CHAPTER FIVE FUNDAMENTAL CONCEPTS OF STATISTICAL PHYSICS "

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1 CHAPTE FIVE FUNDAMENTAL CONCEPTS OF STATISTICAL PHYSICS " INTODUCTION In the previous chapters we have discussed classical thermodynamic principles which can be used to predict relationships among the various macroscopic properties of a system. These thermodynamic principles can be used to predict such things as the difference between the heat capacities GT and GZ, and to show how these quantities might vary with other macroscopic parameters, such as pressure and temperature. It is not possible, however, to derive from thermodynamic considerations alone such things as the absolute magnitudes of the heat capacities. The same is also true for the internal energy and the entropy of a system. Only differences, or changes in these quantities can be calculated using the techniques of classical thermodynmaics. We can make significant progress and go beyond the limitation of pure thermodynamics only by making some assumptions regarding the fundamental nature of matter. We did this in chapter two when we made use of one of sciences most fruitful assumptions: that matter is not continuous, but is composed of a large number of discrete particles called molecules. By applying the basic laws of Newtonian physics regarding the motion and interaction of these microscopic particles, we were able to derive an equation of state for an ideal gas along with an expression for the internal energy of that gas. We were then able to derive the heat capacity at constant volume and pressure for this ideal gas. Although our treatment of the motion of molecules in a gas was very elementary, the molecular theory of gases has actually been very thoroughly developed. Progress in the treatment of liquids and solids, however, has been much slower because of the complexity of the molecular interactions within liquids and solids. A typical approach to modeling a system on the atomic scale is to make certain reasonable assumptions about the motion and interaction of the individual molecules. The subject of kinetic theoryattempts to precisely apply the laws of mechanics to individual molecules of a system in an attempt to understand the behavior of the system as a whole. In this approach, it is assumed that the laws of mechanics, deduced from the behavior of matter in bulk, can be applied, without change, to particles like molecules and electrons. This is a formidable task and has many inherent difficulties - not the least of which is keeping up with the position and momentum of each "* of the nearly 10 individual particles found in a cubic centimeter of air at standard temperature and pressure. Statistical thermodynamics (or statistical mechanics), on the other hand, ignores the details of the interactions among individual particles and attempts to use statistical principles to predict the averagebehaviour of the system. In fact, unlike simple kinetic theory, it is the large number of individual elements of the system which makes this particular approach so useful. Both approaches, however, make the basic assumption that the molecules of the system obey the classical laws of mechanics. As progress has been made in both of these areas, it has now become evident that this principle assumption is not correct. Macroscopic predictions based upon these two models, although very promising, have not been in complete accord with experimental facts. This failure of small-scale systems to obey the same mechanical laws as large-scale systems has led to the development of what we now call quantum mechanics. Today, the subject of statistical thermodynamics is best treated from the viewpoint of quantum mechanics, and we will typically make use of quantum mechanical arguments throughout our treatment of statistical thermodynamics. CHAACTEISTIC FEATUES OF MACOSCOPIC SYSTEMS AND THE APPLICABILITY OF STATISTICS The world that we experience through our senses consists of objects that are macroscopic, i.e., large compared to atomic dimensions. In fact, it is almost inconceivable just how many atoms or molecules make up even the smallest visible object. As we begin our study of statistical thermodynamics, we want to investigate how just a few unifying concepts of atomic theory can lead to an understanding of the observed behavior of macroscopic systems, how the quantities which describe the directly measurable properties of such systems are interrelated, and how these quantities can be deduced from a knowledge of atomic characteristics. Although a macroscopic system consists of very many atoms, the laws of quantum mechanics describing the dynamical behavior of atomic particles are well established. In addition, the electromagnetic forces responsible for the interactions between these atomic particles are also very well understood. Ordinarily these are the only forces relevant because gravitational forces between atomic particles are usually negligibly small

