Introduction

        The Kinetic Theory of Matter is little more than a hundred years old. Its basic ideas have been developed in the later half of the nineteenth century and reflect one of the more significant ideas ever developed about the nature of the physical world. Until this time, the basic nature of matter could be described in terms of properties of continuous mediums. All materials could be considered as either solids or fluids with the fluids including both gases and liquids. We use the same descriptions today for studies of the macroscopic properties of matter. The basic properties of solids, such as length varying with temperature, friction, density, etc., all very well worked whether we had a continuous or discrete basis for materials. Similarly, if we explored the nature of fluids, whether a continuous medium, or if discrete particles gave rise to the descriptive properties, the viscosity, volume, pressure and other quantities were all adequate for the kinds of questions asked. The gas laws, Boyle's Law, Charles' Law and Guy-Lussac's Law all described gases very well. They agreed very well with observations.

        The Greeks of course gave us the word "atmos" describing the most elementary form of matter, or the smallest indivisible part of matter. Little did they know, that atoms as we know them, existed. Rather it was the consensus that matter was in fact continuous and the major difference between various forms of matter was the viscosity. That separated solids from fluids. Solids did not flow.

        Several scientists, however, suspected that perhaps there was more to it. They felt that matter was comprised of tiny particles which we today call atoms. Chemists made major contributions to this belief, but there was not physical evidence, nor any idea as to the nature of atoms themselves. Major contributions were made by three scientists, Ludwig von Boltzmann, Willard Gibbs, and James Clerk Maxwell. (Maxwell is the same one that gave us Maxwell's Equations describing all that we know, even today, about the electromagnetism.) These scientists demonstrated, not proved, that discrete particles could, in fact, make up all of matter in such a way that all the macroscopic properties could be accounted for. They showed that it was possible and later, others demonstrated convincingly, that atoms actually did exist. The next unit discusses the nature of atoms and all the weird concepts that the quantum world has associated with it. Many scientists still would not  accept the kinetic theory as being correct and valid only as a mathematical exercise that was "nice", but worthless for practical applications. One such scientist was Ernst Mach. He was a positivist who influenced many others of his day (including Einstein.) He said that if you can't see them (atoms), they don't exist.

         It was Einstein who, explaining the phenomena of Brownian Motion, finally broke this resistance to the existence of atoms. (This was the first of Einstein's three"great" papers of 1905.) Brownian Motion describes the random motion of tiny particles of smoke, dust, or fine pollen grains in thermal equilibrium in a gas or liquid. They are in statistical equilibrium of the atoms and molecules of the fluid in which they are suspended. Atoms
were too small to be seen. (we often measure not quantities directly, but their effects on other quantities in nature.)  It was the English Botanist, Robert Brown who first observed the movement of pollen grains in a gas or liquid. Einstein suggested that the movement was much like a basketball being hit by a gas of tennis balls. Eventually it moves around. By the way, even though the rest of the world seemed convinced,
Ernst Mach maintained his "incorruptible skepticism" to his death.

        In the next chapter we explore the nature of atoms. But at this point we must first convince ourselves how it is reasonable that atoms do exist, and that the large scale, macroscopic properties of matter, can be attributed to the discrete or particulate nature of the world. Lucrecious, the Roman philosopher, pointed out that atoms themselves must exist even if we cannot see them. For a wedding band grows thin over the years and theatoms of gold disappear. The idea of the existence of atoms is not new. Richard Feynman, Nobel Laureate in physics, and arguably one of the greatest lecturers regarding the nature of the world and why it works as it does once said "If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what
statement would contain the most information in the fewest words? I believe
it is the ATOMIC HYPOTHESIS (or the atomic FACT, or whatever you wish to call it) that ALL THINGS ARE MADE OF ATOMS--little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an ENORMOUS amount of information about the world, if just a little imagination and thinking are applied." ["The Feynman Lectures on Physics", lecture 1, p. 2.]

Feynman asks: "How do we KNOW that there are atoms? By one
of the tricks mentioned earlier: we make the HYPOTHESIS that there are atoms, and one after the other results come out the way we predict, as they ought to if things ARE made of atoms." (Note that this is exactly what Popper, as part of his positivist doctrine of "falsification", DENIES that it possible and
proper for scientists to do--i.e., CONFIRM any scientific theory this way!)

Feynman goes on to mention some of the specific pieces of evidence, including the Brownian motion of tiny chunks of matter being bounced around by individual collisions with atoms or molecules--which Einstein finally gave a
completely convincing explanation for in 1905.

