Jürgen Vollmer

Chaos, spatial extension, transport, and non-equilibrium thermodynamics

Abstract The connection between the thermodynamic description of transport phenomena and a microscopic description of the underlying chaotic motion has recently received new attention due to the convergence of ongoing developments in the theory of deterministic chaotic systems, in the foundation of non-equilibrium statistical physics and of non-equilibrium molecular dynamics simulations. An overview of these developments is given with an emphasis on explicit calculations on exactly solvable models, that may serve as paradigms for this approach to model transport.

Propagation effects of current and conductance noise in a model neuron with subthreshold oscillations

We have examined the effects of current and conductance noise in a single-neuron model which can generate a variety of physiologically important impulse patterns. Current noise enters the membrane equation directly while conductance noise is propagated through the activation variables. Additive Gaussian white noise which is implemented as conductance noise appears in the voltage equations as an additive and a multiplicative term. Moreover, the originally white noise is turned into colored noise. The noise correlation time is a function of the systems control parameters which may explain the different effects of current and conductance noise in different dynamic states. We have found the most significant, qualitative differences between different noise implementations in a pacemaker-like, tonic firing regime at the transition to chaotic burst discharges. This reflects a dynamic state of high physiological relevance.

Basin boundary, edge of chaos and edge state in a two-dimensional model

Basin boundaries are the boundaries between the basins of attraction of coexisting attractors. When one of the attractors breaks up and becomes a transient repelling structure the basin boundary also disappears. Nevertheless, it is possible to distinguish the two types of dynamics in phase space and to define and identify a remnant of the basin boundary, the edge of chaos. We here demonstrate the concept using a two-dimensional (2D) map, and discuss properties of the edge of chaos and its invariant subspaces, the edge states. The discussion is motivated and guided by observations on certain shear flows like pipe flow and plane Couette flow where the laminar profile and a transient turbulent dynamics coexist for certain parameters, and where the notions of edge of chaos and edge states proved to be useful concepts to characterize the transition to chaos. As in those cases we use the lifetime, i.e. the number of iterations needed to approach the laminar state, as an indicator function to track the edge of chaos and to identify the invariant edge states. The 2D map captures many of the features identified in laboratory experiments and direct numerical simulations of hydrodynamic flows. It illustrates the rich dynamical behavior in the edge of chaos and of the edge states, and it can be used to develop and test further characterizations.

OSCILLATING PHASE SEPARATION IN MICROEMULSIONS. I. EXPERIMENTAL OBSERVATION

We examine the phase separation of a single phase of water-in-oil microemulsion droplets towards a phase of smaller water droplets coexisting with a water-rich excess phase. This transition is found to oscillate when induced by a continuous temperature increase. A periodic clouding and clearing is observed under the microscope and in the microcalorimeter, allowing to determine the oscillation period from the extinction of transmitted light and from the specific heat. The period depends on the surfactant concentration and increases as a square root of the scan speed. To the authors’ knowledge this is the first time that oscillations in a temperature induced phase separation of a mixture have been observed.

Oscillating phase separation in microemulsions. II. Description by a bending free energy

We propose a mechanism to describe the phase separation of a single phase of water-in-oil microemulsion droplets towards a phase of smaller water droplets coexisting with a water-rich excess phase. The phase separation shows oscillatory behavior when induced by a continuous temperature increase. A periodic clouding and clearing is observed in the extinction of transmitted light which is also reflected in the specific heat. To model this behavior the bending free energy describing the equilibrium phase transition is applied to identify the energy barriers in the dynamics of this transition. They are due to conservation laws preventing the relaxation to a close to equilibrium size distribution of droplets unless volume and surface is redistributed simultaneously for a large number of droplets. By numerical integration of an expression for the time evolution of the size distribution of droplets it is verified that constant heating gives rise to oscillations. Besides clarifying the origin of the oscillations ...

Breath figures: nucleation, growth, coalescence, and the size distribution of droplets.

The analysis of the size distribution of droplets condensing on a substrate (breath figures) is a test ground for scaling theories. Here, we show that a faithful description of these distributions must explicitly deal with the growth mechanisms of the droplets. This finding establishes a gateway connecting nucleation and growth of the smallest droplets on surfaces to gross features of the evolution of the droplet size distribution.

Entropy balance, time reversibility, and mass transport in dynamical systems

We review recent results concerning entropy balance in low-dimensional dynamical systems modeling mass (or charge) transport. The key ingredient for understanding entropy balance is the coarse graining of the local phase-space density. It mimics the fact that ever refining phase-space structures caused by chaotic dynamics can only be detected up to a finite resolution. In addition, we derive a new relation for the rate of irreversible entropy production in steady states of dynamical systems: It is proportional to the average growth rate of the local phase-space density. Previous results for the entropy production in steady states of thermostated systems without density gradients and of Hamiltonian systems with density gradients are recovered. As an extension we derive the entropy balance of dissipative systems with density gradients valid at any instant of time, not only in stationary states. We also find a condition for consistency with thermodynamics. A generalized multi-Baker map is used as an illustrative example. (c) 1998 American Institute of Physics.

Fluctuation-preserving coarse graining for biochemical systems.

Finite stochastic Markov models play a major role in modeling biological systems. Such models are a coarse-grained description of the underlying microscopic dynamics and can be considered mesoscopic. The level of coarse-graining is to a certain extent arbitrary since it depends on the resolution of accommodating measurements. Here we present a systematic way to simplify such stochastic descriptions which preserves both the meso-micro and the meso-macro connections. The former is achieved by demanding locality, the latter by considering cycles on the network of states. Our method preserves fluctuations of observables much better than naïve approaches.

Oscillatory instabilities in phase separation of binary mixtures: Fixing the thermodynamic driving

Binary liquid mixtures can show pronounced oscillations in the differential scanning calorimeter signal for the specific heat and in the turbidity when phase separation is induced by continuously ramping the temperature. For a fixed ramp rate, i.e., a linear temporal drift of temperature, only a small number of oscillations have been observed. In the present manuscript we describe an experimental setup where simultaneous video-microscopy and shadow-graph measurements can be performed on mixtures subjected to an arbitrary temporal temperature evolution. In particular, it can be adjusted to fix the thermodynamic driving force, which characterizes the rate of change of the composition of the coexisting phases. With this novel technique both the number of oscillations and the temperature interval where oscillations are observed increase significantly. This technique can easily be applied to a great variety of binary mixtures, permitting detailed investigations of their phase-separation kinetics under slowly ramping temperature.

Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz Gas

We extend and review recent results on nonequilibrium transport processes described by multibaker maps. The relation of these maps to the dynamics of the Lorentz gas is discussed. Special emphasis is put on the concept of coarse graining and its use in defining the analog of thermodynamic entropy and in deriving an entropy balance. A full analogy with Irreversible Thermodynamics can only be obtained if at certain points we deviate from traditional dynamical system theory, and allow for open boundary conditions which make the system to converge to a ‘forced’ stationary measure. This measure differs from the natural SRB measure which can be realized with periodic boundary conditions only.