2 Chapter Five: Fundamentals of Statistical Physics 2 compared to electromagnetic forces. In addition, a knowledge of nuclear forces is usually not necessary since the atomic nuclei do not get disrupted in most ordinary macroscopic physical systems and in all chemical and biological systems. Thus, our knowledge of the laws of microscopic physics should be quite adequate to allow us, in principle, to deduce the properties of any macroscopic system from a knowledge of its microscopic constituents. One might think, for example, that if we knew the position and momentum of each particle of a system, and the interaction force acting on each individual particle, we could predict the motion of each individual particle as they interact with one another. However, a typical macroscopic system of the type encountered in 25 everyday life contains about 10 interacting atoms. Determining the three-dimensional position and momentum 25 of each of these 10 interacting particles dwarfs the capabilities of even the most fanciful of future computers; furthermore, unless one asks the right questions, reams of computer output are likely to provide no insight whatever into the essential features of a problem. In addition, these "simple" interactions often produce unexpected results. For instance, consider a gas of identical simple atoms (e.g., helium atoms) which interact with each other through simple known forces. It is by no means evident, from simple microscopic information, that such a gas can condense very abruptly so as to form a liquid. Yet this is precisely what can happen. The discovery of concepts sufficiently powerful to gain a clear understanding of the relationships between microscopic interactions and measurable macroscopic parameters clearly represents a major intellectual challenge. But, over the years, clever individuals have been able to utilize relatively simple reasoning to gain substantial progress in understanding macroscopic systems. And, surprisingly, it is the very presence of large numbers of particles that allows us to effectively use statistical methods to gain some insight into how microscopic interactions manifest themselves in macroscopic ways. As an example of how this may be accomplished, we shall look at a few simple examples dealing with gas particles confined inside a container. Fluctuations in Equilibrium Consider a gas of identical molecules, e.g., helium (He) or nitrogen (N 2). If the gas is dilute(i.e., if the number of molecules per unit volume is small), the average separation between the molecules is large and their mutual interaction is correspondingly small. If the gas is sufficiently dilute so that the interaction between its molecules is essentially negligible this gas is called an ideal gas. Each of the molecules of this ideal gas spends most of its time moving like a free particle uninfluenced by the presence of the other molecules or the container walls; only rarely does it come sufficiently close to the other molecules or the container walls so as to interact (or collide) with them. In addition, if the gas is sufficiently dilute, the average separation between its molecules is much much larger than the average de Broglie wavelength of a molecule, so that quantum-mechanical effects are of negligible importance and it is permissible to treat the molecules as distinguishable particles moving along classical trajectories. We now consider such an ideal gas of molecules confined within a container or box which is isolated (i.e., it does not interact with any other system) and which has been left undisturbed for a very long time (thus, the system is in thermodynamic equilibrium). Now imagine that we can use a movie camera to record the motion of the molecules as they move around inside the container without disturbing them in any way. Successive frames would show the positions of each of the molecules at regular intervals separated by some short time 7. As we watch this movie we would observe the gas molecules in constant motion. Any given molecule would move along a straight line until it collides with some other molecule or with the walls of the box and then it would continue moving along some other straight line until it collides again; and so on and so forth. Each molecule would move strictly in accordance with the laws mechanics. Nevertheless, molecules moving throughout the box and colliding with each other would represent a situation so complex that the picture on the screen would appear rather chaotic (unless, of course, is very small). We now focus our attention only on the positions of the molecules and their distribution in space. (We ignore entirely the momentum of the molecules.) We thus define a limited amount of information which we really want to know (i.e., the precise state of the system in which we are really interested). To be more precise, let's consider the box to be divided by some imaginary partition into two equal parts (see Fig. 5.1). We will denote the number of molecules in the left half of the box by 8and the number in the right half by 8. Of course 8 8 œ the total number of molecules in the box. If is large, we would ordinarilyfind that 8 8, i.e., that roughly half of the molecules are in each half of the box (altough this statement is only approximately true). As we watch (5.1)

