Some Basics of Statistical Mechanics

The branch of physics now known as statistical mechanics started out as thermodynamics, the study of heat and related concepts. The relation of thermodynamics to the rest of physics, i.e. the relation of heat and motion, was studied by scientists like James Prescott Joule in the 19th century. Due to their efforts, we have the idea that what they used to refer to as “heat” is a form of energy which is transferred from one object to another, manifesting in ways other than the bulk motion of the objects (in particular, as a change in the “internal energy” of the objects involved) .

Energy, a concept that was already associated to the motion of objects (see also Lagrangians and Hamiltonians), can be transferred from one object to another, or one system to another, and in the case of heat, this transfer involves the concept of temperature. Temperature is what we measure on a thermometer, and when we say something is “hot” or “cold”, we are usually referring to its temperature.

The way by which temperature dictates the direction in which heat is transferred is summarized in the second law of thermodynamics (here we give one of its many equivalent statements):

Heat flows from a hotter object to a colder one.

This process of transfer of heat will continue, decreasing the internal energy of the hotter object and increasing the internal energy of the cooler one, until the two objects have equal temperatures, in which case we say that they are in thermal equilibrium.

But if heat is a transfer of energy, and energy is associated to motion, then what was it, exactly, that was moving (or had the capacity to cause something to move)? What is this “internal energy?” For us, who have been taught about atoms and molecules since childhood, the answer might come rather easily. Internal energy is the energy that comes from the motion of the atoms and molecules that comprise the object. But for the scientists who were developing the subject during the 19th century, the concept of atoms and molecules was still in its very early stages, with many of them facing severe criticism for adopting ideas that at the time were still not completely verified.

Still, these scientists continued to take the point of view that the subject of thermodynamics was just the same physics that had already been applied to, say, the motion of cannonballs and pendulums and other objects, except that now they had to be applied to a very large quantity of very small particles (quantum mechanics would later have much to contribute also, but even before the introduction of that theory the concept of atoms and molecules was already starting to become very fruitful in thermodynamics).

Now we have an explanation for what internal energy is in terms of the motion of the particles that make up an object. But what about temperature? It is possible to explain temperature (and therefore the laws that decide the direction of the transfer of heat) using more “basic” concepts such as Newton’s laws of motion, like we have done for the internal energy?

It was the revolutionary ideas of Ludwig Boltzmann that provided the solution. It indeed involved a more “basic” concept, but not one we would usually think of as belonging to the realm of physics or the study of motion. The idea of Boltzmann was to relate temperature to the concepts of information, probability, and statistics, via the notion of entropy. We may therefore think of this era as the time when “thermodynamics” became “statistical mechanics”.

In order to discuss the idea of entropy, for a moment we step away from physics, and discuss instead cards. It is not cards themselves that we are interested in, but information. Entropy is really about information, which is why it also shows up, for instance, when discussing computer passwords. Cards will give us a simple but concrete way to discuss information.

Consider now, therefore, a deck of 52 ordinary playing cards. A hand, of course, consists of five cards. Using the rules of combinatorics, we can find that there are 2,598,960 different hands (combinations of 52 different playing cards taken five at a time, in any order). In the game of poker, there are certain special combinations, the rarest (and highest-ranking) of which is called the “royal flush”. There are only four possible ways to get a royal flush (one for each suit). In contrast, the most common kind of hand is one which has no special combination (sometimes called “no pair”), and there are 1,302,540 different combinations which fit this description.

Now suppose the deck is shuffled and we are dealt a hand. The shuffling process is not entirely random (not in the way that quantum mechanics is), but there are so many things going on that it is near-impossible for us to follow and determine what kind of hand we are going to get. The most we can do is make use of what we know about probability and statistics. We know that it is more likely for us to obtain a no pair rather than a royal flush, simply because there are so many more combinations that are considered a no pair than there are combinations that are considered a royal flush. There are no laws of physics involved in making this prediction; there is only the intuitive idea that an event with more ways of happening is more likely to happen compared to an event with less ways of happening, in the absence of any more information regarding the system.

We now go back to physics. Let us consider a system made up of a very large number of particles. The state of a single particle is specified by its position and momentum, and the state of the entire system is specified by the position and momentum of every one of its particles. This state is almost impossible for us to determine, because there are simply too many particles to keep track of.

