Cool drinks and the inevitable arrow of time

Page 2 of 2

Let’s see what entropy means in practice by considering why a gas expands to fill a vacuum. A gas is made up of molecules bouncing around randomly. If we were able in some manner to take ‘a picture’ of where each molecule of a gas was located at a particular instant, we would have an image of what physicists call a ‘microstate’. While we can’t really ‘see’ the positions of the molecules of our gas in this manner, a description of gas state can be given in terms of large scale properties such as temperature, pressure and volume – a description given according to these properties is called a ‘macrostate’.

To aid our understanding let's construct an imaginary experiment – what physicists call a ‘toy model’ so we can better understand what’s going on and be clear about microstate and macrostate. Imagine a rectangular container, like a fish tank but closed on all sides – and imagine that it is divided in half by an airtight partition. Since we’re trying to make things simple for ourselves, let’s fill the left half with a gas but imagine our gas only contains ten individual molecules – as opposed to the trillions that would be present in a real experiment with real gas.

At the start of the experiment, all of our ten molecules of gas are on the left of the partition. Now let us remove the partition and wait for a few seconds.

Most of us would agree that we will find that the molecules have spread out through our container – and we would probably all intuitively realise that our imaginary gas in the box will now have a different pressure. We know that before removing the airtight partition that the sealed portion of the box on the left had a particular pressure – and that the sealed portion on the right, being empty, had no pressure. Scientists would say that the macrostate of the box has changed. But what does this tell us about entropy? What about microstate?

Since each of our ten molecules are moving randomly, each now has a 50% chance of being found on either side of the container. Since there are 10 molecules, there are 1024 different combinations of which side of our box we might find our imaginary molecules in. Each one of these represents a different microstate (Remember? Microstate is a description of the location of the molecules in our box). Just for a moment, for those of us slightly less mathematical, let’s have a look at these simple charts

Mirostates possible for a box with only ONE molecule

	molecule	left	right
microstate 1	1	x
microstate 2	1		x

Mirostates possible for a box with only TWO molecules

	molecule	left	right
microstate 1	1	x
	2	x
microstate 2	1	x
	2		x
microstate 3	1		x
	2	x
microstate 4	1		x
	2		x

If we continue the mathematical trend we can see that the amount possible microstates for our molecules grow in a similar manner.

number of molecules	1	2	3	4	5	6	7	8	9	10
number of possible microstates	2	4	8	16	32	64	128	256	512	1024

Returning to our experiment, our ten randomly moving molecules can potentially be found in 1024 different combinations of which side of the container they are located and, crucially, all of them are equally likely.

If we measure the pressure in each half of the container simultaneously, what are we most likely to find? Again, our common sense would tell us that the pressure would be equal on both sides. But let us see how this situation arises from the movements of our ten molecules.

Let us consider first the probability that we will find the pressure unchanged in the left-hand side of the container and perfect vacuum on the right. For this to happen, every single molecule would need to be on the left, which can only occur in one way: with molecule 1 being on the left, molecule 2 being on the left and so on up to molecule 10 being on the left. The probability of this happening is therefore 1 in 1024, or 0.098%. How about finding nine molecules on the left and one on the right? Well, this could happen in ten distinct ways: molecule 1 could be on the right with molecules 2 to 10 being on the left, molecule 2 could be on the right with all the others on the left etc. The probability is therefore 10 in 1024, or 0.98%.

It becomes increasingly laborious to find every microstate corresponding to a particular number of molecules on each side, however, mathematics tells us that our randomly moving gas molecules can arrange themselves in 252 different ways that result in 5 on the left side of our box and 5 are on the right. Therefore, out of the possible 1024 different microstates there are 252 in which the molecules are arranged five on each side of the box. At 24.6% (252/1024) this is the highest probability of all the possible arrangements of the molecules.

Summary of our toy model – and entropy!

From the instant that we removed the airtight partition our system moved from a state of low entropy - all ten gas molecules on one side of the box – to high entropy – 5 gas molecules on each side of the box. This occurred due to a number of different factors

1. a group of molecules in a given volume will move randomly
2. at any given point in time there is a higher probability that randomly moving molecules will be found evenly distributed through a given volume
3. when molecules are evenly distributed through a given volume the system can be said to have attained maximum entropy

In fact, our toy model was slightly misleading, since, with only ten molecules, we expect to see quite significant random fluctuations of the number on each side. But with trillions of molecules, random fluctuations are completely undetectable, and so we see absolute uniformity of pressure. This is true of all isolated systems, and is embodied in the Second Law of Thermodynamics: the entropy of an isolated system will never decrease.

Back to ice clinking in glasses

Although we have been describing a gas, entropy works the same way for any type of material, including your icy drink. While the ice and water in your glass is relatively cold, the temperature of the air around it, especially in the hot summer months, is much warmer. The randomly moving molecules in the air will attempt to spread out to the cooler ice and water. Eventually, the entropy of the ice and water will increase, the ice will melt to water, and finally the water will warm to the same temperature of the surrounding air.

Entropy in other systems

The most profound consequence of the Second Law is that in an expanding universe the entropy of the Universe can never decrease. Can it stay the same? In theory, yes. In practice, however, from the Sun heating the Earth to the heat oozing from your computer, the Universe is full of hotter bodies heating colder bodies. The entropy of the Universe is constantly increasing, and has been since the moment of the Big Bang.

Living things are subject to entropy as well. The body of a living organism is a very low entropy system. Like any low entropy system, any organisms require a constant input of energy to maintain its low entropy state. In a plant, this energy is provided by the Sun. In an animal, it is provided by eating plants or other animals. And, like most machines, living organisms (particularly warm blooded ones) are woefully inefficient – humans convert about 80% of the energy they take in to heat to maintain their bodies at the optimum operating temperature. Since animals produce this heat by burning the energy stored in other low entropy systems (i.e. the bodies of plants and other animals) they are constantly raising the entropy of the universe. They cannot exist any other way.

Another consequence of entropy is with something that is often called the ‘arrow of time’. Most obviously, while we can look in any direction we choose through space, we can only see in one direction through time -we can remember the past but not the future – and each moment, as it passes, is irreversible. Scientists, however, find definitions based on human experience unsatisfactory, and seek an arrow of time based solely on the objective laws of physics. Can they find one? Yes, but only one. The only physical law which describes irreversible behaviour is the Second Law of Thermodynamics (entropy always increases in the forward direction of time). This is known as the thermodynamic arrow of time.

If we wish to explore the relationship between the psychological arrow of time and the thermodynamic arrow of time, then we should avoid such terms as “past” and “future”, which, after all, have no meaning without an arrow of time. Our question regarding the Second Law of Thermodynamics could then be something such as “the human memory can see in the direction in which the entropy of the universe decreases but cannot see in the direction in which it increases.” How can this be explained?

What is astonishing about entropy and the Second Law of Thermodynamics is the scope and influence of a concept which is appears quite simple on the outside, the movement and distribution of gas molecules within a box.

For more information

2nd Law
www.2ndlaw.com/

Joel Leibowitz - Penn State - The Arrow of Time
http://www.rps.psu.edu/time/arrow.html