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Register a new userCatching homologies by geometric entropy
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A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability to single out topological features of networks proceeding in a bottom-up manner: first we consider small size networks by analytical methods and then large size networks by numerical techniques. Two different classes of networks, the random graphs and the scale--free networks, are investigated computing their Betti numbers and then showing the capability of geometric entropy of detecting homologies.
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We unveil the existence of a non-trivial Berry phase associated to the dynamics of a quantum particle in a one dimensional box with moving walls. It is shown that a suitable choice of boundary conditions has to be made in order to preserve unitarity. For these boundary conditions we compute explicitly the geometric phase two-form on the parameter space. The unboundedness of the Hamiltonian describing the system leads to a natural prescription of renormalization for divergent contributions arising from the boundary.
Noethers celebrated theorem associating symmetry and conservation laws in classical field theory is adapted to allow for broken symmetry in geometric mechanics and is shown to play a central role in deriving and understanding the generation of fluid circulation via the Kelvin-Noether theorem for ideal fluids with stochastic advection by Lie transport (SALT).
In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change of temperature corresponds to a change of the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change of the material manifold, i.e. a change of the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change of temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.
This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincar{e} systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.
We consider the geometrization of quantum mechanics. We then focus on the pull-back of the Fubini-Study metric tensor field from the projective Hibert space to the orbits of the local unitary groups. An inner product on these tensor fields allows us to obtain functions which are invariant under the considered local unitary groups. This procedure paves the way to an algorithmic approach to the identification of entanglement monotone candidates. Finally, a link between the Fubini-Study metric and a quantum version of the Fisher information metric is discussed.
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