann entropy, the average energy, and the negative free energy of the state defined by the bosonic, or fermionic, grand partition function can be expressed as spectral actions, and all spectral action coefficients can be given in terms of the modified Bessel functions. In the Fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches to $0$, can be expressed in terms of the Riemann zeta function. This recovers a recent result of Chamseddine–Connes–van Suijlekom."
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
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