entropy
i knew when i started this blog that i would eventually have to discuss the meaning of entropy and not in a semantic sense. be forewarned, this is about to get very technical. usually, i prefer to discuss these concepts in an abstract sense with the mathematics assuming an underyling foundation as opposed to explicitly introducing them-- it just gets in the way for the reader. so, here we go...
Foundational model for the universal environment: The proposed system for the universe rests on subsets of infinite Hilbert spaces-- a mathematical concept for finite sub-systems whereupon every Cauchy sequence [general distribution of probability density] converges to a singular element (be it a point, plane, etc.). Essentially, this establishes that entropy does not exist within the subsystem (open, not closed or isolated since this does not exist in reality) because the norm of their collective differences would eventually approach a limiting factor (usually 0). Within the realm of quantum mechanics, these spaces are mathematically assumed to have measurable states that can be ascribed to a unit vector (in reality, the Heisenberg Uncertainty Principle would negate such a possibility furthering entropy by merely measuring the system). From a real world perspective, a Hilbert space could never exist; mathematically in this context of infinite dimensions (ie- the universe), it will always be isomorphic and will always have an orthonormal basis according to Zorn's Lemma (since it defines a maximal bound and a space is separable IFF it has a countable sequence satisfying the condition which it will). So, a better model might be to use a Banach space which defines infinite-dimensional spaces that contain functions since this is essentially what a macro-model of infinitely separable Hilbert spaces creates. However, either one will suit our needs.
The principles: The next two theories will expound upon our foundational environment and describe the behaviors at play. Perturbation Theory defines a set of approximations that will describe a complicated quantum system relative to a simpler system (ie- the reason we had to define our sub-systems in the preceding paragraph). The second principle is the Schrödinger equation [link provides an excellent graphic of the equation with explanations]
which establishes the time dependence of our quantum mechanical system. Remember that measurable state I mentioned? Well, let's treat the measurement of that state as a vector (could be many different things here: time, momentum, direction-- it's academic since we're examining entropy and any part of a vector can contribute to the overall perturbation of the system). I recognize the implicit limitations of the Perturbation theory within this state since it can only describe idealized states (hence our focus on the sub-systems). The evolution of the theory in this direction would lead us to the realization that our model, if it were adiabatically derived in keeping with the Second Law of Thermodynamics, then all of isolated sub-systems would interact with each other contributing to the overall entropy of the system (the universe). Why? Well, at first glance, two neighboring sub-systems would naturally try to achieve equilibrium. We can intuitively derive that conclusion, but doesn't that contradict entropy? Yes, within an isolated sub-system this could be true, but the overall system's entropy has now increased. This is the Third Part of the Second Law. Unfortunately, it can only apply to isolated systems. It would break down anyway because recent scientific measurements have indicated that the universe is still creating matter [anybody who can find an article that shows this-- it would be appreciated, i read this several years ago, but can't find anything now].
Now to tangible life. Human beings (life) can be thought of as one of these sub-systems and we create many wonderful ordered things-- appliances, buildings, infrastructure, etc. Life in itself must have an innate ability to counteract entropy or else it could not exist (at least not in a way that we can currently comprehend). How would our bodies maintain themselves if our internals could not guarantee their coordination with each other within set physical parameters? As time moves on, we will procreate and establish increased order over exponentially increasing sub-systems (our progeny, inventions, etc.); ironically, this will exponentially contribute to entropy within the overall system. This is because to create order (establish an equilibrium of systems as described in the previous paragraph) we must convert energy to support this new construct. As you know, there is no 100% efficient way to convert energy; most often, this displaced energy is "lost" as heat, hence contributing to the thermodynamic entropy of the overall system. Life itself creates entropy. H(t) in the Schrödinger equation is the total energy of the system which means that all of the potential [break].
Sorry, this is going to get cut short. Please provide comments; I admit that there may be some gaps in logic and I glossed over some other parts, but I have been harangued by my family for the past hour making it a little difficult to concentrate and ultimately forcing me to cut this short. TBC...
Foundational model for the universal environment: The proposed system for the universe rests on subsets of infinite Hilbert spaces-- a mathematical concept for finite sub-systems whereupon every Cauchy sequence [general distribution of probability density] converges to a singular element (be it a point, plane, etc.). Essentially, this establishes that entropy does not exist within the subsystem (open, not closed or isolated since this does not exist in reality) because the norm of their collective differences would eventually approach a limiting factor (usually 0). Within the realm of quantum mechanics, these spaces are mathematically assumed to have measurable states that can be ascribed to a unit vector (in reality, the Heisenberg Uncertainty Principle would negate such a possibility furthering entropy by merely measuring the system). From a real world perspective, a Hilbert space could never exist; mathematically in this context of infinite dimensions (ie- the universe), it will always be isomorphic and will always have an orthonormal basis according to Zorn's Lemma (since it defines a maximal bound and a space is separable IFF it has a countable sequence satisfying the condition which it will). So, a better model might be to use a Banach space which defines infinite-dimensional spaces that contain functions since this is essentially what a macro-model of infinitely separable Hilbert spaces creates. However, either one will suit our needs.
The principles: The next two theories will expound upon our foundational environment and describe the behaviors at play. Perturbation Theory defines a set of approximations that will describe a complicated quantum system relative to a simpler system (ie- the reason we had to define our sub-systems in the preceding paragraph). The second principle is the Schrödinger equation [link provides an excellent graphic of the equation with explanations]
which establishes the time dependence of our quantum mechanical system. Remember that measurable state I mentioned? Well, let's treat the measurement of that state as a vector (could be many different things here: time, momentum, direction-- it's academic since we're examining entropy and any part of a vector can contribute to the overall perturbation of the system). I recognize the implicit limitations of the Perturbation theory within this state since it can only describe idealized states (hence our focus on the sub-systems). The evolution of the theory in this direction would lead us to the realization that our model, if it were adiabatically derived in keeping with the Second Law of Thermodynamics, then all of isolated sub-systems would interact with each other contributing to the overall entropy of the system (the universe). Why? Well, at first glance, two neighboring sub-systems would naturally try to achieve equilibrium. We can intuitively derive that conclusion, but doesn't that contradict entropy? Yes, within an isolated sub-system this could be true, but the overall system's entropy has now increased. This is the Third Part of the Second Law. Unfortunately, it can only apply to isolated systems. It would break down anyway because recent scientific measurements have indicated that the universe is still creating matter [anybody who can find an article that shows this-- it would be appreciated, i read this several years ago, but can't find anything now].Now to tangible life. Human beings (life) can be thought of as one of these sub-systems and we create many wonderful ordered things-- appliances, buildings, infrastructure, etc. Life in itself must have an innate ability to counteract entropy or else it could not exist (at least not in a way that we can currently comprehend). How would our bodies maintain themselves if our internals could not guarantee their coordination with each other within set physical parameters? As time moves on, we will procreate and establish increased order over exponentially increasing sub-systems (our progeny, inventions, etc.); ironically, this will exponentially contribute to entropy within the overall system. This is because to create order (establish an equilibrium of systems as described in the previous paragraph) we must convert energy to support this new construct. As you know, there is no 100% efficient way to convert energy; most often, this displaced energy is "lost" as heat, hence contributing to the thermodynamic entropy of the overall system. Life itself creates entropy. H(t) in the Schrödinger equation is the total energy of the system which means that all of the potential [break].
Sorry, this is going to get cut short. Please provide comments; I admit that there may be some gaps in logic and I glossed over some other parts, but I have been harangued by my family for the past hour making it a little difficult to concentrate and ultimately forcing me to cut this short. TBC...
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