The Apparent Irreversibility of Time Pt. 2: Is Entropy Essential to Time?

Why does the process of time appear so irreversible? This question leads us to the second law of thermodynamics. It states that the entropy of an isolated system always increases or equivalently, that heat always flows from higher to lower temperatures. In order to connect the concepts of time and entropy, we must first dissect this law and define entropy. 

The increase of entropy in a system over time suggests that the number of ways a system can be arranged increases with time. Imagine a system of 20 rubber balls and 20 plastic containers in which rubber balls go into plastic containers. In the system's most orderly state (least uniform), all 20 balls exist in 1 of the 20 containers. Therefore, there would be 20 ways in which the system could by arranged. 20 balls could exist in container 1, or container 2, or container 3, etc. However, in its most uniform state, where each rubber ball can exist in any of the 20 containers, there are 20! (2.43*10^18) possible arrangements for the system. Therefore, when we say that the entropy of every isolated system always increases, we are saying that as time progresses the universe tends towards uniformity and disorder.

In the example which I used in my previous post, gas held in a bottle is released into a room. This is a perfect example of an increase in entropy. The molecules do not dare permanently restrict themselves to the confines of the bottle. The system must always proceed towards uniformity. Of course, in principle it is not impossible for the system to occur in the opposite direction. But in practice, such an occurrence wouldn't happen in a million years. The idea that the second law of thermodynamics is a law of disorder was developed by Ludwig Boltzmann. His ideas were questioned because of the reversibility of Newton's equations. Why couldn't the velocities of the molecules just be reversed and the entropy of the system decrease? To such a question Boltzmann sarcastically replied, "Go ahead and reverse them." 

This bottle of gas scenario can be represented by the following two pictures:

To sum up everything I have been saying so far, the only reason why we know the right picture occurs after the left picture is because of the second law of thermodynamics.

Of course, no definition of entropy would be complete without a mathematical representation. One might think that entropy, which we shall denote as S, should be proportional to the N number of ways a system can be distributed. This would be represented as S = kN, where k is the proportionality constant. In a combination of two isolated systems(microstates), this would mean that the combination(macrostate), S12, would equal kN1 * kN2 =  S1 * S2. However, in order for this concept of entropy to work, the entropy value must be innate to each isolated system(microstate). Therefore, in order to calculate the entropy of a combination of two systems, we must simply add S1 and S2. S cannot be directly proportional to N. Instead, S1 = k*ln(N1). In a combination of two isolated systems, S12 = k * ln(N1 * N2) = k * ln(N1) + k * ln(N2) = S1 + S2. This leads us to the formal mathematical expression for entropy in statistical mechanics:

where kB is the boltzmann constant, and P is the probability that the system is in it's ith microstate. 

So, the entropy of the universe is always increasing. As a result, we are able to differentiate between the past and the future. The world tends towards disorder and uniformity and time is irreversible because of this. But we still have not answered WHY the universe must become more uniform and why heat always flows from higher to lower temperature. If we go back in time the entropy of the universe should be lower than it is now. Consequently, it can be assumed that the universe began with a low entropy. Why is this? Maybe, increasing entropy is a result of the expansion of the universe(cosmic inflation). Or maybe it's the other way around. 

I'll leave you with an interesting proposal. Space has no innate directionality. There is no up, down, or sideways. It also has no discernible center. However, in the presence of a large object such as the earth, we are able to create a perceived directionality. With the Earth as center, it becomes the authority on what we call north, south, east, and west. Perhaps there is a large event which has created a perceivable directionality in time; the big bang. 

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The Apparent Irreversibility of Time Pt. 1: The Problem

Initially, the directionality of time does not appear to be a problem. The past becomes the present and the present becomes the future and we ignore the intricacies of its definition. After all, time is so intrinsic to the universe and our daily lives. Time moves forward. Without that direction we would find it difficult to even call time, time. But, what is time if it lacks a past or a future? What if there is no distinction between the processes of past to future and future to past; is it still "time"?

As you view the motion of this pendulum can you decide whether this video is playing forward in time or in reverse? The problem with the arrow of time is its apparent reversibility. That is to say, if this system is reversible then it can proceed in either the forward or reverse temporal directions without breaking physical law. Lets take a look at another simple system; a planet in orbit.

By reversing this video we would find that at every corresponding position the velocity of the planet would also be reversed (-v) and the acceleration would remain constant. This completely agrees with physical law and in neither clip can we discern whether we pressed play or rewind. Thus, Newton's equation for the force of the gravity (F = GMm/r^2) between these 2 planetary bodies is time reversible and in fact this entire system works just as well in reverse as it does in the forward direction. Actually, all of Newton's laws and equations happen to be time reversible. 

Whether or not these simple systems are time reversible, we are still able to maintain our previous perception of time. There are many events in our daily lives that are apparently irreversible. For example, here's a glass breaking.

And here's it breaking in reverse...

Surely this situation is different since glass does not put itself back together in such a manner by the laws of physics. In a complicated system such as this our original perceptions are justified and we find that there must be some difference between the past and the future. It is a seemingly irreversible event. Now consider another complicated system where gas, which is held in a bottle, is released into a room. As time goes by, the gas exits the bottle and evenly distributes itself throughout the room. Again, it is seemingly impossible for this situation to reverse itself and for all the dispersed molecules of gas to gather and enter the bottle once again. However, this event is dictated entirely by Newton's laws, which are time reversible. How can a situation described by time reversible equations produce such apparently irreversible results? One might even argue that perhaps if we were able to obtain data on each molecule of gas, including its position, velocity, and the results of their elastic collisions, that the entire system could be described in time reversible simple states. 

To determine the validity of this claim let us consider a third event. You stand at the edge of a pool. The water is completely still. You take your finger and touch the water, creating a series of circular ripples emanating outwards. Laws determining the propagation of waves are time reversible and therefore, at a fundamental level this event should be completely reversible. This means it is possible to design boundary conditions at the edges of the pool such that you create waves emanating inward to a single point. Additionally, at the moment those waves converged it would be necessary for your finger to come into contact with the water to absorb the momentum. For all practical purposes, this is impossible. Therefore, in principle these three events are not irreversible. Rather, the chances that the reverse of these situations may occur(in reality) by accident or design are extremely low. Why is this?

The thing is...ALL the fundamental laws of Physics are time reversible. This includes Newton's laws, the laws of electromagnetism, and even the wave equation of quantum mechanics. How can we reconcile this fact with our macroscopic observations and interpretations of time and reality? Why does the process of time and the processes of the universe appear so irreversible? I will attempt to answer this question in my next post...

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