Physics - Cambridge - Statistical Physics

Statistical Physics Dr. A. J. Macfarlane1 Lent 1998 1LATEXed by Paul Metcalfe – comments to soc-archim-notes@lists.cam.ac.uk. Revision: 1.10 Date: 1998-06-03 20:13:39+01 The following people have maintained these notes. – date Paul Metcalfe Contents Introduction v 1 Quantum Statistical Mechanics 1 1.1 Introduction to Quantum Statistical Mechanics . . . . . . . . . . . . 1 1.2 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Towards thermodynamic variables . . . . . . . . . . . . . . . . . . . 5 1.5 Towards applications . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5.1 N particle partition function . . . . . . . . . . . . . . . . . . 7 1.5.2 Extensive and intensive variables . . . . . . . . . . . . . . . 8 1.5.3 Density of states . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5.4 Gas of spinless particles . . . . . . . . . . . . . . . . . . . . 9 1.5.5 Entropy and the Gibbs paradox . . . . . . . . . . . . . . . . . 9 1.6 Harmonic oscillator model . . . . . . . . . . . . . . . . . . . . . . . 10 2 Thermodynamics 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Applications of dE = TdS − PdV . . . . . . . . . . . . . . . . . . 12 2.2.1 Integrability conditions . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Specific heats . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.3 Adiabatic changes . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.4 Entropy of n moles of ideal gas . . . . . . . . . . . . . . . . 14 2.2.5 van der Waal’s equation . . . . . . . . . . . . . . . . . . . . 15 2.2.6 The Joule effect . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Some thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 The second law . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Grand ensemble methods 19 3.1 The formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Systems of non-interacting identical particles . . . . . . . . . . . . . 21 3.2.1 A little quantum mechanics . . . . . . . . . . . . . . . . . . 21 3.2.2 The partition functions . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Degenerate Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.1 Heat capacity at low temperature . . . . . . . . . . . . . . . . 27 3.5 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . 28 iii iv CONTENTS 3.6 White dwarf stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Classical statistical mechanics 31 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Diatomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Specific heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Weak interparticle forces . . . . . . . . . . . . . . . . . . . . . . . . 34 4.6 The Maxwell distribution . . . . . . . . . . . . . . . . . . . . . . . . 35 Approximations 37 Introduction These notes are based on the course “Statistical Physics” given by Dr. A. J. Macfarlane in Cambridge in the Lent Term 1998. These typeset notes are totally unconnected with Dr. Macfarlane. The recommended books for this course are discussed in the bibliography. Other sets of notes are available for different courses. At the time of typing these courses were: Probability Discrete Mathematics Analysis Further Analysis Methods Quantum Mechanics Fluid Dynamics 1 Quadratic Mathematics Geometry Dynamics of D.E.’s Foundations of QM Electrodynamics Methods of Math. Phys Fluid Dynamics 2 Waves (etc.) Statistical Physics General Relativity Dynamical Systems They may be downloaded from http://home.arachsys.com/˜pdm/ or http://www.cam.ac.uk/CambUniv/Societies/archim/notes.htm or you can email soc-archim-notes@lists.cam.ac.uk to get a copy of the sets you require. v Copyright (c) The Archimedeans, Cambridge University. All rights reserved. Redistribution and use of these notes in electronic or printed form, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of the electronic files must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in printed form must reproduce the above copyright notice, this list of conditions and the following disclaimer. 3. All materials derived from these notes must display the following acknowledge- ment: This product includes notes developed by The Archimedeans, Cambridge University and their contributors. 4. Neither the name of The Archimedeans nor the names of their contributors may be used to endorse or promote products derived from these notes. 5. Neither these notes nor any derived products may be sold on a for-profit basis, although a fee may be required for the physical act of copying. 