Physics Hoch An Introduction Concepts and relationships in thermal and statistical physics form the foundation for describing systems consisting of macroscopically large numbers of particles Developing microscopic statistical physics and macroscopic classical thermodynamic descriptions in tandem, Statistical and Thermal Physics: An Introduction provides insight into basic concepts at an advanced undergraduate level Highly detailed and profoundly thorough, this comprehensive introduction includes exercises within the text as well as end-of-chapter problems The first section of the book covers the basics of equilibrium thermodynamics and introduces the concepts of temperature, internal energy, and entropy using ideal gases and ideal paramagnets as models The chemical potential is defined and the three thermodynamic potentials are discussed with use of Legendre transforms The second section presents a complementary microscopic approach to entropy and temperature, with the general expression for entropy given in terms of the number of accessible microstates in the fixed energy, microcanonical ensemble The third section emphasizes the power of thermodynamics in the description of processes in gases and condensed matter Phase transitions and critical phenomena are discussed phenomenologically K12300 An Introduction In the second half of the text, the fourth section briefly introduces probability theory and mean values and compares three statistical ensembles With a focus on quantum statistics, the fifth section reviews the quantum distribution functions Ideal Fermi and Bose gases are considered in separate chapters, followed by a discussion of the “Planck” gas for photons and phonons The sixth section deals with ideal classical gases and explores nonideal gases and spin systems using various approximations The final section covers special topics, specifically the density matrix, chemical reactions, and irreversible thermodynamics Statistical and Thermal Physics Statistical and Thermal Physics ISBN: 978-1-4398-5053-4 90000 781439 850534 K12300_COVER_final.indd 4/6/11 2:55 PM Statistical and Thermal Physics Michael J R Hoch National High Magnetic Field Laboratory and Department of Physics, Florida State University Tallahassee, USA and School of Physics, University of the Witwatersrand Johannesburg, South Africa Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A TA Y L O R & F R A N C I S B O O K Taylor & Francis Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor & Francis Group, LLC Taylor & Francis is an Informa business No claim to original U.S Government works Version Date: 20131107 International Standard Book Number-13: 978-1-4398-5054-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and 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Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my wife Renée Contents Preface Acknowledgments Physical Constants Part I xix xxiii xxv Classical Thermal Physics: The Microcanonical Ensemble Section IA Introduction to Classical Thermal Physics Concepts: The First and Second Laws of Thermodynamics Chapter Introduction: Basic Concepts 1.1 STATISTICAL AND THERMAL PHYSICS 1.2 TEMPERATURE 1.3 IDEAL GAS EQUATION OF STATE 1.4 EQUATIONS OF STATE FOR REAL GASES 11 1.5 EQUATION OF STATE FOR A PARAMAGNET 12 1.6 KINETIC THEORY OF GASES AND THE EQUIPARTITION OF ENERGY THEOREM 13 1.7 THERMAL ENERGY TRANSFER PROCESSES: HEAT ENERGY 19 20 PROBLEMS CHAPTER Chapter Energy: The First Law 23 2.1 THE FIRST LAW OF THERMODYNAMICS 23 2.2 APPLICATION OF THE FIRST LAW TO A FLUID SYSTEM 25 ix x ◾ Contents 2.3 TERMINOLOGY 27 2.4 P–V DIAGRAMS 28 2.5 QUASI-STATIC ADIABATIC PROCESSES FOR AN IDEAL GAS 29 2.6 MAGNETIC SYSTEMS 30 2.7 PARAMAGNETIC SYSTEMS 33 2.8 MAGNETIC COOLING 36 2.9 GENERAL EXPRESSION FOR WORK DONE 37 2.10 HEAT CAPACITY 38 2.11 QUASI-STATIC ADIABATIC PROCESS FOR AN IDEAL GAS REVISITED 41 2.12 THERMAL EXPANSION COEFFICIENT AND ISOTHERMAL COMPRESSIBILITY 42 