Imaging geometry through dynamics: the observable representation

For many stochastic processes there is an underlying coordinate space, $V$, with the process moving from point to point in $V$ or on variables (such as spin configurations) defined with respect to $V$. There is a matrix of transition probabilities (whether between points in $V$ or between variables defined on $V$) and we focus on its ``slow'' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the ``observables,'' and they can be used to recover geometrical features of $V$

Relative momentum for identical particles

Possible definitions for the relative momentum of identical particles are considered

Localization of thermal packets and metastable states in Sinai model

We consider the Sinai model describing a particle diffusing in a 1D random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: the disorder average of the thermal distribution of the relative distance y=x-m(t), with respect to the (disorder-dependent) most probable position m(t), converges in the limit of infinite time towards a distribution P(y). In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint asymptotic distribution of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute in the infinite-time limit the localization parameters Y_k, representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place, and the correlation function C(l) representing the disorder-averaged probability that two particles of the thermal packet are at a distance l from each other. We moreover prove that our results for Y_k and C(l) exactly coincide with the thermodynamic limit of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.Comment: 17 page

Metastable states in glassy systems

Truly stable metastable states are an artifact of the mean-field approximation or the zero temperature limit. If such appealing concepts in glass theory as configurational entropy are to have a meaning beyond these approximations, one needs to cast them in a form involving states with finite lifetimes. Starting from elementary examples and using results of Gaveau and Schulman, we propose a simple expression for the configurational entropy and revisit the question of taking flat averages over metastable states. The construction is applicable to finite dimensional systems, and we explicitly show that for simple mean-field glass models it recovers, justifies and generalises the known results. The calculation emphasises the appearance of new dynamical order parameters.Comment: 4 fig., 20 pages, revtex; added references and minor change

Green's function for the Relativistic Coulomb System via Sum Over Perturbation Series

We evaluate the Green's function of the D-dimensional relativistic Coulomb system via sum over perturbation series which is obtained by expanding the exponential containing the potential term $V({\bf x)}$ in the path integral into a power series. The energy spectra and wave functions are extracted from the resulting amplitude.Comment: 13 pages, ReVTeX, no figure

Spectral properties of zero temperature dynamics in a model of a compacting granular column

The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero 'temperature') the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.Comment: 34 pages, 4 figures, 3 table

Dirac and Weyl Equations on a Lattice as Quantum Cellular Automata

A discretized time evolution of the wave function for a Dirac particle on a cubic lattice is represented by a very simple quantum cellular automaton. In each evolution step the updated value of the wave function at a given site depends only on the values at the nearest sites, the evolution is unitary and preserves chiral symmetry. Moreover, it is shown that the relationship between Dirac particles and cellular automata operating on two component objects on a lattice is indeed very close. Every local and unitary automaton on a cubic lattice, under some natural assumptions, leads in the continuum limit to the Weyl equation. The sum over histories is evaluated and its connection with path integrals and theories of fermions on a lattice is outlined.Comment: 6, RevTe

Quantum-classical transition in Scale Relativity

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds - ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^mu - dx^mu) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a non-relativistic-motion approximation of the Dirac equation.Comment: 28 pages, no figur

Noise-induced switching between vortex states with different polarization in classical two-dimensional easy-plane magnets

In the 2-dimensional anisotropic Heisenberg model with XY-symmetry there are non-planar vortices which exhibit a localized structure of the z-components of the spins around the vortex center. We study how thermal noise induces a transition of this structure from one polarization to the opposite one. We describe the vortex core by a discrete Hamiltonian and consider a stationary solution of the Fokker-Planck equation. We find a bimodal distribution function and calculate the transition rate using Langer's instanton theory (1969). The result is compared with Langevin dynamics simulations for the full many-spin model.Comment: 15 pages, 4 figures, Phys. Rev. B., in pres

Switching between different vortex states in 2-dimensional easy-plane magnets due to an ac magnetic field

Using a discrete model of 2-dimensional easy-plane classical ferromagnets, we propose that a rotating magnetic field in the easy plane can switch a vortex from one polarization to the opposite one if the amplitude exceeds a threshold value, but the backward process does not occur. Such switches are indeed observed in computer simulations.Comment: 4 pages, 4 figures, submitted to Phys. Rev. Let