# Chapter 9: Statistical Mechanics

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1 Chapter 9: Statistical Mechanics Chapter 9: Statistical Mechanics Introduction Statistical Mechanics The Hamiltonian Phase Space Trajectories and Integrals of the Motion Dynamical Variables and Mean Values Ensembles of Systems Statistical Ensembles The Distribution Function Composite Systems The Mean and Variance of Additive Quantities The Measurability of Thermodynamic Quantities INTRODUCTION Classical thermodynamics defines the thermodynamic functions and their interrelationships, but does not provide a recipe for calculating them. The fundamental equation must be measured experimentally and the other constitutive equations calculated from it. To complete the theory we need a method for computing the fundamental equation of a system from the microscopic behavior of the physical particles it contains. It should be obvious that a statistical theory is needed to do this. Thermodynamic quantities such as the energy are not distributed among the atoms or molecules of a system in any fixed way, but are continuously interchanged as the particles interact with one another. The value of, say, the energy of a group of atoms that is embedded in some larger aggregate changes with time even when the aggregate is isolated and at equilibrium. It fluctuates about some mean value. Since the microscopic fluctuations are local and rapid, they are not usually noticeable on a macroscopic scale. It is the mean values rather than the instantaneous values of the thermodynamic variables that are reflected in the macroscopic behavior of the system. Given that the thermodynamic variables are associated with statistical averages of the microscopic behavior, it is important to be precise about the sort of average that is meant. Recall that the thermodynamic quantities were defined so that they would have an operational significance in the macroscopic sense. The definition of each of the thermodynamic quantities includes a prescription for measuring it through a set of macroscopic operations. The statistical definitions of the thermodynamic quantities must be consistent with these operational definitions. page 111

2 The experimental measurement of a thermodynamic quantity is ordinarily done by performing a series of experiments on a system or, equivalently, on different systems that are nominally identical. The experimenter anticipates that the individual experiments will yield slightly different results, and reports the average of these. The scatter in the results is due in part to the imprecision of the experiment, and in part to statistical fluctuations in the system. The statistical scatter ordinarily decreases with the size of the system, so the macroscopic properties of large systems have well-defined values that can be measured reproducibly. Much of the utility of classical thermodynamics depends on the possibility of selecting a small set of macroscopic parameters (the complete set of constitutive variables) whose values fix the values of all other macroscopic variables of interest. For this procedure to be physically useful the inherent microscopic fluctuations in the system must be such that the following two statements hold. (1) All examples of a macroscopic system that have the same values of the macroscopic constitutive coordinates have the same mean values of all other macroscopic variables. (Here, as in the following, "all" means "almost all". We can always admit the possibility of pathological examples of a system if it is sufficiently unlikely that we shall ever encounter one.) (2) A single measurement of a macroscopic variable is almost certain to yield a value that is incrementally close to its mean value. If statement (1) were false for all reasonable choices of the constitutive coordinates it would not be possible to define a useful macroscopic thermodynamic state. The fact that the classical constitutive equations successfully forecast the behavior of physical systems argues that statement (1) is usually true. However, there are possible counter-examples. One of these may be the plastic solid. Some very good theoreticians have spent a great deal of effort in the search for a constitutive equation that predicts the plasticity of a material from the current values of a manageable set of macroscopically measurable parameters (the "mechanical equation of state"). Their limited success may simply show that plastic deformation inherently depends on the microscopic details of the grain and dislocation configuration. If statement (2) were false the fundamental equation would be relatively uninformative. We usually wish to use it to predict the outcome of a given experiment. Its ability to do this is predicated on the sharpness of the properties of the system in a given macroscopic state. The thermodynamic properties are generally as sharp and reproducible as one might wish. However, there are counter-examples. The classic one is the "critical opalescence" of a liquid near its critical point. When the liquid is very near its critical point its density fluctuates widely with the result that light is scattered and the liquid is nearly opaque. The following chapters give the statistical theorems that justify statements (1) and (2) and permit the fundamental equation to be calculated, at least in principle. We shall page 112

