The Boltzmann model fails to establish equilibrium, derive the correct entropy or provide a direction of time. However, a model proposed in a 1906 paper by Marian Smoluchowski provides an alternative which accomplishes all three. Using an approach similar to Einstein’s 1905 paper on Brownian motion, Smoluchowski proposed treating each molecule in an ideal gas as a Brownian particle whose motion is described by a random walk. The gas is composed of N such particles, whose probability distributions are superimposed to get the composite behavior of the gas. For each molecule, the step size is set equal to the mean free path and the step interval to the mean time between collisions. In the simplest form of the model, the direction of each subsequent step is selected at random and equally likely to be in any direction.
Starting from an arbitrarily selected time t0, the probability distribution of the displacement of the molecule at any future time will approach a Gaussian distribution for large N, where N is the number of steps taken since t0. The displacement r(t) is defined as the radial distance from the origin to the position of the molecule at time t0 + t. The variance will increase linearly with time, meaning that the standard deviation will increase in proportion to √t. The increase in the standard deviation is not reversible, which can be seen by envisioning an arbitrary reversal of the molecule’s trajectory. While this would retrace the last step, at that point the direction of the next step will be determined by random selection, as before. Continuing with the random walk will result in a resumption of the increase in the standard deviation. It might be argued that the molecule can retrace all of its steps to get back to where it started, but this would involve arbitrarily discarding the random walk model and substituting a deterministic model in its place.
The Smoluchowski model has a definite time directionality, since the dispersion as measured by the standard deviation can only increase with time, in accordance with the second law. While the underlying ideal gas model is strictly deterministic, the random walk model partly incorporates probability. However, it is apparent that the time directionality derives from the laws of motion underlying the deterministic ideal gas model of kinetic theory.
To see how this works at the molecular level, see the diagram above, where a is the origin and b is the location of the molecule after N steps. The subsequent step will find the molecule at somewhere on the circumference of the circle centered on b. With equal likelihood that the molecule will go in any direction, it is more probable the it will land on the portion of the circumference outside of circle a than the portion within circle a. This is true for every step of the random walk, so that the tendency toward dispersal will always exceed the tendency toward contraction.
Revised 8/7/22