6. You must cause any edited versions to carry prominent notices stating that you edited them and the date of any change. THESE NOTES ARE PROVIDED BY THE ARCHIMEDEANS AND CONTRIB- UTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABIL- ITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ARCHIMEDEANS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSE- QUENTIAL DAMAGES HOWEVER CAUSED AND ON ANY THEORY OF LI- ABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUD- ING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THESE NOTES, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAM- AGE. Chapter 1 Quantum Statistical Mechanics 1.1 Introduction to Quantum Statistical Mechanics Statistical mechanics deals with macroscopic systems of many particles. Consider an isolated system S of gas in a vessel whose walls neither let heat in or out and is subject to no mechanical action. No matter how it is prepared, it is an experimental fact that S reaches a steady state (a state of thermodynamic equilibrium) in which a set Σ of (rather few) thermodynamic variables are constant. Σ includes the pressure P , volume V , temperature T , total energy E and entropy S. Any state of S (before or after thermodynamic equilibrium is reached) contains particles with positions and momenta changing in time. We cannot possibly analyse this in detail — even if we knew the forces and initial conditions, and could solve the resulting system there is no hope that we could organise usefully the vast amount of data. At best, we aim to treat the possible states of motion of S by some averaging or statistical procedure that allows us to predict values of the variables in Σ that are constant in states of thermodynamic equilibrium of S. We use quantum mechanical ideas to approach the subject. Consider • microstates of S, the stationary quantum states |i〉. • macrostates, which correspond to states of thermodynamic equilibrium. The latter are not states in the quantum mechanical sense, but do involve the vast numbers of microstates. The ergodic hypothesis (which is provable for some systems) is that S passes in time through all its possible states compatible with its state of thermodynamic equilib- rium. This allows us to replace time averages for a single system S with averages at a fixed time over a suitable ensemble E of systems, each identical to S in its macroscopic properties. The most probable state of E reveals which microstates of S contribute to its macroscopic states of thermodynamic equilibrium and yield excellent approximations to the values of its thermodynamic variables. Let E, N be the total energy and total number of particles of S respectively. One meets three types of ensemble. • microcanonical : each member of E has the same values of E and N . 1 2 CHAPTER 1. QUANTUM STATISTICAL MECHANICS • canonical : each member of E has N particles but E is not fixed, although the average total energy is time independent. • grand : neither E nor N is fixed, although the averages are. For the latter two, fluctuations about the average values are found to be very small and all three types of ensemble give the same thermal physics. This course studies mainly canonical ensembles. We will do systems with identical bosons/fermions via the grand ensemble for technical simplicity. The fact that averaging procedures give reliable estimates of values of physical variables and apparently different procedures yield equivalent results is due to the effect of the very large number of particles involved. For instance, consider the probability pm of obtaining N/2 +m heads from N coin tosses. pm = 2−N ( N N 2 +m ) ∼ √ 2 πN e− 2m2 N , using Stirling’s formula1 for N � m � 0. If N = 1023 and m = 1017 then the exponential term e−1011 is effectively zero. 1.2 Canonical ensemble We study a system S ofN particles in a volume V withN and V fixed. The microstates of S are the (complete orthogonal) set |i〉 with energy eigenvalues E i. We assume the spectrum is discrete but with possible degeneracy. We associate with S a canonical ensemble E . This is a very large number A of distinguishable replicas of S, S1, . . . ,SA. Suppose that in a possible state of E there are ai members of E in the microstate |i〉. Then ∑ i ai = A and ∑ i aiEi = AE. (1.1) The average energy of the members of E is thus the fixed value E. Given a set {ai} (a configuration of E) there are W (a) = A!Q i ai! ways of realising it. Proof. We can distribute a1 systems in the microstate |1〉 over E in ( A a1 ) ways. Then we can distribute the a2 states in |2〉 in ( A−a1 a2 ) ways (and so on). Thus W (a) = ( A a1 )( A− a1 a2 ) · · · = A!∏ i ai! . We assign equal a priori probability to each way of realising each possible con- figuration of E and thus there is a probability proportional to W (a) of realising the configuration {ai}. 