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Chaotic Dynamics in Nonequilibrium Statistical Mechanics

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

For most of its history, non‐equilibrium statistical mechanics has producedmathematical descriptions of irreversible processes by invoking one or another stochasticassumptions in order to obtain useful equations. Central to our understanding of transport influids, for example, are random walk processes, which typically are described by stochasticequations. These in turn lead to the Einstein relation for diffusion, and its generalizations to other transportprocesses. This relation, as formulated by Einstein, states that the mean square displacementof a diffusing particle grows linearly in time with a proportionality constant givenby the coefficient of diffusion. If we assume that such a description applies tomechanical systems of many particles, we must explain the origins of irreversibility indeterministic – and time reversible – mechanical systems, and we mustlocate the source of stochasticity that is invoked to derive transport equations. For...

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Abbreviations

Chaotic systems:

The time evolution of a deterministic mechanical system defines a trajectory in the phase space of all the generalized coordinates and generalized momenta. Consider two infinitesimally separated points that lie on two different trajectories in this phase space. If these two trajectories typically separate exponentially with time, the systems is called chaotic provided the set of all points with an exponentially separating partner is of positive measure.

Chaotic hypothesis:

The hypothesis that systems of large numbers of particles interacting with short ranged forces can be treated mathematically as if the system were chaotic with no pathologies in the mathematical description of the systems' trajectories in phase space.

Dynamical systems theory:

The mathematical theory of the time evolution in phase space, or closely related spaces, of a deterministic system, such as a mechanical system obeying Hamiltonian equations of motion.

Ergodic systems:

A mechanical system is called ergodic if a typical trajectory in a phase space of finite total measure spends a fraction of its time in a set which is equal to the ratio of the measure of the set to the total measure of the phase space.

Escape rate formula:

Consider a chaotic dynamical system with a phase space constructed in such a way that the phase space has some kind of an absorbing boundary. The set of points, \( { {\mathcal{R}} } \), in the phase space such that trajectories through them never escape through the absorbing boundary either in the forward or the backward motion is called a repeller . One can define a set of Lyapunov exponents , \( { \lambda_i({\mathcal{R}}) } \) and a Kolmogorov–Sinai entropy, \( { h_\text{KS}({\mathcal{R}}) } \) for motion on the repeller. Dynamical systems theory shows that the rate of escape, γ, of points, not on the repeller, through the boundary is given by

$$ \gamma = \sum_{i}\lambda^{+}({\mathcal{R}}) - h_\text{KS}({\mathcal{R}})\:, $$
(1)

where the sum is over all of the positive Lyapunov exponents on the repeller.

Gaussian thermostats:

A dynamical friction acting on the particles in a mechanical system which keeps the total energy, or the total kinetic energy of the system at a fixed value. It was invented by Gauss as the simplest solution to the problem of finding the equations of motion for a mechanical system with a constraint of fixed energy.

Gelfand triplet:

An operator with right and left hand eigenfunctions, possibly defined in different function spaces, and an inner product of one function from the right space and one from the left space. Generally one of these spaces contains singular functions such as Schwartz distributions and the other contains sufficiently smooth functions so that the inner product is well defined. The term rigged Hilbert space is also used to denote a Gelfand triplet.

Hyperbolic dynamical system:

A chaotic system where the tangent space to almost all trajectories in its phase space can be separated into well‐defined stable and unstable manifolds, that intersect each other transversally.

Kolmogorov–Sinai entropy per unit time:

A measure of the rate at which information about the initial point of a chaotic trajectory is produced in time. The exponential separation of trajectories in phase space, characteristic of chaotic motion, implies that trajectories starting at very close-by, essentially indistinguishable, initial points will eventually be distinguishable. Hence as time evolves we can specify more precisely the initial point of the trajectory. Pesin has proved that for closed, hyperbolic systems, the Kolmogorov–Sinai entropy is equal to the sum of the positive Lyapunov exponents. The Kolmogorov–Sinai entropy is often called the metric entropy.

Lyapunov exponents:

Lyapunov exponents, \( { \lambda_i } \), are the rates at which infinitesimally close trajectories separate or approach with time on the unstable and stable manifolds of a chaotic dynamical system. For closed phase spaces, that is, no absorbing boundaries present, Pesin theorem states that for hyperbolic dynamical system the Kolmogorov–Sinai entropy, \( { h_\text{KS} } \) is given by the sum of all the positive Lyapunov exponents.

$$ h_\text{KS}= \sum_i \lambda_i^{+}\:. $$
(2)
Mixing systems:

Mixing systems are dynamical systems with stronger dynamical properties than ergodic systems in the sense that every mixing system is ergodic but the converse is not true. A system is mixing if the following statement about the time development of a set A t is satisfied

$$ \lim_{T\rightarrow\infty}\frac{\mu(A_T\cap B)}{\mu(B)} = \frac{\mu(A)}{\mu({\mathcal E})}\:, $$
(3)

where B is any set of finite measure, and \( { \mu({\mathcal{E}}) } \) is the measure of the full phase space. This statement is the mathematical expression of the fact that for a mixing system, every set of finite measure becomes uniformly distributed throughout the full phase space, with respect to the measure μ.

Normal variables:

Microscopic variables whose values are approximately constant on large regions of the constant energy surface in phase space.

Pseudo‐chaotic systems:

A pseudo‐chaotic system is a dynamical system where the separation of nearby trajectories is algebraic in time, rather than exponential. Pseudo‐chaotic systems are weakly mixing as defined by the relation

$$ \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T \mskip2mu\mathrm{d}\tau\left[\mu(A_{\tau}\cap B)-\frac{\mu(A)\mu(B)}{\mu({\mathcal E})}\right]=0\:. $$
(4)
Sinai–Ruelle–Bowen (SRB) measure:

SRB measures for a chaotic system are invariant measures that are smooth on unstable manifolds and possibly singular on stable manifolds.

Stable manifold:

A stable manifold about a point P in phase space is the set of points that will approach P at time t approaches positive infinity, that is in the infinite future of the motion.

Transport coefficients:

Transport coefficients characterize the proportionality between the currents of particles, momentum, or energy in a fluid, and the gradients of density, fluid velocity or temperature in the fluid. The coefficients of diffusion, shear and bulk viscosity, and thermal conductivity are transport coefficients, and appear as coefficients of the second order gradients in the Navier–Stokes and similar equations.

Unstable manifold:

An unstable manifold about a point P in phase space is the set of points that will approach P as time approaches negative infinity, that is, as one follows the motion backwards in time to the infinitely remote past.

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Acknowledgments

I would like to thank Henk van Beijeren for reading a draft of this article and for his very helpfulremarks. I would also like to thank Rainer Klages for his new book (Reference [12]), which was very helpfulwhen preparing this article.

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  1. Institute for Physical Science and Technology and Department of Physics, University of Maryland, College Park, USA

    J. Robert Dorfman

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Dorfman, J.R. (2009). Chaotic Dynamics in Nonequilibrium Statistical Mechanics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_66

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