An exploration of energy, thermodynamic laws, and ergodicity.
Downhill skiing, basketball coaching, coffee-mixing, and smoking examples are provided to explore these physics concepts.
About the author: Jeff Hulett is a career banker, data scientist, behavioral economist, and choice architect. Jeff has held banking and consulting leadership roles at Wells Fargo, Citibank, KPMG, and IBM. Today, Jeff is an executive with the Definitive Companies. He teaches personal finance at James Madison University and provides personal finance seminars. Check out his new book -- Making Choices, Making Money: Your Guide to Making Confident Financial Decisions -- at jeffhulett.com.
This article was originally published April 12, 2021. It has been revised by the author over time.
Physicists have a way of making physics seem very complicated. Great stroyteller physicists like Richard Feynman were masters of reducing physics to its practical, easier to understand components. Unfortunately, the great storytellers physicists like Dr. Feynman are the great exception.
For physicists, classical (macro) physics is considered old-school Newtonian physics. Generally, classical physics plays second fiddle to the Einstein-ushered quantum mechanics (micro) physics. However, classical physics is very relevant to our everyday life. Also, the rules deduced by Newton, Boltzmann, and others were the basis for the explosion of quantum mechanics of today. Having a working understanding is helpful. The following are examples that will help ground the reader in classical physics. The intent is to help the reader connect classical physics to more typical everyday systems. The examples are provided with math to reinforce the physics intuition. However, much of the physics intuition may be achieved via the article language and infographics.
The physics of downhill snow skiing -
For our example, think of downhill skiing as a system, such as:
system input -> preparing to ski down the hill,
system process -> the act of skiing down the hill, and
system output -> completing the ski run.
Two essential physics laws are explored for our example:
First Law of Thermodynamics. Energy cannot be created or destroyed; it can only be converted from one form to another. ("1lotd")
Second Law of Thermodynamics. Disorder, characterized by a quantity known as entropy, always increases. ("2lotd")
For the skiing system example, there are 3 kinds of relevant system energy types:
P - Potential energy (Ep),
K - Kinetic energy (Ek), and
H - Friction or Heat (Eh).
These energy types describe the energy involved at each stage of the ski run.
The downhill skiing system is described in 2 stages:
The first stage, "On the hill" -
This describes skiing on a mountain slope. This stage primarily considers potential and kinetic energy, mostly leaving out heat energy - also known as friction. On the hill, a small amount of energy is transferred to heat. Please note: "On the hill" is shown in stages as the skier goes down the hill - α, β, or γ in the graphic.
The second stage, "Bottom of the hill," -
This describes finishing the ski run at the bottom of the hill. The vast majority of energy is transferred as heat at the end of the ski run. Please note: "Bottom of the hill" is shown in stage δ when the skier finishes the run at the bottom of the hill.
"On the hill" Stage
This is located as stages α, β, or γ in the graphic.
Location on the hill - this drives the ratio of kinetic to potential energy. As the skier goes down, the ratio increases. That is:
limit as the skier completes their run
where n = a point in the process of skiing down the hill, n monotonically increases as the skier gets closer to the end of the run, e.
p = potential energy, k = kinetic energy, t = total energy.
The ratio reaches infinity once the skier reaches the bottom of the hill (I.e., potential energy is exhausted and all non-heat energy has been transferred to kinetic energy) As a reminder, for simplicity I leave out E(h) since heat (friction-based) energy is mostly irrelevant during the ski run.
Steepness of the hill - this drives the total energy available for both kinetic and potential energy transformation. That is, Et = Ekn + Epn, subject to hill steepness. In the next graphic, a steeper hill translates to a wider bar, Et.
Consider time only as a reference point. Notice a“t” subscript for time is not used, only the “n” subscript is used for the skier's ski run location. The location on the hill helps describe the entropic transformation (entropy or 2lotd). That is, moving from lower entropy (top of the hill) to higher entropy (bottom of the hill). The transition from potential energy to kinetic energy, and then finally to heat is the second law of thermodynamics in action. Notice the same amount of energy still exists, but the final heat sink is an expression of the universal disorder described as entropy. This is consistent with the conservation of energy (1lotd) as total energy is fixed. Admittedly, energy is fixed in this skiing example where we assume the skier or many skiers are able to finish this run. At the bottom of the article, we discuss the concept of ergodicity. This handles the degree to which the skier or many skiers may or may not be able to complete the ski run.