43 PROBLEMS CHAPTER Chapter Entropy: The Second Law 47 3.1 INTRODUCTION 47 3.2 HEAT ENGINES—THE CARNOT CYCLE 47 3.3 CARNOT REFRIGERATOR 50 3.4 ENTROPY 52 3.5 ENTROPY CHANGES FOR REVERSIBLE CYCLIC PROCESSES 54 3.6 ENTROPY CHANGES IN IRREVERSIBLE PROCESSES 56 3.7 THE SECOND LAW OF THERMODYNAMICS 58 3.8 THE FUNDAMENTAL RELATION 59 3.9 ENTROPY CHANGES AND T–S DIAGRAMS 59 3.10 THE KELVIN TEMPERATURE SCALE 61 3.11 ALTERNATIVE STATEMENTS OF THE SECOND LAW 61 3.12 GENERAL FORMULATION 64 3.13 THE THERMODYNAMIC POTENTIALS 67 69 PROBLEMS CHAPTER Contents ◾ xi Section IB Microstates and the Statistical Interpretation of Entropy Chapter Microstates for Large Systems 75 4.1 INTRODUCTION 75 4.2 MICROSTATES—CLASSICAL PHASE SPACE APPROACH 76 4.3 QUANTUM MECHANICAL DESCRIPTION OF AN IDEAL GAS 79 4.4 QUANTUM STATES FOR AN IDEAL LOCALIZED SPIN SYSTEM 81 4.5 THE NUMBER OF ACCESSIBLE QUANTUM STATES 83 PROBLEMS CHAPTER 91 Chapter Entropy and Temperature: Microscopic Statistical Interpretation 95 5.1 INTRODUCTION: THE FUNDAMENTAL POSTULATE 95 5.2 EQUILIBRIUM CONDITIONS FOR TWO INTERACTING SPIN SYSTEMS 96 5.3 GENERAL EQUILIBRIUM CONDITIONS FOR INTERACTING SYSTEMS: ENTROPY AND TEMPERATURE 101 5.4 THE ENTROPY OF IDEAL SYSTEMS 103 5.5 THERMODYNAMIC ENTROPY AND ACCESSIBLE STATES REVISITED 107 PROBLEMS CHAPTER 111 Chapter Zero Kelvin and the Third Law 115 6.1 INTRODUCTION 115 6.2 ENTROPY AND TEMPERATURE 116 6.3 TEMPERATURE PARAMETER FOR AN IDEAL SPIN SYSTEM 117 Appendix B ◾ 401 GAUSSIAN APPROXIMATION TO THE BINOMIAL DISTRIBUTION For large N, we now show that the binomial distribution can be well approximated by the Gaussian function for values of n not too far from the peak We use a Taylor expansion to approximate the probability distribution and again find it convenient to consider ln P(n) because this is a more slowly varying function than P(n) and the expansion may be expected to converge rapidly Retaining terms up to second order, the expansion has the form ⎛ d ln P (n) ⎞ ln P (n) = ln P (〈n〉) + ⎜ ⎝ dn ⎟⎠ n ⎛ d ln P (n) ⎞ (n − 〈n〉) + ⎜ (n − 〈n〉)2 + ⎝ dn2 ⎟⎠ 〈n 〉 (B6) The first derivative of ln P(〈n〉) evaluated at (n) is zero, and the second derivative is given by ⎛ d ln P (n) ⎞ 1 N ⎜⎝ dn2 ⎟⎠ = − 〈n〉 − (N − 〈n〉) = − 〈n〉(N − 〈n〉) = − Npq 〈n 〉 (B7) Substituting in the Taylor expansion and then taking antilogarithms give the Gaussian distribution form P (n) = P (〈n〉)e −(n − 〈n 〉) / Npq The coefficient P(〈n〉) may be obtained from the normalization condition by integrating over the range to N For sufficiently large N, the upper limit may be extended to infinity because the integral will converge With the standard integral form given in Appendix A, we obtain finally in this approximation, P (n) = e −(n − Np) / Npq 2p Npq (B8) 2 This has the Gaussian distribution form P (x ) = 1/ 2ps e −( x − x ) / 2s , and we identify the dispersion as s = Npq in agreement with the value obtained directly from the binomial distribution For large N, the Gaussian approximation is very close in form to the exact binomial distribution Figure B1 for N = 30 shows the exact binomial distribution P(n) versus n as the plotted points and the 402 ◾ Appendix B 0.18 0.16 0.14 P(n) 0.12 0.10 0.08 0.06 0.04 0.02 0.00 10 15 n 20 25 30 FIGURE B1 The binomial distribution for N = 30 and p = q = 0.5 shown as plot- ted points and the Gaussian approximation as the full curve Gaussian approximation as the curve The agreement is seen to be very good, showing that the Gaussian approximation works well even for fairly small N It is clear that for very large values of N, comparable with Avogadro’s number, the Gaussian approximation gives an excellent fit to the binomial distribution Appendix C: Elements of Quantum Mechanics The time-dependent Schrödinger equation for a particle of mass m moving in a fixed potential V(r) is − 2m ∇2 Ψ(r , t ) + V (r )Ψ(r , t ) = i ∂Ψ(r , t ) ∂t (C1) The wave function Ψ(r, t) gives the probability amplitude of finding the particle