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Originally Posted by OngBonga
As for entropy, I'd like to know something. If I have some food colouring, and put a droplet into a tub of water, we can expect to see the colouring spread out. Eventually it will be evenly distributed, it will be of equal concentration. When that point is reach, would it be possible (in theory, not practise) to use retroanalysis of the motion of the particles to determine exactly how much food colouring was added, when, from what height etc, or is all this information lost?
Classically, this is a perfectly deterministic system, all information is preserved. Theoretically, you could do exactly what you propose.
Practically, since it takes time to solve each particle's information, and you must take finite steps to approximate the next positions... The task is beyond monumental for any macroscopic system. This demonstrates the need for the statistical approach of Thermodynamics.
Note that a classical particle is a point mass. It is NOT a wave-particle. It is an idealized infinitesimally small localized mass which is NOT a black hole. Classical means without invoking any QM or GR.
If we start to think about the system in terms of QM, then we have to deal with wave functions and the uncertainties in position and momentum make this problem intractable to analyze in that context. We are moved to talk about the combined wave functions of all the particles in the system, and we call this the state function. This encapsulates all the properties of the system, insofar as they can be observed. Solving this explicitly (not an approximation) for more than a handful of particles is already getting obscenely time intensive. We are getting better and better at making computer approximations which are accurate to within experimental uncertainties.
It is the cutting edge of physics to explore this "meso-scale" region where QM rubs up against Thermodynamics.
Originally Posted by OngBonga
It seems to me that either a) eqilibrium is appraoched but never reached, or b) information is lost.
Where is the flaw in my thinking?
I don't even see how the two are related. Entropy isn't a "loss of information." It is "unusable" energy. Think of it as if something is moved by an "accidental" process to a low-energy state, then that energy becomes "unusable," since it can not be extracted by our "intentional" process.
The total entropy of any closed system can not go down. If we are very clever, we can theoretically do things which hold the amount of change in entropy to 0.
Equilibrium is EVERYWHERE. Anything which is not accelerating is experiencing a static equilibrium of forces. Anything which is going through a constant mishmash of motion that all kinda blurs together such that it kinda isn't changing over time is in a state of dynamic equilibrium.
Asymptotic behavior is common among solutions to differential equations. The equations of physics are differential equations. So it's cool that this is on your mind. It shows an intuitive understanding of the solutions to Diff Eqs. Oscillations are also common among solutions. Mix these together and you get dampened oscillations - oscillations which decrease over time and approach an asymptote.
Practically, QM steps in before infinite time has passed.
Consider the charge on a capacitor in a simple circuit with a Voltage source, a Resistor, and a Capacitor. The voltage across the capacitor will approach the value of the voltage source, but the resistor slows the process down. As the process is slowed, the charge collecting on each capacitor plate reduces the potential difference between the capacitor plate and the voltage source. So the accelerating force acting on the charges (electrons) decreases as the process occurs. This gives rise to asymptotic behavior. But QM steps in and says... hehe, I'm only going to allow you to take charges in units of 1e-, so as soon as you're at least that close to the charge required for equilibrium, then you're close enough.
Does that make sense?
Since certain quantities which describe matter come in discrete lump sizes, if you are within a lump size of equilibrium, you are there. So the "infinite time to reach the asymptote" doesn't practically apply with quantized values.
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