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I am devising a magic system that requires the user to output their energy (say, metabolismic calories) as a trigger, thus letting the sea of mana do the rest. Still, the sea of mana would multiply the energy input (user's output) to a usable level. Regarding the scale, how would I measure the ratio of a user's output to the total resultant output (user's output plus mana's augmentation) based on the value that I named "Negative Mana Resistance"?

It is easier to think of, but when it comes to the math, it shows its difficulty. The idea is that the more a user performs mana bending, the more the sea of mana (which permeates everywhere on the planet's surface, and probably the entire planet) understands their intention, and so the enhancement being done by the sea of mana increases (both in quantity and quality), hence the name "Negative Mana Resistance". As the mana resistance of an individual goes lower, it reaches a negative value, at which point the amount of energy the user spends would be enhanced rather than partially transfered to the mana manifestation (some people, most of them actually, have positive resistance; that is, the energy they output would be partially transferred to the process of mana bending; thus, the force they expend to perform mana bending would be more than the actual mana manifestation triggered, thus rendering their ability generally similar to ordinary human). How would it be translated into an equation? I assume that this "understanding by the sea of mana" must be quantized somehow, but how would it work? Perhaps with accumulated exposure period?

Note: The tag for refers to the question's requirement to propose a quantitative system of magic that could be determined should the effect of the magic be studied scientifically.

Note 2: To summarize, I need a basic framework of principles that could account for:

  • Exposure time
  • Accuracy
  • Baseline resistance
  • Tendency to reduce resistance value based on exposure time and its relationship with accuracy
  • Resultant resistance

Note 3: For answer(s) that satisfy the question (or most of it), I'll give a full credit in-universe; it would be nice to include a way to refer to the system you've invented or conceived in your answer, like: Master X, in his/her book "Whatever the book named", year XXXX; or: AnswererName's Law of Reality-Bending

NB: Please tell me if this question is off topic or if it lacks something, or anything I could do to improve it or to make it on-topic, as personally I don't know where to ask for this, as I thought it would be ridiculous to ask it on Mathematics SE.

Note 4: Just to clear up things, but in the question, it is suggested that the one having negative mana resistance will be superior in magic, as they basically could outputs more with littlest input from them, and 0 resistance actually means input that a user spent would be equal to the resultant mana manifestation.

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  • $\begingroup$ As to Notes 1 and 2, would you be amenable to answers which rely on Sanderson's First Law of Magic rather than cold equations? "The ability for an author to resolve conflict with magic is directly proportional to the reader's ability to understand it." Boiling it down to equations ensures everyone understands its limits, but it also makes it really hard to avoid loopholes which ruin your story. Its easier to have fundamental principles sans. equations and let your position as an author handle the rest. $\endgroup$ – Cort Ammon May 10 '15 at 17:35
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    $\begingroup$ If equations are required, is there a domain that you prefer the model to derive from? Systems which are intended to be explored scientifically are rarely invented out of thin air, there's usually a physical basis for them. Would electrical terminology work best for you? Social studies? Chemical Engineering? Chemistry? $\endgroup$ – Cort Ammon May 10 '15 at 17:36
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    $\begingroup$ One final question: given that energy is a conserved concept in physics, and you're doing energy like multiplication with mana, does that mean your mana is conserved? Casting clearly converts mana into more traditional energy forms, but is there anything which converts traditional energy into mana? (And mind you, if you're not careful, thermodynamics will complain if you convert waste energy into mana without careful consideration) $\endgroup$ – Cort Ammon May 10 '15 at 17:41
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    $\begingroup$ One consequence of "continued use of mana makes subsequent use easier" is that magic use can easily become less and less controllable, leading to a runaway thaumaturgic reaction, resulting the in the magical equivalent of a nuclear bomb. Very hard on both the magician and those around him/her. $\endgroup$ – WhatRoughBeast May 11 '15 at 5:01
  • $\begingroup$ @CortAmmon ehm, that's a collaborative project, and the system is explained to me (by my friend) vaguely. He largely appreciates help to systemize the system he imagined. For the story, I believe there's nothing that could be ruined, as he had mentioned that in his world, not everything could be solved by "a wish that could wipe reality etc"., and there has to be a limit $\endgroup$ – Hendrik Lie May 11 '15 at 5:12
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This answer is based on the world described in my other answer. This one is focused on the portion of the interaction between the sea of mana and the individuals that science will quantify and model quickly. It leaves open the idea that there are other ways to interact with the sea, and only concerns itself with what will get discovered quickly by science.