3 do this twice, first for systems whose particles behave classically and then for quantized systems. 9.2 STATISTICAL MECHANICS The Hamiltonian Let an isolated system contain N particles that obey the laws of classical mechanics. The behavior of the system can be predicted (at least in theory) if the position and momentum of each of the particles are known at any instant of time. Both the past and the future are determined by the mechanical equations of motion. The N vector positions and momenta of a classical system of N particles constitute a complete set of constitutive coordinates. They determine the value of the Hamiltonian function, H, which fixes the total energy and the equations of motion. The Hamiltonian is defined by the function N H({p},{q}) = p k 2 2m k + V(q 1,...,q N ) k=1 = K + V 9.1 where K is the kinetic energy, which depends on the momenta only, and V is the potential energy, which usually depends only on the relative positions of the particles. The Hamiltonian is a function of 6N variables: the three components of position and the three components of momentum of each of the particles (there are of the order of in a macroscopic system). For notational convenience we shall usually write the function H in the simpler form H({p},{q}) = H(p,q) 9.2 where p represents the 3N variables, p i, that give the momenta of the N particles and q represents the 3N variables q i that give the coordinates. x 1 q 3 p 3 p 1 q x 1 3 p 2 q2 x 2 Fig. 9.1: Vector positions and momenta of three particles in space. page 113

4 The value of the Hamiltonian is the total energy of the system. Its partial derivatives are the 6N equations of motion that determine the evolution of the system. They are called the Hamilton equations of motion: dq i dt dp i dt = q i = H p i 9.3 = p i = - H q i 9.4 Given values of the N positions, {q}, and momenta, {p}, at some reference time the equations of motion can be integrated forward or backward to find the 6N functions q i and p i that give the positions and momenta at any other time. The Hamiltonian function is, hence, the fundamental equation of the system of particles Phase Space The independent variables q i and p i define a 6N-dimensional space that is called the phase space, Í N, of the system. A point (p,q) in Í N corresponds to an instantaneous state. The equations of motion generate a curve in Í N which is called the trajectory of the system. The trajectory is specified by the one-parameter set of equations p i = p i (t) 9.5 q i = q i (t) 9.6 and gives the instantaneous state as the system evolves. In classical mechanics the variables p and q are continuous, and a state (p,q) is defined by confining the values of the variables to the differential intervals (p i,p i +dp i ) and (q i,q i +dq i ). The state occupies a 6N-dimensional differential volume in Í N dpdq = (d 3N p)(d 3N q) = dp 1...dp 3N dq 1...dq 3N 9.7 about the point (p,q). The volume is the product of momentum and distance and, therefore, has the dimensions of action. It is convenient to choose the dimensions of p and q so that their product is dimensionless. It is further useful to choose the units so that a unit of volume in phase space contains exactly one state. While the "volume" of a classical state is not defined, the uncertainty principle of the quantum theory places an ultimate limit on how fine-grained the phase space can be. The uncertainty principle states that a component of momentum, p i, and its conjugate position, q i, cannot be simultaneously measured with a precision greater than the uncertainty Îp i Îq i h 9.8 page 114