1logn! ∼ n logn− n+ 1 2 log 2πn 1.2. CANONICAL ENSEMBLE 3 We will associate the state of thermodynamic equilibrium of S (for the fixed values N , V , E) with the configuration of E that is most probable in the presence of the constraints (1.1). To compute this suppose A � 0 such that ai � 0 (negligible probability attaches to configurations where this fails). For large n, Stirling’s formula allows logn! ∼ n (logn− 1) and so logW ∼ A logA−A− ∑ i ai (log ai − 1) = A logA− ∑ i ai log ai. We seek to maximise this subject to the constraints (1.1). We use two Lagrange multipliers α and β and solve 0 = ∂ ∂aj ( logW − α ∑ i ai − β ∑ i aiEi ) and so log aj + 1 + α+ βEj = 0. Thus aj = e−1−α−βEj . (1.2) We eliminate α via A = ∑ i ai = e −1−αZ , defining the canonical partition func- tion Z = ∑ i e−βEi ≡ ∑ Ej Ω(Ej)e−βEj , (1.3) where Ω(Ej) is the degeneracy of the energy level Ej . The fraction of members of E in the microstate |i〉 in the macrostate of thermody- namic equilibrium is ρi = ai A = 1 Z e−βEi. (1.4) This is the Boltzmann distribution, and may be thought of as the probability of finding |i〉 in the state of thermodynamic equilibrium. We define the average 〈X〉 of a physical variable X taking the value X i in the state |i〉 by 〈X〉 = ∑ i ρiXi. (1.5) For instance 〈E〉 = 1A ∑ i aiEi = E (reassuringly). Z is very important. It leads directly from quantum mechanical data to calculation of the thermodynamic variables for S in thermodynamic equilibrium. For instance, E = −∂ logZ ∂β . (1.6) 4 CHAPTER 1. QUANTUM STATISTICAL MECHANICS Here, holding V fixed corresponds to keeping all the E i fixed. For A large (as in all cases of interest) the most probable state is overwhelmingly so. It gives in effect the average over all possible states of the ensemble. The idea of associating average values with actual physical predictions depends on the possible variances being negligible. We can calculate the variance in the energy in a similar way. Note thatE = − 1Z ∂Z∂β and so ∂E ∂β = − 1 Z ∂2Z ∂β2 + 1 Z2 ( ∂Z ∂β )2 = −〈E2〉+ 〈E〉2 = − (∆E)2 . For typical large systems E ∝ N , and as E depends smoothly on β we expect ∂E ∂β ∝ N as well. Thus |∆E| E ∝ √ N N = N− 1 2 , which is very small for very large N . 1.3 Temperature Given two systems Sa and Sb with fixed volumes Va, Vb and numbers of particles Na, Nb respectively we place them in contact to form a composite system Sab such that energy can pass from one to the other and a state of thermodynamic equilibrium is reached. We make a canonical ensemble Eab for Sab by distributing A replicas of Sa and A replicas of Sb independently across the A members of Eab so that each member of Eab has a component of type Sab. Suppose that Sa has microstates |i〉 of energyEai, Sb has microstates |σ〉 of energy Ebσ and Sab has microstates |i, σ〉 of energy Eai + Ebσ .2 Suppose the |i〉 occurs ai times, |σ〉 occurs bi times and |i, σ〉 occurs ciσ times in Eab. Then we have ∑ i ai = A ∑ σ bσ = A (1.7) ai = ∑ σ ciσ bσ = ∑ i ciσ ∑ iσ ciσ = A (1.8) We demand that the average energy or Eab be fixed at AE = ∑ i aiEai + ∑ σ bσEbσ (1.9) AE = ∑ iσ ciσ (Eai + Ebσ) . (1.10) Now the configuration {ciσ} arises in 2If the interaction of particles of Sa with those of Sb is negligible, except insofar as necessary to allow a state of thermal equilibrium to be reached. 1.4. TOWARDS THERMODYNAMIC VARIABLES 5 W (c) ∝ A!∏ iσ ciσ! (1.11) ways and the configuration {ai}{bσ} arises in W (a)W (b) ∝ A!∏ i ai! A!∏ σ bσ! (1.12) ways. We can calculate the most probable distribution of {c iσ} by maximising (1.11) subject to (1.8) and (1.10). We get ciσ = e−1−αab−β(Eai+Ebσ) ρiσ = ciσ A . We can also maximise (1.12) subject to (1.7) and (1.9) to get ai = e−1−αa−βEai ρi = ai A bσ = e−1−αb−βEbσ ρσ = bσ A . Defining Zab(β) = ∑ iσ e−β(Eai+Ebσ) Za(β) = ∑ i e−βEai Zb(β) = ∑ σ e−βEbσ we see that Zab(β) = Za(β)Zb(β) and ρiσ = ρiρσ. A common value for β characterises the state of thermal equilibrium of Sab and it is natural to assume that β is some function of temperature T and that the average energy (the energy) of a system is a function of N , V and T . We define T by β = 1kT where k is a constant. As this argument applies to any two systems, k should be a universal constant (Boltzmann’s constant) the same for all systems when a universal definition of T is used. We will define T for a standard system (a thermometer) and for other systems by putting them into a state of thermal equilibrium with our standard system. Recall the ideal gas law, PV = NkT for a sample of N molecules.3 k is the same because equal volumes of all gases at fixed temperature and pressure are observed to have the same number of molecules. T is in degrees Kelvin. 