Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Hamiltonian dynamical systems approach to nonequilibrium statistical mechanics

• Thermostated dynamical systems approach to nonequilibrium statistical mechanics

• The red thread through this book

In this chapter we analyze a paradigmatic model for deterministic diffusion, which is a chaotic map acting on the real line. In Section 2.1 we explain the basic idea of deterministic diffusion starting from the famous drunken sailor's problem and introduce our model. Section 2.2 outlines a method of how to exactly calculate the parameter-dependent diffusion coefficient for this dynamical system. This physical quantity turns out to be a fractal function, a result that is expected to be typical for a wide class of dynamical systems. To the end of this section we supply some intuition about the physical origin of this fractality.

The research presented in this chapter has been performed together with J.R.Dorfman [Kla95; Kla96b; Kla99a]. For a mathematical underpinning of these results see Refs. [Gro02; Kel07]; at the beginning we partially follow Ref. [Kla96b].

Here we put the analysis started in the previous chapter on more mathematical grounds. In Section 3.1 we rigorously define a slightly generalized model, which consists of the simple map studied before amended by a second parameter mimicking the action of a bias. We then introduce the quantities we want to compute, namely the cumulants which yield in particular the current and the diffusion coefficient as functions of the two control parameters. In Section 3.2 we outline two further methods which both enable one to compute exactly the values of these quantities for our model. Section 3.3 is devoted to an analysis of the phase diagram of this system. Its fundamental structure is explained by simple heuristic arguments. We then numerically evaluate the current and the diffusion coefficient on the basis of these methods and discuss the results with respect to the phase diagram. In Section 3.4 we establish a link between these simple maps and ratchets. The latter systems are usually modeled by overdamped Langevin equations and have been widely studied both in theory and in experiments.

This chapter profoundly reflects a cooperation with J.Groeneveld [Gro02], to whom the author is very much indebted for many invaluable explanations

We proceed with our introduction to elementary deterministic transport processes by studying the consequences of deterministic chaos for a simple model of diffusion-controlled chemical reaction. In Section 4.1 we motivate the use of two-dimensional reactive-diffusive multibaker maps starting from a more physically realistic model, which is a reactive-diffusive particle billiard. In Section 4.2 we review the diffusive properties of multibakers. We then demonstrate that under parametric variation the diffusion coefficient of such a model exhibits a self similar-like structure analogous to the ones discussed in the previous two chapters. In Section 4.3 we focus on the reactive properties of our model. Particularly we derive the dispersion relation of the chemiodynamic modes, explicitly construct the phase space distribution of these modes and define the reaction rate by comparison with phenomenology. We then show that the parameter-dependent reaction rate behaves in a highly irregular manner displaying a complicated scenario of nonequilibrium phase transitions.

The work presented in this chapter originated in collaboration with P.Gaspard [Gas98c].

So far we have focused on purely deterministic systems, where for a given equation of motion the initial condition reproducibly determines the outcome of the dynamics. In this chapter we turn our attention to the interplay between deterministic chaos and random perturbations. As an example we choose again our standard model whose deterministic transport properties have been analyzed in Chapters 2 and 3, that is, a piecewise linear map acting on the real line. In Section 5.1 we study the impact of spatial disorder onto deterministic diffusion in this map [Kla02b] while Section 5.2 deals with the case of perturbing the system by time-dependent noise [Kla02c].

After having explored the most elementary types of dynamical systems generating deterministic transport, which are piecewise linear chaotic maps defined on a periodic lattice, we now investigate nonlinear generalizations of such models. These systems may exhibit intermittency and complicated bifurcation scenarios under parameter variation. In this chapter we show that such mechanisms lead to transitions from normal to anomalous diffusion, where the mean square displacement of an ensemble of moving particles is not linear in time anymore.

In Section 6.1 we introduce the climbing sine map, which can be extracted from a driven nonlinear pendulum equation via discretization of time. We find that this model generates fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. These structures are analyzed by applying the Taylor-Green-Kubo formula for diffusion discussed in the previous chapter. The final part of this section provides a more detailed analysis of the bifurcation scenario exhibited by this system. In Section 6.2 we study a paradigmatic model for intermittent transport, the Pomeau-Manneville map lifted onto the real line. This model exhibits subdiffusive behavior, which is analyzed within the framework of continuous time random walk theory, fractional diffusion equations and a generalized Taylor-Green-Kubo formula.

This chapter reflects the author's collaboration particularly with N.Korabel and A.V.Chechkin [Kor02; Kor04b; Kor04a; Kor07].