at position r at time t For the purposes of this book, we are generally interested in time-independent or stationary states Writing the wave function as a product of a spatial part and a time-dependent part as Ψ(r, t) = ψ (r)e−iωt and substituting into Equation C1 lead to the timeindependent Schrödinger equation − 2m ∇2 y (r ) + V (r )y (r ) = E y (r ) (C2) In more compact form, we have ψ (r) = Eψ (r), with the Hamiltonian operator defined as H = − ( / 2m)∇2 + V (r ) The energy eigenvalues E = ħω are obtained by solving this equation for a given potential function V(r) Note that | Ψ(r , t )|2 = | y (r )e −iw t |2 = | y (r )|2, showing that the eigenstates are stationary states with a time-independent probability density for finding the particle at any given point 403 404 ◾ Appendix C PARTICLE IN A BOX EIGENSTATES AND EIGENVALUES Consider a particle moving in one dimension in a potential well V(x) = with infinite walls at x = and x = L This is the one-dimensional box situation, and the corresponding time-independent Schrödinger equation is d 2y (x ) 2mE = y (x ) = − k 2y (x ), dx (C3) with 2mE / = k The general solution to Equation C3 may be written as y (x ) = C1e ik x + C2e −ik x Using the boundary conditions ψ (0) = gives C1 = −C2, and it follows that y (x ) = C1 sink x At the other boundary of the well, we require ψ (L) = 0, which implies sink L = 0, and this leads to k L = np or k n = np / L, with n = 1, 2, 3, … The boundary conditions lead to quantization of κ and E The situation is similar to the case of vibrational standing waves on a string stretched between two fixed points or nodes The energy eigenvalues for the particle are En = k n2 n2h2 = 2m 8mL2 (C4) The probability of finding the particle in the box is given by ∫ 0L |y (x ) |2 dx = and inserting y (x ) = C1 sink x = C1 sin(np x /L) in the integral gives the constant C1 = / L For a particle in a three-dimensional box with infinite potential barriers at the sides, the energy eigenvalues may be obtained in similar fashion to the one-dimensional box case, and we obtain En = h2 ⎛ nx2 n2y nz2 ⎞ , + + 8m ⎜⎝ L2x L2y L2z ⎟⎠ (C5) with nx , n y , nz = 1, 2, 3, For a cubical box Lx = Ly = Lz = L and putting V = L3, Equation C5 becomes En = h2 (nx2 + n2y + nz2 ) 8mV / (C6) Appendix C ◾ 405 THE HARMONIC OSCILLATOR The potential function for a one-dimensional harmonic oscillator is V (x ) = 12 kx 2, with k the effective spring constant for a particular system such as a diatomic molecule The displacement x from the origin can take positive or negative values, and the potential has the form of a parabolic well with a minimum at x = The energy levels for a particle of mass m moving in this static potential are given by the time-independent Schrödinger equation d 2y (x ) ⎛ 2⎞ + kx y (x ) = Ey (x ) ⎝2 ⎠ 2m dx 2 − (C7) Finding solutions to the harmonic oscillator Equation C7 is not straightforward, and we simply outline the procedure Further details can be found in texts on quantum mechanics For a given energy, the wave function will fall off rapidly with increasing x because of the strong dependence of V(x) on x It is convenient to rearrange Equation C7 in the following form: ⎛ d2 2⎞ ⎜⎝ 2m dx + E − kx ⎟⎠ y (x ) = (C8) In the asymptotic large x limit, Equation C8 can be written to a good approximation as ⎛ d 2⎞ ⎜⎝ 2m dx − kx ⎟⎠ y (x ) = (C9) This is similar in form to the differential equation [(d 2/d x ) − (x − 1)] f(x) = 0, which has the following solution f (x ) = Ae −(1/ 2) x , where A is a constant We expect the solution to Equation C9 to have the Gaussian form y (x ) = Ce −(1/ 2)a x , with C and α constants to be determined Substituting ψ(x) in Equation C9, cancelling the common exponential factor and the constant C lead to (a − mk / )x − a = In the large x limit, the first term is dominant, and we require that the coefficient of x2 is zero, giving a = mk / 406 ◾ Appendix C For small x, a solution to the Schrödinger equation may be obtained