ergodicity

The Encyclopedia of Mathematics (2002) defines ergodic theory as the “metric theory of dynamical systems. The branch of the theory of dynamical systems that studies systems with an invariant measure and related problems.” This modern definition implicitly identifies the birth of ergodic theory with proofs of the mean ergodic theorem by von Neumann (1932) and the pointwise ergodic theorem by Birkhoff (1931). These early proofs have had significant impact in a wide range of modern subjects. For example, the notions of invariant measure and metric transitivity used in the proofs are fundamental to the measure theoretic foundation of modern probability theory (Doob 1953; Mackey 1974). Building on a seminal contribution to probability theory (Kolmogorov 1933), in the years immediately following it was recognized that the ergodic theorems generalize the strong law of large numbers. Similarly, the equality of ensemble and time averages – the essence of the mean ergodic theorem – is necessary to the concept of a strictly stationary stochastic process. Ergodic theory is the basis for the modern study of random dynamical systems, e.g., Arnold (1988). In mathematics, ergodic theory connects measure theory with the theory of transformation groups. This connection is important in motivating the generalization of harmonic analysis from the real line to locally compact groups.

cosmickneehighs asked:

hello hi minifics? ocd alison pls

She always washes the dishes
                                                 (by hand)
                                                                  washes them twice,
                                                                                        three times,
                                                                                        four - 
but by the fifth time, if
they’re
not
clean,
   she isn’t going
               to wash them
                     anymore; she
                   asks Beth to 
             break them
   and make them
disappear.

(But Beth puts them aside in the dishwasher - to be washed,
                                                                                  rinsed, and
                                                                                  sanitized
for future use, though the numbers never line up, so Alison must
D   E   S   T   R   O   Y   them herself.)

So Beth sits by, watching the dishwasher go unused, rubbing lotion between Alison’s fingers at night because the soap chafes them raw (beneath the rubber gloves) - and goes out to buy new plates whenever they run short.

It’s a Minific Kind of Night!

4

CSA: The Ergodicity Exhibition

Developed from their Evolo skyscraper competition entries, Ergodicity, an exhibition hosted by Canterbury School of Architecture, presented thesis work from eleven Graduate Diploma students.

With over 70 percent of the worlds growing population soon to live within major cities, the exhibition reconsiders the effect of increasing densities. Projects developed their research and design to accommodate for a variety of topics affecting our urban areas today, including: population increase, the rising demand for resources, pollution, waste management, and the digital revolution.

The projects which were shown covered a wide range of locations and programmatic responses, but as a collective all questioned ‘what role can the Skyscraper play in improving our urban areas?’

Responses included approaches such as Tiny Tokyo by Carma Masson, a mixed-use community micro scraper based in the business district of central Tokyo. Tiny Tokyo re-evaluates the approach towards designing skyscrapers, using them as a tool for reviving local heritage and culture, whilst introducing relevance for the people they are designed for, rather than designing them as a corporate tool. 

The future of our history is a concept which has been explored within Luke Hill’s project titled Dis.Assemble. This project involves a complex network composed of 6 miles of disused rail systems buried deep beneath London’s streets which provides a subterranean industrial waste facility: its sole intention to ‘Dis.Assemble’ materials produced by the metropolis above.

Unused space has also been explored within Jake Mullery’s SYMCITY thesis, describing an architectural construct that occupies the ‘dead’ space between existing skyscrapers. 

A comedic thesis by Paul Sohi told the story of one man’s life growing and living in a world of 10 billion people, where 90% of society lives in urbanised cities. The comic explores what such a world would be like.

The launch night was attended by many and with special guest Peter Wynne Rees, chief planner for the City of London, the exhibition was an opportunity to showcase the work of students at the Canterbury School of Architecture ahead of the end of year summer show which starts on the 31st of May.

-Text+photography by Taylor Grindley

Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies. […] Why is the difference between ensemble and time averages of such importance? Well, basically, because when you assume the processes to be ergodic, ensemble and time averages are identical. Let me give an example even simpler than the one Peters gives:

Assume we have a market with an asset priced at 100€. Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be 100€ – because we here envision two parallel universes (markets) where the asset price falls in one universe (market) with 50% to 50 €, and in another universe (market) it goes up with 50% to 150€, giving an average of 100€ ((150+50)/2). The time average for this asset would be 75€ – because we here envision one universe (market) where the asset price first rises by 50% to 150€, and then falls by 50% to 75€ (0.5*150).

From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen. Assuming ergodicity there would have been no difference at all.

The anti-black swan: oversignifying unlikely events and large deviations is as dangerous as undersignifying?

http://www.youtube.com/watch?v=f1vXAHGIpfc Time for a Change: Introducing irreversible time in economics Ole Peters

An exploration of the remarkable consequences of using Boltzmann’s 1870s probability theory and cutting-edge 20th Century mathematics in economic settings. An understanding of risk, market stability and economic inequality emerges.

