The Small-Mass Limit and White-Noise Limit of an Infinite Dimensional Generalized Langevin Equation

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no global Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain \(L^1\) convergence in the white-noise limit.     

Invariance properties of the Dirac equation with external electro-magnetic field

In this paper, we attempt to obtain the nature of the external field such that the Dirac equation with external electro-magnetic field is invariant. The Poincaré group, which is the maximal symmetry group for field free case, is constrained by the presence of the external field. Introducing infinitesimal transformation ofx and ψ, we apply Lie’s extended group method to obtain the class of external field which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic field, though only potentials explicity appears in the equation. Presented at the Sixth Chittagong Conference on Mathematical Physics, January 2001.     

Index Theory, Gerbes, and Hamiltonian Quantization

We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms (Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS construction to the bundle gerbe approach discussed recently by Carey and Murray, including an explicit computation of the Dixmier-Douady class. An advantage of our method is that it can be applied whenever one has a form of the APS theorem at hand, as in the case of fermions in an external gravitational field. Received:     

Breakdown of Stability of Motion in Superquadratic Potentials

Based on the KAM theory, investigation of the equation of motion of a classical particle in a one-dimensional superquadratic potential well, under the influence of an external time-periodic forcing, raised a hope that all the solutions are bounded. Indeed, due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to infinity as the amplitude increases. Therefore, because of this relationship between the frequency and the amplitude, intuitively one might expect that all resonances that could cause the accumulation of energy would be destroyed, and thus all solutions would stay bounded for all time. More formally, according to Moser's twist theorem, this could mean the existence of invariant tubes in the extended phase space and therefore would result in the boundedness of the solutions. Actually, the boundedness results have been established for a large class of superquadratic potentials, but in general, the above intuition turns out to be wrong. Littlewood showed it by creating a superquadratic potential in which an unbounded motion occurs in the presence of some particular piecewise constant forcing. Moreover, Littlewood's result holds for a larger class of forcings. Here it is proven for the continuous time-periodic forcing. For this purpose a new averaging technique for the forced motions in superquadratic potentials with rather weak assumptions on the differentiability of the potentials has been developed. Received: 1 February 1995 / Accepted: 15 March 1997     

The relation between metric and spin-2 formulations of linearized Einstein theory

A twenty-dimensional space of charged solutions of spin-2 equations is proposed. The relation with extended (via dilatation) Poincaré group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the nonsingular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electromagnetic potentials. The notion ofasymptotic conformal Yano-Killing tensor is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities.     

On the Free Energy of a Gaussian Membrane Model with External Potentials

We consider a class of d-dimensional Gaussian lattice field which is known as a model of semi-flexible membrane. We study the free energy of the model with external potentials and show the following:     

Path integral for Koenigs spaces

I discuss a path-integral approach for the quantum motion on two-dimensional spaces according to Koenigs, for short “Koenigs spaces”. Their construction is simple: one takes a Hamiltonian from a two-dimensional flat space and divides it by a two-dimensional superintegrable potential. These superintegrable potentials are the isotropic singular oscillator, the Holt potential, and the Coulomb potential. In all cases, a nontrivial space of nonconstant curvature is generated. We can study free motion and the motion with an additional superintegrable potential. For possible bound-state solutions, we find in all three cases an equation of the eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly. The text was submitted by the authors in English.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Nonlinear Elastic Free Energies and Gradient Young-Gibbs Measures

We investigate, in a fairly general setting, the limit of large volume equilibrium Gibbs measures for elasticity type Hamiltonians with clamped boundary conditions. The existence of a quasiconvex free energy, forming the large deviations rate functional, is shown using a new interpolation lemma for partition functions. The local behaviour of the Gibbs measures can be parametrized by Young measures on the space of gradient Gibbs measures. In view of the unboundedness of the state space, the crucial tool here is an exponential tightness estimate that holds for a vast class of potentials and the construction of suitable compact sets of gradient Gibbs measures.     

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.     

Stochastic diffusion in a periodic potential

This paper deals with two equations for classical stochastic diffusion in a potential. First, the full Fokker-Planck equation in phase-space for a Brownian particle in a periodic potential and linearly coupled to an external field is considered. The solution discussed previously by the author and co-worker is improved upon. An alternative approximation is introduced. Then, the simpler Smoluchowski equation, which is derivable from the Fokker-Planck equation under suitable conditions, is solved using Hill's determinant method. Finally a WKB-type method is proposed to solve the Smoluchowski equation for a general class of potentials.     

