Emilio Cortés

Thermal relaxation of systems with quadratic heat bath coupling

We consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates. Performing an explicit elimination of the bath variables we arrive at an equation of evolution for the system variables alone. In the weak coupling limit we show that the equation is of the generalized Langevin form, with fluctuations that are Gaussian and that obey a fluctuation-dissipation relation. If the system-bath coupling is linear in the system coordinates the resulting fluctuations are additive and the dissipation is linear. If the coupling is nonlinear in the system coordinates, the resulting fluctuations are multiplicative and the dissipation is nonlinear.

On extremal paths for stochastic processes that involve step potentials and the generalized Hamilton-Jacobi equation

From the behavior of extremal paths across a boundary, defined by a finite step potential, we obtain by means of a least action principle, a first integral of the Euler-Lagrange equation (a conservation principle). This first integral gives the geometrical properties which are refraction and reflexion laws for extremal paths across the boundary and also it leads to a generalized Hamilton-Jacobi equation for the extremal action.

Ratchet motion induced by a correlated stochastic force

We apply a rigorous formalism we have just worked out (Cortes and Espinosa, Physica A 267 (1999) 414) about escape rates and the Hamilton–Jacobi equation, to study the ratchet motion of a Brownian particle and calculate the probability current in a periodic non-symmetric potential subject to correlated fluctuations. We are able to obtain the current behaviour as a function of the correlation time parameter and compare with other results in the literature.

Escape rates over potential barriers: variational principles and the Hamilton–Jacobi equation

We describe a rigorous formalism to study some extrema statistics problems, like maximum probability events or escape rate processes, by taking into account that the Hamilton–Jacobi equation completes, in a natural way, the required set of boundary conditions of the Euler–Lagrange equation, for this kind of variational problem. We apply this approach to a one-dimensional stochastic process, driven by colored noise, for a double-parabola potential, where we have one stable and one unstable steady states.

Störmer problem restricted to a spherical surface

In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic dipole, we propose to study the restricted motion of the particle in a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic St?rmer problem within a sphere, and although this problem no longer represents the real St?rmer problem, it shows the complex behaviour of this magnetic field through the classical dynamics equations that can be formally integrated. We start from a Lagrangian approach which allows us to analyse the dynamical properties of the system, such as the role of a velocity dependent potential, the symmetries and the conservation properties. We derive the Hamilton equations of motion, which in this restricted case can be reduced to a quadrature. From the Hamiltonian function we find, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. This equivalent potential function, which is a double well potential, allows us to get a clear description of the dynamics of the system. Then we obtain, by means of numerical integration, different plots of the trajectories in three-dimensional graphs in the sphere. This restricted case of the St?rmer problem is still nonlinear, with complex and interesting dynamics and we believe that it can offer the student a better grasp of the subject than the general three-dimensional case.

On the transient stochastic dynamics driven by Gaussian colored noise of systems with time-dependent control parameters. The effect of initial conditions

On a recent work we studied the transient stochastic dynamics driven by Gaussian white noise (GWN), of linear systems with time-dependent control parameters, by means of the connection between the nonlinear relaxation times (NLRT) and the quasideterministic (QD) approach. In the present study we make an extension of the analysis to the Gaussian colored noise (GCN) problem. Here, we first calculate the characteristic time associated with the decay of the unstable state of such linear systems, when the control parameter is a linear function of time of the form a(t) = bt − a0, with b > a0 > 0, which is continuously swept from below to above a threshold t = (a0/b). This type of linear modulation is known as the ramp model. Then, going further, we consider a general case where the control parameter is modulated by a family of functions a(t) = btδ ∮ a0, with δ > 0. The effects of the coupling between the initial state, at time t = 0, of the system with the noise are specially emphasized. The results of NLRT for the ramp model and the general case are compared.

Extremal trajectories for stochastic equations obtained directly from the Langevin differential operator

Shows that the differential operator for the extremal trajectory of a stochastic process can be connected directly to the systematic part of the differential operator that defines the stochastic equation. By assuming linearity in this operator and Gaussianity for the fluctuation, the author is able to write these relations for Markovian as well as non-Markovian processes.

Path integral method applied to the Brownian rotation of a symmetric top

The conditional probability density function in angular velocities space is obtained in an exact and closed expression for a symmetric top undergoing brownian motion. The distribution turns out to be non-gaussian. We obtain the distribution function by shifting the problem of solving stochastic differential equations to the problem of solving ordinary differential equations. This is done using the method of path integrals developed by Feynman and Hibbs for quantum mechanics.

Störmer problem restricted to a spherical surface and the Euler and Lagrange tops

In a recent work, Cortes and Poza (2015 Eur. J. Phys. 36 055009) analysed, in full, the dynamics of a charged particle in the field of a magnetic dipole restricted to a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic Stormer problem on a sphere. Here, we started from a Lagrangian approach: we derived the Hamilton equations of motion and observed that in this restricted case the equations can be reduced to quadratures, and they were integrated numerically. From the Hamiltonian function we found, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. In the present work we start from a change of variable to the cosine of the polar angle. In terms of this variable we obtain an equation that turns out to be the same as the one of a particle in a quartic potential. Then, we can actually solve the equations of motion for the polar angle using Jacobi elliptic functions, and for the azimuthal angle we use the same integrals which were expressed by Jacobi in terms of theta functions, both in the Euler and Lagrange tops. In this restricted Stormer problem, the student at undergraduate or graduate level will have a good example of an integrable nonlinear physical system in which, after analysis of its complex dynamics, one can obtain an analytical solution by means of some special functions of mathematical physics. Additionally, one discovers that the equations of motion of this restricted case of a charge in a magnetic dipole field have the same mathematical structure as the corresponding equations of other well known integrable classical dynamical systems.

First integral of the Euler-Lagrange equation and boundary conditions for stochastic processes: Bistable potential driven by colored noise

Abstract Starting from the variational problem of finding the maximum probability path, which is obtained by applying a condition of a mobile end point of the extremal path, we set a first integral of the Euler-Lagrange (EL) equation; this constant of motion, together with appropriate boundary conditions, leads to an initial value problem. We consider the one-dimensional stochastic process of a particle on a bistable potential, driven by colored noise, and obtain the escape rate across the barrier. By solving numerically the initial value problem, without any approximation, we plot the τ-dependence of the extremal action for the particle to go from the bottom to the top of the potential. The behavior of S ( τ ) for this problem has not been reported in the literature, and it is qualitatively similar to a result we obtained recently for the oscillator potential.