by multiplying the Gaussian function by a polynomial in x of the form H n (x ) = (a0 + a1 x + a2 x + + an x n ), with the order n to be determined for each eigenstate Note that successive terms in the polynomial have even and odd parity, and because the Gaussian function has even parity under a sign change of x, the parity of the wave function is determined by the parity of the nonzero terms in the polynomial The polynomials Hn(x) are known as Hermite polynomials The simplest solution is obtained by retaining only the zeroth-order term in the polynomial, giving y (x ) = a0Ce −(1/ 2)a x Substituting in Equation C8 and using a = mk/ , we obtain the eigenvalue E0 = k /m = w (C10) We have put w = k /m , which is the angular frequency of the classical harmonic oscillator Equation C10 gives the ground state energy for the quantum mechanical harmonic oscillator Following a similar procedure to the above, we obtain the next eigenvalue using the first-order term in − (1/2 )a x the polynomial, corresponding to odd parity, so that y (x ) = a1 xCe and find E1 = w (C11) This shows that the first excited state is at an energy w above the ground state Proceeding in this fashion, we can obtain successive eigenvalues by choosing alternating even and odd parity terms in the polynomial function The next eigenvalue is found using the zeroth- and second-order terms (a0 + a2 x ), and this leads to E2 = 25 w We find that the energy eigenvalues are given in terms of the quantum number n by the expression En = ⎛n + ⎞ w , ⎝ 2⎠ where n = 0, 1, 2, 3, (C12) The eigenfunctions are obtained using the normalization condition ∫ ∞ −∞ y ( x ) dx = Appendix C ◾ 407 For the ground state wave function, we obtain, for example, the Gaussian function 1/ ⎛ a ⎞ −(1/ 2)a x y (x ) = ⎜ e ⎝ p ⎟⎠ (C13) This form may be readily verified using the integral for the Gaussian function given in Appendix A Wave functions for the excited states are usually expressed in terms of the appropriate Hermite polynomial and are given in texts on quantum mechanics STATE VECTORS AND DIRAC NOTATION The quantum state of a system is specified by a state vector, which is independent of the basis states used to describe the system Projections of the state vector onto the basis states give the components of the state vector in a particular basis or representation much like the components of classical vector with respect to a particular set of axes The Dirac “bra-ket” notation provides a convenient and compact way for specifying the quantum state of a system In general, we specify a state vector in this notation by means of the ket ø , where 𝜙 specifies the particular eigenstate for the system considered For the one-dimensional particle in a box case, for example, we can specify a given state as n using the single quantum number n For three-dimensional N particles, we require 3N quantum numbers to specify the state vector, which is written as n1x , n1 y , n1z ; n2 x n2 y n2 z ; ; nNx , nNy , nnz Appendix D: The Legendre Transform in Thermodynamics INTRODUCTION TO THE LEGENDRE TRANSFORM The three thermodynamic potentials H, F, and G that are introduced in Chapter can be shown to take their particular forms by making use of the Legendre transform For a function F(x) of a single variable x, Legendre transforms allow us to represent the function F(x) by another function L(s), where s is a variable given by s = dF(x)/dx, which is the slope of the original function at a given point The Legendre transform L(s) of F(x) is defined by the relation L(s) = F (x(s)) − s(x )x(s), (D1) with x(s) the value of x for which the slope s is obtained This unusual form can be understood by making use of the geometrical representation shown in Figure D1 in which s(x) is the slope of the tangent to the curve at a point x(s) and L(s) is the intercept for this tangent Legendre transforms are applied to functions which are convex (with ∂2 F / ∂x > 0) for which the slope increases with