Thermodynamic Cross-Effects from Dynamical Systems

We give a thermodynamically consistent description of simultaneous heat and particle transport, as well as of the associated cross-effects, in the framework of a chaotic dynamical system, a generalized multibaker map. Besides the density, a second field with appropriate source terms is included in order to mimic, after coarse graining, a spatial temperature distribution and its time evolution. A new expression is derived for the irreversible entropy production in a steady state, as the average of the growth rate of the relative density, a unique combination of the two fields.Comment: 4 pages, 2 postscript figure

A MultiBaker Map for Thermodynamic Cross-Effects in Dynamical Systems

A consistent description of simultaneous heat and particle transport, including cross effects, and the associated entropy balance is given in the framework of a deterministic dynamical system. This is achieved by a multibaker map where, besides the phase-space density of the multibaker, a second field with appropriate source terms is included in order to mimic a spatial temperature distribution and its time evolution. Conditions are given to ensure consistency in an appropriately defined continuum limit with the thermodynamic entropy balance. They leave as the only free parameter of the model the entropy flux let directly into a surroundings. If it vanishes in the bulk, the transport properties of the model are described by the thermodynamic transport equations. Another choice leads to a uniform temperature distribution. It represents transport problems treated by means of a thermostatting algorithm, similar to the one considered in non-equilibrium molecular dynamics.Comment: 18 pages, 3 postscript figure

Escape-rate formalism, decay to steady states, and divergences in the entropy-production rate

In summer 1997 we were sitting with Bob Dorfman and a few other friends interested in chaotic systems and transport theory on a terrace close to Oktogon in Budapest. While taking our (decaf) coffee after a very nice Italian meal, we discussed about logarithmic divergences in the entropy production of systems with absorbing boundary conditions and their consequences for the escape-rate formalism. It was guessed at that time that the problem could be resolved by a careful discussion of the physical content of the absorbing boundary conditions. To our knowledge a thorough analysis of this long-standing question is still missing. We dedicate it hereby to Bob on occasion of his 65th birthday.Comment: 16 pages; RevTex 4 with graphicx package; eps-figure

Dynamics of "leaking" Hamiltonian systems

In order to understand the dynamics in more detail, in particular for visualizing the space-filling unstable foliation of closed chaotic Hamiltonian systems, we propose to leak them up. The cutting out of a finite region of their phase space, the leak, through which escape is possible, leads to transient chaotic behavior of nearly all the trajectories. The never-escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically. It is an approximant of the full Hamiltonian foliation, the better the smaller the leak is. The escape rate depends sensitively on the orientation of the leak even if its area is fixed. The applications for chaotic advection, for chemical reactions superimposed on hydrodynamical flows, and in other branches of physics are discussed

Tipping phenomena in typical dynamical systems subjected to parameter drift

Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed

Defining climate by means of an ensemble: why it it possible

Presentation given at the 21st EGU General Assembly (EGU2019), published on the Proceedings from the conference held 7-12 April, 2019 in Vienna, Austria, id.17414.Any prediction for the weather of a time instant in the far future is uncertain due to possible model errors, the unknown forcing, and internal variability. Since the latter is inherent to the Earth system, it is more useful to consider the statistical (probabilistic) description of all possibilities permitted by the system’s dynamics (in a particular model under some particular forcing), which is the basic idea behind defining climate. Regardless of how one chooses initial conditions to construct an ensemble representing an initial probability distribution, this initial choice is forgotten during time evolution, and the distribution (“spread”) of the evolving ensemble members converges to a unique, so-called natural probability distribution (supported by a dynamical, so-called snapshot or pullback attractor; it may depend on time if the forcing is not stationary). Among others, this is the case for a set of initial conditions consistent with observations. We find in an intermediate-complexity general circulation model, the Planet Simulator of the University of Ham- burg, as well as in a toy model, that the rate of the convergence (equivalently, that of the loss of memory about initial conditions) is approximately exponential. Since independence of initial conditions is approximately reached within a finite time (a few decades in the Planet Simulator in the presence of a mixed-layer ocean), we propose to identify climate with the natural probability distribution. We use the relationship between the El Niño–Southern Oscillation and the Indian monsoon in the Max Planck In- stitute Grand Ensemble to illustrate that evaluating a statistical quantifier, the correlation coefficient, with respect to time in a single ensemble member may provide with a very poor approximation to the same quantifier evalu- ated with respect to the natural probability distribution. In particular, conclusions about its time evolution can be opposite

Chaotic particle dynamics in viscous flows: the three-particle stokeslet problem

It is well known, that the dynamics of small particles moving in a viscous fluid is strongly influenced by the long-range hydrodynamical interaction between them. Motion at high viscosity is usually treated by means of the Stokes equations, which are linear and instantaneous. Nevertheless, the hydrodynamical interaction mediated by the liquid is nonlinear; therefore the dynamics of more than two particles can be rather complex. Here we present a high resolution numerical analysis of the classical three-particle Stokeslet problem in a vertical plane. We show that a chaotic saddle in the phase space is responsible for the extreme sensitivity to initial configurations, which has been mentioned several times in the literature without an explanation. A detailed analysis of the transiently chaotic dynamics and the underlying fractal patterns is given

Finite-size particles, advection, and chaos: A collective phenomenon of intermittent bursting

We consider finite-size particles colliding elastically, advected by a chaotic flow. The collisionless dynamics has a quasiperiodic attractor and particles are advected towards this attractor. We show in this work that the collisions have dramatic effects in the system's dynamics, giving rise to collective phenomena not found in the one-particle dynamics. In particular, the collisions induce a kind of instability, in which particles abruptly spread out from the vicinity of the attractor, reaching the neighborhood of a coexisting chaotic saddle, in an autoexcitable regime. This saddle, not present in the dynamics of a single particle, emerges due to the collective particle interaction. We argue that this phenomenon is general for advected, interacting particles in chaotic flows.FAPESPCNPqOTKA[T72037], HungaryMedical Research Council (MRC)[G0502236], UKCollege of Physical Sciencies, University of Aberdee