5 where h is Planck's constant. Hence a distinguishable state of the system occupies a minimum volume in phase space Îp 1...Îp 3N Îq 1...Îq 3N h 3N 9.9 If we measure q in units of, the DeBroglie wavelength of the particles, and p in units of h/, then the product is dimensionless and the uncertainty principle reads Îp i Îq i There is then one definable state per unit volume in Í N. dimensionalization unless otherwise specified. We shall use this Trajectories and Integrals of the Motion The trajectories of the system are curves through Í N that are solutions to the equations of motion. The trajectories cannot intersect, since otherwise the path of the system from the state (p,q) at the intersection would not be determined by the equations of motion. The trajectories must satisfy other constraints as well if they are to be consistent with the definition of the system and the laws of mechanics. The laws of mechanics assert the constancy of the total energy, E, linear momentum, P, and angular momentum, L. These equations are called constants of the motion (or integrals of the motion), and give rise to the three constraints H(p,q) = E 9.11 k p k = P 9.12 k (p k x q k ) = L 9.13 Equations have the consequence that the trajectories lie on 6N-7 dimensional hypersurfaces of constant E, P and L in Í N. The definition of the system usually imposes additional constraints since it specifies the values of at least some of the macroscopic parameters. These are analogous to the verbal and operational constraints of classical thermodynamics. The macroscopic parameters that are fixed are also constants of the motion, and only those trajectories that preserve their constant values are allowed. For example, if the system is fluid and isolated its volume and chemical content have definite values. The volume defines the permissible range of the variables q i ; none can lie outside the physical dimensions of the system. The trajectory must lie entirely within this restricted subvolume of Í N. The page 115

6 composition fixes the dimensions of Í N and also sets the relative proportions of the distinguishable particles that are present Dynamical Variables and Mean Values Let R be a quantity that is determined by the microstate of the system, R = R(p,q) 9.14 R might, for example, be the total kinetic energy of the particles in the system or the energy or pressure of a fluid subsystem. A quantity of the type R is called a dynamical variable. The mean value of the dynamical variable R along a trajectory of the system is denoted R or R. It is given by the time average R = lim t 1 t 0 t R(p(t'),q(t')) dt' 9.15 Since the behavior of a system is determined by its states (p,q) it must be possible to associate each macroscopic thermodynamic parameter with an appropriate dynamical variable. Letting R be such a variable, equation 9.15 suggests that its time average should be equal to the value of the thermodynamic parameter, since R would be the average result of a large number of hypothetical measurements of R that are taken without perturbing the dynamical state. But this view is necessarily oversimple, for two reasons. First, the integral in 9.15 is taken along a single trajectory. Since a trajectory is a one-dimensional path through a 6N-dimensional space there are very many possible trajectories that are consistent with the macroscopic constraints on the system. An isolated classical system evolves along a particular one of these, which has a particular value of the time average of the dynamical variable R. Unless all possible trajectories have the same mean values of all of the relevant dynamical variables, identifying the macroscopic thermodynamic parameter with the value of R determined by equation 9.15 would have the consequence that its macroscopic value would depend on the precise microscopic trajectory taken by the system. The macroscopic constitutive coordinates would have to be sufficient to determine the precise trajectory, and they are not. This problem became apparent during the early development of statistical thermodynamics. It was addressed in the theoretical work leading to and supporting the ergodic theorem, which states that almost all of the permissible trajectories of an isolated macroscopic system have almost identical mean values of any given dynamical variable. When this theorem holds we need not be concerned about the particular trajectory as long as it is reasonably typical. In fact, the ergodic theorem has the stronger consequence that page 116

7 we can ignore the trajectories entirely and replace the time average that is defined in equation 9.15 by the phase average that will be defined below. The development of the ergodic theorem is worth studying for its own sake, since it generated some of the most elegant mathematics in theoretical physics (the best mathematical treatment I know is by A.I. Khinchin, Mathematical Foundations of Statistical Mechanics; the most readable is by R. Tolman, The Principles of Statistical Mechanics, which also includes a comprehensive discussion of almost everything else we shall consider here). However, a more modern view of the connection between microscopic and macroscopic variables suggests that the ergodic theorem is not really needed. The reason is suggested by the second problem that is inherent to the thermodynamic interpretation of equation The second problem arises from the impossibility of making measurements on a system without changing it. The observations that must be made to measure a thermodynamic variable require that the isolation of the system be broken so that an appropriate probe can be introduced. The perturbation is, ideally, small in the macroscopic sense, but must nonetheless alter the microscopic state of the system and change its trajectory. It follows that repeated thermodynamic measurements are not taken at sequential points along a given trajectory. They rather sample isolated points on an unknown set of distinct trajectories. Since a small perturbation in (p,q) may set the system off on a very different evolutionary path, and since molecular relaxation times are very short compared to reasonable intervals between macroscopic measurements, the different trajectories need not even be very close to one another in phase space. Hence even in the context of classical mechanics there are inherent arguments against equating thermodynamic parameters with the time averages of dynamical variables. An alternate strategy that is more easily justified was suggested by Gibbs, who associated the thermodynamic quantities with phase or ensemble averages. 9.3 ENSEMBLES OF SYSTEMS Statistical Ensembles Thermodynamic measurements are, in fact, done on systems whose internal states are unknown beyond the requirement that they be consistent with the macroscopic constraints that were imposed when they were assembled. It follows that the outcome of a measurement of a dynamical variable is random in two senses: both the trajectory of the system and the sampling point along the trajectory are unknown. Moreover, we are not normally interested in a particular system, but in the outcome of a hypothetical experiment on an arbitrarily selected example of the system. We speak of "the properties of water", not the properties of a particular sample of water. page 117

