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Register a new userDistribution of Fluctuational Paths in Noise-Driven Systems
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Dynamics of a system that performs a large fluctuation to a given state is essentially deterministic: the distribution of fluctuational paths peaks sharply at a certain optimal path along which the system is most likely to move. For the general case of a system driven by colored Gaussian noise, we provide a formulation of the variational problem for optimal paths. We also consider the prehistory problem, which makes it possible to analyze the shape of the distribution of fluctuational paths that arrive at a given state. We obtain, and solve in the limiting case, a set of linear equations for the characteristic width of this distribution.
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Fluctuations in systems away from thermal equilibrium have features that have no analog in equilibrium systems. One of such features concerns large rare excursions far from the stable state in the space of dynamical variables. For equilibrium systems, the most probable fluctuational trajectory to a given state is related to the fluctuation-free trajectory back to the stable state by time reversal. This is no longer true for nonequilibrium systems, where the pattern of the most probable trajectories generally displays singularities. Here we study how the singularities emerge as the system is driven away from equilibrium, and whether a driving strength threshold is required for their onset. Using a resonantly modulated oscillator as a model, we identify two distinct scenarios, depending on the speed of the optimal path in thermal equilibrium. If the position away from the stable state along the optimal path grows exponentially in time, the singularities emerge without a threshold. We find the scaling of the location of the singularities as a function of the control parameter. If the growth away from the stable state is faster than exponential, characterized by the ability to reach infinity in finite time, there is a threshold for the onset of singularities, which we study for the model.
We report on the onset of anti-resonant behaviour of mass transport systems driven by time-dependent forces. Anti-resonances arise from the coupling of a sufficiently high number of space-time modes of the force. The presence of forces having a wide space-time spectrum, a necessary condition for the formation of an anti-resonance, is typical of confined systems with uneven and deformable walls that induce entropic forces dependent on space and time. We have analyzed, in particular, the case of polymer chains confined in a flexible channel and shown how they can be sorted and trapped. The presence of resonance-antiresonance pairs found can be exploited to design protocols able to engineer optimal transport processes and to manipulate the dynamics of nano-objects.
We study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, and hence fails to be symmetric. However, we demonstrate a scenario where the failure of proposal symmetry is a higher order effect. Hence, from these microscopic dynamics we derive as a limit as the proposal size goes to zero and the number of spins to infinity, a non-local stochastic version of the harmonic map heat flow (or overdamped Landau-Lipshitz equation). The equation is both mathematically well-posed and samples the canonical/Gibbs distribution related to the kinetic energy. The failure of proposal symmetry due to interaction between the confining geometry of the spin system and the colored noise is in contrast to the uncorrelated, white-noise, driven system. Specifically, the choice of projection of the noise to conserve the magnitude of the spins is crucial to maintaining the proper equilibrium distribution. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.
The effect of a change of noise amplitudes in overdamped diffusive systems is linked to their unperturbed behavior by means of a nonequilibrium fluctuation-response relation. This formula holds also for systems with state-independent nontrivial diffusivity matrices, as we show with an application to an experiment of two trapped and hydrodynamically coupled colloids, one of which is subject to an external random forcing that mimics an effective temperature. The nonequilibrium susceptibility of the energy to a variation of this driving is an example of our formulation, which improves an earlier version, as it does not depend on the time-discretization of the stochastic dynamics. This scheme holds for generic systems with additive noise and can be easily implemented numerically, thanks to matrix operations.
We have found experimentally that the shot noise of the tunneling current $I$ through an undoped semiconductor superlattice is reduced with respect to the Poissonian noise value $2eI$, and that the noise approaches 1/3 of that value in superlattices whose quantum wells are strongly coupled. On the other hand, when the coupling is weak or when a strong electric field is applied to the superlattice the noise becomes Poissonian. Although our results are qualitatively consistent with existing theories for one-dimensional mulitple barriers, the theories cannot account for the dependence of the noise on superlattice parameters that we have observed.
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