Nav: Home

The discrete-time physics hiding inside our continuous-time world

April 15, 2019

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. So they often model the dynamics of physical systems as continuous-time "Markov processes," named after mathematician Andrey Markov. Indeed, scientists have used these processes to investigate a range of real-world processes from folding proteins, to evolving ecosystems, to shifting financial markets, with astonishing success.

However, invariably a scientist can only observe the state of a system at discrete times, separated by some gap, rather than continually. For example, a stock market analyst might repeatedly observe how the state of the market at the beginning of one day is related to the state of the market at the beginning of the next day, building up a conditional probability distribution of what the state of the second day is given the state at the first day.

In a pair of papers, one appearing in this week's Nature Communications and one appearing recently in the New Journal of Physics, physicists at the Santa Fe Institute and MIT have shown that in order for such two-time dynamics over a set of "visible states" to arise from a continuous-time Markov process, that Markov process must actually unfold over a larger space, one that includes hidden states in addition to the visible ones. They further prove that the evolution between such a pair of times must proceed in a finite number of "hidden timesteps", subdividing the interval between those two times. (Strictly speaking, this proof holds whenever that evolution from the earlier time to the later time is noise-free -- see paper for technical details.)

"We're saying there are hidden variables in dynamic systems, implicit in the tools scientists are using to study such systems," says co-author David Wolpert (Santa Fe Institute). "In addition, in a certain very limited sense, we're saying that time proceeds in discrete timesteps, even if the scientist models time as though it proceeds continually. The scientists may not have been paying attention to those hidden variables and those hidden timesteps, but they are there, playing a key, behind-the-scenes role in many of the papers those scientists have read, and almost surely also in many of the papers those scientists have written."

In addition to discovering hidden states and time steps, the scientists also discovered a tradeoff between the two; the more hidden states there are, the smaller the minimal number of hidden timesteps that are required. According to co-author Artemy Kolchinsky (Santa Fe Institute), "these results surprisingly demonstrate that Markov processes exhibit a kind of tradeoff between time versus memory, which is often encountered in the separate mathematical field of analyzing computer algorithms.

To illustrate the role of these hidden states, co-author Jeremy A. Owen (MIT) gives the example of a biomolecular process, observed at hour-long intervals: If you start with a protein in state 'a,' and over an hour it usually turns to state 'b,' and then after another hour it usually turns back to 'a,' there must be at least one other state 'c' -- a hidden state -- that is influencing the protein's dynamics. "It's there in your biomolecular process," he says. "If you haven't seen it yet, you can go look for it."

The authors stumbled on the necessity of hidden states and hidden timesteps while searching for the most energy-efficient way to flip a bit of information in a computer. In that investigation, part of a larger effort to understand the thermodynamics of computation, they discovered that there is no direct way to implement a map that both sends 1 to 0 and also sends 0 to 1. Rather, in order to flip a bit of information, the bit must proceed through at least one hidden state, and involve at least three hidden time steps. (See attached multimedia for diagram)

It turns out any biological or physical system that "computes" outputs from inputs, like a cell processing energy, or an ecosystem evolving, would conceal the same hidden variables as in the bit flip example.

"These kinds of models really do come up in a natural way," Owen adds, "based on the assumptions that time is continuous, and that the state you're in determines where you're going to go next."

"One thing that was surprising, that makes this more general and more surprising to us, was that all of these results hold even without thermodynamic considerations," Wolpert recalls. "It's a very pure example of Phil Anderson's mantra 'more is different,' because all of these low-level details [hidden states and hidden timesteps] are invisible to the higher-level details [map from visible input state to visible output state]."

"In a very minor way, it's like the limit of the speed of light," Wolpert muses, "The fact that systems cannot exceed the speed of light is not immediately consequential to the vast majority of scientists. But it is a restriction on allowed processes that applies everywhere and is something to always have in the back of your mind."
-end-


Santa Fe Institute

Related Scientists Articles:

Scientists discover a 2-D magnet
A team led by the University of Washington and the Massachusetts Institute of Technology has for the first time discovered magnetism in the 2-D world of monolayers, or materials that are formed by a single atomic layer.
Scientists present El Nino
The ecological effects of the strong 2015-2016 El Niño. Carbon burial in aquatic ecosystems.
Russian scientists slowed down aging
A group of Russian and Swedish scientists just published a breakthrough paper, reporting results of a joint study by Lomonosov Moscow State University and Stockholm university.
Scientists develop new antibiotic for gonorrhea
Scientists at the University of York have harnessed the therapeutic effects of carbon monoxide-releasing molecules to develop a new antibiotic which could be used to treat the sexually transmitted infection gonorrhea.
Biodiversity needs citizen scientists
Could birdwatching or monitoring tree blossoms in your community make a difference in global environmental research?
Scientists need your help to spot ladybirds
Scientists are calling on people who are out in their garden this summer to take part in The Ladybird Challenge and help discover how far an alien ladybird species in the UK is affecting other insects, including a wasp parasite.
Scientists ID genes associated with educational attainment
A USC co-author of the study says the genes that are correlated with educational attainment are expressed in the brain during prenatal development.
What values are important to scientists?
While many people are marking today scrutinizing the virtues of their Valentines, Michigan State University revealed a first-of-its-kind study on the virtues and values of scientists.
Scientists take nanoparticle snapshots
An international team of researchers led by X-ray scientist Christoph Bostedt of the US Department of Energy's (DOE) Argonne National Laboratory and Tais Gorkhover of DOE's SLAC National Accelerator Laboratory used two special lasers to observe the dynamics of a small sample of xenon as it was heated to a plasma.
Why do scientists chase unicorns?
Scientists chase unicorns because if they could prove the existence of the magical beasts, the world would be a better place.

Related Scientists Reading:

Best Science Podcasts 2019

We have hand picked the best science podcasts for 2019. Sit back and enjoy new science podcasts updated daily from your favorite science news services and scientists.
Now Playing: TED Radio Hour

Changing The World
What does it take to change the world for the better? This hour, TED speakers explore ideas on activism—what motivates it, why it matters, and how each of us can make a difference. Guests include civil rights activist Ruby Sales, labor leader and civil rights activist Dolores Huerta, author Jeremy Heimans, "craftivist" Sarah Corbett, and designer and futurist Angela Oguntala.
Now Playing: Science for the People

#521 The Curious Life of Krill
Krill may be one of the most abundant forms of life on our planet... but it turns out we don't know that much about them. For a create that underpins a massive ocean ecosystem and lives in our oceans in massive numbers, they're surprisingly difficult to study. We sit down and shine some light on these underappreciated crustaceans with Stephen Nicol, Adjunct Professor at the University of Tasmania, Scientific Advisor to the Association of Responsible Krill Harvesting Companies, and author of the book "The Curious Life of Krill: A Conservation Story from the Bottom of the World".