increase in x Note that the Legendre transform may be defined with a change in sign for mathematical reasons and the definition given above is chosen for our applications in thermodynamics Before considering the thermodynamic potentials, we examine the Legendre transform for a function of two variables The procedure can of course be extended to any number of variables, but two 409 410 ◾ Appendix D s(x) = dF(x)/dx x(s)s(x) F(x) L(s(x)) x x(s) FIGURE D1 Graphical illustration of the Legendre transform of a function F(x) of a single variable x The relationship L(s) = F(x(s)) ‒ s(x)x(s), which is given in the text, is readily established from this plot The set of slopes s and intercepts L(s) specifies the function completely variables are often sufficient for our purposes Consider a function F(x, y) of the variables x and y The function can be represented as a surface in a three-dimensional plot If one of the variables is held fixed, the situation is similar to that discussed above and as depicted in Figure D1 More generally, the function can be represented as the envelope of the set of tangent planes to the surface For a given point (x, y), we have the slopes s x = ∂F / ∂x and s y = ∂F / ∂y , and it follows that L(s x , s y ) = F (x , y ) − s x x − s y y (D2) THE LEGENDRE TRANSFORM AND THERMODYNAMIC POTENTIALS In applying the Legendre transformation to thermodynamic relationships for a fluid system, we choose as our function the internal energy E(S, V) expressed in terms of the entropy S and the volume V As discussed in Section 3.12, this choice corresponds to the energy representation for a system with a fixed number of particles N If S is kept fixed, then we obtain the partial Legendre transform as ⎛ ∂E ⎞ L = E −⎜ V = E + PV = H ⎝ ∂V ⎟⎠ S (D3) The identity P = −(∂E / ∂V )S follows from the fundamental relation (Equation 3.18) TdS = dE + PdV , and we have made use of the definition Appendix D ◾ 411 of the enthalpy given in Section 7.1 We see that H is simply the partial Legendre transform of E with S kept constant In Equation D3, the extensive state function S, which is in general not readily controlled, has been replaced by the intensive variable P which is readily controlled as an independent variable The Helmholtz potential F = E − TS as defined in Equation 7.2 is obtained as the partial Legendre transform of E with V held constant We find in this case ⎛ ∂E ⎞ F = E − ⎜ ⎟ S = E − TS ⎝ ∂S ⎠V (D4) Processes in which E can change at constant V correspond to the canonical ensemble case in statistical physics as introduced in Chapter 10 The bridge relation between F and the partition function Z can be seen to follow in a natural way The Gibbs potential G defined in Equation 7.3 is the complete Legendre transform of E corresponding to both S and V being allowed to change ⎛ ∂E ⎞ ⎛ ∂E ⎞ G = E −⎜ ⎟ S − ⎜ V = E − TS + PV ⎝ ∂S ⎠ V ⎝ ∂V ⎟⎠ S (D5) The three thermodynamic potentials given above are of great importance in the development of thermal physics as discussed in Chapters 3, and elsewhere in the book Finally, to obtain the grand potential ΩG = − PV , that is used in Chapter 11 in connection with the grand canonical distribution, we allow the internal energy E to be a function not only of S and V and but in addition of particle number N so that E = E(S, V, N ) The grand canon ical ensemble corresponds to a set of systems each in thermal and diffusive contact with a reservoir at temperature T and with chemical potential μ We therefore allow S and N to vary but keep V fixed In this case the Legendre transform of E is given by ⎛ ∂E ⎞ ⎛ ∂E ⎞ ΩG = E − ⎜ ⎟ S−⎜ N = E − TS − m N ⎝ ∂S ⎠ V , N ⎝ ∂N ⎟⎠ V ,S (D6) From the expressions for F and G it follows that ΩG = F − G = − PV as shown in Chapter 11 Appendix E: Recommended Texts on Statistical and Thermal Physics INTRODUCTORY LEVEL Betts, D.S and Turner, R.E., Introductory Statistical Mechanics, Addison Wesley, Wokingham, 1992 Blundell, S and