# Can Time Flow Backwards? A Thermodynamic Approach

Eugene Ho, Philosophy, Undergraduate, National University of Singapore

The flow of time is said to be asymmetric, progressing from the past, to the present and then to the future, in that direction, and no other. But what accounts for this temporal asymmetry? What is the difference between the past and the future? Thermodynamics contends that the difference is entropy

### Entropy and Thermodynamic Asymmetry

Roughly speaking, the second law of thermodynamics tells us that the entropy of an isolated system only increases and does not decrease over time, whereby entropy is a function of the state of a system associated with the disorder in the arrangement of its particles, given its energy distribution.

Let’s unpack what this means.

Consider the different physical states a cup of iced water left without any external interaction can take, at different times.

Given these three states, S1, S2, and S3, it seems obvious that the chronological sequence of events progresses (from S1 to S2 to S3) in that sequence and not the other way around. We know this because a progression of events in the other direction (from S3 to S2 to S1) is not equally plausible given our daily experiences. This is confirmed by our understanding of thermodynamics which predicts that low-entropy states such as S1 will progress into high-entropy states such as S3 — ice melts when placed in warm water, and the water does not extract heat from ice, spontaneously forming ice cubes when left alone.

Since entropy is the only physically measurable property we know of with which we can distinguish the past from the future, it seems to be a good candidate for providing a physical explanation for why time only flows in one direction — by reducing temporal asymmetry to thermodynamic asymmetry.

According the second law of Thermodynamics, entropy tends to a state of disorder, from low-entropy states to high. We see this in reality – ice cubes and water, as ice cubes (low entropy) tend towards water (high entropy). Since it is simple to understand, we will use this as an example to explain why time flows in one direction.

### The Problem: Time Reversal Invariance

However, even if we equate thermodynamic asymmetry with temporal asymmetry, the question of why there is any asymmetry at all, still remains. If we say that time flows from past to future without reversing, because entropy only increases towards the future, we still need to explain why entropy only increases towards the future in the first place.

This is especially important, considering that the underlying dynamical laws governing the interactions between particles seem to be time reversal invariant — i.e. if one reverses all the corresponding vectors such that the direction and velocity of each molecule’s interactions are reversed, so do the sequence of events. This is further supported by mathematical idealisations of dynamical systems as the Poincaré Recurrence Theorem, which observes that isolated finite systems can be expected to eventually return to a state arbitrarily close to an identical initial state infinitely many times, given an infinite amount of time. While in practice we do not observe entropy decreasing over time, this means that it is in theory possible that a physical process can progress from S3 to S1, despite what thermodynamics tells us.

Given that we want to provide a physical, scientific explanation for temporal asymmetry in terms of thermodynamic asymmetry, we now need to explain our observations of entropy increase over time, where the reverse is theoretically possible.

However, the underlying laws of Thermodynamics are time reversal invariant – which simply means that ice cubes-water can return close to their original state (ice cubes), infinitely many times, if an infinite amount of time is given (entropy decrease).

### Proposal: Phase Space?

Since the underlying dynamical laws governing the movement of particles tell us that entropy decrease is possible, this means that the axiom that entropy always increases is not absolute. Perhaps in this light, the second law of thermodynamics should be understood not as a universal law, but as a probabilistic law. If so, this would explain why we experience time as flowing in one direction, while successfully accounting for the phenomenon of entropy decreasing.

Boltzmann proposes that we can understand this by modelling all possible configurations of the thermodynamic processes of a particle system using phase space, whereby a phase space for N particles has 6N dimensions — 3 spatial dimensions and 3 velocity dimensions per particle.

States such as S1, S2, and S3 can be given macroscopic descriptions such as iced water, cool water, and lukewarm water, describing the contents of the glass in terms of its macrostate.

However, with phase space, we can (in theory) describe the underlying microscopic configuration of each and every particle in the glass, since we are modelling each and every particle’s position and velocity. This means that every point in phase space represents one possible configuration of all the microstates of the isolated system as a whole. If we assign an equal probability of each point in phase space, we can thus understand the volume of phase space occupied by a certain macroscopic description as representing the probability that such a macrostate obtains.

Using this model, we can observe that high-entropy macrostates such as S3 occupy a far larger volume of phase space in contrast to low-entropy macrostates such as S1, since every particle has approximately the same energy, and thus, any particle can take the place of any other particle in S3. In contrast, S1, whose arrangement is limited by the stipulation that half the particles must be rigidly ordered such that they take the form of ice cubes,  has a far smaller number of possible microscopic configurations which can fulfill its macroscopic description, and thus occupies a far smaller volume of phase space.

Since there are abundantly many more possible microstates from each resulting interaction which corresponds to net entropy increasing, it follows that for any given state of an isolated system, subsequent entropic fluctuations overwhelmingly result in entropy increasing. This explains why we can expect S2 to almost always progress to S3, instead of S1.

If the phase space model is a good explanation for why the universe we live in increases in entropy with time, then perhaps we have found an answer for why time only flows in one direction.

We can explain this phenomenon with the phase space model. In short, phase space states that ice cubes occupy a far smaller volume phase space compared to water. This because in water, particles are not arranged, and can take the place of another particle. Compared to ice cubes, where particles are rigidly ordered, water has a far more arrangement of particles compared to ice cubes, and thus occupy more phase space. In cool water, a state between ice cubes and water, the particles interact more often, and thus tends to a state of water than ice cubes.

