But between two temporal points, one can, and usually must, travel only in one direction: from earlier to later, and never from later to earlier. How is this distinctive feature of time to be understood?
Ludwig Boltzmann looks to the second law of thermodynamics to understand time’s directionality. He argues that the move from order to disorder does not occur with time, but rather that this move is time, or rather, that it determines the direction of time.
Boltzmann further asks about the implications of time’s directionality. If one can travel through time in only one direction, then is the set of possible events limited or conditioned by this fact? In his words,
Is the apparent irreversibility of all known natural processes consistent with the idea that all natural events are possible without restriction? Is the apparent unidirectionality of time consistent with the infinite extent or cyclic nature of time? He who tries to answer these questions in the affirmative sense must use as a model of the world a system whose temporal variation is determined by equations in which the positive and negative directions of time are equivalent, and by means of which the appearance of irreversibility over long periods of time is explicable by some special assumption. But this is precisely what happens in the atomic view of the world.
The universe, which according Boltzmann is on average in equilibrium, can contain local regions which, apparently spontaneously, move themselves into a state of disequilibrium. His view of the universe is, in part, his gas theories writ large. His conception of the universe could be understood as vaguely analogous to Brownian motion: galaxies vibrating rather randomly and spontaneously.
From this notion of the universe, Boltzmann sees the directionality of time, and perhaps time itself, as arising from pockets of disequilibrium. Time, and its direction, is the flow from a state of more disequilibrium to a state of less disequilibrium.
Boltzmann is, in effect, reversing the priority of the second law of thermodynamics: It is not the case, according to him, that a system moves within time, or through time, from a state of less entropy to a state of more entropy; rather, he seems to be saying that it is the move from one state to another which gives directionality to time, and perhaps even gives rise to time itself.
One can think of the world as a mechanical system of an enormously large number of constituents, and of an immensely long period of time, so that the dimensions of that part containing our own “fixed stars” are minute compared to the extension of the universe; and times that we call eons are likewise minute compared to such a period. Then in the universe, which is in thermal equilibrium throughout and therefore dead, there will occur here and there relatively small regions of the same size as our galaxy (we call them single worlds) which, during the relative short time of eons, fluctuate noticeably from thermal equilibrium, and indeed the state probability in such cases will be equally likely to increase or decrease. For the universe, the two directions of time are indistinguishable, just as in space there is no up or down. However, just as at a particular place on the earth’s surface we call “down” the direction toward the center of the earth, so will a living being in a particular time interval of such a single world distinguish the direction of time toward the less probable state from the opposite direction (the former toward the past, the latter toward the future). By virtue of this terminology, such small isolated regions of the universe will always find themselves “initially” in an improbable state. This method seems to me to be the only way in which one can understand the second law — the heat death of each single world — without a unidirectional change of the entire universe from a definite initial state to a final state.
Boltzmann argues that the direction of time is relative: to call a particular movement through time “forward” is merely a matter of perspective; another observer might call it “backward.” The direction of time is an indexical: to move “toward the future” or “toward the past” is like moving “to the right” or “to the left” — one observer’s “right” is another observer’s “left.”
As Lawrence Sklar has pointed out, Boltzmann even leaves room for the scenario in which different local pockets of disequilibrium might have their own times running in opposite directions. This would be within the larger framework of the universe as a whole, which Boltzmann seems to view as literally timeless, having no time because it is in a state of equilibrium.
Time, then, according to Boltzmann, is secondary, and contingent upon the primary appearance of local pockets of disequilibrium. Boltzmann has some explaining to do: how and why would these local pockets of disequilibrium randomly and spontaneously appear?
Obviously no one would consider such speculations as important discoveries or even — as did the ancient philosophers — as the highest purpose of science. However it is doubtful that one should despise them as completely idle. Who knows whether they may not broaden the horizon of our circle of ideas, and by stimulating thought, advance the understanding of the facts of experience?
On Boltzmann’s account, it could be that a pocket of disequilibrium would appear in the midst of a universe in equilibrium: this would constitute time moving backward, as it were. If that pocket of disequilibrium, in its process of moving toward equilibrium, would halt, that would amount to time standing still. If that process of moving toward equilibrium were subject to occasional relapses in which there was a temporary movement back to disequilibrium, this would amount to time repeatedly changing its direction.
Further, Boltzmann’s view might entail that within the one universe there could be various disparate pockets of disequilibrium, and these sundry pockets could have times which are moving in different directions. Simply put, in one part of the universe, time could move in a direction opposite to the way it’s moving in another part of the universe.
That in nature the transition from a probable to an improbable state does not take place as often as the converse, can be explained by assuming a very improbable initial state of the entire universe surrounding us, in consequence of which an arbitrary system of interacting bodies will in general find itself initially in an improbable state. However, one may object that here and there a transition from a probable to an improbable state must occur and occasionally be observed. To this the cosmological considerations just presented give an answer. From the numerical data on the inconceivably great rareness of transition from a probable to a less probable state in observable dimensions during an observable time, we see that such a process within what we have called an individual world — in particular, our individual world — is so unlikely that its observability is excluded.
Boltzmann’s notion that time is not merely the framework within which the second law of thermodynamics plays out, but rather that the very movement from disequilibrium to equilibrium is itself time, and sets the direction of time, is thought-provoking and worth consideration.
His claim, however, that pockets of disequilibrium might suddenly, spontaneously, and randomly appear within a universe which is in equilibrium is a bold claim, for which he offers no argumentation, the vague analogy to Brownian motion, and other phenomena within physical chemistry notwithstanding.
