The creator of Dr Pepper, Dr Alderton
First, some physical intuition. I pour some Dr Pepper into a cup. What shape does the fluid take on? There are really three fluids - Dr Pepper (largely water, basically incompressible), carbon dioxide (which takes the shape of tiny, interacting bubbles) and ambient air. There are many forces - solid resistance, buoyancy force, skin and interaction forces on the bubbles, gravity and (since the Dr Pepper and carbon dioxide are colder than the ambient temperature) thermal forces. The short run question of what happens to the fluids is complicated and depends on many tiny factors.
Despite this, the long run solution is easy - basic fluid statics tells us that the Dr Pepper will take the form of the cup and basic thermostatics tells us it will be the same temperature as the ambient atmosphere.
John von Neumann
How do we capture this intuition that - roughly - in the short run history matters but in the long run only structure matters? For many years, physicists and mathematicians have turned to Ergodic Theory to answer this question. Ergodic theory doesn't exactly have a great reputation.
Many people - including high powered top level experts - think that not only does ergodic theory require the formal manipulation skills of a von Neumann, the geometric insight of a Clerk Maxwell and the engineering experience of a Shannon - it doesn't even solve the problem.
But really ergodic theory is very simple - except for all the parts that are hard. Shannon's paper can be polished off in a couple days, and (with all due respect to Joe Doob) it's not clear that there is more to the theory than that.
You don't want to take a few days. Well, here's the few minutes version.
A connected and a disconnected network
We start with the intuitive idea of a network. We call the nodes the state. There are finitely many states and the each have a name. From a given state, there is a rule to transfer to one of the other states to which that node is connected. The rule and the network together are called the system. In theory the rule can be anything, for instance it might be "always go as far down as possible" where down is defined geometrically or topologically. The rules can be probabilistic.
A system is called "ergodic" if the long run amount of time spent at each node is independent of which node you start at. The idea of state gives us the short run detail dependence and the ergodicity gives us long run structure dependence.
For my deliberately dumb "go as far down as you can" rule on the above connected network, I have six possible runs
Pink, Purple, Purple, Purple...
Brown, Purple, Purple, Purple...
Blue, Black, Purple, Purple, Purple...
Orange, Purple, Purple, Purple...
Black, Purple, Purple, Purple...
Purple, Purple, Purple...
No matter where I start, the long run relative frequency \(f_{purple} = 1\) and all others are \( 0 \). Therefore, this dumb system is ergodic. If we try the same thing on the disconnected network:
Black, Brown, Pink, Pink, Pink...
Brown, Pink, Pink, Pink...
Purple, Orange, Orange, Orange...
Blue, Orange, Orange, Orange...
Orange, Orange, Orange...
For the first two starting places the relative frequency of pink goes to one, for the second three, the relative frequency of orange goes to one. This dumb system is non-ergodic. But notice it is two ergodic pieces. In general, a non-ergodic system can be severed into ergodic components (in this case, the two connected subnetworks).
The underlying being connected isn't in general sufficient for being ergodic. On the above left graph, sever the Orange-Purple connection and follow the "go down" rule (question to check if you understand: what are the two ergodic subsystems?). It turns out* kinds of rules that are of physical interest are usually of the form "Given that I am on state N, I go down each connection NM with a certain probability \( p_{NM}\neq 0 \)". For such a rule, being connected is sufficient for ergodicity**. So in this informal blog post I'll choose rules and networks such that connectedness and ergodicity are equivalent.
The ergodic distribution tells us the long run behavior of the system, but it also teaches us about the medium run behavior. We know that if the frequency at a state is "too low" (compared to the ergodic frequency), then we will see a flow into that state. This is more or less a definition of what a flow is!
This is all well and good, but what does it have to do with physics? A continuous system is ergodic if the one can cut up the possible states of the system into a discrete ergodic system. Let's make a pair of networks out of a physical model - a billiards model. I mentally divide a square billiard table into four regions A, B, C and D
Being in a region isn't sufficient to fix the dynamics - I need to know the velocities. I think that it's obvious that velocity digitizes into four chunks based on the number of regions away from the starting region you end up in after a time step. So the states are really:
A0, A1, A2, A3
B0, B1, B2, B3
C0, C1, C2, C3
D0, D1, D2, D3
Each 0 connects only with itself (remember, the billiard isn't necessarily staying still, it could be through all four blocks in one tick). There's a cycle A1 connects to B1 connects to C1 connects to D3 connects to C3 connects to B3 connects to A1. There are three cycles of length two, A2 connects to C2 connects to A2, B2 connects to D2 connects to B2 and A3 connects to D1 connects to A3. This is illustrated below
This particular digitization of the underlying continuous system isn't ergodic. If you start off with A0, then \( f_{A0} =1 \), if you start off with a non-zero velocity state, then \( f_{A0} =0 \). That's enough to show that this isn't an ergodic system.
It turns out that there is no nontrivial digitization of this system that is ergodic. That's because this system is exactly solvable... and I won't tell you why that's connected to ergodicity***.
Let's put a circular block in the middle of the square. Now the graph isn't disconnected. A particle that started four blocks per tick can have it's angle of attack by hitting the circular block to now be 3 blocks per tick (that is, it may be turned around because 3=-1). I don't know if this particular graph is really ergodic and I'm not going to check. In a tour de force, Yakov Sinai proved that this system has an ergodic digitization. This shows that the system is itself ergodic.
That means if the particle isn't in, say, C enough (compared to the ergodic distribution) we will see a flow towards C, just as in the discrete case. This is how ergodicity connects to physical quantities.
Finally: wasn't that Black Thought freestyle great?
*By the magic of symbolic dynamics
** By the magic of Markov Chains
*** I'll let wikipedia do it
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