3 Chapter Five: Fundamentals of Statistical Physics 3 the movie, we see the molecules colliding occasionally with each other or with the walls, some of them enter the left half of the box, while others leave it. The number 8 of molecules actually located in the left half of the container fluctuates constantly in time, but these fluctuations (for large Ñare usually small enough so that 8does not differ too much from /#. You might wonder if it is possible for all the molecules to be found on one side of the box at a particular instant in time. There is, in fact, nothing that prevents all molecules from being in the left half of the box (so that 8 œ, while 8 œ0). But just how likely is it that such an event might occur? To gain some insight into this question, let us ask in how many ways the molecules can be distributed between the two halves of the box. We shall call each distinct way in which the molecules can be distributed between these two halves a configuration. First, consider a single molecule. This molecule can be found in the box in two possible configurations, i.e., it can be either in the left half or the right half. Since the two halves have equal volumes and are otherwise equivalent, we suspect that the molecule is equally likely to be found in either half of the box. If we consider # molecules, each one of them can be found in either of the 2 halves. Hence the total number of possible configurations (i.e., the total number of possible ways in which the 2 molecules can be distributed between the # two halves) is equal to # # œ # œ % since there are, for each possible configuration of the first molecule, 2 possible configurations of the other. If we consider 3 molecules, the total number of their possible configurations is equal to œ2 3 œ8. Similarly, if we consider the case of 4 molecules, the number of possible configurations is given by 2 % œ "'. (These 16 configurations are listed explicitly in the appendix in Table 5.1.) [NOTE: In our treatment of this simple situation, we assume that the likelihood of finding a particular molecule in any half of the box is unaffected by the presence there of any number of other molecules. This should be true if the total volume occupied by the molecules themselves is negligibly small compared to the volume of the box.] In the general case of molecules, the total number of possible configurations is 2 2 â 2 œ2. Note that there is only one way of distributing the N molecules so that all N of them are in the left half of the box. This arrangement represents only one special configuration of the molecules compared to the 2 possible configurations of these molecules. Hence we would expect that, among a very large number of frames of our movie, on the average only one out of every 2 frames would show all the molecules to be in the left half. If c denotes the fraction of frames showing all the molecules located in the left half of the box, i.e., if c denotes the relative frequency, or probability, of finding all the molecules in the left half, then c œ " # Similarly, the case where no molecule at all is in the left half is also very special since there is again only one such configuration of the molecules out of 2 possible configurations. Thus the probability c! of finding no molecule located in the left half should also be given by c! œ " # More generally, consider a situation where 8of the molecules of the gas are located in the left half of the box and let us denote by VÐ8Ñthe number of possible configurations of the molecules in this case. [That is, VÐ8Ñ is the number of possible ways the molecules can be distributed in the box so that 8of them are found in the left half of the box.] Since the total number of possible configurations of the molecules is 2, one would expect that, among a very large number of frames of our movie, on the average VÐ8Ñout of every 2 such frames would show 8molecules to be in the left half of the box. If c 8 denotes the fraction of frames showing 8molecules located in the left half, i.e., if c 8 denotes the relative frequency, or probability, of finding 8molecules in the left half, then c V œ Ð8Ñ # 8 Example: Consider the special case (Fig. 5.1 and Table 5.1) where the gas consists of only four molecules. Suppose that a movie of this gas consists of a great many frames. Then we expect that the fraction c 8 of these frames showing 8 molecules in the left half (and correspondingly 8 œ 8molecules in the right half) is given by: 1 4 6! % 16 $ " 16 # 16 c œ c œ c œ c œ c œ (5.2) (5.3) (5.4)

4 Chapter Five: Fundamentals of Statistical Physics 4 Figure 5.1 Four Particles in a Divided Box Three pictures of four particles in a box at different instances of time. The box is divided into two equal parts and the number of particles in each half of the box is indicated below that half of the box. The short line segment eminating from each particle indicates the direction of the particle's velocity. Table 5.1 Enumeration of the 16 possible ways that 4 particles can be located in the two halves of a container n n' C(n) L L L L L L L 3 1 L L L L L L 3 1 L L L 3 1 L L 2 2 L L 2 2 L L L L 2 2 L L 2 2 L L 2 2 L 1 3 L L 1 3 L The first four columns indicate the location of each of the four particles in the box. Each is labeled as to which side of the box the particle is in. The columns designated by 8 and 8 w indicate the total number of particles in each half of the box, and the column designated by GÐ8Ñ indicates the number of different microstateswhich correspond to a given macrostate.

5 Chapter Five: Fundamentals of Statistical Physics 5 As we have seen, a situation where 8= (or where 8= 0) corresponds only to a single possible molecular configuration. More generally, if is large, then VÐ8Ñ << 2 if 8 is even moderately close to (or even moderately close to 0). In other words, a situation where the distribution of molecules is so nonuniform that 8<< /2 (or that 8 >> /2) corresponds to relatively few configurations. A situation of this kind, which can be obtained in relatively few ways, is rather special and is accordingly said to be relatively nonrandomor orderly; it occurs relatively infrequently. On the other hand, a situation where the distribution of the molecules is almost uniform, so that 8 8, corresponds to many possible configurations; indeed, as is illustrated in Table 5.1, VÐ8Ñ is a maximum if 8 œ 8 œ /2. A situation of this kind, which can be obtained in many different ways, is said to be randomor disordered; it occurs quite frequently. In short, the case where the molecules are distributed more randomly (or uniformly) occur more frequently than the less random ones. The physical reason is clear: All molecules must move in a very special way if they are to concentrate themselves predominantly in one part of the box; similarly, if they are all located in one part of the box, they must move in a very special way if they are to remain concentrated there. To gain a greater appreciation for just how often we might expect to find all the molecules on one side of our container, let us consider some specific examples. If the gas consisted of only 4 molecules, all of them would, on the average, be found in the left half of the box once in every 16 frames of our movie. A fluctuation of this kind would thus occur with moderate frequency. On the other hand, if the gas consisted of 80 molecules, all of these would, on the average, be found in the left half of the box in only one out of 2 10 frames of our movie. In other words, if we could view a million frames every second, we would have to watch our movie considerably longer than the age of the universe before we would have a reasonable chance of obtaining just one frame which would show all the molecules in the left half of the box. [NOTE: There are about seconds 10 in a year and the estimated age of the universe is of the order of 10 years.] Finally, suppose that we consider as a 3 realistic example a box having a volume of 1 cm and containing air at atmospheric pressure and room temperature. Such a box contains about molecules. A fluctuation where all of these are located in one half of the box would, on the average, appear in only one out of 2 10 frames of our movie. Fluctuations where not all, but a predominant majority of the molecules are found in one half of the box, would occur somewhat more frequently; but this frequency of occurrence would still be exceedingly small. Hence we arrive at the following general conclusion: If the total number of particles is large, fluctuations corresponding to an appreciably nonuniform distribution of these molecules occur almost never. Now, let's again consider our isolated ideal gas which has been left undisturbed for a long time. The number 8 of molecules in one half of the box will fluctuate in time about the value /2 which occurs most frequently. The frequency of occurrence of any particular value of 8decreases rapidly the more 8differs from /2, i.e., the greater the difference? 8 where?8 8 2 (5.5) Indeed, if is large, only values of 8with? 8 << occur with significant frequency. Positive and negative values of?8would, of course, occur equally often. The time dependence of 8for the case where œ % and œ %! are shown for a particular sequence of time in Figure 5.3. If were much greater than 40, we would expect that 8Îwould be almost always 1/2, with very small fluctuations about this value.

