All posts by jayne

Conway’s game of life

One of the cooler things I discovered recently was that when you google search “Conway’s game of life” google automatically begins to play the game of life in the background of the search results page. It is rather mesmerizing,

The game of life is played on a grid of little squares; squares have two configurations “alive” and “dead” which are usually represented by using two different colours. A single game consists of a initial seed or starting configuration and a set of rules which govern how the grid updates:

(1) Any live cell will remain alive if it has 2 or 3 live neighbors; otherwise it will die.

(2) Any dead cell with 3 live neighbors will become alive.

The rules are roughly motivated by physical considerations; if the population density is too low or too high then individual members of a colony will die before reproducing. However three alive cells may reproduce to create a extra adjacent live cell.

The game is extremely popular because the seemingly overly simple rules can generate extremely complicated patterns. Recently it was demonstrated that certain initial conditions of the game of life can give rise to emergent behavior in the sense that we can aggregate large sections of the grid (consisting of multiple cells) and treat this section as an individual “macroscopic cell” which behaves non-trivially. There are no simple set of rules for how these macroscopic cells behave; all update

rules continue to apply to the small indivisible cells which constitute them. Nevertheless it is possible to  initialize the game of life so that the macroscopic squares effectively appear to evolve according to conditions (1) and (2). There is quite a spellbinding example of this available here. Check it out as it is unbelievable at first sight.

Emergent time coordinate

Recently I have been very interested in the concept of a thermal time. The basic premise is that unidirectional time coordinate we experience may not be a fundamental property of some GUT (grand unified theory) scale physics but is instead an emergent phenomenon. At first this sounds somewhat anarchical as physical laws mostly focus on predicting the state of a system at some future time and almost invariably involve some kind of description of the behaviors of a system under dynamical evolution.  The idea is at once fascinating but also unconventional enough to be something you need a really good reason for doing. If you are genuinely curious try reading Rovelli and Connes [1].

The story begins with the Tolman Ehrenfest effect. Static observers who measure Temperature in gravitational fields find that it depends on the gravitational potential at the point where the measurement is made so that


is a constant (interestingly this scaling is usually associated with the relation between the coordinate time and the proper time measured by a clock traveling along a world line) [2,3].  In other words a long column of fluid in thermal equilibrium will naturally have a temperature gradient with hotter liquid at the bottom.  The difference is tiny; nevertheless it implies general relativity is deeply intertwined with the temperature of your beer tower.

What if at some level statistical physics and thermodynamics are fundamental and what we perceive as time is an emergent property. We can postulate that the universe is described by a generally covariant theory which treats all coordinate directions  evenhandedly and time is an artifact of the thermal state that we are immersed in.

There is no place for a preferred time variable in general relativity; furthermore if we look at the physically observable quantity measured by a clock it is in fact the proper time along a world line.  The coordinate time which parameterizes the field variables,

and trajectories of relativistic particles is just a variable. Indeed there is one equivalent definition of t for each Lorentz transformation,
because the equations of motion are manifestly Lorentz covariant it really doesn’t matter which you choose. They are all just reparameterizations.

Intriguingly thermal states break Lorentz invariance. They pick out a preferred coordinate system; in fact a thermal bath preferences the Lorentz frame in which it is at rest. This thermal state may be used to define a preferred physical time. Yet the background theory remains generally covariant.

In thermal equilibrium our thermal states can be described by Gibbs states; these encode information about the Hamiltonian and the dynamical properties of the system are attributed to this thermal state rather than direct Hamiltonian evolution. To some extent it is valid to assume you are always working in a thermal regime as you stereotypically don’t have access to the full microscopic state of a system and hence you simply reconstruct the microscopic state from macroscopic observations

[1] Rovelli and Connes [arXiv: 9406019v1]

[2] Rovelli and Smerlak [arXiv:1005.2985v5]

[3]Tim-Torben Paetz [#://]




The Zeno Paradox

The Zeno Paradox

For fun this week we have a short story from Dag on The Zeno Paradox.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face, eyes focus on some memories swirling in the dark behind the window. This is the place where men come to absolve their sins before disappearing into the night.

The bar door swings open. A man in a trench coat steps in, pauses to look around. His long shadow stretches towards the lonely guy as if trying to tighten its icy fingers around his throat. The barman gives the newcomer a quick glance only to get back to his world of endless nights when time stands still like the rows of bottles behind him.

“I hate rain”, he mutters to himself.

The newcomer sits in front of the lonely guy.

“William?”. The guy takes a long drag of his cigarette, savours the smoke for awhile, turns his head towards the newcomer and exhales straight into his face.

“Who’s asking?”, he says.

“Wilson. Do you have it?”

William doesn’t reply immediately. He pours himself a glass of whiskey, double shot, looks through it at Wilson, puts it on the table, adds more and then gulps it down like it is his last.

“Yeah,” says William, inhaling the cigarette.”I have it,” he adds, exhaling a thin streak of smoke.

“Give it to me.” Wilson’s voice sounds greedy. William looks straight into his eyes and says almost caringly,

“I’ll give it to you but you must listen to my story first.” “Keep it short, pal,” replies Wilson.

“I loved Gail more than anything, more than myself. I first saw her in a small dancing studio at night. It was raining like today.” William’s voice becomes shaky. He takes another shot of whiskey.

“She was practicing some moves in front of a big mirror. She looked so beautiful, like out of this world. Her body moved across the dance floor with a grace I’d never seen before. I was standing there, glued to that big window and I knew that Gail was the woman I wanted to be with.” He grabs Wilson by the arm and says feverishly “Can you understand that? Can you?!”. Wilson shakes off the hand.