The lecture presents two problems from economics: the leverage problem “by how much should an investment be leveraged”, and the St Petersburg paradox. Neither can be solved with the concepts of randomness prevalent in economics today. However, owing to 20th-century developments in mathematics these problems have complete formal solutions that agree with our intuition. The theme of risk will feature prominently, presented as a consequence of irreversible time.

Our conceptual understanding of randomness underwent a silent revolution in the late 19th century. Prior to this, formal treatments of randomness consisted of counting favourable instances in a suitable set of possibilities. But the development of statistical mechanics, beginning in the 1850s, forced a refinement of our concepts. Crucially, it was recognised that whether possibilities exist is often irrelevant – only what really materialises matters. This finds expression in a different role of time: different states of the universe can really be sampled over time, and not just as a set of hypothetical possibilities. We are then faced with the ergodicity problem: is an average taken over time in a single system identical to an average over a suitable set of hypothetical possibilities? For systems in equilibrium the answer is generally yes, for non-equilibrium systems no. Economic systems are usually not well described as equilibrium systems, and the novel techniques are appropriate. However, having used probabilistic descriptions since the 1650s economics retains its original concepts of randomness to the present day.

The solution of the leverage problem is well known to professional gamblers, under the name of the Kelly criterion, famously used by Ed Thorp to solve blackjack. The solution can be phrased in many different ways, in gambling typically in the language of information theory. Peters pointed out that this is an application of the ergodicity problem and has to do with our notion of time. This conceptual insight changes the appearance of Kelly’s work, Thorp’s work and that of many others. Their work - fiercely rejected by leading economists in the 1960s and 1970s - is not an oddity of a specific case of an unsolvable problem solved. Instead, it is a reflection of a deeply meaningful conceptual shift that allows the solution of a host of other problems.

The transcript and downloadable versions of the lecture are available from the Gresham College website:

Is ergodicity at the root of all macroeconomic opinions?

Schools of macroeconomic thought differ widely in their policy preferences to achieve social optima. A broad chiasm exists between Keynesians and neoclassical economists with respect to monetary policy and fiscal policy preferences. While the following description is a summary, it will suffice to illustrate how different views on ergodicity explain the differences in these schools of thoughts.

Keynesians and allies believe that there are economic conjectures whereby monetary intervention can generate real growth (situations where the output gap is significant and inflation is below target for example). Neoclassical economists and their monetary allies believe that the gravity of market forces is so powerful that monetary surprises cannot yield real economic benefits.

On the monetary debate, neoclassical economists & monetarists believe that economies are ergodic as market forces ensure price adjustments that maintain the economy at potential at most times and thus any gains due to a monetary surprise today will be balanced by a price change that will annihilate those nominal gain. Keynesians and allies believe that a short-term gain will forever alter the development path of an economy, hence initial conditions matter. Depending on each perspective the economy either has a long run steady state or a path that can be altered at each short-term junction. While neoclassical economics believes in the ergodicity of economic systems, Keynesians and associates believe in path dependence.

With respect to the fiscal debate, neoclassical economists believe that changes in government expenditures cannot efficiently modulate economic activity and change potential output because agents’ behaviour is altered by the expectations of a balancing fiscal change in the future. Since the government must over time keep a reasonable balance, a tax cut that leads to a deficit heralds higher future taxes and leads agents to save the tax cut (Ricardian equivalence). Keynesians on the other hand feel that short-term stimuli may create a boost in the economy’s growth path whose value exceeds the amount of the stimulus. 

Who should we believe? Both schools of thought have a point. Unlike natural systems ergodicity does not apply always and everywhere with the same power. The challenge of wise economic management lies in the ability to recognize with a certain degree of certainty when a change in expected policy can yield positive results from those instances where a change in policy simply changes the timeframe of economic consequences.

2

Finished up Lude and Thumper. decided to just google search other people’s mods for Don’t Starve and based her on one of those. I’m happy with the outcome.

I also made Johnny’s hair darker. When I’ve made most of the main cast (Navidson and the rest) I’m planning on probably redrawing a couple of these in Illustrator. It’s been too long since I’ve had an art project. Maybe I’ll recreate a few scenes too.

butchrobot replied to your post “All these people are talking about the effect of money on politics as…”

who is TAS i want to see this youtube video

President TAS? You remember; he issued an average of 3.7 million executive orders every day and got that constitutional amendment passed that made presidential terms 3 weeks long? He’s the one the TAS Jupiter Atmosphere Colony I live in was named after? He passed the Comprehensive Systemic Reforms that are the reason we all live in an C*-communist ergodic pseudomatriarchy now?