6 Chapter Five: Fundamentals of Statistical Physics 6 Figure 5.2 Fifteen Frames of a Computer Simulated Movie of Four Particles Moving in a Container, Separated into Two Equal Halves, Showing the Number of Particles in Each Half and the Velocity Direction of Each Particle Fig. 5.2 Computer-made pictures showing 4 particles in a box. The fifteen successive frames (labeled 4 œ!ß"ß#ßâß"%ñare pictures taken a long time after the beginning of the computation with assumed initial conditions. The number of particles located in each half of the box is printed directly beneath that half. the short line segment emanating from each particle indicates the direction of the particle's velocity.

7 Chapter Five: Fundamentals of Statistical Physics 7 Figure 5.3 A Plot Showing the Number of Particles (and elative Number of Particles) in the Left Half of the Container as a Function Time for œ % and œ %!. Fig. 5.3 A plot of the number of particles (and the relative number of particles) in the left half of the box as a function of the frame index 4 or the elapsed time > œ The number 8of the 4th frame (or the relative number) is indicated by a horizontal line extending from 4to 4 ". The top pair of graphs are of the case where œ %, while the bottom pair of graphs are for the case where œ %!. You will notice that the information for the case where œ % is identical to that shown in Figure 5.2 for the first 15 frames.

8 Chapter Five: Fundamentals of Statistical Physics 8 Conclusion The microscopic state, or microstate of a gas confined within a container can be described in greatest detail by specifying the position and velocity of each molecule of the system at any particular time (giving the maximum possible information about the gas molecules at that particular instant). From this microscopic point of view, the motion of the motion of the gas particles appears very complex since the locations and the momenta of each individual molecule are constantly changing from one instant to another. From a large-scale or macroscopicpoint of view, however, one is not typically interested in the behavior of each and every molecule, but in a much less detailed description of the gas. Thus the macroscopic state, or macrostate, of the gas, for example, might be quite adequately described by specifying merely the numberof molecules located in any part of the box at any particular time. [NOTE: To be specific, we could imagine that the container is subdivided into many equal cells, each having the same volume and each cell's volume being large enough to contain many individual molecules. The macroscopic state of the gas could then be described by specifying the number of molecules located in each cell of the container.] From this macroscopic point of view, an isolated gas which has been left undisturbed for a long time represents a very simple situation since its macroscopic state does not tend to change significantly over time. Indeed, suppose that, starting at some time > 1, we observed the gas over some moderately long period of time T by taking a movie of it. Alternatively, suppose that, starting at some other time > 2, we again observed the gas over the same time Tby taking a movie of it. From a macroscopic point of view these two movies would ordinarily look indistinguishable. For the case where we have divided our container into two equal parts, we would find that in each case the number 8of particles in the left half of the box would ordinarily fluctuate about the same value /2, and the magnitude of the observed fluctuations would ordinarily also look alike. Disregarding very exceptional occurrences the observed macrostate of the gas is thus independent of the starting time of our observations; i.e., we can say that the macrostate of our gas does not tend to change in time. In particular, the value about which 8fluctuates (or, more precisely, its averagevalue) does not tend to change in time. A system of many particles (such as our gas) whose macroscopic state does not tend to change in time is said to be in equilibrium. This seems to be consistent with out understanding of the concept of thermodynamic equilibrium as presented in the first part of this course as we attempted to describe classical thermodynamics, and thus justifies our attempt to describe a complex molecular system using simple statistical arguments. " Much of the material in this chapter (in particular the figures) is taken from the first chapter of Berkeley Physics Course, Vol. 5, "Statistical Physics", by F. eif, McGraw-Hill, 1967.