Ergodic System
An Ergodic System is a temporal stochastic process which has the same statistical behavior averaged over time as over the system's entire possible state space.
- See: Stochastic Process, State Space, Measure Theory, Thermodynamics, Ergodic Theory, Ergodic Hypothesis, Ergodic State..
References
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/ergodicity Retrieved:2020-3-28.
- In probability theory, an ergodic system is a stochastic process which proceeds in time and which has the same statistical behavior averaged over time as over the system's entire possible state space. The modern, formal statement of ergodicity relies heavily on measure theory.
The idea of ergodicity was born in the field of thermodynamics, where it was necessary to relate the individual states of gas molecules to the temperature of a gas as a whole and its time evolution thereof. In order to do this, it was necessary to state what exactly it means for gases to mix well together, so that thermodynamic equilibrium could be defined with mathematical rigor. Once the theory was well developed in physics, it was rapidly formalized and extended, so that ergodic theory has long been an independent area of mathematics in itself. As part of that progression, more than one slightly different definition of ergodicity and multitudes of interpretations of the concept in different fields coexist.
For example, in classical physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics , the relevant state space being position and momentum space. In dynamical systems theory the state space is usually taken to be a more general phase space. On the other hand in coding theory the state space is often discrete in both time and state, with less concomitant structure. In all those fields the ideas of time average and ensemble average can also carry extra baggage as well—as is the case with the many possible thermodynamically relevant partition functions used to define ensemble averages in physics, back again. As such the measure theoretic formalization of the concept also serves as a unifying discipline.
The term ergodic is commonly thought to derive from the Greek words (ergon: "work") and (hodos: "path", "way"), as chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics . At the same time it is also claimed to be a be a derivation of ergomonode, coined by Boltzmann in a relatively obscure paper from 1884. The etymology appers to be contested in other ways as well.
- In probability theory, an ergodic system is a stochastic process which proceeds in time and which has the same statistical behavior averaged over time as over the system's entire possible state space. The modern, formal statement of ergodicity relies heavily on measure theory.
2017
- Nassim Nicholas Taleb. (2017). "The Logic of Risk Taking." Blogpost.
- QUOTE: ... As we saw, a situation is deemed non ergodic here when observed past probabilities do not apply to future processes. There is a “stop” somewhere, an absorbing barrier that prevents people with skin in the game from emerging from it –and to which the system will invariably tend. Let us call these situations “ruin”, as the entity cannot emerge from the condition. The central problem is that if there is a possibility of ruin, cost benefit analyses are no longer possible.[i] …