In a paper in 1876 Loschmidt gave a recipe of how one can prepare an initial condition with decreasing entropy for any system which follows a motion with increasing entropy. One need only reverse all its velocities at a certain instant of time: v(t) -> -v(t) This procedure is equivalent to time reversal and the argument is usually called the reversibility paradox because the equations of classical mechanics are invariant under time reversal while the Boltzmann equation is not. The procedure of time reversal vio- lates the hypothesis of "molecular chaos" since it is like a film which runs backwards.

All molecules which have just had a collision will collide again and thus their velocities are correlated. This paradox represented a severe objection to the mechanical interpre tation of the second law of thermodynamics. Apparently there are just as many initial conditions which lead at least for a short time to a decrease in the entropy of the system as there are initial conditions leading to an increase in the entropy. Why do we never observe a decrease in entropy for large isolated systems? For very small systems one can easily observe a decrease of entropy in the form of statistical uctuations, e.g. density fluctuations, local pressure uctuations or local temperature uctuations in very small regions of a gas. For very large systems this is not the case because the small local fluctuations average out. Statistical physicists have coined the term typicality for the usual behaviour of macroscopic systems. If one pours, for instance, a dye into a liquid it will gradually spread through the whole liquid. This behaviour is typical whenever you make such an experiment. It is easy to tell the sequence in which snapshots of a spreading dye were taken, even after their original order has been deranged. How does this unidirectional behaviour in time come about? For a very large system, by far the largest number of states corresponds to equilibrium - and quasi-equilibrium-states. The latter are states which differ very little from the equilibrium state with maximum en tropy and cannot be distinguished macroscopically from the equilibrium state. In our example of the liquid containing a dye they correspond to a practically uniform distri bution of the dye through the whole liquid with very small local intensity uctuations of the dye. With increasing size of the system, the preponderance of the equilibrium - and quasi-equilibrium-states becomes ever more overwhelming. If for every possible state of a large system we put a marked sphere into an urn and afterwards drew spheres from the urn indiscriminately, we would practically always draw an equilibrium - or quasi- equilibrium-state. The transition from nonequilibrium to equilibrium thus corresponds to a transition from exceptionally rare nonequilibrium-states to extremely probable states. This is Boltzmann's statistical interpretation of the second law.