A misconception in ergodicity: Identify ergodic regime not ergodic process

Preamble 

    Figure 1: Two observable's approach to
 ergodicity for Bernoulli Trials. 

Ergodicity appears in many fields, in physics, chemistry and natural sciences but in economics to machine learning as well. Recall that, ergodicity in physics and mathematical definition diverges significantly due to Birkhoff's statistical definition against Boltzmann's physical approach. Here we will follow Birkhoff's definition of ergodicity which is a statistical one. The basic notion of ergodicity is confusing even among experienced  academic circles. The primary misconception is that ergodicity is attributed to a process, a given process being ergodic. We address this by pointing out that ergodicity appears as a regime or a window so to speak for a given process's time-evolution and it can't be attributed to an entire generating process.    

No such thing as ergodic process but ergodic regime given observable

A process being ergodic is not entirely true identification. Ergodicity is a regime over a given time window for a given observable derived from the process. This is the basis of ensemble theory from statistical physics.  Most of the processes generates initially a non-ergodic regime given an observable.  In order to identify an ergodic regime,  we need to define for a discrete setting : 

1. the ensemble (sample space) : In discrete dynamics we also have an alphabet that ensemble is composed of.
2. an observable defined over the sample space.
3. a process (usually dynamics on the sample space evolving over-time).
4. a measure and threshold to discriminate ergodic to non-ergodic regimes. 

Interesting thing is that different observables on the  same ensemble and the process may generate different ergodic regimes.  

 What are the processes and regime mathematically?

A process is essentially a dynamical system mathematically.  this includes stochastic models and as well as deterministic systems sensitive to initial conditions. Prominently these both combined in Statistical Physics. A regime mathematically implies a range of parameters or a time-window that a system behaves very differently. 

 Identification of ergodic regime

Figure 2: Evolution of
time-averaged OR observable.

The main objective of finding out if dynamics produced by the process on our observable enters or in an ergodic regime is to measure if ensemble-averaged observable is equivalent to time-averaged observable value. Here equivalence is a difficult concept to address quantitatively. The simplest measure would be to check if $\Omega = \langle A \rangle_{ensemble} - \langle A \rangle_{time}$ is close to zero, i.e., vanishing. $ \Omega$ being the ergodicity measure and $A$ is the observable with different averaging procedure. This is the definition we will use here. However, beware that in the physics literature there are more advanced measures to detect ergodicity, such as considering diffusion-like behaviour, meaning that the transition from non-ergodic to ergodic regime is not abrupt but have a diffusing approach to ergodicity.  

Figure 3: Evolution of
time-averaged mean.

In some other academic fields approach to ergodic regime has different names not strictly but closely related, such as in chemical physics or molecular dynamics, equilibration time, relaxation time, equilibrium, steady-state for a given observable, in statistics Monte Carlo simulations, it is usually called burn out period. Not always, but in ergodic regime, observable is stationary and time-independent. In Physics, this is much easier to distinguish because time-dependence, equilibrium and stationarity are tied to energy transfer to the system. 

Ergodic regime not ergodic process : An example of Bernoulli Trials

Apart from real physical processes such as Ising Model, a basic process we can use to understand how ergodic regime could be detected using Bernoulli Trials. 

Here for a Bernoulli trials/process, we will use random number generators for a binary outcome, i.e., RNG Marsenne-Twister  to generate time evolution of an observables on two sites:  Let's say we have two sites  $x, y \in \{1, 0\}$.  The ensemble of this two site system $xy$ is simply the sample space of all possible outcomes $S=\{10, 11, 01, 00\}$. Time evolution of such two site system is formulated here as choosing $\{0,1\}$  for a given site at a given time, see Appendix Python notebook. 

Now, the most important part of checking ergodic regime is that we need to define an observable over two side trials. We denote two observable as $O_{1}$, which is an OR operation between sites, and $O_{2}$ is averaged over two sites. Since our sample space is small, we can compute the ensemble average observables analytically:

• $O_{1}  =  (x+y)/2$  then  $10, 11, 01, 00  ;  (1/2 + 2/2 + 1/2 + 0 ) /4 = 0.5$
• $O_{2}  =  x OR y$ then   $10, 11, 01, 00  ;  ( 1  + 1 + 1 + 0 )/4 = 0.75$   

We can compute the time-averaged observables over time via simulations, but their formulation are know as follows: 

•  Time average for $O_{1}$ at time $t$  (current step)  is   $ \frac{1}{t} \sum_{i=0}^{t} (x_{i}+y_{i})/2.0$
• Time average for $O_{2}$ at time $t$  (current step)  is   $ \frac{1}{t} \sum_{i=0}^{t} (x_{i} OR y_{i})$.

One of the possible trajectories are shown in Figure 2 and 3. For approach to ergodicity measure, we shown this at Figure 1. Even though, we should run multiple trajectories to have error estimates, we can clearly see that ergodicity regime starts after 10K steps, at least. Moreover, different observables have different decay rates to ergodic regime.  From preliminary simulation, it appears to be OR observable converges slower, though this is a single trajectory.

Conclusion

We have shown that manifestation of the ergodic regime depends on the time-evolution of the observable given a measure of ergodicity, i.e., a condition how ergodicity is detected. This exposition should clarify that a generating process does not get an attribute of "ergodic process" rather we talk about "ergodic regime" depending on observable and the process over temporal evolution. Interestingly, from Physics point of view, it is perfectly possible that an observable attains ergodic regime and then falls back to non-ergodic regime.