Since the days of Newton the world had been viewed as strictly deterministic. While Newton only knew about gravity and had no knowledge of the other three known forces, he was convinced that the world was completely deterministic. Given knowledge about a system, we should be able to calculate the entire future (as well as the past) exactly. Any deviations were merely due to lack of complete knowledge about the systems. But we have seen that probability is crucial in describing what goes on and that really large systems are not so deterministic. The best we can do is describe averages and the probability plays a major role in this description.

We have to ask ourselves just how probaility and determinism are related and what role probability plays in how the world is structured. We make use of probability theory to demonstrate what state (or set of characteristics) a particulate system is most likely found in. Indeed, we do not see its significance on the everyday scale we observe. But it is a necessary tool to describe how the laws of nature dictate that the particles making up our world almost "must" be in certain states.

The Five States of Matter

        We will look at a basic model of matter and then discuss in a rather detailed manner, the kinetic structure of an "ideal" gas. Do not be alarmed by the seeming complexity of some of the derivations. They are not included here to threaten the reader, nor is it expected that the reader should be able to derive or replicate such results. Rather, some details are provided so that the reader can be convinced that the results are valid and make sense, as well as to appreciate the significance of what the world is really like. The processes involving statistical mechanics provide the means for averaging over a large number of particles, many of the discrete quantities of speed and energy, resulting in large scale effects of temperature and pressure. The journey through this "material" itself is a process, but one we can handle, and one that is enjoyable.

        First we look at the several states of matter. To do this, let us consider two kinds of mechanical energy. Potential energy is the energy of position. Suppose we lift a book up off of the table. If we drop it, it gains speed and makes a disturbing sound as it meets the table again. It had energy because of its position relative to the table. When it was on the table it had no potential energy, but when lifted it had positive potential energy. When that book was released from a height above the table, it gained speed. Because it had speed in motion it had a special kind of energy, called kinetic energy or energy of motion. These two basic kinds of energy, and their relative strengths, can be used in a simplistic model of the states of matter.

        Let us model the five states of matter. The first is a solid. We are familiar with solids. They are rigid and do not deform very easily. If we represent it as a periodic structure of atoms, then there must be very little motion (kinetic energy) of the atoms within the material and a relatively high degree of potential energy (holding them in place.) Because the atoms themselves, whatever their nature, do not readily move around, although they may vibrate back and forth in position, we say that they have a relatively low degree of kinetic energy. Thus we can characterize solids as a set of particles (call them atoms) in a fairly structured arrangement. In this system we find a relatively high degree of potential energy holding the particles in position and a low degree of kinetic energy of motion.
 
 
 
 
 
 
 
 
 
 
 
 

Figure 4-1. Solid Phase - Potential energy dominates  over the kinetic energy of motion. Particles are held rigidly in place although free to vibrate in place. Remember that this is strictly a model.
 

        Liquids on the other hand have some degree of potential energy in that the particles making it up do not have complete freedom of motion throughout the system, but instead are held soemwhat in place. Yet, the particles do move about. We find here competition between the two kinds of energies, potential and kinetic energy. While we have not studied in detail a complete description of mechanical energy (potential and kinetic) it will be adequate at this point to say that potential energy is the amount of energy an object possesses strictly because of its position. In matter, individual components (atoms) are held in place by their potential energy. And the kinetic energy is the energy of motion.

This is a reasonable way of looking at it if we want to accept the "caloric" model of energy, in that we can add or subtract discrete amounts and those amounts are distributed in potential and kinetic amounts. We know today that the caloric model is not quite right, but it is a simple model that works quite well to explain rather gross aspects of energy itself. It is not our intent here to describe everything about energy and the energy content of matter so let it suffice to look at it this way.
 
 
 
 
 
 
 
 
 
 

Figure 4-2. Liquid State - kinetic and potential energies are in competition for dominance in the system. Some motion and vibration are allowed, but certainly not completely random movement. Particles are still constrained.
 
          The Gaseous phase can be considered essentially the opposite of the solid. Here we find the particles in what we call random motion, not at all structured in position. They are constrained only by a container. They are free to move back and forth in the system, colliding with each other in a random manner. Their velocities and momentum are determined by the temperature of the system. This means we have to define temperature. Together with the probability description, temperature and another quantity entropy, are two state variables that describe a system. We will have to explore how they are related to energy.
 
 
 
 
 
 
 
 
 

Figure 4-3. Gaseous Phase - Characterized by a high degree of kinetic energy (relatively) and low degree of potential energy.
 

