1. Introduction

One of the central problems in the foundations of statistical mechanics is the problem of reconciling the expressly time-asymmetric behaviour of irreversible thermodynamic processes with the underlying time-symmetric dynamics. How can it be that, at the macroscopic level, we can observe time-asymmetric phenomena such as ice cubes melting in glasses of water, but not spontaneously forming from glasses of water, gases mixing and coming to an equilibrium temperature, but not unmixing, while the dynamical laws that govern the microscopic constituents of such systems are incapable of distinguishing past and future? How can temporal symmetry give rise to temporal asymmetry?

One popular answer has been that the imposition of an initial low-entropy condition at the time of the universe's origin, buttressed by an argument about the statistical properties of statistical mechanical systems, can solve the above problem; it can explain why ice melts in warm glasses of water, and why they don't spontaneously form from warm glasses of water (see Goldstein Reference Goldstein, Bricmont, Dürr, Galavotti, Ghirardi, Petruccione and Zanghì2001, Feynman Reference Feynman1967, Boltzmann Reference Boltzmann and Brush[1898] 1964, Sklar Reference Sklar1993 and Price Reference Price1996 for various forms of this argument and criticism). In this paper, I question whether the imposition of a low-entropy macro-condition at or near the time of the big bang alone can solve the problem. In particular, I will respond to the arguments presented by Albert (Reference Albert2000), though the general argument will hold against anyone who thinks that the low-entropy initial condition of the universe can explain thermodynamic irreversibility.

2. The Problem: Irreversibility from Reversibility

Let us assume for the purposes of this paper that whatever theory correctly describes the microscopic constituents of thermodynamic systems, it is time-reversible and that the vast majority of micro-states that are compatible with any macro-condition will evolve towards an equilibrium state, where the thermodynamic observables do not change with time.Footnote ¹ A dynamical theory is temporally symmetric (or, equivalently, time-reversible) just in case, for any sequence of states {S(t)} allowed by the theory, the time-reversed sequence {S(−t)} is allowed as well, so the theory is incapable of picking out a privileged temporal direction.Footnote ² How is the macroscopically observed approach to equilibrium, with its expressly temporally asymmetric behaviour, to be reconciled with the temporally symmetric dynamical laws that determine its microscopic behaviour?

As a concrete example, suppose that there is a glass of water with a half-melted ice cube in it (suitably isolated from its environment). Given this present macro-condition, we can follow its underlying (Newtonian) dynamics either to predict or to retrodict its future or past macro-condition, respectively. In each case, this dynamics would indicate that the system spontaneously evolved to its present nonequilibrium state from an equilibrium one, and that it will return to an equilibrium state in the future. Based solely on the uniform probability distribution over this macro-condition and the dynamics that underlie thermodynamic systems, it would appear that any nonequilibrium macro-condition one comes across popped into existence as an enormously improbable fluctuation from a past equilibrium macro-condition (contrary to our best recollections), and will return to an equilibrium macro-condition in the future.

It would seem that we often have records of past nonequilibrium conditions: we remember the unmelted ice cube being in the glass five minutes ago. But can our memories or records of the evolution of the ice cube be taken as veridical? Given that we take our memories and records of past events to be describable in statistical mechanical terms and are also governed by time-reversible dynamics, the above concerns apply equally well to our own memories. Just as, on the basis of Newtonian mechanics, we could retrodict that the ice cube arose as a spontaneous fluctuation from an equilibrium state, so can we retrodict that our current memories of the ice cube most likely arose out of a spontaneous fluctuation as well. In fact, taking our memories as statistical mechanical systems, it would appear that all our memories arose spontaneously from equilibrium states, and should not be taken as veridical. A sceptical disaster looms.

3. The Big Bang to the Rescue?

The concern is alleviated, on many accounts, by postulating the existence of a highly nonequilibrium condition at some point in the early universe. For unless we appeal to such a highly nonequilibrium state at some point in the distant past, at the moment of (or a short time after) the big bang, it would be overwhelmingly likely that the past nonequilibrium states that we recall, that we have records of, that we posit, were preceded by states evolving in the past temporal direction closer and closer to equilibrium states. And so without the posit of a highly nonequilibrium state shortly after the big bang, it would seem highly improbable that anything that we take ourselves to know about the past, either through memories or records of past events, would be true.

