Table of contents

Markovian dynamics are the most elemental and prevalent form of stochastic dynamics in science and engineering. The statistical evolution of given Markovian dynamics is governed by its master equation, and the integrability of the master equation's positive-valued stationary solutions determines the dynamics' statistical behavior: integrability implies that the dynamics converge to steady state, and non-integrability implies that the dynamics are transient. This letter establishes that all the positive-valued stationary solutions of the master equation represent Poissonian steady states. Poissonian steady states generalize the notion of 'regular' steady states, and quantify the stationary structures of general Markovian dynamics. In particular, we unveil and quantify the stationary structures of transient Markovian dynamics.

The mathematical properties of the face-centred cubic lattice Green function

$$\begin{equation*} \fl G(w) \equiv {1\over {\pi ^3}}\int _{0}^{\pi }\int _{0}^{\pi }\int _{0}^{\pi } {{{\rm d}\theta _1\,{\rm d}\theta _2\,{\rm d}\theta _3}\over {w-{\rm c}(\theta _1)\,{\rm c}(\theta _2)- {\rm c}(\theta _2)\,{\rm c}(\theta _3)-{\rm c}(\theta _3)\,{\rm c}(\theta _1)}} \end{equation*}$$

and the associated logarithmic integral

$$\begin{eqnarray*} &&\fl S(w) \equiv {1\over {\pi ^3}}\int _{0}^{\pi }\int _{0}^{\pi }\int _{0}^{\pi } \ln [ w-{\rm c}(\theta _1)\,{\rm c}(\theta _2)-{\rm c}(\theta _2)\,{\rm c}(\theta _3)\nonumber\\&&\ms -\,{\rm c}(\theta _3)\,{\rm c}(\theta _1)]\,{\rm d}\theta _1\,{\rm d}\theta _2\,{\rm d}\theta _3 \end{eqnarray*}$$

are investigated, where c(θ) ≡ cos (θ) and w = w₁ + iw₂ is a complex variable in the w plane. In particular, the theory of Mahler measures is used to obtain a closed-form expression for S(w) in terms of ₅F₄ generalized hypergeometric functions. The method of analytic continuation is then applied to this result in order to prove that

$$\begin{equation*} \fl S(3)= -{14\over 15}\ln 2+{8\over 5}\ln 3-{8\over 135}\,\, {_5F_4} \left[\begin{array}{@{}cccccc@{}}4\over 3,&{3\over 2},&{5\over 3},&1,&1;&\cr &&&&&1\cr 2,&2,&2,&2;&\end{array}\right] . \end{equation*}$$

Next the relation dS/dw = G(w), where w ∈ (3, +∞), is used to derive the simple formula

$$\begin{equation*} G(w)={1\over w} \left\lbrace {_2F_1}\left[{1\over 6},{1\over 3};1;{{27(w+1)}\over {4w^3}} \right]\right\rbrace ^2 , \end{equation*}$$

where w lies in a restricted region $\mathcal{R}_1$ of the cut w plane. The limit function

$$\begin{equation*} G^{-}(w_1)\equiv \lim _{\epsilon \rightarrow 0+} G(w_1-{\rm i}\epsilon ) =G_{\rm R}(w_1)+{\rm i}G_{\rm I}(w_1) \end{equation*}$$

is also evaluated in the intervals w₁ ∈ ( − 1, 0] and w₁ ∈ (0, 3]. It is shown that G_R(w₁) and G_I(w₁) can be expressed in terms of ₂F₁[z(w₁)] hypergeometric functions, where the independent variable z(w₁) is a real-valued rational function of w₁. Finally, new formulae are derived for the number of random walks r_n on the face-centred cubic lattice which return to their starting point (not necessarily for the first time) after n nearest-neighbour steps.

We introduce a physical approach to social networks (SNs) in which each actor is characterized by a yes–no test on a physical system. This allows us to consider SNs beyond those originated by interactions based on pre-existing properties, as in a classical SN (CSN). As an example of SNs beyond CSNs, we introduce quantum SNs (QSNs) in which actor i is characterized by a test of whether or not the system is in a quantum state |ψ_i〉. We show that QSNs outperform CSNs for a certain task and some graphs. We identify the simplest of these graphs and show that graphs in which QSNs outperform CSNs are increasingly frequent as the number of vertices increases. We also discuss more general SNs and identify the simplest graphs in which QSNs cannot be outperformed.

In a recent paper, Laurençot and van Roessel (2010 J. Phys. A: Math. Theor.43 455210) studied the scaling behaviour of solutions to a two-species coagulation–annihilation system with total annihilation and equal strength coagulation, and identified cases where self-similar behaviour occurs, and others where it does not. In this paper, we proceed with the study of this kind of system by assuming that the coagulation rates of the two different species need not be equal. By applying Laplace transform techniques, the problem is transformed into a two-dimensional ordinary differential system that can be transformed into a Lotka–Volterra competition model. The long-time behaviour of solutions to this Lotka–Volterra system helps explain the different cases of existence and nonexistence of similarity behaviour, as well as why, in some cases, the behaviour is nonuniversal, in the sense of being dependent on initial conditions.