        There are two other phases to consider although they do not at all represent what we consider the normal states of matter. The fourth phase of matter develops when enough energy is given to the system so that electrons are literally stripped off of the atoms themselves and the matter exists as a plasma. It is called a Plasma and is the kind of "gas" we find within the interior of the sun and other typical stars. This plasma obeys normal gas laws, but the actual form of the gas does not consist of atoms, but rather of a gas of electrons and ions (atomic remains when electrons are stripped off) can be said that there is actually one additional state of matter. (Perhaps we might even want to add another in between they plasma and this state when we learn more about the actual nature of atoms and nuclei themselves.) This state could be called Primordial Soup, a system of particles, no longer atoms or even nuclei of atoms, but a mixture of the basic building blocks of the "things" that go together to make the simplest elements of matter. We explore matter in much detail in future chapters and look at it's ultimate nature. We will recall this soupy state later.
 

Ideal Gas Model

        Let us develop a model in more detail for an "ideal" gas and show how the link is formed between the basic atomic structure of the gas and the fluid characteristics we commonly observe. Remember that the big idea developed with the statistical mechanics is that it shows how a particulate picture of matter on the microscopic level (atoms) can give us the macroscopic properties commonly used to describe that state. It demonstrates that atoms themselves can exist and give these results. To do this we first establish the basic requirements of the model of an "ideal" gas. (Rigorous details may be found in any classical Physics text.)

        The ideal gas consists of a system of hard spheres that move about in random motion. They form a gas that is dilute; that is, there are not so many particles that they interact with each other in such a way that the pressure and other quantities are affected. They collide with the walls of the container and each other in an elastic way. i.e., there is conservation of energy in the collisions. We need to consider the gas on a statistical average where individual particles lose their identity. In fact, at any instant, a particular particle may be going faster or slower than the average with more or less energy and momentum than the average. We will see what the statistical averages tell us about the gas, and how much individual particles deviate from the average values of the quantities that describe the gas.

As per a Newtonian description we would have to characterize all the particles and add their effects together. This sort of determinism no longer works. Indeed, according to chemists, a mere 22.4 litres (a little bigger than a five gallon jug of water that fills most water coolers) contains Avogadro's number of particles.This is roughly 6 X 10^24 particles or molecules of a gas. A reasonably sized classroom may contain  a thousand of these "jugs" or a total of 6 X 10^27 molecules of the gas. That's quite a few atoms!

Now, all the computers in the world would not have enough memory. there are something like a billion computers and each had 100 GB (GigaBytes) of memory that still is not enough to track even a fraction of the particles.So Newton's completely deterministic world fails on the microscopic level.

Figure 4-4. Model of an Ideal Gas. We ignore collisions between particles for the following derivation, but their effect is merely averaged in all three directions.
 
 
 

The actual derivation is more complex than the following, but this non-rigorous development will help us to explain how tiny particles interact and affect large scale results. The reader does not have to even understand the derivation itself but rather know it is straightforward and done in freshman physics texts. Let L be the length of the container. Then the rate of bombardment is proportional to N particles in the box and their speed. We can write the number of collisions with a wall in a given time interval by:
 

We assume that each atom or particle that impacts the wall gives some momentum to the wall:

where m is the mass and v is the speed. Momentum is a basic characteristic of motion, relating both the mass and velocity of an object. It is interesting to note that Newton's second law defines an applied force in terms of the time rate of change of the momentum of an object upon which it is applied. It can be shown that the force on the wall is merely the change in momentum times the rate at which particles collide with the wall.

Force can be written:

Then

where <1/2 mv²> is the average kinetic energy of the particles in the system. Kinetic energy is the energy an object possesses due to its motion. Energy is the ability to do work. Because an object is in motion it is capable of doing work.

The correct result looks like:

as averaged in three directions, x, y, and z. These are statistical averages. This model does not prove that atoms exist, but it does demonstrate that it is possible to explain all macroscopic phenomena by assuming that matter is made of atoms and-molecules.

        Pressure is defined in an elementary manner as the Force applied over an Area:

We should recognize that Area X Lengthis merely volume:

P = 2/3 N/V <1/2 mv2>  (4-7)
or
P V = 2/3 N <1/2 mv2> (4-8)

We see that this looks familiarly like the "Ideal Gas Law". Recall, from earlier science experiences, Boyles Law, which showed that pressure was inversely proportional to volume. i.e., at constant temperature as the pressure increases, the volume decreases. Thus, as the volume increases, the pressure decreases and so forth. Charles Law states that at constant volume as Pressure is increased, so is the temperature and vice versa. Combining all of the gas laws into a general form is the Ideal Gas Law:

P V = n k T  (4-9)
 

where n is the number of moles (a chemical term for a certain amount of material) and k i23 a constant, called Boltzmann's Constant (k = 1..38 x 10 ) This gas law had been used to describe actually observed relations between the variables of Pressure, Temperature and Volume. They are macroscopic characteristics. We can call these state variables.
 

