There are two things that I associate with the Fall season: Halloween and raking leaves.  While the former brings out my inner child who yearns for costumes and candy, the latter is the focus of this article, as it provides a wonderful illustration of the second law of thermodynamics.

Ah, the second law... Personally, I have never been particularly drawn to the study of thermodynamics, but I have always had an affinity for the second law and all of its fascinating philosophical implications.

The zeroth law of thermodynamics discusses thermal equilibrium, and in so doing, gives meaning to the term temperature.  The first law describes the conservation of energy, and dictates that energy cannot be created nor destroyed.  The second law says, in a variety ways, that no system can be 100% efficient, and as such, every system has losses. 

If you are curious, the reason that there is a zeroth law is that it was developed decades after the first two, but precedes them in terms of scientific principle, in that it lays a foundation for further study.  Rather than shift laws one and two up to two and three, it was decided that they be left alone, and that the new law be referred to as law zero.  This was a great choice, because it gave scientists the opportunity to use the term "zeroth", and as a result, sound very intelligent.

In summary, law zero says there will be lunch, law one says that it will not be free, and law two says that actually, you have to tip.

While the zeroth and first laws of thermodynamics set up a structure for studying energy and temperature, the second law gets the wheels turning, by offering something somewhat unusual for science: direction. 

The simplest statement of the second law is that heat travels spontaneously from hot bodies to cold ones.  That is, the heat from a steaming cup of coffee leaves the liquid and travels to the cooler surrounding air in the room.  This simple statement, so obvious in nature, gives rise to some of the most profound concepts in science. 

No one would think for a moment that additional heat from the room would spontaneously transfer to an already scolding cup of coffee.  That would be like an already wealthy person taking additional funds from someone who is financially constrained.  Though such an occurence is not unsual within society, it is a road seldom taken by our equitable mother nature.  The second law states that such a clear disregard for equilibrium is so improbable, that it is illegal.  In nature, everything searches for equilibrium. 

Let us take the air in a classroom for example.  What are the chances that all of the particles that make up this volume of air will spontaneously shift to one side of the room, leaving a vacuum on the other side of the room (and causing terrible consequences for those students left in a vacuum of space)?  The probability of this occuring is so low, that we may simply conclude that it is nil.

Let us now reverse the problem.  If one were to manually displace all of the air particles in the classroom to one side, what would happen if they were set free?  If the air were left to its own devices, it would immediately try to balance itself in the given space.  The air would naturally disperse to fill the volume with air of equal pressure and temperature (assuming that the boundaries encompassing the space are uniform).

One might refer to the original state, where all of the particles are grouped together at one side, as 'organized' or orderly.  The second situation, where the air is evenly dispersed, could then be deemed disorganized or disorderly.  We could even call this dispersed situation 'random', or even better, 'chaotic'.

It turns out that scientists like to measure everything, including randomness.  In thermodynamics, the quantity of chaos is called entropy.  A very chaotic, disorganized state, is one with high entropy, whereas a very organized system has low entropy.  Entropy is a measure of energy per unit temperature, or Joules per degree Kelvin (J/K).

Returning to the dispersing air within a room, we could conclude that the original, organized state had little entropy, while the random state, that it naturally tended towards, is one of high entropy.  So, the notion that heat moves from hot bodies to cold ones leads directly to the idea that the entropy of a system increases with time.  A kindergarten teacher can leave a room full of children calmly sitting at tables colouring, and return to complete pandemonium in just a few minutes.  A closed system tends towards chaos: it is the second law.

So, how does this relate in any particular way to the Autumn season?  I'm glad you asked.

Last week, I was raking leaves with my two-year-old daughter.  After organizing the leaves into distinct piles, it was time for her to jump in them.  After a half an hour of fun, I wanted to place the leaves in bags - my daughter, a lovely but at times possessive little girl, objected to the removal of her piles.  I conceded, and left the piles of leaves as they lay, with the intention to return to bag them at a later time, when one particular little girl was napping.

When I did return with bags, I noticed that the orderly piles had become randomly distributed around the lawn by wind.  The organization of the leaves decreased, that is, the system of leaves increased in entropy.  The opposite would never happen.  One would never expect the wind to organize a randomly dispersed collection of leaves into clean piles.

As I raked the leaves into piles, again, and finally disposed of them, I thought of the second law of thermodynamics.  The only way to bring about an increase in the organization of a system is to interfere with it.  However, we must realize that when we do so, we become part of the system.  The total system, consisting of the leaves, the rake, and my body, experiences an overall increase in entropy.  I may, philosophically speaking, decrease the entropy of the leaves, but my concerted, non-spontaneous effort to do so requires an input of energy on my part, which causes my body temperature to increase.  Overall, there is a net increase in entropy; there always is.