8 To phrase the sampling problem that is implied by both these statements as a problem in statistics we imagine an ensemble of systems that contains an arbitrarily large number of systems that are identically constituted and subject to the same macroscopic constraints. A measurement of a property is then done by selecting a system at random and determining the instantaneous value of the appropriate dynamical variable. A sequence of measurements is generated by repeating the random selection and measurement. This procedure is evidently an accurate description of a series of measurements done on different examples of a given system. By the arguments of the previous section, it is also a plausible description of a series of measurements on the same system; each measurement perturbs the system in an unknown way and, in effect, creates a new example of the system The Distribution Function To treat the statistics of measurement on an ensemble of systems we define the differential probability, dw(q,p), that an arbitrarily chosen example of the system is in the state specified by (q,p). The differential probability is written in terms of the distribution function, (p,q), dw(q,p) = (q,p)dqdp 9.16 where dqdp is a shorthand notation for (h) -3N d 3N qd 3N p, the differential volume of phase space that was defined in equation 9.7. The distribution function, (q,p), is the fraction of the members of the ensemble whose states lie in the differential volume of phase space located at (q,p). The distribution function is normalized according to the integral Í (q,p)dqdp = where the integral is taken over the part of Í N for which the density function is defined. The probability that the dynamical variable, R(q,p), of a system selected at random has the particular value, R', is P{R=R'} = R=R' (q,p)dqdp = Í (R-R') (q,p)dqdp 9.18 where the integral is taken over all states (q,p) for which R(q,p) = R'. The function, (R- R') is the Dirac -function, which is a mathematical device for selecting those values out of the volume integral. The mean value of R(q,p) in a large series of such measurements is R = Í R(q,p) (q,p)dqdp 9.19 Equation 9.19 defines the phase average or ensemble average of the variable R(q,p). It is the quantity that is properly associated with the macroscopic variable, R. page 118

9 9.3.3 Composite Systems Much of the fundamental reasoning used in the development of classical thermodynamics is based on hypothetical experiments in which an isolated system is decomposed into subsystems that interact with one another. The subsystems are assumed to be independent in the sense that each can be assigned a definite thermodynamic state that is characterized by its own constitutive coordinates. When the systems interact they perturb the thermodynamic states of one another. When the molecular constitution of the system is taken into account there are restrictions on how the subsystems may be chosen; the subsystems are molecular aggregates that must have no more than a weak interaction with one another. To specify the conditions that are satisfied by weakly interacting subsystems consider a composite system that contains two subsystems. If the subsystems have individual thermodynamic states it must be possible to express the dynamical variables of each in terms of its coordinates and momenta alone. That is, writing (q 1,p 1 ) for the coordinates and momenta of the N 1 particles in system 1 and (q 2,p 2 ) for the coordinates and momenta of the N 2 particles in system 2, the dynamical variables of system 1 must satisfy constitutive equations of the form R 1 = R 1 (q 1,p 1 ) 9.20 that are independent of (q 2,p 2 ), and the dynamical variables of system 2 must obey similar equations in (q 2,p 2 ). In particular, the Hamiltonian of a subsystem must depend only on its own coordinates. For system 1, H 1 = H(q 1,p 1 ) = k p 1 k 2 2m k + V({q 1 }) 9.21 The interaction between the two systems is weak in the sense that the potential, V, of one does not depend explicitly on the configuration of the other. When these conditions are satisfied the subsystems are said to be statistically independent. Weakly interacting systems influence one another by transferring quantities, such as energy, that can pass across the boundary between them. The rate of transfer is assumed sufficiently slow that each subsystem has a definite content at any instant of time and is in a trajectory that is consistent with its content. The model is a reasonable one for systems that are separated by walls whose molecular composition can be ignored. Given the limited range of molecular interaction, it is also a reasonable one for subvolumes of large systems that are themselves large on the atomic scale. In this case only the interfacial layer of molecules interacts with both subsystems and the total consequence of the interaction can be gathered into a surface effect. page 119