         If we relate this to the microscopic picture derived from the model introduced already, we get the result which links the macroscopic properties to the microscopic ones:

From this we can see that temperature T is not a form of energy itself, but is a "measure" of the average kinetic energy that the particles in the gas have. We need at this point to more deeply examine what we mean by this statement.

Temperature is not the energy of the system. But it tells us something about the energy of the system. If we have a hot tub full of bubbling hot water (not boiling, but hot enough for us to relax in it, certainly above body temperature, say 105 degrees F) we can say that it does have an amount of energy. It is possible, under some circumstance, to extract that energy to do work. If on the other hand we have a cup of hot water (say it is at 190 degrees F) , ready to insert a tea bag into, we know it also has energy in it. The total amount of energy (call it heat energy) of the hot tub is a lot more than that of the cup of water prepared for tea, even though the tea water is hotter. Temperature is NOT energy.

If U is the total energy of the system, then:
 

which makes sense as the total number of particles X the average energy per particle. Then we can also relate the Pressure and Volume to the energy content of the gas:

        This indicates, with the microscopic perspective that in fact the Pressure and volume are related to the internal energy of the system. In fact, we can relate the macroscopic as well as microscopic state variables of the system to energy. That very useful concept, suspected in part, is confirmed by linking the microscopic and macroscopic worlds.

        An extension of this model involves considerations of the relationship between temperature and the average kinetic energy of the system. Recall that the model is based on the collisions of spherical objects. We examined the motion in one dimension, but in fact the particles randomly move in three directions, x, y, and z dimensions. The 3/2 kT is considered, on the average, to be split equally in each of the directions, or 1/2 kT for each direction. This is called the Equipartition of Energy. In calculating other quantities for the ideal gas (some that we do not consider here) such as Specific Heat Capacity, we find that monatomic gases (in which the molecular structure of the components is that of one atom) very closely approximate the values predicted by the model. In diatomic systems (particles are molecules with two atoms) there is additional freedom of motion. Not only can the particles translate back and forth in the x, y, and z directions, but they can vibrate back and forth along an axis joining the molecules, and can rotate too. The Theory of Equipartition of Energy permits additional energy for these extra degrees of freedom. Experiment agrees well with the model here. It also does for polyatomic (many atom) models too.
 
 
 
 
 
 
 
 

Figure 4-5. Diatomic Model of an Ideal Gas. Additional Degrees of freedom must be considered for vibration and rotation. Because the moment of inertia is small for rotation about an axis joining the two molecules, the contribution is small compared to that of rotation about perpendicular axes.

A Degenerate Model

        Let us develop our model further and explore a special outcome that is not described by observable properties. This is where probability and that elusive quantity, entropy come in. Suppose that the particles of out system have two possible states. (Real systems are more complex and have more possible states.) This two state system could easily be represented by a system of coins. Each coin on a table can be either heads up or heads down. Those are the only allowed conditions. We call such a system a binary system. We need to look at the ways we can combine particles in combinations of heads and tails. Suppose we have just 2 particles in our system. Let arrows represent heads up or down and the subscripts the respective particles. We can have the following possible combinations of heads and tails:
 
 
 
 
 

Figure 4-6. A Two Coin System. Coins may be heads up or heads down. The arrows represent status of Heads, up or down.

         There are four different ways to combine 2 coins, but there is only 1 way to have both heads and only 1 way to have both tails. There are 2 ways to have one head and one tail. Suppose we have a system with three coins. As we explore the .various combinations, let us consider a quantity called Head excess. This is the number of heads minus the number o tails. onsider the following three coin system. We now show how the three coins may be combined to give combinations of heads up and down. The number of heads and tails are shown and the values of head excess, s, are given in the right hand column. Charles Kittel (see Rittel and Hroemer, Thermal Physics) develops a similar model using dipole magnets (North and South Pole) magnets.
 
 

        From looking at these arrangements, we can see that it is three times more likely to have an arrangement with a head excess of +1 or -1 than of + 3. Note also, that there are exactly 8 total ways to combine them. There is only 1 way to get all heads. There is only one way to get all tails or 0 heads. But there are 3 ways to combine them with 2 heads and 1 tail. And there are 3 ways to combine them with 1 head and 2 tails. Recall that for the 2 particle system there are exactly 4 ways to combine the particles. Note that this is a system described by statistics and probability.

