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The discrete-time physics hiding inside our continuous-time world (phys.org)
106 points by dnetesn 8 days ago | hide | past | web | 85 comments | favorite





This immediately makes me think of Sabine Hossenfelder's book "Lost in Math".

Just because the math is pretty doesn't mean the universe actually works that way. Maybe it does, maybe it doesn't. Sure seems like the universe doesn't seem to care much about our math.

I think the authors of the paper are being perfectly straightforward (e.g., they explicitly include the phrase "in a certain very limited sense" RE: time proceeding in discrete time steps), but the title of the article itself feels a bit misleading.


Good luck telling in advance which math is pointless guessing and which math is the groundwork for the next revolution in our understanding.

Which neither means that beautiful math is necessarily describing the world inaccurately.

Taken to the extreme, the best description of the world is a long list of all events and facts. No beauty in that. Now science tries to compress this list of events to a simple model that explains it best. Like in statistics, one should aim for the simplest model predicting most of the data.

I'd argue that if you think of science as an attempt to compress all the data in the world, then OP's statement is obvious, and at the same time there's good reason to value beautiful (i.e. short & simple) explanations higher provided they generalize well.


You say "best", except a long list of events and facts has no predictive power, no way to infer what the future will be, only describes what the present is. The whole point of a model is to be able to predict the future. So in that sense, a long list of all events and facts is the worst description.

Not sure why you’re being downvoted; what you’re saying is true. A model whose length is the same as the data it is describing has no predictive power. A perfect model what be the size of the minimum program that prints out all of the events in the universe (that is, the program whose size matches the Kolmogorov complexity of the universe).

> Sure seems like the universe doesn't seem to care much about our math.

Well, some mathematical truths are so universal even the universe can't escape them. the incompleteness theorems [0] come to mind.

Regarding the article, I think most physicists consider the continuous models they use to attack the problems at hand a hack to make the mathematics consumable. I don't think they consider the universe itself to be non-discrete.

[0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...


I am a physics in academia and no, most physicists do not think that (especially given that most physicists are not quantum foundations physicists, and even then). It is an interesting question, and if I am forced to guess on its answer I would even agree with you, but good science would be to not speculate the way you imply.

Perhaps those who expect quantum computers to scale to millions of qubits do. The amplitudes involved seem to defy discreteness...

There are workarounds for that: for instance, do not use such a poorly conditioned basis to describe the system. In particular, problems in physics that appear only in one particular basis are probably human-created notation issues, not something fundamental, as physics should be basis independent.

Could you give examples on how the incompleteness theorem explains physical phenomena? I would be curious.

I would suggest Aaronson's "NP-complete problems and physical reality" for this type of thinking (public PDF on his academic page). It is short and has a ton of examples. I really like the style, but it is not a common view among working physicists (yet? It is just not particularly practical for the moment).

> Just because the math is pretty doesn't mean the universe actually works that way.

Interesting thought ... is mathematics really orthogonal to the universe or does it "result from" or fall out as a result of the universe within which it was described?

Getting philosophical, I know, but if you follow, then perhaps the degree to which you don't have to twist mathematics to describe the universe might suggest you are in fact on the right path....


Are we not looking at a mathematical result that constraints what the universe "can do"?

Agreed. Easy and intuitive as Euclidean geometry it is not right. It is not easy ask anyone doing proof in that paradigm.

Getting more mtr complicate not meaning they are right.


Reminds me of a haiku i wrote a long while ago:

—-

Movement and measure

How to discern life through time

Everything is change

—-

Time and the inherent math of the universe is amazing to think upon, especially when you look back upon the wisdom of ancient humans who were able to perceive cycles of the stars that were many thousands of years long whilst human conscious life is so fleetingly short.

Its interesting that we do not try to ask HOW civilizations past could have been aware of cycles that last for tens of thousands of years.

And now humans are so myopic on what is in front of them, we hardly look up.


> Its interesting that we do not try to ask HOW civilizations past could have been aware of cycles that last for tens of thousands of years.

That’s probably because they weren’t. What “cycles of the stars that were many thousands of years long“ were ancient peoples aware of?


Precession of the equinoxes? For Earth, this is circa 25,000 years, although I don't believe there's a reasonably accurate measurement of this value in Antiquity.

I'm not sure that meets the criteria for a "cycle of the stars" since it's a planetary change, but I guess for Hipparchus it probably did look like a celestial change. It definitely doesn't match the claim that "we do not try to ask HOW civilizations past could have been aware", though. We know exactly how Hipparchus recognized this. Nonetheless, this is a reasonable response to my question.