Albert (Reference Albert2000) is a strong proponent of this view, and recognizes these issues. Albert acknowledges that this cosmological posit (in his terminology the ‘past hypothesis’) cannot directly entail that the ice cube was fully unmelted five minutes ago or that it did not arise as a spontaneous fluctuation from a prior equilibrium state, but he thinks that the posit, in conjunction with the macro-state of the rest of the present universe, can guarantee the veracity of our memories of such events and furthermore can show that the apparent history of the ice cube is in some sense typical.

To explicate his view, Albert considers the “pinballish device” replicated in Figure 1. At the bottom of the device sit several glasses of warm water, some of which contain half-melted ice cubes. On the basis of the microscopic dynamics of the system in question, we would expect that the ice cubes arose from an equilibrium state in the past. However, if we posit that ten minutes ago the ice cubes were fully unmelted and at the top of the pinballish device when they fell, then things will come out right. Here is Albert's claim:

It is (to begin with) certainly not the case that this last posit will make either the present macro-condition or the five-minutes-ago macro-condition overwhelmingly probable: this posit (as a matter of fact) will make no particular present or five-minutes-ago macro-condition overwhelmingly probably. What it will do (rather) is to make certain prominent thermodynamic features of the present and five-minutes-ago macro-conditions overwhelmingly probable (their average temperatures, for example, and the degree to which what ice there is in them is melted, and so on), but it will clearly assign similar probabilities to a rather wide variety of quite distinct five-minutes-ago macro-conditions (macro-conditions associated with the ice cubes having landed in quite different sets of glasses, for example). What we have, though, in this last posit, and what we were lacking in the previous one, is a probability-distribution relative to which what we remember of the entirety of the last ten minutes, and what we know of the present, and what we expect of the future, is (you might say) typical. (Albert Reference Albert2000, 84)

Figure 1.

Albert's pinballish device. At present, there are partially melted ice cubes in some of the glasses.

Nothing here seems particularly objectionable, save for the vagueness of Albert's notion of typicality. Let us specify exactly what Albert is claiming. As he notes, based on the posit to the effect that ten minutes ago the ice cubes were all at the top of the device, we might expect to find a variety of different configurations of ice cubes now (that is, some glasses of water might be empty, some might contain more than one ice cube, etc.), only one of which is actually realized. Each of these configurations of ice cubes in the glasses, along with their thermodynamic variables, constitutes a distinct macro-condition of the system. But we do find that some gross thermodynamic features will be consistently attributable to the system based on the initial posit, features that will remain over many runs of the experiment, where we drop a number of ice cubes from the top of the pinballish device. Albert's notion of typicality is thus summed up as follows:

So far, so good. In fact, based on the present macro-state of all the other glasses, one can infer whether or not any unmelted ice cubes were present ten minutes ago in any particular remaining glass. So it would appear that the initial posit, the macro-description of the rest of the system and the standard statistical rule, allow one to trace the thermodynamic history of the ice cube in the glass.

But Albert quickly moves to cosmological considerations. He immediately notes that the story he has just told gets everything right (regarding the typicality of the present ice cube situation), but that at times before the time indexed by the initial posit, things will still go horribly wrong, since it would seem that that situation must have arisen as a spontaneous fluctuation from equilibrium. And so one must accept the granddaddy of all initial posits, the ‘past hypothesis’ to the effect that the universe began in a highly nonequilibrium state (Albert Reference Albert2000, 85).

What the past hypothesis gives us then, in accordance with the description of typicality above, is a probability distribution over some vague set of macro-conditions, characterized by a nonequilibrium state somewhere in the distant past, which is supposed to restrict the set of macro-conditions to those that make it overwhelmingly probable that the universe evolved pretty much in the same way that we take it to have evolved. Furthermore, given the present state of the universe, one should be able to determine (more or less) the thermodynamic evolution of any system of interest. The past hypothesis allows us to trust our memories, any records of the past we may have, and to validate the posit that the ice cube, half-melted and sitting in a glass of water, was fully unmelted ten minutes ago and didn't arise as a spontaneous fluctuation from an equilibrium state. Or so Albert claims.