      We define a quantity, called the degeneracy of the system, and denote it by d(N,s) which describes the number of ways to arrange the system of N particles and a certain head excess, s. The variables enclosed in parentheses are descriptors and determiners of the functional relation, degeneracy. In real thermal systems it describes the number of states with the same energy, or more realistically, the same energy and within a small range of this value as well. Let us look at Figure 4-7 again, but this time, list the degeneracy as well. At this point it is appropriate to change the terminology to use spin excess instead of head excess since it is the proper term to ascribe properties of atoms. Atoms are characterized by a condition called spin in which the electrons act as if they were, in fact, spinning on their axes in an upwards or downwards direction. So while this model is simplistic, it does have real world applications which are quite sophisticated.
 
 

It can be shown by mathematical induction that the number of ways to combine heads and tails is:

 

         So, a 2 particle system has 2² or 4 ways and a 3 particle system has 2³ or 8 different arrangements. We can also show that a system to generate the possible combinations can be developed from the binomial  expansion. The  reader can consult any Algebra text or set of Mathematical Tables to verify this result. The exclamation mark after a number means factorial, or the number itself multiplied continuously by one less until 1 is reached. For example, the quantity 4! means 4 X 3 X 2 X 1 = 24. Additionally, the factorial of zero is 1 by definition.

        At this point it might be easy to get frustrated, lost or discouraged. Do not let this happen. The reader is not expected to perform the math, but rather to follow along so that there is some credibility to the results. Moreover, it should become apparent that the actual mathematical details are not so difficult that they can only be performed by extraordinary persons. It just takes practice and perseverence and time. It may seem unnecessary to go through this tedious process, but developing the role of degeneracy is crucial to explaining simple ideas like why the very air we breathe is distributed in a random fashion and not configured in some remote corner of the room. It explains in terms of statistics and probability how the world really is! Let us use the binomial expansion (basic algebra) to relate real world systems to a quantifiable state:
 
 
 
 
 
 
 

        Replace the X and Y by heads up and heads down possibilities for the particles in the system:
 

The first term represents the number of ways that zero heads up can be found, only 1. The second term represents the number of ways that two heads can be arranged. And so on. If g represents the total number of ways to arrange the system, then g is the sum of the terms. For a ten particle system we can expand the terms and get (the reader may consult any Algebra text for assistance in the calculation):

        Each successive term represents the number of ways to combine that many heads. There is 1 way to get 0 heads, 10 ways (each of the particles, one at a time) to get 1 head, 45 ways to get 2 heads, 120 ways to get 3 heads, 210 ways to get 4 heads, 252 ways to get 5 heads, 210 ways for 6 heads, and so on. The sum of these arrangements is:
 

Figure 4-9. d(N,s) versus spin excess for a 10 coin system where coins may be found heads up or down. Note the symmetrical shape of the curve.

         Thus there are 1024 total ways to combine ten coins. (2¹⁰ = 1024) Note here that the probability to find the system in a particular arrangement is merely 1 divided by the number of ways to achieve that arrangement. For the 10 coin system, there is only 1 of 1024 ways to have either no heads and just 1 way to get all heads. That is a very small probability, about 1 of 1024 Additionally, note that there are 252 ways to get 5 heads. That's about:

        This indicates about a 1 of four chance to get five heads. Suppose that we consider the chances of finding the system with anywhere from 4 to 6 heads we have:

ways to find the system arranged. The probability is 672/1024 or about 66% probable that one of those values will arise. Now, note that for 3 to 7 heads is:
 

ways to find the system arranged. Thus the probability is  912/1024 or about 89% that the system will be in one of these ways.

        For a very large system of particles, such as the gas of air molecules in the room, we will note that the curve becomes more and more sharply peaked. (Avogadro's Law tells us that there are about 10²⁶ molecules of a gas in this room.) The mathematical calculations demonstrating this effect are complex and not necessary at this point.

The results, however, show that any particular value of degeneracy, d(N,s) can be approximated by:
 

Editing Done to Here
where d(N,0) is the degeneracy or number of ways we can arrange the binary system with zero spin excess.(This is the situation of equal numbers of heads and tails.) Note that this is the most probable configuration for the system since the probability for finding a certain configuration is the inverse of the degeneracy. Don't let the mathematical form trouble you since we will not use it for calculations. You are not expected to remember its notation.