Further reading

• Practical Understanding of Ergodicity :  Elementary ergodicity and some basic references.
• Is ergodicity a reasonable hypothesis?  : Boltzmann's definition of ergodicity.
• Scaling of ergodicity in binary systems : An idea on extending Bernoulli trials ergodicity to N-dimensions (sites).
• Effective ergodicity in single-spin-flip dynamics PRE Approach to ergodicity in magnetic systems, extents to neural networks. 
• IsingLenzMC R package : Effective ergodicity convergence R utilities and Ising-Lenz 1-D Monte Carlo.
• Diffusive behaviour of ergodicity convergence in Ising-Model.

Appendix: Code

Bernoulli Trial example we discussed is available as a Python notebook on github here. 

Please cite as follows:

 @misc{suezen22ergoreg, 

     title = {A misconception in ergodicity: Identify ergodic regime not ergodic process}, 

     howpublished = {\url{http://science-memo.blogspot.com/2022/05/ergodic-regime-not-process.html}, 

     author = {Mehmet Süzen},

     year = {2022}

}  

A practical understanding of ergodicity

Preamble

In this tutorial, we will provide a brief introduction on what does ergodicity means and how to detect when a dynamical system behaves ergodically. The very definition of ergodicity is not uniform in the literature, especially definitions of Boltzmann and Birkhoff are not the same. So, one should talk about a set of ergodicity, i.e., ergodic theorems, rather than a single one. We follow the basic definition that, the system is ergodic for a given observable when the ensemble-averaged value of the observable is the same as its time-averaged value. But beware that this definition is a purely statistical definition and does not reflect Boltzmann's ergodicity from Physics.

Boltzmann: Father of ergodicity
(Wikipedia)

What are the ensemble and time averaging?

The ensemble is a fancy word. It doesn't mean a group of musical instruments but it is short for the statistical ensemble. It is developed by Gibbs, an American theoretical physicist. Essentially all possible state of a physical system. In statistics, this actually has another name, a sample space.
For example, sample space for the outcome of two fair coins tossed at the same time would be
$$ \Omega = \{HH, HT, TH, TT\}$$
Let's say we represent them as bits $H=1, T=0$ and we want to compute an observable $\mathscr{A}$, the sum of outcomes, in the entire ensemble would read.
$$\Omega_{\mathscr{A}} = \{2,1,1,0\}$$
And the ensemble average, arithmetic mean, would be $\langle \mathscr{A} \rangle_{ensemble} = 1.0$. A time-averaging can be computed via an experiment, so-called trials. Let's say we had 6 trials
$$ \Omega_{time} = \{HH, TH, HH, HT, TH, HT\}$$
and time-averaged observable will be $\langle \mathscr{A} \rangle_{time} = 1.33$. Note that larger the trials time-averaged values approach to ensemble averaged ones.

Note that this sounds very naive and useless example, but if we have very large sample space with the complicated setting, so-called in thermodynamic ensembles and limits, computing time-averaged values are much easier, whereby computing ensemble average is intractable, most of the case in statistical physics.

Connection to the law of large numbers 

Statistically  inclined readers might catch that the above definition sounds like the law of large numbers. It is indeed the strong form of the law of large numbers and it is a special case of an ergodic theorem.


Conclusions: Why would I care about ergodicity?

Apart from the intellectual appeal, ergodicity is very important in statistical physics. Almost all computations rely on the ergodic theorems. So-called N-body systems are simulated based on this for computing physical properties. Ergodicity pops up everywhere from economics to deep learning.

Further Reading with Notes

The literature is vast in ergodicity. Due to its mathematical nature, text on ergodicity can easily be non-accessible for an average scientist, well, even for an average mathematician.

• Modern Ergodic Theory, Joel L, Lebowitz and Oliver Penrose link (excellent for an introduction)
• Computational Ergodic Theory, Choe, link (advanced text)
• An introduction to Chaos and Nonequilibrium Statistical Mechanics, link (explains why Boltzmann's ergodicity is different)
• Deep Learning and complexity: link (ergodicity in spectra, different kind of ergodicity)
• Ergodicity and economics: link (ergodicity in risk decisions, different kind of ergodicity)

Please cite as follows:

 @misc{suezen23lpc, 

     title = {A practical understanding of ergodicity}, 

     howpublished = {\url{http://science-memo.blogspot.com/2020/01/a-practical-understanding-of-ergodicity.html}}, 

     author = {Mehmet Süzen},

     year = {2023}

}  

Postscript notes

• Sample space and ensemble are not synonymous. Ensemble is a special type of sample space where by all unique combinations are exhausted for a given system, i.e., defined event set.
• Origins of ergodicity goes back to Boltzmann but Gibbs's ensembles also provides a conceptual parallels. An ensembles could be thought as a distinct measurement protocols. Then the question of ensemble equivalence is similar to to question of ergodicity, i.e., in a strict mathematical sense and still an analogy that if two protocols are equivalent, because technically a time-averaging would also curate a different set.  Only distinction is that time-averaging is tied to a a given ensemble. 
• Law of large numbers and ergodicity:  There is a deep connection between law of large numbers and ergodicity.  The ergodicity of von Neumann-Birkhoff means visiting all possible states are considered. Given advanced level  language used in definitions of ergodicity and law of large numbers, invoking measure theory made difficult to understand that two concepts overlaps for practical purposes. However, both appears to have multiple different inceptions. Strong form of law of large numbers overlaps with  von Neumann-Birkhoff ergodicity.