Noting a long-term cycle of equinox precession is impressive but also not terribly surprising. If you can note the change, you can reasonably assume it's cyclical (because the alternative is that it's going to suddenly stop moving at some point in the future).


The Mayan Long Count calendar? If I remember right, it's method of counting leap years is more accurate than the modern Gregorian calendar.

The Mayan calendrical system uses a "year" of 360 days without attempting to account for leap days. Essentially, the long count (the classic 13.0.5.7.6 style system) is a base-20 count of the number of days from the epoch, with one of the digits being base 18 instead. The other components are the 260-day year of 20 "months" of 13 days, the 365-day year of 18 months of 20 days (with 5 extracalendrical days), (and together these two things create a 52-year cycle that's the "Short Count" calendar), a week-ish system (of 9 days), a lunar cycle count, a correction to the lunar cycle because it's annoyingly 29.5 days long, and a 819-day cycle we don't understand yet.

Their calendrical system did not attempt to correct for leap years, but we do know that they knew the length of the year to roughly the same precision as the Gregorian approximation.


>Scientists believe that time is continuous, not discrete—roughly speaking, they believe that it does not progress in "chunks," but rather "flows," smoothly and continuously.

Is this true? I didn't think there was any kind of consensus on this topic. I also thought some scientists believe the Planck time(/length) is the smallest unit of possible time i.e. they would say time is discrete


Just because Planck times are the smallest possible measurement of time intervals doesn't mean that time progresses in discrete Planck time chunks. That's obvious enough, but maybe seems unmeasurable ... and so what difference does it really make?

A bigger way to think about it is ... is the a universe like a movie where each still image lasts a Planck-time (or some other smaller chunk)? Is the entire state of the universe encoded in stasis for that instant, and then boom ... it's time for the next "Planck frame".

It's really hard to resolve that kind of universal time chunkiness with the ordinary time dilation effects that can be observed at relativistic energies. Whose frame of reference counts? We do know that space-time can be warped ... it's harder to imagine that this kind of warping and curving is discrete. That's one reason why many believe it's a smooth flow.


Yeah, but not just unmeasurable by like a guy with a measuring device, unmeasurable in the sense that it _can't effect anything at all_, right? (if it caused an effect of any sort you could eventually measure or observe in any way... nope).

To say that time is "really" continuous under such circumstances seems like a weird philosophical claim.


Thank you, you eloquently expressed the point I was trying to make. To say "we know that quantity Y can never be measurable or observable below discrete length X... but also, Y is continuous and can be subdivided below X, just trust us on this one" doesn't sound like science. Philosophical at best.

Philosophy is not just saying random stuff and "trust us".

And inversely, science is not just measuring -- that's the naive positivist (and later Popperian) view of what scientists do.

For example:

As Thomas Kuhn (1961) argues, scientific theories are usually accepted long before quantitative methods for testing them become available. The reliability of newly introduced measurement methods is typically tested against the predictions of the theory rather than the other way around. In Kuhn’s words, “The road from scientific law to scientific measurement can rarely be traveled in the reverse direction” (1961: 189). For example, Dalton’s Law, which states that the weights of elements in a chemical compound are related to each other in whole-number proportions, initially conflicted with some of the best known measurements of such proportions. It is only by assuming Dalton’s Law that subsequent experimental chemists were able to correct and improve their measurement techniques (1961: 173).

https://plato.stanford.edu/entries/measurement-science/#EpiM...


All good points. It's not simple or mechanical, and it's often erroneous, but the realm of science is still the realm of that which can be observed and measured, and refuted or confirmed by observation and measurement. Trying to figure out how to do that is what makes it science.

It's possible the Uncertainty Principle is wrong. One can come up with theories that it is wrong. And what sorts of observations might hypothetically observe or refute them. As far as I know, there is no serious science doubting HUP at present. (Which doesn't prove it's right, it's just the current state of things).

But simply speculating about what things might exist that current scientific understanding says scientific laws would prevent us from observing or measuring or seeing any effects of whatsoever -- is still not science. It's not that you can't have a scientific theory which current measurements seem to refute and still press for it (your Dalton's law example). It's that you can't have a scientific theory with no even theoretical way to refute or confirm it by observation. That's not a scientific theory.


>Philosophy is not just saying random stuff and "trust us".