"Bottom of the hill" Stage
Ultimately, the potential energy is transformed into kinetic energy, then, the kinetic energy is transformed into heat energy (aka, work or friction) and released into the atmosphere.
This is located as stage δ in the graphic.
Location bottom of the hill - this drives the ratio of kinetic and potential energy to heat energy. When the skier is at the bottom of the hill, the amount of energy dedicated to potential or kinetic approaches 0, and heat energy approaches total energy. That is:
(limit as the skier completes their run)
That is, once you get to the bottom of the hill, the hill flattens and your movement from gravity ends. This occurs because your energy is transformed to heat from the friction associated with a flattened hill.
Energy is conserved as suggested by the 1lotd. However, in the skiing context, heat is a lot less useful than being at the top of a hill with skis! Think of the heat dissipation at the end of your run as "disorder." At the top of the hill, the structure of your skis, your ski training, and the grooming of the ski slope are more "ordered." Thus, the increasing heat at the bottom of the hill is associated with higher entropy or disorder in the skiing system, so the 2lotd is realized.
Another name for heat is work. So, once you get to the bottom of the hill the energy is transferred to heat via friction. Also, if you wish to keep going on a flat surface, you will need to work via your own internal energy and legs to move your skis forward. Again, energy is conserved, it is just transformed from the hill's gravity to your legs! If you work, you will get hungry from the calories demanded by your body. When you eat to satisfy your hunger, you are transferring energy, in this case, calories, from the food you eat to your body. Again, energy is conserved. The 1lotd works! Also, it takes work to prepare the food and calories. So some of the total energy is in the the form of food preparation heat energy. The 2lotd works! You can see how you can follow the energy transformation cycle from one participant to the next.
A basketball example: When my daughter was a teenager, she had a basketball coach named Marie Williams. Coach Marie was famous for yelling “use it” as motivation, whenever something negative happened in a game. This could be a bad referee call, a steal, a hard foul, etc. Her point was to create kinetic energy from potential energy. So if you get a bad call from a ref, “use it” as potential energy and transform it into kinetic energy to be successful in the next play. I doubt Coach Marie ever knew she was using physics principles for coaching basketball! By the way, this picture is of Jenny Crouse, she is not my daughter. Though, I suspect Ms. Crouse regularly "uses it" to drive hoops success!
Ergodicity
Ergodicity is how entropy is operationalized into the many systems of our lives. [i] While entropy characterizes the disorder to which all systems are subject, ergodicity is the speed at which entropy overcomes the participants of a system. This skiing narrative provides a useful example. The total energy and the transformation between energy components described by the equation will work for every skier participant. That is, ASSUMING they do not have something happen that prevents them from finishing the skiing system. This is a big assumption! For skiing, something bad would be a massive knee injury or a big storm ending your ability to ski. You could get bored with skiing and decide to do something else.
Ergodicity is related to entropy. Erodicity is the degree to which ruin, or some "game over" challenge prevents us from finishing the system we started. Generally, these challenges are a random surprise. Meaning, that when we started the system, we did not expect the challenge to prevent us from completing the system. Again the system here is going down a ski slope. The degree to which there is some chance of ruin or otherwise keeping us from finishing what we started is the degree to which the ski system is non-ergodic. Most, if not all human activities are more non-ergotic. Compared to many natural activities that are more ergodic.
In the late 1800s, Ludwig Boltzmann developed the statistical mechanic's rigor to entropy. The rigor is captured in this deceivingly simple formula:
S = k log W
This hugely influential equation is memorialized on Boltzmann's gravesite in Austria. In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a certain kind of thermodynamic system can be arranged.
