The origins of entropy

Posted on 8 September 2016

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One of the unsolved mysteries of modern physical cosmology is explaining how the universe began in a low entropy state. From within the context of physics, an explanation can only be traced backward in time. To look deeper, one must seek an explanation for the origin of entropy itself. This, as it turns out, is intimately tied to the origins of space and time. We are prior to physics per se, and investigating a level that is more fundamental, which can be traced all the way back to the first distinction.

The seed of time and space are implicit in the first distinction. This then provides the framework for the possibility of different states in time and space, by means of subsequent distinctions. And entropy is indeed a formal or mathematical notion within that context.

Even in traditional classical statistical mechanics, entropy is defined mathematically as a comparison of the number of (unobservable) microstates of a system that give rise to the same (observable) macrostate. Note that this definition is based on the assumption that the actual state of a system is characterized by (unobservable) distinctions (associated with microstates) that have been ignored / conflated / confused to yield the same (observable) macrostate. The key underlying distinction here is between observable and unobservable states of a system: If we could observe all microstates, there would be no meaning associated with entropy. Entropy arises when we describe a system by means of unobservable microstates, and the entropy of a macrostate is a measure of how many microstates are conflated in the same macrostate.

Now, I would propose that this distinction between observable and unobservable states is also implicit in the first distinction: The first distinction distinguishes a present object, which is observable, from its absence, which is unobservable. However, to get entropy requires, as you propose, some additional structure, so that there is the possibility of counting numbers of unobservable states corresponding to a given observable state. The larger the number, the larger the entropy. And, indeed, in the beginning, there is just one observable and one unobservable state, so the universe begins in a low entropy state.

The possibility of increase in entropy arises only through the process of making additional distinctions that allow for a universe to have a larger variety of unobservable states. In addition, there also needs to be a subsequent process of ignoring or conflating those distinctions, so that the distinctions between these states can not be observed, and get identified or conflated into a single macrostate that is observable.

Entropy thus arises by a two-fold process of imagining distinctions and then conflating them. But one has not completely conflated them, since they remain part of the theoretical description of microstates. They just become empirically unobservable. This is essentially a kind of symmetry: the description of the system can be transformed between any of the microstates without changing the observable macrostate. The observable macrostate is thus an invariant of the symmetry transformation between microstates.

Now, to define the notion of entropy, we first need to define some more fundamental notions:

1. A set of multiple objects (e.g., a set of coins, or a set of logical statements), called a system.

2. A set of possible states for each object (e.g., heads/tails for each coin, or true/false for each statement).

3. From (1) and (2) above, we can define a set of possible states of the system (e.g., for a three-coin system the possible states are HHH, HHT, HTH, THH, THT, TTH, HTT, TTT, where H=heads and T=tails).

4. A set of possible generic states of the system, which is meant to describe a general property of the system, e.g., a generic state of the three coin system is the total number of heads, which can be 0, 1, 2, or 3.

5. A function from the possible states of the system to the set of possible generic states of the system. Note that this function ignores or erases distinctions between possible states of the system to produce generic states (e.g., The state HHH maps to 3, the states HHT, HTH, THH all map to 2, the states THT, TTH, HTT all map to 1, and the state TTT maps to 0.

Now, with all that in place, given a particular generic state, we can count the number of possible states that correspond to it, and define entropy as a quantity proportional to this number. Thus, the generic state 3 corresponds to just one state HHH, while the generic state 2 corresponds to three states HHT, HTH, THH. When this number is large, it means that the generic state discards more distinctions (higher entropy), and when this number is smaller, it means that the generic state retains more distinctions (lower entropy).

All of the above can be translated or expressed directly in the language of distinction, because it involves just sets and elements and numbers. I’ve not done so here, though, because it would make the concepts less transparent to readers unfamiliar with that notation.

Following is an example expressed in terms of the Laws of Transformation. http://www.integralscience.org/lot.html

Let’s say we have distinguished three spaces, and each space can be marked by O or    , indicating it has one of two states.

Then the possible states of the three-space system can be expressed as

{O,O,O}
{  ,O,O}
{O,  ,O}
{O,O,  }
{O,  ,  }
{  ,O,  }
{  ,  ,O}
{  ,  ,  }

To use the earlier terminology, these are microstates of the system. We then group these into generic states, as follows:

{O,O,O}

{  ,O,O}
{O,  ,O}
{O,O,  }

{O,  ,  }
{  ,O,  }
{  ,  ,O}

{  ,  ,  }

To use the earlier terminology, these are the macrostates of the system. We assign to each macrostate a number corresponding to the number of microstates in the group:

{O,O,O} -> O

{  ,O,O}
{O,  ,O} -> OOO
{O,O,  }

{O,  ,  }
{  ,O,  } -> OOO
{  ,  ,O}

{  ,  ,  } -> O

The entropy of a given macrostate (group of states) is proportional to the assigned number. The macrostates assigned O have the lowest entropy, while the macrostates assigned OOO have high entropy.

This example shows explicitly how the creation of distinctions results in a set of distinguished microstates of a system, and then through a process of ignoring some of the distinctions between the microstates, those distinctions become unobservable, and the system is seen only as having a smaller set of observable macrostates. Each macrostate corresponds to a number of microstates (and its assigned number is an invariant of a symmetry transformation between its microstates). The more microstates a given macrostate has, the more distinctions were ignored to conflate those microstates into a single macrostate, corresponding to a higher entropy state.

As mentioned earlier, the seed of time arises with the first distinction. However, this description of entropy suggests that the arrow of time (the increase in entropy that is associated with the asymmetry between past and future) is associated with ignoring distinctions that have been made. Specifically, when distinctions between microstates have been ignored in a non-uniform way, this favors the appearance of some macrostates over others, assuming that all microstates are equally probable.

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