However, we may be able to determine properties of the system without having to look at every single particle. Such properties may involve the total energy, pressure, volume, and so on. These properties determine the “macrostate” of a system. The “true” state that may only be specified by the position and momentum of every single particle is called the microstate” of the system. There may be several different microstates that correspond to a single macrostate, just like there are four different combinations that correspond to a royal flush, or 1,302,540 different combinations that correspond to a no pair.

Let the system be in a certain macrostate, and let the number of microstates that correspond to this macrostate be denoted by \Omega. The entropy of the system is then defined as

\displaystyle S=k_{B}\text{ln }\Omega.

where k_{B} is a constant known as Boltzmann’s constant. We may think of this constant and the logarithm as merely convenient ways (in terms of calculation, and in terms of making contact with older ideas in thermodynamics) to express the idea that the higher the number of microstates, the higher the entropy.

Now even though the system may not seem to be changing, imperceptible to us, there may be many things that happen on a microscopic level. Molecules may be moving around in many directions, in motions that are too difficult for us to keep track of, not only because the particles are very small but also because there are just too many of them. This is analogous to the shuffling of cards. All that we have at our disposal are the tools of probability and statistics. Hence the term, “statistical mechanics“.

What have we learned from the example of the shuffling of cards? Even though we could not keep track of things and determine results, we could still make predictions. And the predictions we made were simply of the nature that an event with more ways of happening was more likely to happen than an even with less ways of happening.

Therefore, we have the following restatement of the second law of thermodynamics:

The entropy of a closed system never decreases.

This simply reflects the idea that under these processes we cannot keep track of, the system is more likely to adopt a configuration with more ways of happening, compared to one with less ways of happening. In other words,it will be in a macrostate that will have more microstates. Microscopically, it may happen that “miraculously” the entropy increases; but given how many particles there are, and how many processes happen, this is extremely unlikely to be a sustained phenomenon, and macroscopically, the second law of thermodynamics is always satisfied. This is like obtaining a royal flush on one deal of cards; but if we are going to reshuffle multiple times, it is unlikely that we keep getting royal flushes for a sustained period of time.

The “closed system” requirement is there to ensure that the system is “left to its own devices” so to speak, or that there is no “outside interference”.

Considering that the entirety of the universe is an example of a “closed system” (there is nothing outside of it, since by definition the universe means the collection of everything that exists), the second law of thermodynamics has some interesting (perhaps disturbing, to some people) implications. What we usually consider to be an “ordered” configuration is very specific; for example, a room is only in order when all of the furniture is upright, all the trash is in the wastebasket, and so on. There are fewer such configurations compared to the “disordered” ones, since there are so many ways in which the furniture can be “not upright”, and so many ways in which the trash may be outside of the wastebasket, etc. In other words, disordered configurations have higher entropy. All of these things considered, what the second law of thermodynamics implies is that the entropy of the universe is ever increasing, moving toward an increasing state of disorder, away from the delicate state of order that we now enjoy.

We now want to derive the “macroscopic” from the “microscopic”. We want to connect the “microscopic” concept of entropy to the “macroscopic” concept of temperature. We do this by defining “temperature” as the following relationship between the entropy and the energy (in this case the internal energy, as the system may have other kinds of energy, for example arising from its motion in bulk):

\displaystyle T=\frac{\partial E}{\partial S}

Although we will not discuss the specifics in this post, we make the following claim – the entropy of the system is at its maximum when the system is in thermal equilibrium. Or perhaps more properly, the state of “thermal equilibrium” may be defined as the macrostate which has the most amount of microstates corresponding to it. This in turn explains why heat flows from a hotter object to a cooler one.

We have now discussed some of the most basic concepts in thermodynamics and statistical mechanics. We now briefly discuss certain technical and calculational aspects of the theory. Aside from making the theory more concrete, this is important also because there are many analogies to be made outside of thermodynamics and statistical mechanics. For example, in the Feynman path integral formulation of quantum field theory (see Some Basics of Relativistic Quantum Field Theory) we calculate correlation functions, which mathematically have a form very similar to some of the quantities that we will discuss.

In modern formulations of statistical mechanics, a central role is played by the partition function Z, which is defined as

\displaystyle Z=\sum_{i}e^{-\beta E_{i}}

where \beta, often simply referred to as the “thermodynamic beta”, is defined as

\displaystyle \beta=\frac{1}{k_{B}T}.

