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.
The joy of reading role-playing games
You don’t have to actually play a role-playing game for it to fire your imagination, so why don’t RPG manuals count as books?
By Damien Walter

I’m a lifelong fan of role-playing games, but I rarely play them.  Dungeons & DragonsCall of CthulhuVampire: The MasqueradeCyberpunk 2013. Traveller.  I’ve been enchanted by the words and illustrations, and drawn into the imaginary worlds of as many RPGs as novels.  So I’m always surprised, and a little dismayed, when RPGs are left out of the popular discussion about books and reading.

Though the term didn’t exist back when I was a teenager, squatting on comic-book floors to thumb through expensive hardback editions, RPGs are an example of the kind of literature described by Espen J Aarseth as “ergodic”.  These are books, like digital literature, computer-generated poetry and MUDs, where a “nontrivial effort is required to allow the reader to traverse the text”.  And they are more common than you might think, especially in geek culture.  Game books that allow you to “choose your own adventure” are ergodic, as are fantasy novels with extensive maps and world-building notes.  But the RPG handbook pushes ergodic reading to its limit.

By putting aside simple narrative storytelling and replacing it with detailed description, the RPG offers the total immersion in an imaginary world so valued by geek readers.  The elaboration of leading characters, political factions and major historical events is sometimes a very dry exercise in world building, but done with enough skill it can spark a deeply satisfying response.

I’ve read far, far more RPGs than I’ve ever actually played.  And they’ve definitely helped spark my imagination, taught me some things about storytelling, and shaped certain mindsets (like the “mania for cataloguing the fantastic” and a “weird fetish for systematisation” I share with China Miéville).


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.

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.

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

Though the term didn’t exist back when I was a teenager, squatting on comic-book floors to thumb through expensive hardback editions, RPGs are an example of the kind of literature described by Espen J Aarseth as ‘ergodic’. These are books, like digital literature, computer-generated poetry and MUDs, where a ‘nontrivial effort is required to allow the reader to traverse the text’. And they are more common than you might think, especially in geek culture. Game books that allow you to ‘choose your own adventure’ are ergodic, as are fantasy novels with extensive maps and world-building notes. But the RPG handbook pushes ergodic reading to its limit.

Reading Roleplaying Game Books

From Damien Walter’s The joy of reading role-playing games at The Guardian. Yes, The Guardian.

I’m usually not a fan of ergodic literature- I’d much rather the author use their words without having to overly format the text to tell me the story, but I gotta be honest- Danielewski is killing it with House of Leaves. Very well done. 


This work is called 815.556 Celsius, and it’s a 3D illustration of the book Fahrenheit 451, and it’s using an ergodic literature and metafiction to interpret the text in a 3D form.

I got a lot of ideas like this on the past but it wasn’t until I got here I was able to do it (…) it’s hard, you know. 

What did you enjoy the most about the course?

Working alongside actual illustration practitioners and getting to see their own practice and the full processes, and even some of their books of sketches as well, and how they developed their ideas.

Which ones are your favourite pubs in London?

Well, when you’ve got money you can go to The Peckham Pelican, which is just across the road from the Camberwell College, but also I recommend to go to Wetherspoons, because it’s a good mix of people and it’s also cheap.

allways sixteen.

i don’t read most books more than once, but it’s time for me to read only revolutions a fourth time. poetic and perpetual, chaotic and reciprocal, the novel rewards the persistent and creative. it’s abstract and ergodic if you resist it. succumb to it and you understand it, admire it, trail it. chase after sam and hailey chasing after each other, [r]evolving from vanity and condescension but ultimately catapulting into pure adoration.

dust cares only for dust.

read eight pages from hailey’s perspective, then flip the book over and read eight pages from sam’s perspective. rinse and repeat. mark z danielewski’s visual stimulation enhances the content and vice versa, manifesting a whole that exceeds its parts and transcending gimmickry.

recommended to those looking for a challenging yet completely unique and ultimately gratifying reading experience. also recommended to my friends.
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[1507.00175] TRIQS/CTHYB: A Continuous-Time Quantum Monte Carlo Hybridization Expansion Solver for Quantum Impurity Problems

[ Authors ]
Priyanka Seth, Igor Krivenko, Michel Ferrero, Olivier Parcollet
[ Abstract ]
We present TRIQS/CTHYB, a state-of-the art open-source implementation of the continuous-time hybridisation expansion quantum impurity solver of the TRIQS package. This code is mainly designed to be used with the TRIQS library in order to solve the self-consistent quantum impurity problem in a multi-orbital dynamical mean field theory approach to strongly-correlated electrons, in particular in the context of realistic calculations. It is implemented in C++ for efficiency and is provided with a high-level Python interface. The code is ships with a new partitioning algorithm that divides the local Hilbert space without any user knowledge of the symmetries and quantum numbers of the Hamiltonian. Furthermore, we implement higher-order configuration moves and show that such moves are necessary to ensure ergodicity of the Monte Carlo in common Hamiltonians even without symmetry-breaking.