“Take it easy, man” he says dryly.

“We were like Bonnie and Clyde. Lovers, friends. It was a blast but nothing good lasts for long in this twisted universe. Gail fell terminally ill.” William stops, lights another cigarette. Smoke seems to make it easier for him.

“I couldn’t watch her body wasting away”, he pauses, eyes fixated on the swirling cigarette smoke.

“Have you heard about the Zeno paradox?” “No” replies Wilson.

“Zeno claimed that nothing moves because to get from A to B you need to cover half the distance, then the half of the half and so on. Every half requires a finite time to travel but there are infinitely many of them so you won’t cover the distance in a finite time.”

“Nonsense” says Wilson.

“Yeah…. Infinitely many pieces can give you a finite thing”, William pauses, “Not in the quantum world.”

“What do you mean?” William gets Wilson’s attention.

“In the quantum world there is no reality. Observation creates it and this means you can manipulate reality by simply looking at physical systems” William puts out the cigarette. “If you observe them frequently enough you can freeze them forever.”

“That’s how the machine works?” interrupts Wilson.

“Yeah, something like that.”

“Where are the blueprints?” Wilson’s eyes flicker with greed.

“I haven’t finished yet.” William lights another cigarette. “I thought I could keep Gail in a state of suspension until they found a cure.”


“I asked her to dance for me one more time and…” he swallows tears.

“What?” asks Wilson impatiently, pouring William another drink. William ignores it.

“Then I set this… machine… in motion.” William’s voice quivers again. He gulps down the glass of whiskey and goes motionless like a mechanical toy with a discharged battery.

“And?” Wilson prompts him.

“At first it worked beautifully. Gail’s body froze in time… She looked so beautiful.”

“And?!” asks Wilson’s impatiently.

“A few days later I noticed some small changes in her face. Blemishes.” He pauses. “The blemishes started to become fuzzy and larger, slowly transforming Gail’s body into… into…” William swallows hard, his Adam’s apple forcing its way up and down like a piston of a worn out engine, “into something undefined, smeared in space.” William’s hand wipes some invisible grease off his face.

“Couldn’t you stop the machine?” interrupts Wilson.

“It was too late. I would have had to reverse the whole time evolution but I didn’t have enough computational power.” William takes out a notebook. “Here’s the blueprint for the machine.” He throws it on the table. “Can I go now?”

“Where is she now? I mean Gail” asks Wilson ignoring William’s question.

“I’d like to believe that she’s become entangled with the rest of the universe” he pauses, looks into the night behind the window. “And that one day I’ll be able to bring her back, see her dancing again…”

Wilson picks up the blueprint and puts it into an internal pocket of his trench coat.”You know I can’t let you go. We need your expertise. Without you it would take us too long to build the machine.” Wilson wraps his fingers around William’s arm. “Just don’t do anything stupid.”

William looks at Wilson and smiles, his eyes hidden in the shadow.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face. A fuzzy, slowly expanding blemish appears at the corner of his eye.

Check out more at Quantum Shorts 2013: Zeno Paradox 

Second law of thermodynamics and heat engines

The second law of thermodynamics is believed to be one of the most fundamental postulates of modern physics. Its most famous formulation is as the statement: the entropy of an isolated system never decreases; and it can be conveniently used as an excuse to never clean your room. This argument is ingeniously simple. Entropy of a macroscopic state is a measure of the number of possible corresponding microscopic configurations. In this case the macrostate is ‘a messy room’ and each microscopic configuration labels the positions of every object in that room such as the 10 books and a computer currently cohabiting your bed. Note that cleaning up a room involves ordering the objects in it. Since there are many possible configurations of these objects for which we say the room is messy but there may be only a small number of orderings for which we consider it clean; your efforts locally decrease the entropy of the room. However we know by the second law of thermodynamics that while we are doing the work to reordering the room we dissipate heat to the environment causing the universe to become more entropic globally; or in other words cleaning up your room contributes to the heat death of the universe and should only be undertaken at your own peril. Check out xkcd’s ideal room configuration.

This illustrates a perhaps more physical reinterpretation of the second law of thermodynamics; it is often restated as the impossibility of convert heat directly into work which ties into the following construction of a Heat engine. In this graphical representation we can see that not all the heat Q H transferred from the hot reservoir can be converted directly into useful work, only a fraction Q H – QC  can be accessed; the remainder must be dissipated to the cold reservoir thereby increasing the entropy of the cold reservoir. If we could completely convert Q H  directly into useful work then we could use this work to reduce the entropy (for example we could erase some information), at the same time we would have QC equal to zero so there would be no counterbalancing increase in the entropy of the cold reservoir. This would lead to a global entropy decrease thereby violating the second law. In our example of a messy room we realize that you are functioning as the hot reservoir and the universe (empty space has a temperature of about 3 Kelvin) is functioning as the cold reservoir. We can clearly see that it is sensible to define the amount of useful work which can be extracted from a system a concept called the Helmholtz free energy F. Making it possible to succinctly summarize everything we have said above by  the statement that  the change in Helmholtz free energy of the system can be at most equal to the average work done on a system: <W>  ≥  ΔF. The averaging over the work distribution in this statement arises because in finite systems we  can have quite large statistical fluctuations. This is an extremely powerful statement and a key tool in many active fields of research particularly in its relation to Szilard engines and generalized forms of the second law like Jarzynski equality which extends the second law of thermodynamics (relationship between free energy and work) to higher moments of the work distribution (the average work is the first moment).