Motivated by recent results in mathematical virology, we present novel asymmetric $\mathbb {Z}[\tau ]$-integer-valued affine extensions of the non-crystallographic Coxeter groups H₂, H₃ and H₄ derived in a Kac–Moody-type formalism. In particular, we show that the affine reflection planes which extend the Coxeter group H₃ generate (twist) translations along two-, three- and five-fold axes of icosahedral symmetry, and we classify these translations in terms of the Fibonacci recursion relation applied to different start values. We thus provide an explanation of previous results concerning affine extensions of icosahedral symmetry in a Coxeter group context, and extend this analysis to the case of the non-crystallographic Coxeter groups H₂ and H₄. These results will enable new applications of group theory in physics (quasicrystals), biology (viruses) and chemistry (fullerenes).

A group analysis of a system describing an ideal plastic flow is made in order to obtain analytical solutions. The complete Lie algebra of point symmetries of this system are given. Two of the infinitesimal generators that span the Lie algebra are original to this paper. A classification into conjugacy classes of all one- and two-dimensional subalgebras is performed. Invariant and partially invariant solutions corresponding to certain conjugacy classes are obtained using the symmetry reduction method. Solutions of algebraic, trigonometric, inverse trigonometric and elliptic type are provided as illustrations and other solutions expressed in terms of one or two arbitrary functions have also been found. For some of these solutions, a physical interpretation allows one to determine the shape of feasible extrusion dies corresponding to these solutions. The corresponding tools could be used to curve rods or slabs, or to shape a ring in an ideal plastic material by an extrusion process.

In this paper, we present an extensive analytical study of continuous-time quantum walks (CTQWs) occurring on two-dimensional lattices under various boundary conditions, focusing on the M × N lattice wrapped on Möbius strips and Klein bottles, which are featured by the twisted boundary conditions. We find that the eigenvalue spectra and transport efficiency (both quantum mechanical and classical) for the two structures show comparable results with those for other well-known two-dimensional lattices, including rectangles, cylinders and tori. We also demonstrate that the behaviors of CTQWs depend on the initial node of the excitation and the network size, both on short and long timescales. In addition, we discover the asymmetric behaviors of limiting probabilities for Möbius strips and Klein bottles, which are quite different from each other and are also compared to those discovered in other two-dimensional networks. Our work provides a comprehensive understanding of recent results about CTQWs on two-dimensional lattices, and sheds light on quantum dynamics on lattices, especially those with different boundary conditions.

The parametrized relativistic quantum mechanics of Stueckelberg (1942 Helv. Phys. Acta15 23) represents time as an operator, and has been shown elsewhere to yield the recently observed phenomena of quantum interference in time, quantum diffraction in time and quantum entanglement in time. The Stueckelberg wave equation as extended to a spin-1/2 particle by Horwitz and Arshansky (1982 J. Phys. A: Math. Gen.15 L659) is shown here to yield the electron g-factor g = 2 (1 + α/2π), to leading order in the renormalized fine structure constant α, in agreement with the quantum electrodynamics of Schwinger (1948 Phys. Rev.73 416).

The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization.

Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck's constant ℏ > 0, meaning that the classical and quantum world views must actually coexist. Traditionally, canonical quantization procedures postulate a direct linking of various c-number and q-number quantities that lie in disjoint realms, along with the quite distinct interpretations given to each realm. In this paper we propose a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory letting them coexist as required. This proposal also shines light on alternative linking assignments of classical and quantum quantities that offer different perspectives on the very meaning of quantization. In this paper we focus on elaborating the general principles, while elsewhere we have published several examples of what this alternative viewpoint can achieve; these examples include removal of singularities in classical solutions to certain models, and an alternative quantization of several field theory models that are trivial when quantized by traditional methods but become well defined and nontrivial when viewed from the new viewpoint.

A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in $\mathbb {Z}_{n_{i}}$, i = 1, ..., k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces of dimensions n₁, ..., n_k. The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.

The quantum model for a damped harmonic oscillator linearly coupled to a reservoir of continuous frequency modes has been exactly diagonalized either with or without the rotating wave approximation. The exact dynamics, formally described in terms of the coupling function, is analyzed for the Drude model. In this scenario, we focus on a class of couplings recovering the Drude form as a limiting case. The exact time evolution of the expectation values of the position, momentum and number operators are described through Fox H-functions, as well as the correlation function of the reservoir. In correspondence with the critical values of the frequency, discontinuities appear in the long time scale dynamics, arbitrarily slowing down the relaxations of the above observables. This effect allows one to enhance arbitrarily the lifetime of the excitations, giving relevant applications in the study of the quantum domain of opto-mechanical and nano-mechanical resonators. The critical frequency approach constitutes a further method of control in the scenario of environment-induced decoherence via reservoir engineering.