4. The Past Hypothesis Scrutinized

This inference, if sound, would be miraculous. The informality of the argument aside, it appears implausible that the mere stipulation of a nonequilibrium state of the universe somewhere in the distant past could justify my memory of an unmelted ice cube ten minutes ago, somehow make it altogether improbable that the ice cube formed as a spontaneous fluctuation ten minutes ago, and testify to the veracity of any records to that effect. Extrapolating Albert's notion of typicality to the case of the universe, let us see how probable the history of our ice cube seems.

Consider as the event space all the microstates on the energy hypersurface of the universe; use the standard statistical measure (Albert's statistical postulate), and define the following propositions:Footnote ³

We then look to establish that

where C is a positive constant. In words, we look to show that conditionalizing on the present macro-condition of the universe and those present states compatible with the past hypothesis is more likely than some threshold probability such that we can justifiably infer that the ice cube was indeed less melted in the past. This relation amounts to a necessary condition on Albert's proposed explanation.Footnote ⁴ Indeed, if the past hypothesis fails to establish the inequality above, then its explanatory value is of little or no worth.

Substituting the definition of conditional probability, we find that (1) can be expressed as

We can simplify the above equation by noting that almost all unmelted ice cubes ten minutes ago evolve to presently half-melted ice cubes by dropping the H term from the expression that conjoins it with U (in any case this won't alter the inequality):

Resubstituting the definition of conditional probability, we can rewrite the equation as

This can be simplified by noting that the terms M that appear on the right side of the equation do no work and can be dropped. This is because the macro-state of the present universe is exhaustively described by the conjunction H & M, and any apparent correlations between the macro-state of the rest of the universe and the state of the ice cube are, by the reversibility argument, almost certainly the result of a spontaneous fluctuation and not in any way correlated with the past, unmelted, state of the ice cube. Thus, we can rewrite the above as

The right side of the equation now places a strong lower bound on the inequality, since the measure associated with a half-melted ice cube on the event space is presumably orders of magnitude larger than that associated with an unmelted ice cube. Accordingly, we can drop the constant C:

(1) is thus equivalent to saying that the initial, nonequilibrium, state of the universe, given that there was an unmelted ice cube in the glass of water ten minutes ago along with the present macro-description of the universe, is much more probable than its likelihood given that there is a half-melted ice cube in the glass now.

So does the past hypothesis solve our trouble? Recall the problem with which we began. It appeared that, no matter how far from equilibrium we find a thermodynamic system, the underlying dynamics dictated that in both temporal directions the system would move towards an equilibrium state. In fact, based on the underlying dynamics and a uniform statistical distribution, nothing in the present situation could ever imply that the system was, or ever will be, further from equilibrium than it is now. More to the point, there is nothing in the present state of affairs that could, in itself, ever provide any grounds for believing that the universe was ever further from equilibrium than it is now. Albert (Reference Albert2000, 93) clearly recognizes this, calling it the fundamental insight of Boltzmann and Gibbs. But if this is the case for our present macro-condition (the ice half-melted in the glass of water), then surely, mutatis mutandis, this applies to the fully unmelted ice cube ten minutes ago. Nothing in that macro-condition could ever count as evidence for the universe having been further from equilibrium than it was ten minutes ago. And so, looking at (3), we are forced to conclude that conditionalizing on the highly nonequilibrium state of the early universe adds nothing to what we've been looking for—a reason to think that the ice cube was previously less melted than it is now. Albert's explanation seems to be dead from the start.

One might object that even though no particular nonequilibrium state can in itself increase the probability that the universe's entropy was even lower, the existence of low entropy states like an unmelted ice cube does increase the probability of a low entropy past relative to higher entropy states such as a half-melted ice cube. But it is hard to see how that is going to help, since the inequality is quite strong in the sense that the left side of (3) needs to be orders of magnitude greater than the right side. This worry can be made more precise by considering (3) without the conditionalization on the macro-state of the rest of the universe (that is, $P(B\vert U) \gg P(B\vert H) $). Here one might be inclined to think that the big bang state is better correlated with the unmelted ice cube than the half-melted cube is, on the order of $P(H) / P(U) $.

At best, this is an unargued for conjecture, and Albert provides no substantial reason to think that this is true. In fact, one can argue that it is clearly false, for the only way that the inequality can hold is if the existence of a presently half-melted ice cube, given the past hypothesis, virtually guarantees that it was unmelted ten minutes ago.