When I originally introduced the second law of thermodynamics, I stated it another way: no system can be 100% efficient.  It turns out that a system that gains entropy has an inherent inefficiency associated with it.  And since every system gains entropy with time, no system is perfectly efficient.  This gives rise to yet another statement of the second law: there is no such thing as a perpetual motion device. 

Every system has at least some loss associated with it.  No contact is perfectly frictionless.  A pendulum swaying back and forth will, after enough time, come to rest.  Its mechanical energy will eventually all be converted into heat.  The notion that no body can undergo perpetual motion is easy to relate to - we all need to rest.  One spinoff that is not so rosy, but is equally true, is that nobody can live forever.

But you know, the second law can work for us under certain conditions... The next time someone calls you lazy, just remind them that there is no such thing as a perpetual motion device.

The final statement of the second law is easily the most compelling, as it imposes a constraint on time itself: time can only flow in one direction.   

Imagine you are shown two pictures of a billiards table.  The first picture shows fifteen coloured balls organized into a triangular shape with the white ball on the opposite side of the table, while the second has all of the balls placed randomly around the table.  You could confidently ascertain that the second picture was taken after the first without even watching the game.  How did you know?  Since an increase in randomness occurs naturally as time moves forward, the second law goes so far as to stipulate the direction of time's arrow.

Most principles in science have a certain symmetry to them, that is, they look the same no matter how you view them.  The second law of thermodynamics is special, because it implies that things move in one direction only.  The notion of permanent change is rather unsettling, but we must accept the fact that our universe will have more entropy tomorrow than it did today.  The second law proves that in life, there is no undo button.

The harsh reality is that the decisions we make in life cannot be unmade.  And, if left unchecked, things naturally tend towards a big mess.  It may take months to paint a masterpiece, carefully placing each grain of colour in just the right spot.  It takes only minutes for the painting to be ruined by rain if left outside. 

Thankfully, we can restore order to chaotic situations if we are willing to put in the work.  Such actions raise our body temperature, which is welcomed now, but is embraced even more during the season that lurks around the corner.

A few days ago, at around 8:00 pm, my 17-month-old daughter was standing in the living room, when she suddenly began to spin herself in circles in an effort to induce dizziness and fall down.  She seemed to enjoy it a lot.  After a few nice, relatively harmless tumbles, she eventually made herself so dizzy, that she fell, smacking her face directly onto the hardwood floor.  At that moment, the “witching hour” ended with a brief fit of crying, and it was time for bed.

As a parent of a toddler, I am beginning to see the importance of recognizing toddler fatigue.  The witching hour is an unofficial domain of time where an otherwise normal child goes bananas.  Imagine how a prison inmate gradually loses his mind in isolation over a period of months, and condense that experience into one hour, and you begin to depict the decline of a toddler in the evening.  For my daughter, this strange yet entertaining timeframe seems to occur between the end of dinner and bedtime.


After my daughter’s spinning/dizzy/falling routine, I began to see how her experience related to an elastic material in gradually increasing tension.  Her final outburst of tears was representative of the ultimate failure of a metal rod, when it finally cracks into two pieces.  If only I could predict it with accuracy... Well, with the help of introductory material science, I think I can.

Material science is a very complex field, and has been the most important driver for the technological progress of man.  Wood has always been useful for shelter, aluminium made the bicycle feasible (high strength to density ratio), carbon fibers have been used in various sport and aeronautic applications (higher strength to density ratio), and carbon nanotubes may one day make the Space Elevator a reality (perhaps the highest strength to density ratio possible). 

Most materials are elastic in nature; that is, they deform proportionally as increasing tensile load is applied across them.  Let us focus on metals, as these materials are representative of toddlers, particularly when they are stressed, or tired.

Metals dominate the periodic table as toddlers overwhelm their parents.  Metals conduct both heat and electricity very well, and toddlers are tiny balls of energy.  Like small children, metals come in many varieties, each with their own unique overall properties.  It is the similarities between the mechanical properties of metals and the evening behaviour of young children that I would like to focus on.

Many different metals are used for practical applications: iron, copper, bronze, aluminium, titanium, etc.  Each of these metal types composes the majority of several different alloys, like steel, which is about 99% iron, with some carbon mixed in.  The alloys are usually prepared in a high temperature state, and are cured (transformed into a solid state) at a lower temperature for a certain duration.  This process is actually quite complex; it is an art, like cooking. 

