This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems.
This revealed that one of Boltzmann's key assumptions, molecular chaos, or, the Stosszahlansatz, that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics.
One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.
Furthermore, due to CPT symmetry, reversal of the direction of time is equivalent to renaming particles as antiparticles and vice versa.
[4] Abstract mathematical tools used in the study of dissipative systems include definitions of mixing, wandering sets, and ergodic theory in general.
One approach to handling Loschmidt's paradox is the fluctuation theorem, derived heuristically by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain value for the dissipation function (often an entropy like property) over a certain amount of time.
[citation needed] Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus.
The fluctuation theorem considers the probability density for all of the trajectories that are initially in an infinitesimally small region of phase space.
The difference is that the role of measurement is obvious in Maxwell’s demon, but not in Loschmidt’s paradox, which may explain the 40-year gap between both explanations.
Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy starting point: the Big Bang.