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Register a new userElectromagnetism on Anisotropic Fractals
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Martin Ostoja-Starzewski
Publication date
2011
fields
Physics
and research's language is
English
Authors
Martin Ostoja-Starzewski
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We derive basic equations of electromagnetic fields in fractal media which are specified by three indepedent fractal dimensions {alpha}_{i} in the respective directions x_{i} (i=1,2,3) of the Cartesian space in which the fractal is embedded. To grasp the generally anisotropic structure of a fractal, we employ the product measure, so that the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving the {alpha}_{i}s. First, a formulation based on product measures is shown to satisfy the four basic identities of vector calculus. This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Amp`ere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwells electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.
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We consider the relativistic Vlasov-Maxwell and Vlasov-Nordstrom systems which describe large particle ensembles interacting by either electromagnetic fields or a relativistic scalar gravity model. For both systems we derive a radiation formula analogous to the Einstein quadrupole formula in general relativity.
Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${mathbb R}^N$, for any integer $Nge1$. It is defined by $zeta_A(s)=int_{A_{delta}}d(x,A)^{s-N},mathrm{d} x$ for all $sinmathbb{C}$ with $operatorname{Re},s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{delta}$ is the $delta$-neighborhood of $A$, where $delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $zeta_A$ is equal to $overlinedim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $zeta_A$ located on the critical line ${mathop{mathrm{Re}} s=overlinedim_BA}$, provided $zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $tildezeta_A(s)=int_0^{delta} t^{s-N-1}|A_t|,mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $tildezeta_A$ computed at $D=dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Bakers theorem from the theory of transcendental numbers.
We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $zeta_A(s):=int_{A_delta} d(x,A)^{s-N}mathrm{d}x$, where $delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $mathbb{R}^N$. The abscissa of Lebesgue convergence $D(zeta_A)$ coincides with $D:=overlinedim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the critical line ${Re(s)=D}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $tto0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(mathcal M+O(t^gamma))$ as $tto0^+$, with $gamma>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(log t^{-1})+O(t^gamma))$ as $tto0^+$, where $G$ is a nonconstant periodic function and $gamma>0$. In both cases, we show that $zeta_A$ can be meromorphically extended (at least) to the open right half-plane ${Re(s)>D-gamma}$. Furthermore, up to a multiplicative constant, the residue of $zeta_A$ evaluated at $s=D$ is shown to be equal to $mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line ${Re(s)=D}$.
An integrable anisotropic Heisenberg spin chain with nearest-neighbour couplings, next-nearest-neighbour couplings and scalar chirality terms is constructed. After proving the integrability, we obtain the exact solution of the system. The ground state and the elementary excitations are also studied. It is shown that the spinon excitation of the present model possesses a novel triple arched structure. The elementary excitation is gapless if the anisotropic parameter $eta$ is real while the elementary excitation has an enhanced gap by the next-nearest-neighbour and chiral three-spin interactions if the anisotropic parameter $eta$ is imaginary. The method of this paper provides a general way to construct new integrable models with next-nearest-neighbour interactions.
Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index associated with a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits striking similarity to the one associated with a Dirac particle in supersymmetric quantum mechanics.
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