Material scientists who prepare the recipes for these alloys must consider the effects that time spent at various temperatures will have on the atoms of the metal.  In the end, the alloy will exhibit many different qualities.  Its mechanical properties are mainly defined by its “stress-strain” curve, which the remainder of this article will ponder.

If a metallic rod of a given length were gripped tightly at both of its end and gradually pulled apart, two measurements could be taken at any point in time.  We could measure the applied load (tensile force) passing through the rod.  This value, in units of Newtons [N], when divided by the area of cross-section of the material in square metres, will tell how much tensile stress is in the metallic rod at that moment in Pascals [Pa].  At the same moment in time, we can verify the corresponding elongation of the rod.  If the rod was initially 2 inches long, and after some time, was stretched to 2.001 inches, the elongation would be 0.001”, and would correspond to a strain of 0.001"/2" = 0.0005 and percent elongation of 0.05%. 

If we take many readings during what is often referred to as a “pull-test”, we can place them on a graph, and call it the stress-strain curve for the given metallic alloy that was subjected to the test.  A typical stress-strain curve for a ductile metallic alloy is shown below.

Figure 1: Typical Stress-Strain Curve for a Ductile Metallic Alloy

There are several observations we can make from the stress strain curve shown in Figure 1.  The elastic region is that between points 1 and 2.  This region is defined by its linear shape.  The constant slope in the elastic range is the Elastic Modulus, E, of the material.  Something critical happens at point 2: the material yields.  The stress of the material at this point is known as its yield strength.

After this point, the ductile nature of the material can be seen (a brittle material snaps in two shortly after it yields, and has a simpler shape).  The behaviour of the inelastic or plastic range of the material is seen between points 2 and 4.  The yield strength is like the point of no return for the material; if the stress is released before this point is reached, it returns elastically to its original length, whereas any deformation beyond the yield strength results in permanent deformation.  If the stress is released at point 3, the rod will be slightly stretched (0.2%) in its unstressed state.  The elongation of the rod is greater for a given stress increase in the plastic range than in the elastic range.  Eventually, the applied stress is too great for the rod, and it fails, breaking in two.  The stress at which this occurs is known as the ultimate strength of the material (point 4 on the graph).

For a toddler, between say, 7:00 pm and 8:00 pm, a similar graph exists: it is the graph of child chaos versus time.  Below is the rough chaos-time curve for my daughter.    

Figure 2: Typical Chaos-Time Curve for a Toddler

The shape of the chaos-time curve for a toddler in the time domain between the end of supper (7:00 pm) and bedtime (8:00 pm), known as the witching hour, is clearly similar to that of the stress-strain curve.  Note that the minimum value (unstressed state) of chaos does not correspond to zero chaos, but rather some inherent chaos level that the parent is accustomed to. 

After 7:00 pm, the child begins to get more hyper at an astonishing rate.  By 7:10, she is running laps around the living room.  At around 7:15 (point 2), she yields, at her yield chaos value, Cy.  Beyond this point, her chaos increases in a less predictable way; she gets loopy.  She invents new words, new games, as she exhibits her ductile qualities.

At around 7:30 pm (point 3), she begins to spin herself in circles.  Now would be a good time to consult the chaos-time curve (CTC) and observe that the end may be near – she could go at any time.  We are in ultimate chaos territory.  Now would be a good time for the parent to intervene, so that ultimate chaos occurs in a more controlled environment (bath time?).  A few nights ago, ultimate chaos, Cu, occurred around 8:00 pm, like clockwork, but my daughter was not in a bath, and was instead sprawled on the floor wishing this daily acid trip would stop.

Every child is different, and if you have one, you might want to consider charting his or her CTC.  You could then calculate his/her Elastic Modulus in units of Chaos per minute (C/min), and both yield and ultimate chaos values.  These are all unique to toddlers.

Fortunately, children are so ductile that they bounce back the next day.  Unlike a material, which, when extended beyond its elastic range undergoes permanent deformation, the changes our children experience are temporary.  As a parent, the best we can do is to manage this timeframe well.  We can choose an appropriate environment to put our children in during the witching hour, or maybe treat ourselves to a stiff drink and just enjoy the show. 

In the end, it is usually quite entertaining.  And, we can seek comfort in the knowledge that they will wake up tomorrow morning in their regular state of chaos, which has by now become welcome.  Maybe we can figure a way to be busy during tomorrow’s critical chaos-time zone and have our spouse or a babysitter be the one to supervise it.

We are not all material scientists, but many of us are parents of toddlers at some point in time.  We should not treat our young children as metal alloys, but we can appreciate that as the sun sets, and the moon becomes bright in the sky, our children begin to behave like metal bodies in tension.