        The purpose of even writing out the complex equation is to show the reader what happens with a large system. The real world, for example, is a very large system. There are about 10^26 molecules of N and 0 in the air in a typical room. (The more mathematically inclined reader can show themselves the following result.) A measure of the spread of the curve is the ratio of s (spin excess) to the number of particles with that value. This can be shown to vary as
 
 
 
 

        Consequently, if N ~ 10²⁶ then the width of the curve is 10^-26 i.e., the curve is very sharply peaked. The probability of finding the system deviating from the random distribution is 1 out of 10 ^N! For Figure 4-9, N is only 10. That is mathematically complicated enough for us at this point. We can then extrapolate our simple results to an extremely large number. See Figure 4-10 for an actual experiment performed with ten coins, tossed 1024 times.  (Actually, ten students tossed ten coins 100 times each and the instructor tossed them the last 24 times.) Note the close correlation to the theoretical outcomes as shown in the Figure 4-9 system.
 
 

       It is interesting to note how constraining the random system for a large number of particles really is. The overwhelming probability that the system will be in the most probable state, that of random distribution explains why particles spread themselves evenly throughout the medium. If in a room we allow a puff of smoke to be released, in a short time it disperses throughout the room. A perfume sprayed in one corner of a room is soon smelled in the opposite corner. It has been said that when Caesar was assassinated, the gasp of air exhaled as he spoke "Et to Brute" in fact consisted of air molecules that we inhale each time we take a breath. The following calculation shows us how completely air would mix.

        Consider that the atmosphere of the earth decreases by a factor of 2 every 5,000 metres (that's about 18,000 feet) in altitude. By 30,000 metres it has decreased to an almost insignificant value. (That's 1/2 to the 6th power or 1/64th of the sea level value, less than 1%.) Most of the air is below this height. The volume of the air is computed in a straight forward manner L~r adding this height to the radius of the earth (6.4 X 10⁶ m) and subtracting out the volume of the Earth: (This is simply subtracting the volume of the Earth from the volume of the Earth with its atmosphere.)
 
 

The total volume of the atmosphere is about 1.55 X 10¹⁹ cubic metres. If a human breath has 1 million billion billion molecules in it, 10²³ molecules, then each cubic metre has about 10⁴ molecules of Casear's last breath in it.  Particles distribute themselves in the random configuration because it is the most probable configuration. It is highly improbable to find a system in any other configuration. This is where probability comes in. What would you bet on?

A Practical Meaning of Never

In 1898 Boltzmann pointed out that "One should not imagine that two gases in a .1 litre container, initially unmixed, will mix, then after a few days separate, then mix again, and so forth. On the contrary, one finds ... that not until a time enormously long compared to 10 to the billionth years will there be any noticeable unmixing of the gases. One may recognize that this is practically equivalent to never..." So for Caesar's final gasp to be mixed with the rest of the atmosphere and then unmix is highly unlikely. In fact, if we take two systems, alike in probability, the total number of states accessible to the system of the whole, or the degeneracy, is the pr2quct of the individual g's. So if two systems each have 10 particles, the basic form of the degeneracy curve is very very sharply peaked and the very random, mixed state is most probable. Secondly, a small -flfviation from this state, say a fraction equivalent to 10 will depend on that relationship given above in the exponential form (Equation 4-20). It can be shown at this reduces the value of the accessible states to e^-93 of its original (maximum) value. This is about 10^-69 of the most probable configuration. We, can say that we will "never" see such a deviation, not even 1 part in 100 billion. Suppose we could change the system 1024 times in a second (rather fast, wouldn't you say?) For 10 particles we are talking about 10 X I.2o- 10 changes per second. Each_j:giange tTyld tabs? 10 seconds. We would have to wait 10 X 10 -10 seconds to deviate by as much as 1 pa in 100 billion: Since the age of the universe is only 10~ seconds old, we can say with confidence that the- event will NEVER occur. This is the practical equivalence of never.

