Irreversibility and the Approach to Equilibrium

  • Franz Schwabl
Part of the Advanced Texts in Physics book series (ADTP)


In this chapter, we will consider some basic aspects related to irreversible processes and their mathematical description, and to the derivation of macroscopic equations of motion from microscopic dynamics: classically from the Newtonian equations, and quantum-mechanically from the Schrödinger equation. These microscopic equations of motion are time-reversal invariant, and the question arises as to how it is possible that such equations can lead to expressions which do not exhibit time-reversal symmetry, such as the Boltzmann equation or the heat diffusion equation. This apparent incompatibility, which historically was raised in particular by Loschmidt as an objection to the Boltzmann equation, is called the Loschmidt paradox. Since during his lifetime the reality of atoms was not experimentally verifiable, the apparent contradiction between the time-reversal invariant (time-reversal symmetric) mechanics of atoms and the irreversibility of non-equilibrium thermodynamics was used by the opponents of Boltzmann’s ideas as an argument against the very existence of atoms1. A second objection to the Boltzmann equation and to a purely mechanical foundation for thermodynamics came from the fact — which was proved with mathematical stringence by Poincaré — that every finite system, no matter how large, must regain its initial state periodically after a so called recurrence time.


Phase Space Boltzmann Equation Time Reversal External Perturbation Recurrence Time 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Franz Schwabl
    • 1
  1. 1.Physik-DepartmentTechnische Universität MünchenGarchingGermany

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