DOI

1

Anomalous long-range correlations at a non-equilibrium phase transition - Gerschenfeld, A. and Derrida, B.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 45, (2012)

Abstract : Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time-time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of these cumulants away from the transition and show that they become singular as the transition is approached, matching with what we already knew in the critical regime.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 45, (2012)

Abstract : Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time-time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of these cumulants away from the transition and show that they become singular as the transition is approached, matching with what we already knew in the critical regime.

DOI

2

Hamiltonian formulation of surfaces with constant Gaussian curvature - Trejo, Miguel and Ben Amar, Martine and Mueller, Martin Michael

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 42, (2009)

Abstract : Dirac's method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phase-space variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry-reparametrization invariance-which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions that are identified as the components of the acceleration of the curve. The formulation leads to four first-order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization, only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 42, (2009)

Abstract : Dirac's method for constrained Hamiltonian systems is used to describe surfaces of constant Gaussian curvature. A geometrical free energy, for which these surfaces are equilibrium states, is introduced and interpreted as an action. An equilibrium surface can then be generated by the evolution of a closed space curve. Since the underlying action depends on second derivatives, the velocity of the curve and its conjugate momentum must be included in the set of phase-space variables. Furthermore, the action is linear in the acceleration of the curve and possesses a local symmetry-reparametrization invariance-which implies primary constraints in the canonical formalism. These constraints are incorporated into the Hamiltonian through Lagrange multiplier functions that are identified as the components of the acceleration of the curve. The formulation leads to four first-order partial differential equations, one for each canonical variable. With the appropriate choice of parametrization, only one of these equations has to be solved to obtain the surface which is swept out by the evolving space curve. To illustrate the formalism, several evolutions of pseudospherical surfaces are discussed.

DOI

3

The spectrum of large powers of the Laplacian in bounded domains - Katzav, E. and Adda-Bedia, M.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 41, (2008)

Abstract : We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one-dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non-integer N and for 3D Laplacian problems.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 41, (2008)

Abstract : We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one-dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non-integer N and for 3D Laplacian problems.

DOI

4

Eigenproblems of large powers of the Laplacian in bounded domains - Ramani, A. and Grammaticos, B. and Pomeau, Y.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 40, F391-F396 (2007)

Abstract : We present a method for computing the spectrum of large powers of the Laplacian in a bounded domain restricting ourselves to the one- and three-dimensional cases. Since it does not seem possible to obtain information on the eigenvalues directly from the transcendental equation that gives the spectrum, we introduce a Wallis-inspired method. We obtain the expansion of the eigenfunction and the eigenvalues in power series where the inverse of the power at which the Laplacian is raised plays the role of the small parameter. We compare these eigenvalues to those obtained through a simple variational approach and remark that the latter offers an excellent approximation to the exact result.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 40, F391-F396 (2007)

Abstract : We present a method for computing the spectrum of large powers of the Laplacian in a bounded domain restricting ourselves to the one- and three-dimensional cases. Since it does not seem possible to obtain information on the eigenvalues directly from the transcendental equation that gives the spectrum, we introduce a Wallis-inspired method. We obtain the expansion of the eigenfunction and the eigenvalues in power series where the inverse of the power at which the Laplacian is raised plays the role of the small parameter. We compare these eigenvalues to those obtained through a simple variational approach and remark that the latter offers an excellent approximation to the exact result.