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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