The partition function is a very convenient way to package information about the system we are studying, and many quantities of interest can be obtained from it. One of the most important ones is the probability P_{i} for the system to be in a macrostate with energy E_{i}:

\displaystyle P_{i}=\frac{1}{Z}e^{-\beta E_{i}}.

Knowing this formula for the probabilities of certain macrostates allows us to derive the formulas for expectation values of quantities that may be of interest to us, such as the average energy of the system:

\displaystyle \langle E\rangle=\frac{1}{Z}\sum_{i}e^{-\beta E_{i}}.

After some manipulation we may find that the expectation value of the energy is also equal to the following more compact expression:

\displaystyle \langle E\rangle=\frac{\partial \text{ln }Z}{\partial \beta} .

Another familiar quantity that we can obtain from the partition function is the entropy of the system:

\displaystyle S=\frac{\partial (k_{B}T\text{ln }Z)}{\partial T} .

There are various other quantities that can be obtained from the partition function, such as the variance of the energy (or energy fluctuations), the heat capacity, and the so-called Helmholtz free energy. We note that for “continuous” systems, expressions involving sums are replaced by expressions involving integrals. Also, for quantum mechanical systems, there are some modifications, as well as for systems which exchange particles with the environment.

The development of statistical mechanics, and the introduction of the concept of entropy, is perhaps a rather understated revolution in physics. Before Boltzmann’s redefinition of these concepts, physics was thought of as studying only motion, in the classical sense of Newton and his contemporaries. Information has since then taken just as central a role in modern physics as motion.

The mathematician and engineer Claude Elwood Shannon further modernized the notion of entropy by applying it to systems we would not have ordinarily thought of as part of physics, for example the bits on a computer. According to some accounts, Shannon was studying a certain quantity he wanted to name “information”; however, the physicist and mathematician John von Neumann told him that a version of his concept had already been developed before in physics, and was called “entropy”. With Neumann’s encouragement, Shannon adopted the name, symbolically unifying subjects formerly thought of as separate.

Information theory, the subject which Shannon founded, has together with quantum mechanics led to quantum information theory, which not only has many potential applications in technology but also is one of the methods by which we attempt to figure out deep questions regarding the universe.

Another way in which the concept of entropy is involved in modern issues in physics is in the concept of entropic gravity, where gravity, as expressed in Einstein’s general theory of relativity, is derived from more fundamental concepts similar to how the simple statistical concept of entropy gives rise to something that manifests macroscopically as a law of physics. Another part of modern physics where information, quantum mechanics, and general relativity meet is the open problem called the black hole information paradox, which concerns the way in which black holes seemingly do not conserve information, and is a point of contention among many physicists even today.

Finally, we mention another very interesting aspect of statistical mechanics, perhaps, on the surface, a little more mundane compared to what we have mentioned on the preceding paragraphs, but not the slightest bit less interesting – phase transitions. Phase transitions are “abrupt” changes in the property of an object brought about by some seemingly continuous process, like, for example, the freezing of water into ice. We “cool” water, taking away heat from it by some process, and for a long time it seems that nothing happens except that the water becomes colder and colder, but at some point it freezes – an abrupt change, even though we have done just the same thing we did to it before. What really happens, microscopically, is that the molecules have arranged themselves into a some sort of structure, and the material loses some of symmetry (the “disordered” molecules of water were more symmetric than the “ordered” molecules in ice) – a process known as symmetry breaking. Phase transitions and symmetry breaking are ubiquitous in the sciences, and have applications from studying magnets to tackling the problem of why we have observed so much more matter compared to antimatter.

References:

Thermodynamics on Wikipedia

Statistical Mechanics on Wikipedia

Entropy on Wikipedia

Partition Function on Wikipedia

Entropy in Thermodynamics and Information Theory on Wikipedia

Quantum Information on Wikipedia

Black Hole Information Paradox on Wikipedia

Phase Transition on Wikipedia

Symmetry Breaking on Wikipedia

It From Bit – Entropic Gravity for Pedestrians on Science 2.0

Black Hole Information Paradox: An Introduction on Of Particular Significance

Thermal Physics by Charles Kittel and Herbert Kroemer

Fundamentals of Statistical and Thermal Physics by Frederick Reif

A Modern Course in Statistical Physics by Linda Reichl

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