1.4 Towards thermodynamic variables Recall that in quantum mechanics, spinless particles in a cube of side L have energies En = � 2 2mL2 |n|2, where n ∈ N3. Note that E ∝ V − 2 3 and we generalise this to Ei = Ei(V ). Consider Sgas in thermodynamic equilibrium in a container of volume V and change V by a small amount by slowly moving one of its walls. 3This applies to all gases at sufficiently low density. 6 CHAPTER 1. QUANTUM STATISTICAL MECHANICS The Ei will change according to Ei �→ Ei + ∂Ei∂V δV and changes in which the ρi do not change will be examined first. The change in energy is supplied by the work done on S by a piston applied to the right hand wall. For slow motion the system passes through successive positions of thermal equilibrium and then the applied force just balances the pressure force PA. Thus the work done to the system is −PAδl = −PδV . δE = δW = −PδV . Now E = ∑ i ρiEi and so P = − ∑ i ρi ∂Ei ∂V = 1 β ∂ ∂V logZ. (1.13) Consider next changes in which the ρi are allowed to vary. Then δE = −PδV +∑ iEiδρi, and note that ∑ i δρi = 0. Define the entropy S = k A logWmax that is, W at thermal equilibrium = k A ( A logA− ∑ i ai log ai ) = k A ( A logA− ∑ i ai log ρi − logA ∑ i ai ) = −k ∑ i ρi log ρi at thermal equilibrium. (1.14) We see that δS = −k ∑ δρi ( log ρi + ρi 1 ρi log ρi ) = −k ∑ δρi (−βEi − logZ + 1) = 1 T ∑ i Eiδρi and thus ∑ iEiδρi = TδS. We have defined S such that δE = TδS − PδV or in terms of exact differentials, dE = TdS − PdV. (1.15) We might regard this as arising from the comparison of ensembles with infinitesi- mally different states of thermal equilibrium. (1.15) states the First Law of Thermody- namics4, written in the form dE = δQ+ δW, (1.16) where δQ is the heat supplied to S and δW is the work done on S. 4You can’t win... 1.5. TOWARDS APPLICATIONS 7 (1.15) also shows that T = ∂E∂S and P = − ∂E∂V . For a gas of spinless particles, E = ∑ i Eiρi ∝ V − 2 3 and so P = 23 E V , giving PV = 23E. Combined with the result for a perfect gas, E = 3 2NkT (see later), we have Boyle’s law PV = NkT . We want to be able to calculate S from the partition function. Now, at thermal equilibrium, S = −k ∑ i ρi log ρi = −k ∑ i ρi (−βEi − logZ) = 1 T E + k logZ. Define the free energy F = E − TS, then F = − 1β logZ. (1.17) Also, S = − 1 T ∂ logZ ∂β + k logZ = ∂ ∂T (kT logZ) = −∂F ∂T . (1.18) We have a very tangible definition of S, as proportional to logWmax, where Wmax is the number of microstates of S contributing to the state of thermal equilibrium. Given the partition function (1.3), we can calculate the thermodynamic variables in a state of thermal equilibrium, P (by (1.13)), E (by (1.6)), F (by (1.17)) and S (by (1.18)). 1.5 Towards applications 1.5.1 N particle partition function Consider a composite system with Hamiltonian H = H1 +H2 and H1 and H2 inde- pendent (they commute). Let H1|a〉 = E1a|a〉 and H2|α〉 = E2α|α〉. Then as H1 and H2 commute, H |a, α〉 = Eaα|a, α〉, where Eaα = E1a + E2α. Then the 2 particle partition function is Z = ∑ a,α e−βEaα = ∑ a e−βE1a ∑ α e−βE2α = Z1Z2. Let Z ≡ ZN describe the sum over states of a gas of N very weakly interacting particles. If these are supposed independent (distinguishable) then Z = z N , where z is the “one particle partition function”. This allows easy calculations, but fails for systems of identical (indistinguishable) bosons or fermions. For the latter it seems best to use grand ensemble methods. Later we will consider a gas of diatomic molecules, H = HT +Hrot +Hvib, where HT describes the centre of mass motion, Hrot describes rotation about the centre of mass and Hvib describes vibration along the axis. We see that the one particle partition function is z = zT zrotzvib. 8 CHAPTER 1. QUANTUM STATISTICAL MECHANICS 1.5.2 Extensive and intensive variables Extensive variables are proportional to the amount of matter in a system at fixed tem- perature, whereas intensive variables are independent of the amount of matter. In general, N , V , S and E are extensive and T and P are intensive. Think initially of a system made up of two independent subsystems. Then V = V1+V2. In a state of thermal equilibrium the systems have the same β (and hence same T ), and so T is intensive. In a state of equilibrium there is mechanical equilibrium and so the systems have the same pressure, and so P is intensive. As Z = Z1Z2 we see that E = E1 + E2, S = S1 + S2 and so on. A logical ex- tension of this argument gives that V , E and S are proportional to N and so extensive. This can fail if volume energies do not swamp surface energies or if intermolecular forces are neither weak nor short range. 1.5.3 Density of states Consider a spinless particle in a box of side L. Instead of using solutions ψ of the Schro¨dinger equation such that ψ = 0 on the walls we use periodic boundary condi- tions, ψ(x+ Li)