Imagine that I walk into an otherwise empty room with an apparently half-melted ice cube sitting in a glass of water. I need not infer, even as a matter of everyday reasoning, that it was unmelted ten minutes ago. There is virtually an infinity of other histories the ice cube could have, both in accord with the second law and those exhibiting anti-thermodynamic pasts: someone could have left the room just moments before I entered, having placed the glass of water with the half-melted ice cube (fresh out of the freezer) in the room. The ice cube need not have ever been in a more unmelted state.

So if (3) is to hold, the macro-condition of the rest of the universe must do nontrivial work in guaranteeing that the inequality is satisfied, in a manner similar to the way that the macro-states of the other ice cubes fixed the thermodynamic history of the first ice cube in the pinballish device; that is, they serve as records of the fact that the ice cube was previously unmelted. Returning to the example of the ice cube in an empty room, it is clear that this move fails as well. There is, presumably, nothing in the present macro-state of the universe that can tell me whether or not a half-melted ice cube was dropped in the room just moments before I entered. More generally, there may not be anything about the present observable universe that serves as a reliable indicator or record of a system's thermodynamic history, either because no lasting records were formed, or because any apparent records underdetermine its history, or because any such records have ‘washed out’, in the sense that they did not last (came to an equilibrium state) up to the present. Appealing to the rest of the universe does not seem to help.

This criticism can be expanded to less exotic scenarios. Consider a case where I have a clear memory of an unmelted ice cube and a half-melted cube before me now. Does this situation license the inference that the ice cube was unmelted ten minutes ago because there are records of this past state of affairs? It does not, since the entire probabilistic derivation could be run anew, this time taking H to be the phase description of the claim that the ice cube is presently half-melted along with my memory of the unmelted ice cube, and U to be the description of the unmelted ice cube ten minutes ago and the formation of my memory. But now we are in the same situation as before. Is there anything about the present state of the rest of the universe that, along with the past hypothesis, fixes the history of the ice cube and memory system by serving as a record of this new state described by U?

Even if there is, one can easily incorporate it into our description of the system of interest and ask whether there is anything about the rest of the universe that fixes that composite system's history, and so on. If and only if this regress can continue all the way to the point where there is no more “rest of the universe” to consider is Albert's proposal successful in vitiating the reversibility objection. (Note that this is exactly the case when considering the pinballish device.)

But the regress cannot continue in this way, because many facts about the past state of the universe are forever lost to us. Indeed, the regress halts when there is nothing about the present macro-state of the rest universe that can fix the history of the system under consideration. The present macroscopic description of the universe underdetermines its history, and there may be nothing (or at any rate, nothing left of any past macro-conditions) that would serve to ground the history of the system of interest.

Perhaps it is instructive to return to Albert's “pinballish” device and identify in what ways this model differs from the full scale past hypothesis that Albert offers as analogous to this case, since in that instance it appeared that Albert's proposal had a certain plausibility. Specifically, we can identify three elements that differ between the case of the whole universe and the pinballish device:

1. 1. The difference in the times indexed by the past hypothesis and the present are much greater than the time required for an ice cube to melt. Conversely, the pinballish device involves time scales on the order of the time required for an ice cube to melt.

2. 2. In the case of the pinballish device, the glasses of water cannot interact with one another, while presumably the subsystems of the universe can.

3. 3. The past hypothesis refers to a vague macro-state in the early universe that is insufficient to generate specific predictions about the macro-states of present subsystems, while the pinballish device allows us to nearly deterministically expect that we will find, at present, half-melted ice cubes in the glasses at the bottom of the device.

These three elements are salient in explaining why the pinballish machine's “local” past hypothesis does prevent the reversibility objection from going through, and why it fails to be convincing when considering subsystems of the universe. Returning to (2), we can now understand B to be the initial state of the pinballish device where, say, 12 ice cubes are at the top of the apparatus (call this the ‘local past hypothesis’), and M is the present macro-description of the rest of the glasses, some of which contain half-melted ice cubes. Here the three elements described above conspire to allow this inequality to be satisfied.