Ok, enlighten me. I've read philosophical works where that certainly seemed to be the case. There's also this weird thing where only a specific lineage of schools of thought and certain ancient Western philosophers are "real philosophy" and others (e.g. anyone with a non-Anglo last name) are "not real philosophers". No True Scotsman at its finest.

>And inversely, science is not just measuring -- that's the naive positivist (and later Popperian) view of what scientists do.

In graduate school, I worked at a lab in Stanford (I didn't attend there, but was fortunate to have spent some time there). Your statement would certainly be news to the PI I worked for, who was a heavy Popperian. You can't falsify the existence of sub-Planck lengths or times, that's for sure. Science was absolutely nothing more than that to him. And while you can certainly claim an Appeal to Authority fallacy, Stanford, at least, considered my PI competent enough to award him a PhD in a scientific discipline, so I'm not sure how much I can agree with your post.


>To say that time is "really" continuous under such circumstances seems like a weird philosophical claim.

Actually what you say is the weird philosophical claim.

It's like saying Australia didn't really exist when we couldn't observe it (or see any effects from it reach us).

The thing is, to say it could be continuous even if that's not observable might be wrong. But not because it can't be both continuous and not observable: just because it can also be discreet and not observable, and we can't tell which of two it is for sure.

Whether we (or any observer imaginable) can observe effects or not, doesn't mean it's not one or the other.


>It's like saying Australia didn't really exist when we couldn't observe it (or see any effects from it reach us).

This is a straw man, and I think you know that. It's more like saying Atlantis doesn't exist, and, well, to the best of our knowledge - it doesn't. I'd be happy to re-evaluate upon receiving evidence that Atlantis may have existed, just as QM physicists should be happy to re-evaluate when sub-Planck-scale phenomena can possibly be observed using a theory which is also consistent with the rest of our observations of the world.

Oh and an even better analogy than Atlantis - it's like someone saying Australia didn't really exist when we couldn't observe it, and claiming they would prove it by sailing from Chile to Africa without encountering any large land masses.

So no, he's not making any weird philosophical claim at all.


>This is a straw man, and I think you know that. It's more like saying Atlantis doesn't exist, and, well, to the best of our knowledge - it doesn't. I'd be happy to re-evaluate upon receiving evidence that Atlantis may have existed

I don't see much difference. If you're "happy to re-evaluate" (e.g. open to the possibility) then the answer (even re: Atlantis") is "I don't know if it exists", not "doesn't exist".


Fortunately it's not a philosophical claim - after all even if it was demonstrated experimentally that time moved in frames like a movie, it could still be a movie played on a projector whose reels spun in continuous time. There's no way to know "what it ultimately is," only the most ultimate thing seen so far.

Talking about what might exist that the laws of nature as we currently understand them would prevent us from ever having any evidence whatsoever of... it's def not a scientific claim or inquiry, sounds philosophical to me. If it can't be observed, like it's even theoretically impossible to observe it or have evidence of it, ever... it's not science.

> If it can't be observed, like it's even theoretically impossible to observe it or have evidence of it, ever... it's not science.

More than that. It simply doesn't exist in any meaningful sense and it's a waste of time to even think about it.

What cannot be observed in principle has exactly zero impact on the universe. That's pretty much a definiton of "not existing".


>Just because Planck times are the smallest possible measurement of time intervals doesn't mean that time progresses in discrete Planck time chunks. That's obvious enough, but maybe seems unmeasurable ... and so what difference does it really make?

Does it mean that time does NOT progress in discrete chunks, or can not? I don't believe so.


I was gonna say the same thing. I thought the HUP suggested the Planck time as the smallest discrete unit of time.

Uncertainty meant discrete unit of ... limit is different from existing of the limit as an unit or time/velocity/position must exist in that limit. It is just observation cannot break that unit.

Not sure


"Observation" in the sense of _any effect at all_.

> It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems,[8] and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.[9] It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.[10][note 1]

https://en.wikipedia.org/wiki/Uncertainty_principle


I think this is highly interesting, but while reading the comments here, I think one should be very careful about the actual content of the paper. It is deriving results about the (phenomenological) master equation. It has been some time since I have studied this, but we are not talking about some first principles theory here. It is possible to derive master equations in some limit from a fundamental theory, but the setting here is purely phenomenological. I.e. the hidden states would be anyway present in the fundamental theory and would be lost due to some integration and limit process in order to arrive at the master equation.

> Scientists believe that time is continuous, not discrete—roughly speaking, they believe that it does not progress in "chunks," but rather "flows," smoothly and continuously.