In the previous chapters we have shown that low-dimensional periodic arrays of scatterers with moving point particles, modeled by time-discrete chaotic maps, typically exhibit transport coefficients that are fractal functions under variation of control parameters. This chapter demonstrates how to directly apply the knowledge gained by the analysis of simple maps to physically more realistic systems, which in this case are Hamiltonian particle billiards. We exemplify this connection by proposing a systematic scheme of how to approximate fractal diffusion coefficients in terms of a hierarchy of correlated random walks. In Section 7.1 we develop the theory for our standard model, the diffusive piecewise linear map L introduced in Chapter 2. In Section 7.2 we apply a suitably adapted version of this approach to diffusion in the periodic Lorentz gas [Kla02d].

The periodic Lorentz gas exhibits an irregular diffusion coefficient, as we have seen in the previous chapter. However, the irregularities were so small that they were hard to detect without further scrutiny. Here we may turn the problem around: Having explored the origin of fractal transport coefficients enables us to use our understanding in order to design systems that exhibit irregular transport coefficients in an especially pronounced manner. This strategy may be particularly useful for practical applications of this phenomenon.

A first example along these lines is provided by a suitable modiffication of the periodic Lorentz gas, the ower-shaped billiard, which we introduce in Section 8.1. Computer simulations indeed reveal a non-trivial parameter dependence of the diffusion coefficient in this model. In Section 8.2 we apply the approaches outlined in the previous chapter in order to analyze the irregularities of the diffusion coefficient. Section 8.3 concludes by establishing crosslinks to theory and experiments of molecular diffusion in porous crystalline solids.

This chapter mainly emerged from a collaboration particularly with T.Harayama, who suggested studying flower shapes [Har02]; for related work see Refs. [Kla00a; Kla02d].

We conclude our discussion of diffusion in particle billiards and Part 1 of this book by applying the theory outlined before to granular particles bouncing inelastically on a vibrating corrugated floor. We call this non-Hamiltonian system the bouncing ball billiard. In Section 9.1 we motivate our model and briefly review the role of resonances for a ball bouncing on a at vibrating plate. We then show computer simulation results for the diffusion coefficient as a function of a suitable parameter and explain its largest irregularities. Section 9.2 contains a more refined analysis of parameter-dependent diffusion in terms of the Taylor-Green-Kubo approach, which corroborates the existence of irregularities on fine scales. In Section 9.3 we discuss relations between the bouncing ball billiard and experiments on vibratory conveyors as they are frequently encountered in industrial applications.

The research outlined in this chapter reflects collaborations with L.Mátyás and I.F.Barna [Mát04a; Kla04b].

In this chapter we introduce some basic concepts, models and notations that are used throughout Part 2. In Section 10.1 we outline what thermostats are and explain why they are indispensable for studying steady states under typical nonequilibrium conditions. Section 10.2 briefly reminds the reader of the Langevin equation. Its derivation exemplifies two extreme cases of modeling thermal reservoirs, that is, either in terms of purely Hamiltonian dynamics or by using a dissipative, irreversible and stochastic equation of motion. In Section 10.3 we elaborate on the general functional form of the velocity distributions of a subsystem and thermal reservoir in thermal equilibrium. Section 10.4 contains a detailed definition of the periodic Lorentz gas, the main model of Part 2, which we will study under application of different thermostats.

We are now prepared to introduce a very popular deterministic and time-reversible modeling of a thermal reservoir, which became known as the Gaussian thermostat. In Section 11.1 we define this thermostating scheme for the periodic Lorentz gas driven by an external electric field. In Section 11.2 we summarize what we consider to be the most important features of the resulting model from a dynamical systems point of view. The properties we report are to a large extent typical for Gaussian thermostated systems altogether. Specifically, we review a connection between thermodynamic entropy production and the average phase space contraction rate as well as a simple functional relationship between Lyapunov exponents and transport coefficients. Furthermore, there exists a fractal attractor in the Gaussian thermostated Lorentz gas, which changes its topology under variation of the electric field strength. Correspondingly, the electrical conductivity is an irregular function of the field strength as a control parameter, which takes up the thread on fractal transport coefficients of Part 1 in this book. We also briefly elaborate on the existence of linear response in this model.

We construct and analyse a second fundamental form of deterministic and time-reversible thermal reservoir known as the Nosé-Hoover thermostat. To motivate this scheme we need a Liouville equation that also holds for dissipative dynamical systems, which we introduce in Section 12.1. After defining this thermostat in Section 12.2 we focus on the chaos and transport properties of the Nosé-Hoover thermostated driven periodic Lorentz gas by characterizing the associated NSS, see Section 12.3. For this purpose we first check whether there holds an identity between the average phase space contraction rate and thermodynamic entropy production as was obtained for the Gaussian thermostat. We then discuss computer simulation results for the attractor and for the electrical conductivity of the Nosé-Hoover driven periodic Lorentz gas by comparing them to their Gaussian counterparts. In Section 12.4 we briefly elaborate on subtleties resulting from the construction of the Nosé-Hoover thermostat, and we outline how this scheme can be generalized leading to a variety of additional, similar thermal reservoirs.

