Microscopic irreversibility and chaos

© 2006 American Institute of Physics, S-0031-9228-0608-210-X The origin of irreversible behavior is a persistent theme in physics. Since the fundamental microscopic laws, including both Newton’s laws and quantum mechanics, are reversible (except for the weak interactions), the fact that most macroscopic systems behave irreversibly has long been recognized as an important issue. The problem of understanding macroscopic irreversibility was solved more than 100 years ago by Ludwig Boltzmann, who recognized that systems evolve toward more probable states, namely, those that have a larger number of microscopic configurations for a given macroscopic state. This is such a powerful tendency that on the macroscopic level, fluctuations that go against it are so improbable as to be negligible. The result is a probabilistic explanation of macroscopic irreversibility and the second law of thermodynamics. (See the article by Joel Lebowitz in PHYSICS TODAY, September 1993, page 32.) Nevertheless, Boltzmann did not explain the microscopic chaotic dynamics that leads to macroscopic irreversibility. Consider an imaginary gas of hard spheres that elastically collide with each other and obey the laws of Newtonian mechanics. It is perfectly conceivable that many microscopic states will never be visited, even in the age of the universe, and that some improbable states will persist, so the exploration required to achieve macroscopic irreversibility is not guaranteed. For example, a hard-sphere gas in a box in which all particles move parallel to the x direction so that they do not collide could persist forever. But, it is known that the hard-sphere gas is chaotic. Thus, on the average, a small perturbation to an initial configuration of particles becomes amplified exponentially over time. Chaos ensures that evolution to a representative sample of microstates occurs,1 and that reversal of the velocities of all the particles does not in practice lead to a time-reversed motion. Therefore, both the microscopic and macroscopic behaviors of statistical systems are irreversible. The strength of this sensitivity to initial conditions due to chaotic dynamics may be characterized by what are known as the Lyapunov exponents. They give the rates of exponential growth of the vector difference between two nearby trajectories along different directions in phase space. In some fields of physics outside the domain of equilibrium statistical mechanics, similar considerations apply. For example, consider turbulence, which can produce irreversible mixing of a localized impurity in a fluid. The evolution of the fluid is governed by the Navier–Stokes equations, a set of deterministic nonlinear differential equations for the velocity field. Turbulence is generally understood to involve chaos arising from the nonlinear dynamics of these equations. (See the article by Gregory Falkovich and Katepalli Sreenivasan in PHYSICS TODAY, April 2006, page 43.) Therefore, a statistical description is necessary.