The jury may still be out on that. A few of the quantum gravity theories (like loop quantum gravity, causal dynamical triangulation, or causal sets) propose a discrete spacetime as a consequence of background independence.

Edit: formatting


Interesting. Though I didn't read the underlying papers.

I'd still like to know what either "flows" or "chunks".

Time? Sure, ok, time flows, or time chunks. But what exactly is it?

Time has been one of my favorite topics to read about in the last 20 or so years, and I still feel like we have hardly made any progress in figuring it out.


It seems to me there's a bait and switch of sorts that we play on kids. We tell them science can answer their questions about the world, or that they can use science to answer those questions, but it kinda can't. Keep digging and pretty soon you're lost in not-quite-accurate metaphors that may have nothing to do with reality and "we can tell you how it behaves, it's [pages of equations], isn't that good enough?"

It may be (probably is) that "what is this" or "why does it do that" are just meaningless questions, in the sense a child (or adult who hasn't re-calibrated their expectations of what science can tell them) means them. Which is interesting itself, I suppose, but at that point just go study philosophy and art, if that's why you were into science in the first place—and I think it's why everyone at least begins to be interested in science.

[EDIT] in particular it's frightening how fast kids' questions stymie an adult because the adult realizes that they're only a step or two away from "why is there something instead of nothing, and also what is something?" and... good lord, at that point seriously just go become religious or get really into philosophy or something, because you're outta luck, kid, sorry, we don't have any actual answers, just observations about what all this stuff does, but no we can't tell you what "stuff" or "does" are.


Noam Chomsky usually has this point, that the human mind seems limited in its "native understanding" to a mechanistic view of the world - that is the only kind of model that is truly satisfactory to us.

He also points out that, ever since Newton, the dream of mechanistic world was shattered, and with it, any hope that science would help us "make sense of the world" in a truly satisfying sense. Instead, we have become content with modelling the world mathematically and predicting how it will evolve.

Of course, space-time and quantum mechanics have driven additional nails in the coffin of a truly understandable world.


>Of course, space-time and quantum mechanics have driven additional nails in the coffin of a truly understandable world.

I'd think the opposite. The fact the world can work at least as strangely as relativity and quantum mechanics suggest, and we still figured those out as much as we have so far, reassures me in our ability to figure out the world we exist in.


What I meant was: quantum mechanics and relativity have shown that not only is the physical world slightly different from the kind of world our mind can picture (e.g. "mechanistic, but with the universal attraction force"), it is in fact nowhere near our built-in mechanistic model, and we can never hope to build a real intuition about the world.

However, we can still hope to model and predict it more accurately than we would have ever hoped before, so there is that :)


It's also frightening how quickly a philosophy professor can get stymied by a really confident or particularly stupid undergrad. I don't think science or philosophy should claim to answer all questions. Many questions are ambiguous, badly formed, misleading, nonsensical, and I don't think a robust account of knowledge is one that ends with a stack of turtles.

Science cannot answer all of our questions, but it can teach us how to ask better ones.


Yes! I do know this one. And it is very interesting to me.

Watch this video: https://www.youtube.com/watch?v=-6rWqJhDv7M

I myself lean to the Aristotelian definition: count of change.

After all, all our clocks are built on this definition. If there were a lone particle traveling in an infinite universe, it'd be nonsensical to talk about its speed because there'd be nothing to relate the distance traveled to (and no observer). Indeed, whenever we talk about "time" and related quantities like "speed", we always compare the observed object to some other, existing reference.

So I think of the interaction of particles/matter as being something fundamental, whereas "time" is a synthetic human notion.


I too subscribe to the idea of time as change, i.e. if all motion stops -- including the motion of subatomic particles -- it is the same as time stopping.

You have your one immutable[0] particle that is left at the tail end of the heat death of the universe. It's not going anywhere, because there is nowhere to go. It's not doing anything because there's nothing to do. It is a lone observer, with nothing to observe.

Except! There would still be virtual particles creating and annihilating. Let's assume that there is an interval between these events (e.g., they aren't constantly present).

Would Aristotle say that no time passes between events? I would tend to say that time yet passes even during the periods when nothing else exists and nothing is happening.

Let's say the second to last things in the universe were another observer with a clock who could time and observe these virtual particle events, and this observer was able to time these events at a 1 second interval between events (duration of event is irrelevant).

And then that second-to-last observer and the clock suddenly annihilate somehow so that only your immutable particle remains. And the virtual particle events still happen at the same interval period.