Nowhere in the description of the equations you are looking for do we talk about things like "difficulty of spells" and whatnot. All of the variables focus on the state of the individual. This is actually a very natural result of a scientific approach. Science will choose to isolate the "physical world" from the individual caster because science likes to divide problems up that way. It will assume the physical world's effects are independent from the caster's effects, so you can come up with any model you please for how people make fireballs or do telekenetics or what not, and then layer on this model for handling the individual "negative mana resistance."

Since this is all about the individual, it is natural to look to psychology for inspiration for the equations. I would easily consider spellcasting a "skill," and as it turns out, there is a very commonly occurring curve in psychology/education/etc. called the logistic function: $f(x)=\frac{L}{1+e^{-k(x-x_0)}}$

One method of measuring "skill" involves assuming that each individual has some "skill value" which scores them on a number line. 0.0 would be the least skilled score someone can achieve, 1.0 is the best, with 0.5 being an average skill. This approach has been studied in great detail in the field of education, and we find that, if you try to order individuals from 0.0 to 1.0 evenly, and then look at the quality of the results of applying a skill, people have a curious tendency to fit on a logicist curve. It just seems to happen.

The layman's version is this:

  • At low skill levels, you really don't understand the material. However, there is some chance of getting the answer right with random luck (ex: on a multiple choice test with A, B, C, D, and E, you have 20% random chance of getting an answer right, even if you don't know a thing about the topic).
  • At some point, you reach a point where the material starts to fit into your mental model of how the world works. There is a rapid rise in effectiveness as you progress along this curve.
  • Eventually you master the topic, at which point additional "skill" doesn't really help, The topic is under your belt.

There are, of course, skills where this is not the case. However, from an educational background, science tends not to try to model those cases. They're just not a good fit for a scientific process. Thus we expect science to find examples where this curve "fits."

So let's add one more term: $f(x)=\frac{L}{1+e^{-k(x-x_0)}} + C$. Now we can map these real life patterns into the equation.

  • $C$ is the "random chance" term. It's how well someone does when they are completely untrained.
  • $L$ is the "maximum effect of skill." The higher this is, the more difference we see between the unskilled and skilled values.
  • $x_0$ is the "inflection point." This is the percentile of the individual who is right in the middle of the process of getting it. It naturally will be a value between 0.0 and 1.0, but it usually is closer to the middle (just to make sure we see a good portion of the whole curve). It is a measure of how hard it is to learn a skill in the sense that a higher value means you have a long period of learning before you finally "get it" and can achieve mastery.
  • $k$ is the "steepness" of the curve. High values of k lead to situations where there is a sudden "dawning understanding" which gives you mastery in one sudden motion. Low values of k lead to situations where you have to work at it for a long time, steadily improving.

Here's the list of things you want to appear in the equation:

  • Exposure time
  • Accuracy
  • Baseline resistance
  • Tendency to reduce resistance value based on exposure time and its relationship with accuracy
  • Resultant resistance (this seems to be an output)

We have 4 variables and 4 things we want to measure. JOY!

