A near universal feature of books and papers addressing the topic of the second law of thermodynamics is the invocation or Loschmidt’s paradox, which is based on the premise that Newton’s laws of motion are time symmetric. This assumption relies on the concept of “mirror symmetry”, which is commonly illustrated by the movie analogy. If we were to view a movie of a particle moving from left to right, it would be indistinguishable from a movie of the particle moving from right to left when the movie is shown in reverse. However, a movie of an expanding gas doesn’t look right if shown in reverse, illustrating the asymmetry of the second law. The contrast between the two movies exemplifies Loschmidt’s paradox, which claims that the irreversible nature of the second law cannot logically be a consequence of the reversible laws of motion.
However, a closer look reveals that we are observing two different phenomena. The first case involves translatory motion, or the motion of a single particle along a one-dimensional trajectory. The second involves the dispersion of particles in three-dimensional space. If the second movie contains a large number of particles, the increasing dispersion will indicate the direction of time. If we gradually reduce the number of particles, at some point the movie will begin to look like a collection of individual particles, each in translatory motion. However, the increase in dispersion will remain even for a gas composed of two particles, although it may not be obvious at first. If we imagine two particles moving relative to one another in free space, they are either approaching one another or moving apart. If they are moving apart, they will continue to do so forever. If they are moving closer, they will pass each other (or collide) in a finite time and then move apart forever. The direction of time is the direction in which the particles eventually disperse.
Loschmidt argued that the laws of motion were reversible since if you arbitrarily reversed the direction of a particle it would retrace its former path to its starting point. The same holds for a gas composed of many particles. However, as noted by both William Thomson and Ludwig Boltzmann, if there is no further interference the particles will again disperse after regaining their initial positions and then will continue to do so. The direction of time is defined by the direction of this spontaneous dispersal. Contrary to Boltzmann’s theory, this has nothing to do with probability. This is apparent from observing ideal gas simulations, which are commonly found on the Internet. These simulations consist of virtual molecules whose trajectories are dictated by Newton’s laws of motion programmed into the simulations. These simulations are strictly deterministic and can be supplied with a variety of initial conditions, virtually all of which will result in the dispersion characteristic of the second law.
But how can mirror symmetry result in irreversibility? Reversibility means that if –t is substituted for t in the equations of motion, the motion will be reversed but otherwise identical. If we describe the velocity in terms of Cartesian coordinates it is clear that reversing time will merely reverse the direction of the x, y and z components of the velocity. However, if we convert to spherical coordinates, the velocity vector will be described in terms of θ, ψ and r. Mirror symmetry is also a feature of the angular velocities of θ and ψ since reversing time converts the rotation from clockwise to counterclockwise. However, reversing the radial velocity is asymmetrical, since if the radial velocity is reversed from positive (the distance from the origin is increasing) to negative (the distance from the origin is decreasing), the particle will always reach a point at which the velocity will once again become positive. If we were to observe this process in a mirror, the distinction between moving closer and moving farther away would not be altered. The plus/minus of translatory motion is mirror symmetrical, while the closer/further motion of radial velocity is not.
The above is analogous to the “Radiation arrow of time” described in Wikipedia: Arrow of time. An example is the circular waves generated by a stone being thrown into a pond. This case involves the dispersal of energy, not matter. An exactly circular pond might reflect the wave back to the source, but the energy would then disperse again, as above.
When contemplating the direction of time it is important to distinguish between the absolute direction and the relative direction. For instance, one can imagine a parallel universe in which time is unfolding in the opposite direction as in our universe. However, this is a meaningless dichotomy, since we can only experience one universe at a time. The question of the direction of time revolves around determining what, other than our subjective experience, distinguishes moving forward in time from moving backward in time. The simple answer is that objects spontaneously disperse only in forward time. Objects can only appear to spontaneously congregate if the movie is run backward, reversing time. While a reversed trajectory is as dynamically plausible as a forward trajectory, a gas returning to its balloon or cream unmixing from a cup of tea clearly contradicts the tendency toward dispersal which characterizes the second law of thermodynamics.
How does dispersal relate to probability and entropy, which form the basis of statistical mechanics? It is true that greater dispersal is usually correlated with increasing probability. However, to treat probability as cause and dispersion as effect is to reverse causality and believe that the immaterial (probability) can physically influence the material (physical motion) in a process akin to telekinesis. The opposite is true since dispersion in space is a consequence of Newton’s First Law of Motion, the law of inertia. The resulting probability gradient is a consequence, not a cause, of this dispersion. By the same token, entropy is not a physical property of nature, but rather a metric by which we can gauge the extent of dispersion. Other metrics can be envisioned for the same purpose. Entropy is a subjective concept with no physical existence, a human artifact.
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Revised 10/23/22