Transient chaos: the origin of transport in driven systems (original) (raw)
Related papers
2004
The escape-rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems, respectively. It has been shown that the key features of the transport properties of both approaches, if modeled by low-dimensional dynamical systems, can conveniently be described in the framework of multibaker maps. In the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit. The limit involves a sequence of coarser and coarser descriptions (projections) until one reaches the level of irreversible macroscopic advection-diffusion equations. The influence of boundary conditions is studied in detail. Only a few of the chaos characteristics possess a meaningful macroscopic limit, but none of these is sufficient to determine the entropy production in a general non-equilibrium state.
Microscopic chaos and transport in thermostated dynamical systems
2003
A fundamental challenge is to understand nonequilibrium statistical mechanics starting from microscopic chaos in the equations of motion of a many-particle system. In this review we summarize recent theoretical advances along these lines. Particularly, we are concerned with nonequilibrium situations created by external electric fields and by temperature or velocity gradients. These constraints pump energy into a system, hence there must be some thermal reservoir that prevents the system from heating up. About twenty years ago a deterministic and time-reversible modeling of such thermal reservoirs was proposed in form of Gaussian and Nose-Hoover thermostats. This approach yielded simple relations between fundamental quantities of nonequilibrium statistical mechanics and of dynamical systems theory. The main theme of our review is to critically assess the universality of these results. As a vehicle of demonstration we employ the driven periodic Lorentz gas, which is a toy model for the classical dynamics of an electron in a metal under application of an electric field. Applying different types of thermal reservoirs to this system we compare the resulting nonequilibrium steady states with each other. Along the same lines we discuss an interacting many-particle system under shear and heat. Finally, we outline an unexpected relationship between deterministic thermostats and active Brownian particles modeling biophysical cell motility.
Transport properties of chaotic and non-chaotic many particle systems
Journal of Statistical Mechanics: Theory and Experiment, 2007
Two deterministic models for Brownian motion are investigated by numerical simulations and kinetic theory arguments. The first model consists of a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks acting as a thermal bath. The second is the same except for the shape of the particles, which is now square. The basic difference in these two systems lies in the interaction: hardcore elastic collisions make the dynamics of the disks chaotic whereas that of squares is not. Remarkably, this difference does not reflect on the transport properties of the two systems: simulations show that the diffusion coefficients, velocity correlations and response functions of the heavy impurity are in agreement with kinetic theory for both the chaotic and non-chaotic model. The relaxation to equilibrium instead is very sensitive to the kind of interactions. These observations are used to think back and discuss some issues connected to chaos, statistical mechanics and diffusion.
Entropy Production for Open Dynamical Systems
Physical Review Letters, 1996
The concept of the conditional probability density is used to define a specific entropy for open dynamical systems exhibiting transient chaos. The production of entropy turns out to be proportional to the difference of the escape rate and the sum of all averaged Lyapunov exponents on the saddle governing the dynamics. The single-particle transport properties do not depend on the microscopic details provided the dynamical systems produce the same entropy. The dimension of the unstable foliation of the saddle is shown to be identical in all microscopic single-particle models of the same transport process. [S0031-9007(96)01271-9]
2010
We show that, in spatially periodic Hamiltonian systems driven by a time-periodic coordinate-independent (AC) force, the upper energy of the chaotic layer grows unlimitedly as the frequency of the force goes to zero. This remarkable effect is absent in any other physically significant systems. It gives rise to the divergence of the rate of the spatial chaotic transport. We also generalize this phenomenon for the presence of a weak noise and weak dissipation.
Structure of motion near saddle points and chaotic transport in hamiltonian systems
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
Generic symmetry and transport properties of near separatrix motion in 11 / 2-degree-of-freedom Hamiltonian systems are studied. First the rescaling invariance of motion near saddle points, with respect to the transformation epsilon-->lambdaepsilon, chi-->chi+pi of the amplitude epsilon and phase chi, of the time-periodic perturbation, is recalled. The rescaling parameter lambda depends only on the frequency of the perturbation, and the behavior of an unperturbed Hamiltonian near a saddle point. Additional rescaling symmetry of the motion with respect to transformation epsilon-->lambda(1/2)epsilon, chi-->chi+/-pi/2 is found for some Hamiltonian systems possessing symmetry in the phase space. It is shown that these rescaling invariance properties of motion lead to strong periodic (or quasiperiodic) dependencies of all statistical characteristics of the chaotic motion near the separatrix on log(10)epsilon with the period log(10)lambda. These properties are examined for dif...
The origin of diffusion: the case of non-chaotic systems
Physica D: Nonlinear Phenomena, 2003
We investigate the origin of diffusion in non-chaotic systems. As an example, we consider 1D map models whose slope is everywhere 1 (therefore the Lyapunov exponent is zero) but with random quenched discontinuities and quasi-periodic forcing. The models are constructed as non-chaotic approximations of chaotic maps showing deterministic diffusion, and represent one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the Ehrenfest wind-tree model). In particular, a simple construction shows that these maps define non-chaotic billiards in space-time. The models exhibit, in a wide range of the parameters, the same diffusive behavior of the corresponding chaotic versions. We present evidence of two sufficient ingredients for diffusive behavior in one-dimensional, non-chaotic systems: (i) a finite size, algebraic instability mechanism; (ii) a mechanism that suppresses periodic orbits.
Control of chaotic transport in Hamiltonian systems
2003
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a ``turbulent'' electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in particular tokamaks, is conceivable.
Weak chaos and anomalous transport: a deterministic approach
Communications in Nonlinear Science and Numerical Simulation, 2003
We review how transport properties for chaotic dynamical systems may be studied through cycle expansions, and show how anomalies can be quantitatively described by hierarchical sequences of periodic orbits.