  • $C$ - related to baseline resistance. There is some baseline ability for casting, and it is represented by $C$.
  • $L$ - tendency for reduced resistance. As you learn more, you move up in the "skill" ranking, and the effect of that is more pronounced if $L$ is large.
  • $x_0$ - Accuracy. The more precise the effect you are looking for, the higher the "skill" where the transition from untrained to trained occurs.
  • $k$ - Exposure Time. Like many things which you can work at until you get it right, the longer your exposure time, the smoother the curve will be ($k$ goes down as exposure time goes up). Just to avoid this weird signage flip, science would probably refer to "exposure rate," which would be the reciprocal of exposure time. That way, $k$ goes up as exposure rate goes up.

One detail: you want to think of things in terms of "negative mana resistance," which is not a typical way of approaching psychology problems. However, if that is the form you want it to be in, it would be trivial to assign $C < 0$ such that an untrained individual spends more trying to channel mana than they get back, but when you cross the x-axis, you start getting more effect from the mana than you spent channeling.

Now different types of spells could be harder or easier to manage. Science would bundle them up into categories, and try to define a logistic curve for each. If you chose a Magic: the Gathering style of spellcasting system, each of the 5 colors (white, black, blue, red, green) would have a different logistic curve, and each person would have a skill value assigned for each of the 5 skills.

When it comes to more complicated skills, this curve would break down because it would become too difficult for science to objectively define the behaviors it wants to see. Accordingly, science would stick to measuring very brute force approaches (i.e. "a solid wall of red-magic force"). The magic equivalent of martial arts may identify that there's more to the problem than that (i.e. they start to uncover the true interactions between the sea of mana and themselves). Each art may define its own approach to quantifying these effects. I guarantee you that the Chinese martial arts will associate this sort of magic with Chi in no time flat, and begin doing really interesting things with it which science initially will claim are impossible (but having to admit that someone might be doing it anyway). Over time, science may refine their laws. The previous answer I gave is completely consistent with modeling the sea of mana as a 3-d array of non-linear elements a. la. computational fluid dynamics. However, the equations you get out of that will be beyond the scope of most readers (read: researchers pay hundreds of thousands of dollars to get software which has thought through those things for it). The equations from the logistic curve should be enough for science.

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  • $\begingroup$ I pick "negative mana resistance" because the original idea of "resistance" of electricity. Ideally, at resistance value of 1, you wouldn't be able to interact with mana at all. But as the resistance goes lower, you'll then be able to interact with mana. The term negative mana resistance refers to when one's resistance went below 0 (at which one could interact with mana with 100% energy efficiency), where at negative value, its energy efficiency will be more than 100% (which you had explained it on your other answer on how it could be possible). 1+ for providing another insight of the possibil $\endgroup$ – Hendrik Lie May 14 '15 at 16:50
  • $\begingroup$ Just a thoughtful comment, as I recently looked back at this: I can't understand the difference between exposure time and exposure rate. Doesn't high exposure time equals to high exposure rate? Please enlighten me :) $\endgroup$ – Hendrik Lie Feb 26 '18 at 17:17
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    $\begingroup$ @HendrikLie If I have an event which happens at a rate of 2 times per minute, then the event occurs once every 1/2 minute. Times are typically measured in seconds or minutes, while rates are measured in 1/seconds or 1/minutes. $\endgroup$ – Cort Ammon Feb 26 '18 at 19:32
  • $\begingroup$ thank you. Presently, your answer seems to work perfectly with my current setting. Accepting it. $\endgroup$ – Hendrik Lie Mar 18 '18 at 19:54
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$\dfrac{E}{\Delta t} = \dfrac{1}{(m_r)^2}m_t$

$E$ = energy output

$m_r$ = mana resistance, where 0 is the lowest (the equation is undefined at that point because with 0 mana resistance you can do anything): The higher the mana resistance the less the sea of mana helps you.

$m_t$ = the strength of your mana trigger

$\Delta t$ = change in time: this is so that releasing energy in a short burst is harder than a releasing energy over a long period of time (think explosion vs burning a log)

Therefore, a normal person has a mana resistance of infinity. A mage has a finite mana resistance.