        Aldous Huxley said that "...six monkeys, set to strum unintelligently on typewriters for millions of millions of years, would be bound in time to write all the books in the British Museum." While Huxley was certainly considered by many to be an authority, does this statement really make sense? He tries to indicate that there is a finite probability that a text could be typed, given enough time. We shal see that his estimates are grossly out of line with reality. How many texts could be "created" by random typing for even a very~elvlery long time? Consider the following. Suppose we have 10¹¹ monkeys (a few more than 6, namely 100 billion, probably more than all the monkeys that ever lived, all together, in the history of the planet earth:) Let them sit at wordprocessors for the duration of the universe, 10¹⁶ seconds. Assume the keyboards have 46 keys and they use only capital characters (for simplicity) and that they type 10 characters per second. That may be fast, and no coffee or banana breaks, but it gives us again some very conservative values. Now a small text may have 54 lines (11 inch paper with 1 inch margin at the top and bottom assuming 6 lines per inch standard spacing) at 60 characters per line (pages are 80 columns wides with 10 columns for margins at each side of the page, left and right) , including punctuation and spaces, and perhaps 100 pages. That would require that 54 X 60 X 100 or 324,000 characters be typed in correct sequence. We know that the sequential probability is merely the product. Thus the probability of typing it correctly would be the chance of hitting the right key (1 of 46) multiplied 324,000 times sequentially.
 

 Or
 
 

         Well, 10¹¹ monkeys typing 10 characters per second for 10 ¹⁶seconds produces (product) 10²⁸ characters. The  probability of typing a correct copy of even a small text would then be:

        This again, is another practical equivalent of never. It should intuitively obvious to the most casual observer that there "ain't no way" a monkey could type any text by accidentt

That Strange Quantity Called Entropy

        Now that we are somewhat comfortable with the concept of accessible states we define a thermal quantity that is perhaps the most important concept in Kinetic Theory. It is called ENTROPY. Entropy itself is intimately related to, and in fact, establishes the link between the microscopic and macroscopic pictures of matter: But it is a somewhat abstract quantity in that it is mathematically defined and can be described with a sense of vagueness. It is technically the logarithm of the number of accessible states to a system. Defined in this way, the logarithm of d(N,s), it is not easy to conceptualize. Remember from our experiences in Algebra that a logarithm of a number is the power to which we must raise the base number so that we get the original number as a result. It loses meaning in the translation from the language of mathematics to English, but we attempt to explain it through application.

         We could count the number of states accessible to a system. This degeneracy, d(N,s), for our binary system for example, could merely be the number of ways that we could arrange the system to achieve a specific number of heads or tails. We also recognized that the most probable way we would find a system was the situation with the greatest degeneracy, or d(N,0). In fact, for systems of large N, the most probable set of states was the state the system would be in because even a minutely small fractional deviation from it would "never" occur. We could look at the randomly distributed particles as achieving the most disorder possible under that particular set of circumstances. We identify a measure of this disorder as the quantity ENTROPY. Entropy, does actually describe ow ordered  or disordered a system is. A system with more accessible states has more entropy and hence more "disorder." We see more about entropy and its relationships with the other thermodynamic quantities of Temperature, Pressure and Energy as we state the basic laws of Thermodynamics.

Laws of Thermodynamics

         Let us take two thermal systems and place them in thermal contact. That means they can exchange energy, but not the number of particles. We saw earlier that the temperature of a system defined in the average kinetic energy of the particles in that system. If the two systems in consideration are in thermal contact, then they will eventually reach a state of equilibrium thermally. The zeroth Law of Thermodynamics states that if two systems are in thermal equilibrium with with a third, they are in thermal equilibrium with each other. If they each have the same temperature as a third system, they have the same temperature as each other. This law describes for us the meaning of thermal equilibrium. It further illustrates the role of exchange of energy for two systems in contact and in terms of entropy implies that each state reaches a maximum of disorder, or the most probable configuration available to the systems. Mathematically, we state:
 
 

        The kinetic energy of the particles, on the average is the same for both systems at the same temperature. This does not mean, however, that the particles in each system are moving at the same speed on the average. Consider two systems where one is made of hydrogen gas and the other of oxygen gas. The individual particles making oxygen have masses eight times heavier than the hydrogen molecules. They consequently go slower than the hydrogen molecules of the same energy.

        One should also recognize, although admittedly the connection is more subtle than obvious, that if a state of a system is described by its temperature and temperature is related to energy, that entropy too, must also be related to these quantities. In fact, those quantities of temperature and energy can be mathematically defined by entropy itself. Recall too, that our simple models of matter described earlier in this chapter, are consistent with this idea. The higher the temperature the more energy the particles had and the faster they moved around. More possible states were accessible to the systems.

        The First Law of Thermodynamics is one of Conservation of Energy. To be able to describe this adequately we must assume that we can isolate our system from the rest of the world. It also means that when we try to describe real world systems we must keep is mind what the system really is. It would be difficult, if not impossible, to isolate any system from the rest of the world. Even our solar system is not isolated from the rest of the galaxy. Nor is our galaxy from the rest o£ the universe. It may be that the only really closed system that cannot interact to some degree with something outside of it is the universe. On a macroscopic level we could say that the energy content of the universe is constant. This was first stated by Rudolf Clausius in 1865.