Would the time interval between virtual particle events somehow be different without the clock/observer as with the sole immutable particle? That doesn't make sense to me, though I will certainly admit that it could very well be!

Aristotle's idea of time is not fully satisfactory to me, though I can't of course rule it out.

[0] Important! Because if the particle is not immutable, then it is waiting for change, which requires time.


> The Planck time is by many physicists considered to be the shortest possible measurable time interval; however, this is still a matter of debate. [0]

So, if this is true, then does it mean that time does not 'flow', but rather 'chunks'?

[0]: https://en.wikipedia.org/wiki/Planck_time


In a very real sense, we may never know. The problem of natural language is that it doesn't really work all that well for scientific purpose: talking about chunks and flows comes with a human's lifetime's worth of thinking they know what those words imply, and so we don't use them except in popular science articles. Time could be discrete, or it could be continuous, but while in normal life the difference between those two is easily determined, this is not the case in physics at all: in order to say "which of the two it is" you first need to have an idea where the boundary between the two would have to be.

(even at the macro level, a river that moves a specified number of molecules per time unit doesn't "flow", it's technically moving discrete "chunks" of water, we just don't say it's moving chunks of water because no one cares about the "well, technically..." in normal conversation. We know what's meant and what to ignore. Physics doesn't have that luxury)

We have no idea where that boundary can even be found; it would certainly have to be below the Planck length, but we have nothing that allows for that kind of precision. The only thing physics can say right now, and possible even ever can, is that nothing we have at our disposal allows us to conclude time "actually is" discrete. We can come up with mathematical frameworks that assume time is a discrete dimension, but even if those yield super accurate predictions that are then verified through experiment, all that does is confirm that a continuous dimension can be reduced to a discrete one without loss of precision.


This is just wrong. The Planck time is just the time scale when quantum and gravity effects become both equally important. The wikipedia article even says so. It's definitely not some sort of time quantum.

The question I've always had about Planck time is about the "clock". If you and I observe something move and we were able to do so at Planck resolution, would our ticks be synchronized?

Can you phrase that question in a way that takes relativity into account? In particular, what do you mean by ticks being synchronized between two observers, when there's no such thing as objective simultaneity?

Two observers in the same inertial frame can share a clock, can't they?

I believe so. I had assumed you were referring to the general case, sorry about the misunderstanding.

My favorite answer is:

“Time is what the clock measures.”


Can you recommend any good books on the topic?

Not OP, but I just fininshed reading Carlo Rovelli's "The Order of Time" and it helped me figure out some pieces of the puzzle a bit better. High level and divulgatory, not an hard science/math book.

Off the top of my head ->

Carroll, From Eternity to Here - reasonably good and quite accessible.

Coveney, The Arrow of Time - better than average

All of my books are packed away right now for a move, so I'm sorry I can't come up with more. Though that Carroll book is really a decent entry point.


I mean, I've always thought that Planck's constant defines the discreteness of time and, as it were, reality. Isn't it rather obvious that this is the case?

"Planck time"..the unimaginably small 5.39 × 10−44 s, is as fundamentally important to the fabric of the universe as it's much more well known "brother", the speed of light in a vacuum c.

Now, we are nowhere near able to measure that short a time frame...we are limited to around 10x-19 or so, but without a doubt time is discrete, we just are unable to get there yet.


I'm no physicist, but the layman explanation I've heard is that the concepts of Planck length and Planck time do not imply that spacetime is discretized into little "voxels"; instead these quantities are just limits on the uncertainty of any possible measurement.

As I understand it there are two contributing facts:

a) the Heisenberg uncertainty principle states that there's a tradeoff between certainty in position vs momentum, so if you're more certain in a particle's position you're less certain of its momentum. (For photons, momentum is proportional to frequency, i.e. wavelength, and frequency is proportional to energy.)

b) By mass-energy equivalence, anything with energy has mass, therefore higher frequency photons are more "massive". A single photon of sufficiently high frequency would form a black hole.

Putting those two together, to measure a distances accurately, you need higher and higher frequency photons with shorter and shorter wavelengths. For example, radar creates blurry images at ~5cm wavelengths, while ordinary photographs can be razor sharp at ~500nm. The Planck length is just the wavelength at which the photon would have so much energy that it would collapse into a black hole and break our current mathematical models. That's why it's nonsensical--with current models--to talk about lengths smaller than a Planck length, but it doesn't mean that space itself is quantized. Similar argument for time.