A cigarette light-up example: As an example, imagine someone lights up a cigarette on one side of a room. Assuming the room is a single, contained space, Boltzmann's formula shows how the atoms of the cigarette smoke will spread to the larger space of the entire open room. The cigarette smoke will initially be concentrated near the smoker's location within the room. In a short period of time, the smoke will become evenly diffused throughout the room. This is why the smell of cigarette smoke quickly spreads across a room. You can thank entropy for permeating smelly cigarette smoke! Aristotle said "Nature abhors a vacuum." This is Aristotle's way of observing that nature desires to evenly spread elements such as cigarette smoke across space.
Boltzmann also developed the initial ergodicity conjecture, flowing from his work on entropy. Ultimately, in 1931, George Birkhoff and John von Neumann are credited with formalizing ergodicity mathematics.
Birkhoff’s Ergodicity equation:
The important intuition is that an ergodic system is a system that can run EQUALLY WELL both over time and can also be completed by many system participants at once. Natural systems are often more ergodic.
A coffee mixing example: Mixing creamer into a coffee cup can be an ergodic system. Imagine a dollop of cool cream is dropped into a hot cup of black coffee. This is a natural system process involving the convection currents that naturally mix cool cream in a hot cup of coffee without a stirrer. An ergodic system suggests there is no difference in the expected outcome if:
Over time -> 1 person prepares 1,000 cups of coffee with creamer over time OR
All at once -> 1,000 people prepare 1 cup of coffee each with creamer at the same time.
For this coffee creamer system, the expected outcome is the same, regardless if the system was completed "Over time" or "All at once." The same outcome is a thoroughly mixed brown cup of coffee. Thus, this coffee mixing system is more ERGODIC.
A non-ergodic outcome is when the Birkoff equivalence, which is the equals sign at the beginning of the equation's second row, changes to an inequality. This is like most human systems, including skiing. Eventually, we will not be able to make the ski run. Maybe we got hurt, maybe a big storm kept us from the slope, maybe we got bored, or maybe we died of old age. Because of these near-term uncertainties:
1 person that skis 1,000 runs over time
- has a different probability of finishing than if -
1,000 people ski a similar run at the same time.
The expected outcome is different. It is more likely the same skier will succumb to some game-over challenge over time. Thus, this downhill skiing system is more NON-ERGODIC.
Ergodicity is relative to the measurer
Notice that in the last section, we used relative terms like "more ERGODIC" or "more NON-ERGODIC" to describe the coffee stirring and skiing systems' outcomes. As these examples suggest, ergodicity is relative to the measurer. People often compare and measure systems to their life needs. Thus, ergodicity is often considered relative to our life span.
In general, we can assume a system that will succumb to disorder ONLY after our life ends is generally ergodic. Although, it never hurts to confirm that a discussion of ergodicity is grounded in a single relative benchmark, like our life span. For example, the convection currents associated with naturally stirring a cup of coffee will probably last as long as the universe lasts. In the last graphic, the "Coffee mix" example is located on the left "Low and Long" green side of the entropy scale. Thus, a natural coffee stirring system is clearly ergodic. Since our ability to ski will decline with injury or age, a downhill skiing system is relatively non-ergodic. In the last graphic, the "Skiing" example is located on the right "High and short" reddish side of the entropy scale.
How you manage ergodicity is essential as to the degree to which an activity is more on the ergodic or on the non-ergodic end of the relative entropy scale.
More non-ergodic: If your desire is to win ski races, you may be more willing to push yourself harder, trading an increased winning probability for a higher non-ergodic outcome like a ski career-ending injury.
More ergodic: On the other hand - if you desire a lifetime of recreational skiing, enjoying the outdoor mountain scenery, along with some healthy exercise - you may be more willing to take it easy on the ski runs to maintain your health, increase resilience, and increase the likelihood you can return to the slope without injury.
Conclusion
This article connected the dots between practical physics concepts, like the laws of thermodynamics, including entropy and ergodicity. Several everyday examples were shown to illustrate the power of physics in our day to day life.
Resources
Please see the articles:
For more examples of managing entropy and ergodicity in your life!
Notes
Please see this reference for more information:
[i] Hulett, The Regenerative Life: How to be an ergodic pathfinder, The Curiosity Vine, 2023