Blundell, K.M., Concepts in Thermal Physics, Oxford University Press, New York, 2006 Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, second edition, Wiley, New York, 1985 Huang, K., Introduction to Statistical Physics, second edition, Chapman and Hall/ CRC Press, Boca Raton, FL, 2010 Kittel, C and Kroemer, H., Thermal Physics, second edition, W.H Freeman, New York, 1980 Mandl, F., Statistical Physics, second edition, Wiley, Chichester/New York, 1988 Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965 Schroeder, D.V., An Introduction to Thermal Physics, Addison Wesley Longman, New York, 2000 Zemansky, M.W and Dittman, R.H., Heat and Thermodynamics, seventh edition, McGraw-Hill, New York, 1997 ADVANCED LEVEL Landau, L.D., and Lifshitz, E.M., Statistical Physics, third edition (trans Sykes, J.B and Kearsley, M.J.), Pergamon Press, Oxford, 1980 413 414 ◾ Appendix E Pathria, R.K., Statistical Mechanics, second edition, Butterworth-Heinemann, Oxford, 1996 Reichl, L., A Modern Course in Statistical Physics, second edition, Wiley, New York, 1998 COMPUTER SIMULATIONS Gould, H, Tobochnik, J and Christian, W., An Introduction to Computer Simulation Methods: Applications to Physical Systems, third edition, Addison-Wesley, Reading MA, 2006 Physics Hoch An Introduction Concepts and relationships in thermal and statistical physics form the foundation for describing systems consisting of macroscopically large numbers of particles Developing microscopic statistical physics and macroscopic classical thermodynamic descriptions in tandem, Statistical and Thermal Physics: An Introduction provides insight into basic concepts at an advanced undergraduate level Highly detailed and profoundly thorough, this comprehensive introduction includes exercises within the text as well as end-of-chapter problems The first section of the book covers the basics of equilibrium thermodynamics and introduces the concepts of temperature, internal energy, and entropy using ideal gases and ideal paramagnets as models The chemical potential is defined and the three thermodynamic potentials are discussed with use of Legendre transforms The second section presents a complementary microscopic approach to entropy and temperature, with the general expression for entropy given in terms of the number of accessible microstates in the fixed energy, microcanonical ensemble The third section emphasizes the power of thermodynamics in the description of processes in gases and condensed matter Phase transitions and critical phenomena are discussed phenomenologically K12300 An Introduction In the second half of the text, the fourth section briefly introduces probability theory and mean values and compares three statistical ensembles With a focus on quantum statistics, the fifth section reviews the quantum distribution functions Ideal Fermi and Bose gases are considered in separate chapters, followed by a discussion of the “Planck” gas for photons and phonons The sixth section deals with ideal classical gases and explores nonideal gases and spin systems using various approximations The final section covers special topics, specifically the density matrix, chemical reactions, and irreversible thermodynamics Statistical and Thermal Physics Statistical and Thermal Physics ISBN: 978-1-4398-5053-4 90000 781439 850534 K12300_COVER_final.indd 4/6/11 2:55 PM ... LANDAU THEORY OF CONTINUOUS TRANSITIONS 186 Part II PROBLEMS CHAPTER 190 Quantum Statistical Physics and Thermal Physics Applications Section IIA The Canonical and Grand Canonical Ensembles and. . .Statistical and Thermal Physics Michael J R Hoch National High Magnetic Field Laboratory and Department of Physics, Florida State University Tallahassee, USA and School of Physics, University... TRANSFORM AND THERMODYNAMIC POTENTIALS 410 Appendix E Recommended Texts on Statistical and Thermal Physics 413 INTRODUCTORY LEVEL 413 ADVANCED LEVEL 413 COMPUTER SIMULATIONS 414 Preface Thermal and statistical