Since the time between the local past hypothesis and the present macro-state is short and commensurate with the time it takes an ice cube to become half-melted from a fully unmelted state, it is overwhelmingly likely that the presently half-melted ice cube in the glass evolved from one of the unmelted cubes described by the local past hypothesis rather than as a spontaneous fluctuation, and we can trace the thermodynamic history of the ice cube from the time of the local past hypothesis to the present. But there is nothing analogous in the case of the global past hypothesis: The ice cube did not begin in an unmelted state 13 billion years ago. Why should I not believe that the presently half-melted cube arose as a spontaneous fluctuation?

In addition, the local past hypothesis asserts that one should expect a total of 12 half-melted ice cubes to be gathered presently in glasses at the bottom of the device. If 11 of the ice cubes are accounted for in M, the remaining glass should have the last half-melted ice cube in it, one that was previously fully unmelted. In this case, M serves to fix the history of our system of interest in a way that the macro-state of the universe cannot. As such, the H and U terms from (2) can be dropped, since they are both contained in the measure associated with B & M, so (2) reduces to

The equality is thus satisfied, in part because of the temporal proximity of the local past hypothesis to the present situation, but also because each ice and water system is isolated from the others, and because the specificity of the local past hypothesis guarantees this to be the case. In a universe such as our own, one does not expect or believe ice cubes to be causally isolated from the rest of the universe, stemming from the time of the big bang. If we see ice cubes in everyday life, we quite reasonably infer that they came into being through some interaction with the rest of the universe, even if there is no trace of the interaction.

Furthermore, the macro-state of the rest of the universe cannot fix the history of our ice cube, even if it is restricted to those microstates compatible with the past hypothesis, in the way the local past hypothesis did. The local past hypothesis reported a very specific macro-condition, one where we knew that all the cubes (exactly 12) would wind up, half-melted, in glasses of water at the present time, one that specified that there could be no causal interaction between the glasses, and one that precluded any of the cubes emerging from a spontaneous fluctuation. Conversely, the full-blown past hypothesis is of a very different sort: the macro-condition it refers to is, at best, a general description of a macro-description that cannot possibly satisfy the aspects of the pinballish device that made the corresponding condition work there.Footnote ⁵

5. Conclusion

It would seem that we are still without the solution to the problem with which we began. Recall that what we wanted was some sort of postulate that made it overwhelmingly likely, given the half-melted ice cube before us and contrary to what the dynamics alone would have us believe, that the ice cube was fully unmelted 10 minutes ago. Further, we hoped to show that this unmelted ice cube did not arise as a spontaneous fluctuation from an equilibrium state. However, what we discovered was that postulating the asymmetric initial condition of the early universe as being relevant to irreversible processes was equivalent to showing that the existence of nonequilibrium states strongly supports the proposition that the universe began in a highly nonequilibrium state. This result indicated a serious problem with the claim that a highly nonequilibrium state at the time of the early universe explains the everyday thermodynamic processes we set out to explain: no present macro-condition can count as evidence that the universe was ever further from equilibrium than it is now, and this goes for any past macro-state as well. Conditionalizing on the macro-condition of the early universe cannot do all the work. We still lack an explanation of irreversible processes that solves the problem of reconciling the reversible dynamics with the apparent actual history of thermodynamic systems, and any reason to think that our records and memories of the past are veridical.

In this paper, I have presented considerations to the effect that the initial nonequilibrium state of the early universe cannot have the explanatory power that it is often thought to have. The postulation of such a state cannot make it overwhelmingly likely that a half-melted ice cube was previously less melted that it is now, nor can it establish the veracity of our records and memories to that effect. While these arguments have been specifically directed against Albert's use of the ‘past hypothesis’ to explain irreversible thermodynamic behavior, the general point holds against all similar claims that thermodynamic irreversibility can be explained by appealing to a low entropy macro-condition of the early universe.

The upshot of this discussion is that any correct explanation of the time-asymmetric behaviour of thermodynamic systems needs more. The nonequilibrium state of the early universe alone cannot explain irreversible behaviour. One must look elsewhere, for example to a branch system approach (e.g., Reichenbach Reference Reichenbach1956, Davies Reference Davies1974), or to independent grounds for believing the veracity of our records (Horwich Reference Horwich1988), or for some other fundamental source of temporal asymmetry that “drives” thermodynamic irreversibility such as the causal asymmetry or radiative asymmetry. The big bang cannot do it all by itself.