Fluctuation-driven directed transport in the presence of Lévy flights

The role of Lévy flights on fluctuation-driven transport in time independent periodic potentials with broken spatial symmetry is studied. Two complementary approaches are followed. The first one is based on a generalized Langevin model describing overdamped dynamics in a ratchet type external potential driven by Lévy white noise with stability index α in the range 1<α<2. The second approach is based on the space fractional Fokker-Planck equation describing the corresponding probability density function (PDF) of particle displacements. It is observed that, even in the absence of an external tilting force or a bias in the noise, the Lévy flights drive the system out of the thermodynamic equilibrium and generate an up-hill current (i.e., a current in the direction of the steeper side of the asymmetric potential). For small values of the noise intensity there is an optimal value of α yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Lévy noise asymmetry and the potential asymmetry. For a sharply localized initial condition, the PDF of staying at the minimum of the potential exhibits scaling behavior in time with an exponent bigger than the −1/α exponent corresponding to the force free case.     

Gauge fields and gravitational field equations

It is shown that the gravitational field equations in free space have a similar form to the free Yang-Mills field equations, where the group SL (2, C) replaces the group SU(2). The Ricci rotation coefficients take the role of the Yang-Mills like potentials, whereas the Riemann tensor takes the role of the gauge fields.     

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L?? ^d determines a lattice spin system with potential A _Φ . The pressure P(A _Φ ) and free energy F _AΦ (β)=?(1/β)P(βA _Φ ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Hölder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.     

Phase transition of trapped interacting Bose gases

We review some recent theoretical work on the phase transition of interacting Bose gases in the presence of external trapping potentials. A general framework for the study of such questions is presented which is based on the application of perturbative momentum-shell renormalization group methods to the trapped gas in the uncondensed phase. After giving an overview of this approach, we first establish its validity by comparing to previous results for homogeneous and harmonically trapped gases. Using this theoretical framework, we then examine various aspects of how external potentials influence the physics of condensation. (i) By studying the case of general power-law potentials and complemented by arguments from variational perturbation theory, it is quantitatively worked out how a growing inhomogeneity of the trapping potential diminishes nonperturbative effects at the transition. (ii) It is shown how by superimposing a weak periodic potentials on the homogeneous system, the characteristic nonperturbative momentum scale of critical interacting Bose gases can be probed. (iii) For a gas in a random potential, it is studied how condensation is affected by the combined influence of disorder effects and particle interactions.     

Many-Body Problem with Quadratic and/or Inversely-Quadratic Potentials in One- and More-Dimensional Spaces: Some Retrospective Remarks

In the context of an ambient space with an arbitrary number $d$ of dimensions, the many-body problem consisting of an arbitrary number $N$ of particles confined by a common, external harmonic potential (realizing a container with soft walls) and interacting among themselves and with the environment with arbitrary conservative repulsive forces scaling as the inverse cube of distances, displays a peculiar behaviour: its effective volume oscillates isochronously without damping. We recently discovered this remarkable phenomenon (valid in the context of both classical and quantum mechanics) and discussed its implications in the context of statistical mechanics and thermodynamics; but after publishing these findings we were informed that essentially analogous results had been previously obtained by Lyndell-Bell and Lyndell-Bell. In the present paper, motivated by the need we felt to acknowledge this fact, we also offer some retrospective remarks on the $N$ -body problem with quadratic and/or inversely-quadratic potentials in one- and more-dimensional space.     

Algebraic treatment of quantum-mechanical models with modified Coulomb potentials

Nonrelativistic models with modified Coulomb potentials are solved by an algebraic method based on SO(4,2) dynamical group. The nonrelativistic model with Yukawa long-range potential and the Stark effect in the nonrelativistic hydrogen atom in the homogeneous and inhomogeneous external electrostatic fields are studied in details. The algebraic method for the Dirac and Klein-Gordon relativistic hydrogen atom as well as relativistic models with the rotationally symmetric modified Coulomb potentials are also discussed.     

Wegner Estimate and the Density of States of Some Indefinite Alloy-Type Schrödinger Operators

We study Schrödinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them, we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. Our estimate is valid for all bounded energy intervals and all space dimensions and implies the existence of the density of states.     

A Proton in the Quark-Meson Coupling Model

We investigate the structure of a proton in free space by using the quark-meson coupling model. In the model, a proton in free space is regarded as a MIT bag with σ, ω and ρ meson fields and the Coulomb potential. With the boundary condition at bag surface, the wave functions of u and d quarks and potentials are calculated self-consistently.