10 When statistically independent subsystems are joined to create a composite system the density function of the composite can be found from the density functions of the subsystems by simple multiplication. Let the i th subsystem be represented by an ensemble with a density function i (q i,p i ). The members of the representative ensemble for the composite system are chosen by randomly selecting one example of each subsystem and joining them together. For simplicity, let there be two subsystems. The set of constitutive coordinates of the composite system is the union of the 6N 1 positions and momenta of system 1 and the 6N 2 positions and momenta of system 2. The probability, dw(q,p), that the composite system has the state (q,p) = (q 1,q 2,p 1,p 2 ) 9.22 is the product of the probabilities that system 1 is in state (q 1,p 1 ) while system 2 is in state (q 2,p 2 ): or dw(q,p) = dw 1 (q 1,p 1 )dw 2 (q 2,p 2 ) 9.23 It follows from the definition of the density function that (q,p) = 1 (q 1,p 1 ) 2 (q 2,p 2 ) 9.24 ln( ) = ln( 1 ) + ln( 2 ) 9.25 The density function of the composite,, is automatically normalized if 1 and 2 are, since Í dqdp = Í 1 (q 1,p 1 ) 2 (q 2,p 2 )dq 1 dp 1 dq 2 dp 2 = Í 1 1 dq 1 dp 1 Í 2 2 dq 2 dp 2 = This result is easily generalized to an arbitrary number of subsystems. If a composite system is made by joining independent subsystems that interact weakly the density function of the composite system is the product of the density functions of the subsystems: (q,p) = k k (q k,p k ) 9.27 Their logarithms add: ln( ) = k ln( k ) 9.28 page 120

11 9.3.4 The Mean and Variance of Additive Quantities Consider a composite system, K, that contains the two subsystems, K 1 and K 2. The set of coordinates, (q,p), of K is then the union of the set (q 1,p 1 ) of K 1 and the set (q 2,p 2 ) of K 2. A dynamical variable, R, of K is additive if R(q,p) = R 1 (q 1,p 1 ) + R 2 (q 2,p 2 ) 9.29 where R 1 and R 2 are the values of R for the subsystems K 1 and K 2 by themselves. Almost all of the dynamical variables of interest to us are of this type. When R is additive its mean value is also additive: For an arbitrary number of subsystems R = Í R dqdp = Í (R 1 + R 2 ) 1 2 dq 1 dq 2 dp 1 dp 2 = R 1 + R R = k R k 9.31 In particular, if the composite system is composed of n identical subsystems for which the expected value of R is R 0, R = nr We now compute the variance of an additive variable, R, which measures the expected value of the difference between its measured value and its mean value. Since the quantity ÎR = R - R 9.33 is equally likely to have either sign, ÎR is zero, and the appropriate measure of the variance is ß = ÎR which measures the expected value of the magnitude of the difference ÎR without respect to its sign. The square of the variance is ß 2 = [R - R ] 2 = R 2-2 R 2 + R 2 = R 2 - R page 121