(Also, the same logic applies to other particles like electrons, protons, and even up to macroscopic items like baseballs; everything has a wavelength...)


> I've always thought that Planck's constant defines the discreteness of time

Most certainly not! Amusingly, this is an incredibly common assumption that undegrad physics students I teach make, probably because of some pop-sci exposure.

All we know is that the Planck constant defines a scale where we do not know what happens. This is a scale at which we can see that the math behind our current theory breaks, but we have absolutely no reason to expect that the way to fix that math is to use some form of discretization related to that constant.

The "quantum" in quantum mechanics really should not be taken that literally. Photons as described by quantum mechanics for instance do not have a discrete spectrum (nothing literally "quantum" there).

There are fascinating conjectures on why maybe the Plank scale should be discrete, but they are way more subtle than "quantum mechanics is discrete" (because it is not always discrete).


> is as fundamentally important to the fabric of the universe as it's much more well known "brother", the speed of light in a vacuum c.

It is not. It would violate special relativity. No such violation has ever been observed.


Like an atom and then its composition.

https://nobeliefs.com/atom.htm


What flip one bit must involve a hidden varable? Is that related it’s being continious to discrete and vice visa?

Some issues in the discussion:

Scope: Many equate maths physics and universe all and only science. Whilst they may be “easy” science not all human observed phenomenon are reduced to them. Even left and right wing ideology?

Process-wise: whether the in principle refutatable may be naive to describe how human work (and likely we do not all do maths in axiomic way). It is a way to make a distinction of science and myth (and in maths axiom is good for proof not necessarily generate maths)


tl;dr: This is a (cute!) technical detail of an approximation technique used in studying the thermodynamics of computation. It has nothing to do with fundamental physics.

Imagine you have a system with N states, represented as a vector of length N. You have a linear operator that transforms that vector into the state in the next step of time. That's basic linear algebra. Now make the linear operator random, so you have stochastic transitions. That makes it into something called a Markov chain.

Now, physicists love differential equations. They've got lot of tools for working with them, and to bring those tool to bear, some folks a while back took a Markov chain and wrote down a set of linear differential differential equations for how the probability distribution over states evolves in time. That's called a "master equation."

Now say that you have an arbitrary function from initial state to final state. Can you describe it as a dynamics governed by a master equation? Not in general. That's not a huge surprise. We know there are lots of things you can't describe with linear differential equations.

If you're working on thermodynamics of computation, though, it would be nice to salvage the master equation framework because its relationship with thermodynamics is well understood, and you don't have to rebuild that. And getting to thermodynamics from straight up stochastic processes is beyond the mathematical abilities of most physicists. That's not a slight on physicists. The mathematical path from a stochastic process to a thermodynamics is an area of deep, difficult specialization in mathematical physics and if you go down that rabbit hole, it will likely be your career (see, for example, Elliott Lieb).

There are two ways to extend this linear world to get better approximations of things:

1. Imagine you have two points on a curve. A linear approximation is drawing a straight line between the two points. To get a better approximation, you take some more points between the two on the curve and draw a sequence of linear going through those points. The equations of each of those lines are going to be different (they have different slopes and intercepts or however you want to parameterize them). Or, in the context of master equations, you insert some additional, "hidden" steps in between your primary time steps with different stochastic matrices.

2. Imagine you have a dynamical system that moves in time steps, and depends on the last two steps for its current move. You can make it into a system that depends only on the last step by expanding the state to include the previous time step as well (that is, instead of the dynamics of x(t), I track the dynamics of (x(t), y(t)) where y(t) = x(t-1)). These are called "hidden variables." It's the same idea as hidden variables in quantum mechanics. Or even in classical mechanics, I can't write down a first order equation for the position of a classical particle...but I can write down a pair of first order equations for the position and momentum.

Lots of folks working on thermodynamics of computation have done both these things to patch their tool.

What this paper does is try to calculate how many hidden steps and how many hidden variables you need to patch the tool for a given arbitrary function that you're trying to approximate, and shows that if you use more hidden variables you need fewer hidden steps and vice versa.

For thermodynamics of computation, they point out that if you are engineering a system with master equation dynamics, you have to pay for the extra hidden states/steps that you need, so the simple statement that invertible functions are thermodynamically free and noninvertible ones require work isn't the only cost accounting to do.

Aside: I have complained before about only whiz bang articles rising on HN. Here we have a straightforward, technical calculation on the front page. Progress!




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