i can’t believe you don’t remember this it was all over the news at the time…

[ Authors ]
Merlijn van Horssen, Emanuele Levi, Juan P. Garrahan
[ Abstract ]
We study the real-time dynamics of a translationally invariant quantum spin chain, based on the East kinetically constrained glass model, in search for evidence of many-body localisation in the absence of disorder. Numerical simulations indicate a change, controlled by a coupling parameter, from a regime of fast relaxation—corresponding to thermalisation—to a regime of very slow relaxation. This slowly relaxing regime is characterised by dynamical features usually associated with non-ergodicity and many-body localisation (MBL): memory of initial conditions, logarithmic growth of entanglement entropy, and non-exponential decay of time-correlators. We show that slow relaxation is a consequence of sensitivity to spatial fluctuations in the initial state. While numerics indicate that certain relaxation timescales grow markedly with size, our finite size results are consistent both with an MBL transition, expected to only occur in disordered systems, or with a pronounced quasi-MBL crossover.

Yay! There’s more!

Will Navidson’s appearance was based primarily on Kyle MacLachlan. I’m sort of kind of a wee bit of a Kyle MacLachlan fangirl/fanboy/fanperson. In his hands is a MagLite and a Hi 8 camcorder. I thought about also drawing his 35mm Nikon strapped to his chest, I might go back and do that.

You may have noticed that Tom looks an AWFUL LOT like Danny DeVito. Now when I first read House of Leaves I was envisioning someone close to Curly Howard but I soon realized that I just can’t picture him telling the old lady-blowjob joke. Danny DeVito on the other hand is a different story… no seriously! I want you to just get your copy of House of Leaves and open it up to Tom’s Story (ITC American Typewriter Font) and just read it in Danny DeVito’s voice. You will be laughing your ass off all night long with that mental image.

It may also be worth mentioning that he was in Hercules, and House of Leaves does occasionally make reference to Greek myths.

And then there is Karen Green. When I’m reading HOL, I typically think of Milla Jovovich as Karen. It’s not exactly easy to convey that in this sort of art style (certainly not as easy to pick out as Danny DeVito) but I’m sure you can see sort of what I mean.

Edit: No, I don’t know what that weird curvy line design thing on Karen’s dress is, just role with it.

2

My paperback of House of Leaves came in the mail today.

“BUT TRISTAN!” All two of my followers exclaim, “YOU ALREADY OWN THE BOOK! A HARDCOVER NO LESS!” That is true. But you see I have a Summer Reading Assignment and I got to actually CHOOSE what book I wanted to read this time. Not only that but I am supposed to annotate the physical book and write an essay on what I read when I get back to school. :D How could I not do one of my most favorite books of all time?

I’m not used to writing in books though, but I know this book front to back at least. Annotating this puppy should be a breeze~~~

Getting this today made me start doodling. I haven’t drawn in FOREVER! It all started with the most complicated thing to ever do in a book like this “It’s time to visualize it.”

Of course the book starts off with Johnny’s introduction. I have had several head canon physical appearances of Johnny but this time around I think I found one I like. 

I envisioned him looking something like Syd Barrett from Pink Floyd. For some reason when I started drawing him I wanted to draw him in a style similar to Don’t Starve.

Then I got the idea of drawing everybody’s favorite ironically blind movie critic, Zampanò! I based his appearance on Jorge Luis Borges (because who else would really fit the bill? Other than TZD but I’ve only ever seen two photographs of him).

I’m not sure where I got the idea for Pelafina’s appearance though. You can see the parallels to Wendy (Don’t Starve) no doubt but I made some changes. I like her.

I’m thinking about doing this for all of the characters in book (Navidson, Karen, Holloway, Tom, Reston, Chad and Daisy, Jed, Wax, Ashley, Hailey, Kyrie, Gnask Man, The Liberty Bell, The “New” Director, and many more!) 

Right now I’m trying to draw Lude and and Thumper. For Lude I’m thinking of basing him off of Neil Patrick Harris/Barney Stinson. As for Thumper… I have no idea. I don’t know who looks like a stripper to me (never thought I’d type that). If you want to suggest someone, feel free.

[ Authors ]
Tamoghna Das, T. Lookman, M. M. Bandi
[ Abstract ]
Two-dimensional (2D) particulate aggregates formed due to competing interactions exhibit a range of non-equilibrium steady state morphologies from finite-size compact crystalline structures to non-compact string-like conformations. We report a transition in heterogeneous microscopic dynamics across this morphological hierarchy as a function of decreasing long-range repulsion relative to short-range attraction at a constant {\it low} density and temperature. Following a very slow cooling protocol to form steady state aggregates, we show that geometric frustration inherent to competing interactions assures non-ergodicity of the system, which in turn results in long-time sub diffusive relaxation of the same. Analysing individual particle trajectories generated by molecular dynamics, we identify {\it caging} dynamics of particles in compact clusters in contrast to the {\it bonding} scenario for non-compact ones. Finally, by monitoring temperature dependence, we present a generic relation between diffusivity and structural randomness of the aggregates, irrespective of their thermodynamic equilibrium.