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  • $\begingroup$ Ehm, to note, mana resistance of 0 means his energy trigger and the amount of energy sea of mana outputs is the same, negative value is allowed as "person with higher NEGATIVE mana resistance is favored to do the magic", but that was a good try. Also change in time means someone's aptitude, as he trains, his Negative Mana Resistance goes higher (negative suffix means that its mana resistance goes lower to negative value) $\endgroup$ – Hendrik Lie May 11 '15 at 5:47
  • $\begingroup$ After considering the answer again, your answer is quite good that it could be an alternative measurement technique, so I'll upvote for that $\endgroup$ – Hendrik Lie May 11 '15 at 7:50
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My method of choice

I apologize if I go through this overly slowly, as I'm trying to work it out for myself as I go along.

Basic model

A person can give an output (an input into the "sea of mana") of energy $I$. The sea then acts as an energy trransfer, and gives an output of energy $O$.

Based on what you said, the more energy a user puts in, the more energy comes out. This means that $O$ is proportional to a power of $I$, or, in mathematical notation, $$O \propto I^n$$ where $n$ is the exponent. It doesn't have to be an integer, though. However, it seems that $O>I$, so $n>1$. You can choose this $n$ and have it work for everyone in the population.

Now we can build in resistance. Let's add in a coefficient, $\alpha$, making the law $$O \propto I^{\alpha n}$$ Less resistance means greater values of $\alpha$, and vice versa. Now, for $O>I$, $\alpha n>1$.

I said earlier that $\alpha n{\color{red}>}1$. This is because $I$ is enhanced by the "sea of mana". However, you said that for negative resistance, the opposite is true: $I<O$. For this to be the case, $\alpha n {\color{red}<}1$.

Here's some behavior of the equation: $$\text{As }\lim_{\alpha n \to 0}, \frac{O}{I} \to \infty$$ $$\text{As }\lim_{\alpha n \to \infty}, \frac{O}{I} \to 0$$

Putting in a constant, $c$, to satisfy the $\propto$, we have

$$O=cI^{\alpha n} \tag{1}$$

Baseline resistance

We have to change our exponential term to account for this, so we add a constant, $\beta$, making the equation

$$O=cI^{\alpha n + \beta} \tag{2}$$

Slight detour

This gives us another variation. Say we keep $\beta$ as our baseline. What if we say $\alpha n=-\gamma$, and that if $| \gamma |>\beta$, the resistance is negative? That works. It might be simpler.

Summary

I haven't actually used a negative sign anywhere, except in the detour. Notice, though, that you don't have to. Exponents make things much more interesting, eh?

Other ideas

  • Have resistance be proportional to a constant, $\delta$, squared ($\text{Resistance} \propto \delta^2$), but for those with negative resistance, make this $i \delta$ (as $i^2=-1$).
  • Have resistance be expressed as $$R=\text{baseline}+\text{specific term}$$ and have the specific term be negative for those with negative resistance.
  • Have resistance be proportional to $\epsilon^{\upsilon}$, where $\epsilon$, $\upsilon$, or both can vary by user. If only $\epsilon$ varies, then positive/negative resistance depends on whether or not $\epsilon>1$. If only $\upsilon$ varies, then positive negative resistance depends on whether or not $\upsilon>0$.
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  • $\begingroup$ What's that N on O>N? Is it related to the following n? $\endgroup$ – Hendrik Lie May 11 '15 at 7:10
  • $\begingroup$ Ehm, from what I could comprehend, then we could tweak the value "a" slightly over time (say, it doesn't have to be static) based on exposure rate (slightly increased, the incremental value perhaps could be denoted in percentage? And those incremental value depends on individual basis?), isn't it? $\endgroup$ – Hendrik Lie May 11 '15 at 7:17
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    $\begingroup$ @HendrikLie Whoops, typo. $N$ should be $I$. Yes, tweaking is possible. $\endgroup$ – HDE 226868 May 11 '15 at 14:56
  • $\begingroup$ If you could provide correlation of exposure time (say, training, actual usage) to your answer, perhaps it could fit the question more closely :) $\endgroup$ – Hendrik Lie May 12 '15 at 14:25
  • $\begingroup$ Oh correct me, but based on my understanding, if one's mana resistance (say, an+$/beta$) is in negative value, O will be larger than it is if one's resistance is positive? $\endgroup$ – Hendrik Lie May 12 '15 at 14:41
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Though my friend's original conception would be that the energy is recycled by souls of the dead (I object it and say that it draws energy from ambient energy of the surrounding, say the sun, or geothermal, or vacuum energy) - Hekdrik Lie