12 where R 2 = Í R2 dqdp 9.36 Let R be an additive variable in a composite system of weakly interacting subsystems which has the additional property that its values in the subsystems are uncorrelated, that is, the measured value R i in subsystem (i) is independent of the measured value R j in the subsystem (j). The values of the R i in weakly interacting subsystems can usually be assumed uncorrelated unless the total value of R is fixed by the constraints on the system (e.g., the energy of an isolated system). If R i and R j are uncorrelated, (ÎR i )(ÎR j ) = ÎR i ÎR j = since both of the mean values on the right-hand side vanish. composite system of n subsystems, It follows that in a (ÎR) 2 = i (ÎR i ) 2 = ij (ÎR i )(ÎR j ) = i (ÎR i ) or, in a simpler notation, ß 2 = i ß i Equation 9.39 has the consequence that the measured value of a dynamical variable in a homogeneous, macroscopic system is almost always very close to the mean value. A macroscopic system can usually be treated as a composite of (n) identical subsystems that are sufficiently large that they interact weakly. Let R be an additive variable. If the system is homogeneous the mean, R 0, and variance, ß 0, of R has the same value in each of the subsystems. According to equation 9.39 the variance of R is ß = n ß Hence the fractional variance in the measured values of R is ß R = which vanishes as ( n ) -1 when n becomes arbitrarily large. ß 0 1n R page 122

13 9.3.5 The Measurability of Thermodynamic Quantities At the beginning of this chapter we set out two rules that should be satisfied to establish contact between classical and statistical thermodynamics: (1) the mean values of the dynamical variables that are associated with macroscopic thermodynamic quantities should be the same for any system that has the same constitution and constraints, and (2) almost all individual measurements of the macroscopic quantities of a given system should produce almost the same values. The results of the previous sections show that both of these conditions are satisfied when the system is macroscopic and can be treated as a composite of a large number of identical, weakly interacting subsystems. The expected value of the dynamical variable, R, that is associated with a particular thermodynamic quantity is just the average of R over the representative ensemble for the system. The representative ensemble includes all possible ways of constructing the system that are consistent with the constraints imposed on it, and is hence the same for every physical example of the system. The expected value of the fractional variation from one measurement to another vanishes with the size of the system; hence repeated measurements of the value of R for a macroscopic system should produce nearly identical results. The correspondence between the statistical properties of representative ensembles and the macroscopic properties of thermodynamic systems holds very widely, and provides a means for calculating the thermodynamic behavior of real systems that we shall develop in the following sections. Note, however, that this approach is based on the assumption that the phenomena of interest are governed by the mean values of the dynamical variables. There are also physical phenomena that are governed by extremal properties. The statistical approach outlines above does not work for these. A specific example is the mechanical yield strength of a crystalline solid, which governs the onset of macroscopic plastic deformation. A crystalline solid begins to deform plastically when the applied stress exceeds the "critical resolved shear stress" at which dislocations can first move freely. The critical resolved shear stress of a crystal is determined by an extremal property. It is the minimum shear stress at which a dislocation can move freely on some crystallographic glide plane, that is, the stress that is just sufficient to overcome the maximum resistance on the weakest glide plane in the crystal. The same comment applies to the fracture strength, which is determined by the normal stress needed to cause unstable propagation of the most severe flaw in the material. It is also determined by an extremal situation. In the early part of this century there were numerous efforts to develop "thermodynamic" theories of yielding and fracture. Most of these were based on the concept that a material would yield or break when the stored elastic energy density exceeded some critical value. These efforts had little success, and we now know that they were fundamentally flawed; the stored energy is determined by a statistical average while page 123

14 yielding and fracture are determined by statistical extrema. On the other hand, researchers have had some success in treating the rate of plastic deformation well beyond yielding in terms of the rate of dissipation of stored energy. This is a more promising area for thermodynamic analysis since the material has generally yielded and the deformation is more plausibly due to the average behavior of the dislocations it contains. page 124

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