        The First Law has an important consequence, largely due to the nature of heat energy. Heat energy, especially in the microscopic sense, is different from other kinds of energy. We see conservation of energy in many everyday situations, but also we see many systems which are not reversible. For example, if a block slides down an inclined plane we can determine the speed at which it hits the bottom. Some of the energy is used in overcoming friction and is used in making heat and perhaps sound. But we have never seen a block reverse its motion and slide up the plane. We have seen ice cubes melt in our drinks of iced tea on a hot summer day,
but we have never seen a glass of tea consolidate some its molecules into ice cubes. Nor do we grow young. We grow old only. There is an irreversibility of time and certain thermodynamic processes. This is implied in the Laws of Thermodynamics, and, especially can only be reconciled and explained (other than described) with the asistance of entropy and the kinetic theory. The Laws of Themodynamics, in principle, can be described from macroscopic properties, but explained in terms of the microscopic model. We could not explain these observable phenomena from the normal experiences of the classical world. It means that the very notion that matter is made of atoms, tiny particles of nature, in fact gives us a basis for many of the macroscopic processes we observe in nature and could not otherwise explain. It gives credence and meaning to much of the physical theory already developed. It "had" to be correct:

        The natural flow of heat, irreversible as we have seen, is encompassed in the Second Law of Thermodynamics. It is stated in many different, but equivalent ways. In an isolated system, heat flows from a hot to a cold body. There is no way to reverse the heat flaw, except through external methods. For example, we tke heat out of our refrigerator. Of course that makes our kitchen warmer as the heat flows from the isolated system of the box itself into the larger, relatively infinite reservoir of the house. It is the same as saying you can't make water flow uphil. It normally flows downhill, but of course with external input of energy we can pump the water uphill. In terms of entropy we state that the entropy of a systems never decreases. It either remains the same or increases in thermodynamic processes.

        You may have heard the phrase, TANSTAAFL, "There Ain't No Such Thing As A Free Lunch." It applies here too. We cant have any system in the real world, since it is impossible to perfectly isolate it from the rest of the universe, as a 100 % efficient system. It is impossible to make a perpetual motion machine. We always have losses.

        On temperature scales we go lower and lower. The Kelvin scale indicates about 273K at room temperature. It starts at absolute zero. The Third Law of Thermodynamics states that you cant ever reach absolute zero. In the laboratory scientists have gone done to small fractions of 1 K absolute, but have never reached zero itself.

        Before we finish our discussions about thermal systems and kinetic theory it is interesting to consider what this theory has to say about the nature of the universe as a whole (not a Black Hole please). To be sure, we would say that the energy .of the universe is constant and is not being created or destroyed on any large scales in the universe. We do see changes occurring, indicating a directionality of processes. Stellar systems, supernovas and galaxies are evolving in a predictable way. The universe is observed to be expanding. Does this mean that there is increasing entropy? Is the universe running dow? Its temperature is decreasing as it is expanding. Present values indicate background radiation, remnants of the Big Bang fireball to be at about 3K. Originally the universe was hotter and more confined. It is expanding and cooling. Some theories suggest that the universe will expand forever. Some even say it really is static. Others say it will expand, stop expanding,
and then contract again to a point. If this happens, the entropy of the universe would have to stop increasing and thus becoming more ordered. The universe, as it contracts, would decrease in entropy. But the third law says we can't get there because at some absolute zero there would only be one state accessible to the system and energy would be at a minimum. We need to keep these ideas in mind as we consider the nature of the universe as a whole later in our study of cosmology.
 
 

QUESTIONS

1. Define Entropy
 

2. Define Temperature
 

3. State the Laws of Thermodynamics
 

4. What is an Ideal Gas?
 

5. What do we mean by never?
 

6. Can you spill a glass of water into a river and recover the same glass of water? Why or why not?
 

7. Explain why the air we breathe is configured randomly about us (in terms of a statistical distribution.)
 
 

BIBLIOGRAPHY

1. Huang, Kerson, Statistical Mechanics, John Wiley, Inc, New York; 1963.

2. Kittel, Charles, Thermal Physics, W.H. Freeman and Co, San Francisco, 1980.

3. Flower,Terrence F., Use of a Probability Model in Statistical Mechanics, U.S. Air Force Academy, Colorado
Spring, 1964.