The approach to nonequilibrium transport outlined in the previous two chapters yields NSS on the basis of non-Hamiltonian equations of motion. This dynamics results from applying deterministic and time-reversible thermal reservoirs, which are designed to efficiently control the temperature of a subsystem under nonequilibrium conditions. The Gaussian and the Nosé-Hoover thermostat constructed before provide two famous examples of such mechanisms. Coupled to the periodic Lorentz gas driven by an external electric field, the resulting NSS have been analyzed with respect to their chaos and transport properties. This chapter provides a shortlist of what we consider to be fundamental properties of Gaussian and Nosé-Hoover dynamics. We put these results into a broader perspective and critically discuss whether these properties might be universal for NSS altogether, irrespective of the type of thermostat used. Our assessment sets the scene for the following two chapters.

So far we have identified important crosslinks between chaos and transport in Gaussian and Nosé-Hoover thermostated models. Some authors conjectured these relations to be universal for dissipative dynamical systems altogether. It is thus illuminating to look at simple generalizations of ordinary Gaussian and Nosé-Hoover dynamics, which we define again for the driven periodic Lorentz gas. We then analyze these thermostating schemes along the same lines as discussed before, that is, by analytically and numerically studying the transport and dynamical systems properties of the respectively thermostated driven periodic Lorentz gas. Finally, we compare the results obtained from our analysis to the ones characterizing conventional Gaussian and Nosé-Hoover schemes as discussed in Chapters 11 to 13.

Gaussian and Nosé-Hoover schemes belong to the class of bulk thermostats, where energy is removed from a subsystem during the originally free flight of a particle. Boundary thermostats, on the other hand, change the energy of a particle only at a collision with the boundaries of the subsystem.

In this chapter we construct and analyze fundamental types of the latter class of thermostats. Section 15.1 focuses on stochastic boundary conditions as have already been mentioned in Section 13.4. In Section 15.2 we show how to make stochastic boundaries deterministic and time-reversible by using a simple chaotic map. This modeling of a thermal reservoir was called thermostating by deterministic scattering. The equations of motion resulting from this method are still deterministic and time-reversible hence sharing fundamental properties with Gaussian and Nosé-Hoover schemes. However, as explained in detail in Section 15.3, both the construction of this deterministic boundary thermostat and the dynamics associated with it are generally very different in comparison to conventional bulk thermostats.

An analysis of models thermostated by deterministic scattering thus enables us to learn more about universalities of nonequilibrium steady states in dissipative dynamical systems. In Section 15.4 we demonstrate this by studying again the transport and dynamical systems properties of a respectively thermostated driven periodic Lorentz gas. Our findings are compared to the ones of Gaussian and Nosé-Hoover thermostats. Section 15.5 briefly reviews results for a hard-disk fluid under nonequilibrium conditions, which is also thermostated at the boundaries.

With this chapter we return to the problem of Brownian motion reviewed in Section 10.2, which we used to introduce the concept of thermal reservoirs. In Section 16.1 we give Brownian motion a different touch by relating it to observations in biology. Our motivation derives from experiments on migrating biological cells, which show Brownian motion-like trajectories. However, the fact that biological entities like living cells are self-propelled, in contrast to passively driven tracer particles in a fluid, suggests that amendments to ordinary Langevin equations are necessary in order to describe such living systems.

This motivates the formulation of active Brownian particles in terms of generalized Langevin equations, which we review in Section 16.2. We argue that limiting cases of such equations contain ingredients that trace back to the conventional Nosé-Hoover dynamics discussed in Chapter 12. This may not come too much as a surprise if one reinterprets the thermal reservoir modeled by Nosé-Hoover dynamics as the internal energy of a migrating cell. That is, the thermal reservoir is now part of a biological entity providing energy for its migration process.

In Section 16.3 we focus on essential features of the velocity distribution functions of active Brownian particles. We argue that the appearance of bimodal profiles may be understood on the basis of Nosé-Hoover dynamics. By employing the general functional forms of the equilibrium velocity distribution functions calculated in Section 10.3 we finally state some necessary conditions for the emergence of such bimodal structures.

The following sections are included:

• Fluctuation relations
  • Entropy fluctuation in nonequilibrium steady state
  • The Gallavotti-Cohen fluctuation theorem
  • The Evans-Searles fluctuation theorem
  • Jarzynski work relation and Crooks relation

• Lyapunov modes

• Fourier's law
  • The basic problem
  • Heat conduction in anharmonic chaotic chains
  • Heat conduction in chaotic particle billiards

• Pseudochaotic diffusion
  • Microscopic chaos and diffusion?
  • Polygonal billiard channels

• ^*Summary

The final chapter consist of three parts: In Section 18.1 we outline what we believe represents a general picture of a chaotic dynamical systems approach to nonequilibrium statistical mechanics. Following the two central themes of this book, we summarize our main results in Section 18.2. Section 18.3 concludes this work with a number of open questions.

The following sections are included:

• Bibliography

• Index