What if you could have both? What if it could be recycled by the souls of the dead, and simultaneously be drawing energy from the surroundings? It turns out combining the two is not as unreasonable as they may first appear, but we need to change the focus slightly. Your question focused on energy, but I would recommend focusing on information instead. The two are closely related, but the bridges you may want to cross are easier crossed with information theory rather than energy balances.

Consider which is more powerful, access to 4MJ of energy (the average energy in a daily intake of food), or the knowledge of where to find a bulldozer with a full tank of gas. Clearly a being with such knowledge could do spectacular feats, especially if nobody else could see the bulldozer for one reason or another.

Maxwell's Daemon is one way of looking at the connection between information and energy, and it is an important one to the world of physics. Basically, the argument is that, once a particle reaches a uniform distribution in a container with two rooms, there is no way to shut a door between them to trap the particle in one chosen room more than 50% of the time. This is considered an absolute rule by current physics. QM bends it, but even QM doesn't break this rule.

But there's a key assumption to the Daemon: the assumption that the particle is randomly distributed at the start of the experiment. What if it wasn't? What if there was structure to the way the particle bounces, which we had simply forgotten. When we go measure it, we see it as a random distribution. But what if the Daemon knew better? What if the Daemon could predict the position of the particle at some time and simply wait for it to get there.

In theory the Daemon could create any steam engine it pleased with this approach. It just may take a lot of dedication. With oracle like knowledge (knowledge not explainable by physics), it could even generate infinite energy. But we're not looking at that extreme, we're looking for something physically realizable.

Landuaer's principle gives us some physically realizable ways to do this. It states that the minimum amount of energy needed to erase one bit of information is kT ln(2) (technicality: The Daemon is erasing one bit of entropy in the universe. This can also be thought of in layman's terms as adding one bit of order to the universe). The hotter you get, the harder it is to erase a bit. In theory, computation done at absolute zero could calculate anything for free, but in reality, there's no degrees of freedom at absolute zero to do calculation with, so we have to just settle for really-really-cold.

A note here: this limit is really really low. At room temperature, it takes a mere 2.85pJ of energy to erase 1 bit of entropy. Our modern computers are a million times less efficient

So what if we could do our calculations in the ultra-cold of space, we could work at 28K instead of 280K (which is room temperature). That gives us an even lower bound of 0.285pJ/bit. That's not much.

Now before we return from Wikipedia and start really tackling the problem, its worth noting that reversible processes can exceed this limit, because reversible processes are theoretically not bound by thermodynamics the same way irreversible processes are. This is the basis for Quantum Computing, but there is, as always a limit. One has to get the data out of the quantum computer, which involves erasing bits. However, you only have to erase enough bits to get the answer out, which is often many fewer bits than you would have needed to solve the problem classically.

And this is where your computronium comes into play: it can do things like this. It can run reversible calculations to emit only the minimum output. In fact, it can emit output undetectably to science, simply by concentrating its classical outputs in locations science is not looking. As long as it outwits the science, it can stay hidden (and note, the shift to terminology describing intelligence is no mistake).

The question is how does it get this information, to avoid being detected. It literally needs to find bits of entropy that are "forgotten," i.e. unusable by science, capture those bits, and ensure science never sees them again. It can't do this with things like solar energy directly - it doesn't have much better of a chance at predicting information in solar photons than science does. However, there is a corner where science doesn't look: death.

