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# IRREVERSIBLE OPEN-SYSTEM MODEL

To provide a physical motivation for the idea that openness necessarily involves time-irreversibility, let us consider another example system drawn from electronic technology, the vacuum thermionic device (``vacuum tube'' or ``valve'') (Langmuir and Compton, 1931; Eastman, 1949). These devices were made by introducing two or more metallic electrodes into a vacuum through which electrons could be transported without dissipation. When a voltage was applied between anode and cathode (and the cathode heated to thermally excite electrons into the vacuum), a nonequilibrium steady state would be established with a nonzero current flowing. Such a nonequilibrium steady state cannot be established in a reversible (or Hamiltonian) system. Consider what would happen if a population of electrons were introduced into some sort of trapping potential in ultrahigh vacuum. The system would effectively be closed, and the motion of the electrons would consist of periodic (thus, reversible) orbits. Of course what happened in the case of the thermionic vacuum tube is that electrons were accelerated by the electrostatic field until they impacted the anode, where they lost their kinetic energy to collisions with the electrons in the metal. Their energy was thus dissipated as heat. However, we can infer a much broader principle from this device: Making contact to a system in such a way as to permit particles to enter and leave (opening the system) in itself introduces irreversibility into the behavior of the system, so long as the contacts have a sufficient number of degrees of freedom and enough indistinguishable particles to behave as reservoirs.

Now, if the openness of the system is to be modeled by boundary conditions applied to the system, these boundary conditions must themselves be time-irreversible. A physically appealing way to achieve such irreversibility is to distinguish between particles moving into the system and those moving out of the system. It is then reasonable to expect that the distribution of particles flowing into the system depends only upon the properties of the reservoirs to which the system is connected, and that the distribution of particles flowing out of the system depends only upon the state of the system. The behavior of the reservoirs is thus analogous to that of an optical blackbody. This picture leads to a fully acceptable model of an open system.

Next: Continuum Formulation Up: Boundary Conditions for Open Previous: Discrete Numerical Model

William R. Frensley
Thu Jun 8 17:53:37 CDT 1995