### Doesn’t Phase Space tell us that our Past is a lie?

However, before we celebrate solving the problem, given that high-entropy macrostates have a larger volumes of phase space, the phase space model does predict that systems like our universe will spend overwhelmingly more time in high-entropy states, such as thermal equilibrium.

This means that while the phase space has the explanatory benefit of predicting that any state described by the universe is overwhelmingly likely to progress into a higher-entropy state as time progresses forwards, if we turn the focus of the phase space model backwards, not on predicting future states, but states prior to any given state, the phase space argument unfortunately uses the very same logic we employed to predict the overwhelming probability that it emerged from a higher-entropy state. In other words, without specifying the initial condition, the phase space model tells us that entropy increases both towards the future and towards the past, resulting in the loss of the thermodynamic asymmetry which motivated us to consider entropy as an explanation for the direction of time in the first place. This can be represented in the graph below:

Thus, accepting the phase space argument also carries positive reasons for rejecting our beliefs that we came from a low-entropy past, because it predicts that for any given state that we know really exists (eg. the state of the world as we know it now), the state described by one’s conscious experience is best explained not by the truth of the Past-Hypothesis (represented in blue) —  that our memories and records, are accurate such that the universe started billions of years ago from a low-entropy state — but by the overwhelmingly more probable claim (if the phase space argument is true) that such a state of affairs including all memories and records of the past originated spontaneously at this very moment, out of a fluctuation from a high-entropy past (represented in red).

Unless we are willing to abandon our commitment to the veracity of our memories and our records regarding the past, this indicates that we should be reluctant to accept the phase space argument as an explanation for why temporal and thermodynamic asymmetry.

By accepting the phase space argument, we now see that most particles are in high entropy states (more water than ice cubes). However, as mentioned earlier, Thermodynamics laws states that the current state can return close to their original state. This causes a conundrum as it could mean that the original state was of high entropy (water instead of ice cubes).

### Defending the Past

One may attempt to ‘fix’ the phase space model by appealing to the already accepted truth of the Past-Hypothesis: The belief that our records of the past are accurate such that we came from a low-entropy past.  If we simply accept the Past-Hypothesis as a given, and stipulate the assumption that the initial boundary condition just is a low-entropy state, then the phase space model successfully predicts the desired account of thermodynamic asymmetry over time.

However, is it epistemically and scientifically responsible to do so?

It certainly seems highly unintuitive to seriously deny the veracity of all our past records, since doing so would result in a kind of radical scepticism which would undermine the scientific method itself, which relies on forming and testing hypotheses based on the observations we have made in the past. But denying a possibility simply because its consequences are unintuitive does not seem to be epistemically responsible either.

Perhaps the solution is to accept the modest epistemic principle that “evidence of the past provides tentative justification and increases the credence of accepting a belief about the past”.

Consider the set of all possible civilisations which can emerge from a high-entropy fluctuation. It seems extremely unlikely that such a civilisation would be one which contains multiple records which corroborate and verify its history in a systematic way, since it is far more likely that such civilisations have little or no coherent records of the past.

Given that we do live in a civilisation which contains multiple records of the past, which corroborate and verify our history in a systematic way, we can weigh the unlikelihood of emerging from a low-entropy past against the unlikelihood that we are the exceptional civilisation which emerged from high-entropy fluctuation with such systematic records.

This then puts us in a position to use our modest epistemic principle based on observable evidence provided by our past records as justification for accepting the Past-Hypothesis.

If so, the defender of the phase space model can thus accept that it was likely that the Big Bang emerged as a result of a high-entropy fluctuation while denying the radical sceptical hypothesis that we have just, at this very moment, emerged out of a high-entropy fluctuation. Thus, he can explain the low-entropy history our records seem to indicate.

A possible defence to the phase space model will be to look at it from the angle of history – that the vast evidence of past records of history and civilisations prove that we came from a low-entropy past (ice cubes). However, one can also view it from the modest epistemic principle, which states that we came from a high-entropy past, which led to the Big Bang, and then eventually coming to our current high-entropy state today.

### What does this mean to me

In this article, I attempted to provide an explanation for the flow of time in terms of thermodynamic asymmetry, while highlighting and addressing problems in with the phase space model in accounting for the world we appear to live in. It certainly seems that time flows in one direction, and if we are able to reduce the flow of time to the mere change of entropy, then with the phase space model we can provide a scientific explanation for why that is so (as long as we accept the Past-Hypothesis).

However, as I wrote this article, I began developing doubts about such an approach. While physicists such as Boltzmann simply equate the flow of time with the change in entropy, I suspect that the very possibility of Poincaré Recurrences which describe entropy decreases as time progresses, indicate that temporal asymmetry should not simply be equated with thermodynamic asymmetry — if entropy increases are only a contingent feature of a succession of states, then I do not think we should treat entropy decreases as time reversals. If so, then while this article has addressed issues in accounting for thermodynamic asymmetry, the question of why time flows still remains unanswered. I hope to attempt to answer this question once more in the future.

The question of why time flows remains unanswered, as we should not merely treat entropy decreases as time reversals.