Science really has no idea what happens at death. We've got some ideas, but generally speaking, lots of information in the body decays faster than we can even consider looking at it. Thus it is not unreasonable to assume that there is some information in or near the soul that escapes science's net in the form of information that is too ephemeral to really pin down with statistics. This information could be captured by the computronium without wasting too much energy. This information could be used to better predict when and where the scientists are looking, keeping the entity hidden.

This entity could then use this information at its maximum effect. For example, if a fireball was desired, it may be able to summon up an information net across the entire world to create a fireball in one location counterbalanced by a millionth of a degree decrease in temperature around the world. It could then take advantage of the fact that it's the only one who knows that millionth of a degree shift occurred to harness the suns energy across the globe to pay the energy price for the fireball. It simply has to know enough about how the world world works to do that process "reversibly."

And that is the limit of this creature that you need to prevent limitless power. It can only exert its infinite will on small things. The larger a task, the more likely it is that the imperfection in how it does reversible computation will crystallize unexpectedly, like whitecaps on an otherwise energy-conserving sea. The entity has to be careful: what are whitecaps to it could be catastrophically powerful waves of force for the denizens of earth.

As for your negative mana resistance? Now its easy because, instead of being a very strict physical law you look for, it's more of a social law. The entity will exert its power where it is most efficient. If you are willing to look the other way, it can do great things, simply because it can get away with doing them affordable. In fact, if you develop enough of a report with the entity, sharing a common language, it may find it in its interests to actually give you energy. It'd be a win-win. You get energy, and it gets a voice which will effectively evangelize the best ways to not look at it, so it can continue to grow.

There are also really interesting religious subtexts here as well. The most obvious is the Taoists, who seek to become Immortal by becoming one with the chaos of the cosmos. Perhaps the Taoists were one such group which attained favor with the entity, and were given protection in return for cultivating a life which brought more information to the entity.

At this point, the physics the scientist sees can be anything you please. The entity would present itself in a form the scientist would understand. The scientist would understand laws of nature, so it would present as having rules. However, as science tries to tap this "limitless" power, the entity would have to shift to avoid loopholes, slowly teaching the scientists how to live in harmony with it.

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  • $\begingroup$ Interesting answer., it provides me with a new insight of how it harness seemingly limitless energy whereas it's not limitless.., considering nobody uses high-yield magic at the same time with everybody else in the planet at once, it makes sense. Basically saying the computronium could exploits energy expenses at the whole planet and concentrate it just because it could "manipulate" informations within it with greater accuracy than what we could? $\endgroup$ – Hendrik Lie May 12 '15 at 14:21
  • $\begingroup$ Also, how could I refer to you in-universe? Is Sir Cort Ammon is sufficient? $\endgroup$ – Hendrik Lie May 12 '15 at 14:30
  • $\begingroup$ @HendrikLie That's the idea. If you look at what actually has to be done, its remarkably similar to what is required by the best scientific definitions of "life" that we have. It's just bigger and has the capability to be more subtle if desired. One could easily make our efforts to understand it be no more than that of a housecat trying to comprehend the construction of the subdivision being developed around its house, or you could make humanity learn and adapt so quickly that the computronium must literally fight for its way of life. It just depends on the story you wish to tell. $\endgroup$ – Cort Ammon May 12 '15 at 15:43
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    $\begingroup$ And I must admit, I giggled at the idea of there being a law named after me! $\endgroup$ – Cort Ammon May 12 '15 at 15:43
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    $\begingroup$ At some point in the next few days, I'll try to put together an equation which could come out of this exploration and post a second answer. There's a few equations I can think of which would tend to lend themselves to a world like this, I just wanted a post which made it clear just how much freedom you can have with the equation and still get away with a vibrant world. $\endgroup$ – Cort Ammon May 12 '15 at 16:14

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