Statistical Physics (Part II) Alexander Shnirman Institute TKM, Karlsruhe Institute of Technology, Karlsruhe, Germany (Dated: June 10, 2018) 1 Contents I. Real systems: systems with interactions 4 A. Classical limit 4 B. Quantum regime 4 II. Thermodynamic perturbation theory 5 III. Virial expansion 6 A. Van der Waals gas 8 B. Quantum corrections for ideal gases 10 IV. Magnetic systems 11 A. Thermodynamics of magnetic systems 11 B. Exchange interaction 12 C. Ising model in 1D, exact solution 14 1. Evaluating partition function 14 2. Thermodynamic results 15 3. Correlation function 17 D. Cluster expansion 18 E. Mean field approximation: ferromagnetic Heisenberg model. 19 1. Free spins in a magnetic field 20 2. Mean field approximation (intuitive way) 20 3. Mean field approximation (formal way) 21 4. Solution of the self-consistency equation 23 5. Critical behaviour 24 6. G(T;H) vs. F (T;M) above and below the transition. 25 V. Landau theory of 2-nd order phase transitions. 27 A. General considerations 27 B. Landau-Ginzburg functional 28 C. Phase transition at T = Tc. Critical exponents. 29 D. Correlations and fluctuations 30 1. Susceptibility 31 2 2. Correlation function. 32 E. Validity of the mean-field approximation, Ginzburg criterion. 33 1. Relative strength of fluctuations 33 2. Fluctuation correction to the heat capacitance 34 3. Lower and upper critical dimensions 37 F. Classification of phase transitions. 37 G. Critical exponents, universality classes 39 H. Scaling hypothesis. Relations between critical exponents. 40 1. Scaling in the mean field theory 40 2. Scaling hypothesis (Widom, 1966) 40 VI. Elements of physical kinetics 42 A. Langevin Equation 42 1. RCL circuits 44 2. Caldeira-Leggett model 44 B. Master equation 47 1. Example: random walk on a 1D lattice 48 2. Fokker-Planck equation, diffusion 50 3. Master equation for a two-level system coupled to a bath 51 4. Fluctuation Dissipation Theorem (FDT) 53 C. Linear response theory 54 1. Linear response and FDT 55 D. Boltzmann equation 57 3 I. REAL SYSTEMS: SYSTEMS WITH INTERACTIONS A. Classical limit Consider a gas of N interacting classical indistinguishable particles. The Hamilton func- tion reads N X p2 1 X H = i + U(r ) + V (r − r ) : (1) 2m i 2 i j i=1 i6=j For the partition function we obttain 1 Z d3N p Z Z = d3N r e−βH(fpig;frig) : (2) N! (2πh¯)3 We can separate the integrals over momenta from those over coordinates: 1 Z d3N p Z Z = e−βHkin(fpig) d3N r e−βHpot(frig) e−βHint(frig) : (3) N! (2πh¯)3 The first integral (over momenta) can easily be calculated using Z 2 dpxdpydpz p 1 −β 2m 3 e = 3 ; (4) (2πh¯) λT q 2π¯h2 where λT = is the thermal length. We obtain, thus mkBT 1 1 Z 3N −βHpot(frig) −βHint(frig) Z = 3N d r e e (5) N! λT The remaining integral over coordinates is very complicated and cannot be evaluated exactly. Thus we have to resort to approximative methods. B. Quantum regime In the quantum case the situation is even worse. The part of the Hamiltonian corre- sponding to the kinetic energy does not commute with the rest [Hkin;Hpot + Hint] 6= 0. Thus e−βH 6= e−βHkin e−β(Hpot+Hint) and Z = Tr e−βH 6= Tr e−βHkin Tr e−β(Hpot+Hint) : (6) We have to find proper approximations. 4 II. THERMODYNAMIC PERTURBATION THEORY Assume the Hamilton operator can be split into a relatively simple "zeroth order" part H0 and a perturbation gV , i.e., H = H0 + gV . Here g is a small dimensionless coupling constant. We have to calculate the density matrix 1 ρ = e−βH (7) Z as well as the (canonical) partition function Z = Tr[e−βH ] (8) and the free energy 1 F = − ln(Z) : (9) β The idea is that these three objects can be expanded in series of powers of g. That is ρ = ρ0 + gρ1 + :::; (10) Z = Z0 + gZ1 + :::; (11) F = F0 + gF1 + :::: (12) Note that Z1, Z2 etc. have a completely different meaning here as compared to the following section about the virial expansion (Sec. III). The zeroth order is easy to calculate: ρ0 = −βH0 −βH0 (1=Z0)e , Z0 = Tr[e ], F0 = −(1/β) ln(Z0). Our task is to calculate the corrections. We investigate the exponent e−τH . In particular we are interested in its value for τ = β. We observe that τ can be thought of as an imaginary time, i.e. e−itH = e−τH for t = −iτ. We have e−τH = e−τH0 S(τ), where S(τ) ≡ eτH0 e−τH : (13) The operator S satisfies the following differential equation @ S(τ) = eτH0 (H − H)e−τH = eτH0 (H − H)e−τH0 eτH0 e−τH @τ 0 0 τH0 −τH0 = e (−gV )e S(τ) = −gVI (τ)S(τ) ; (14) τH0 −τH0 where VI (τ) ≡ e V e . This is nothing but the interaction representation (recall the lecture course QM II). The formal solution (Dyson series) reads τ R −g dτ1 VI (τ1) S(τ) = Tτ e 0 ; (15) 5 where Tτ is the "time" ordering operator. Expanding to the first order we get τ Z S(τ) = 1 − g dτ1 VI (τ1) + :::: (16) 0 Thus we obtain β Z −βH −βH0 −βH0 τH0 −τH0 Z = Tr[e ] = Tr[e S(β)] = Z0 − g dτTr[e e V e ] + ::: 0 β Z −βH0 −βH0 = Z0 − g dτ Tr[e V ] + ··· = Z0 − gβTr[e V ] + :::: (17) 0 This we have obtained the first order correction to the partition function −βH0 Z1 = −βTr[e V ] : (18) It is straightforward to calculate the correction of the first order to the free energy. We have F = −(1/β) ln(Z) = −(1/β) ln(Z0 + gZ1 + ::: ) Z1 gZ1 = −(1/β) ln Z0 1 + g + ::: = F0 − + ··· = F0 + ghV i0 + :::: (19) Z0 βZ0 Here 1 −βH0 hV i0 = Tr[e V ] : (20) Z0 III. VIRIAL EXPANSION This method allows to obtain an expansion of the kind 2 PV = NkBT (1 + Bn + Cn + ::: ) ; (21) which modifies the classical equation of state of the ideal gas PV = NkBT . It usually works 3 βµ for nλT 1 or equivalently z ≡ e 1, that is in the case of diluted gases. The main idea is to use the expansion of the grand canonical partition function ZG via the canonical ones ZN , where N is the number of particles. This expansion reads X Nβµ βµ 2βµ ZG = ZN e = 1 + Z1e + Z2e + :::: (22) N=0 6 If z ≡ eβµ 1 it might be sufficient to retain only the few first terms in the expansion. Then the problem reduces to calculating the canonical partition functions with small numbers of particles, e.g., Z1 and Z2. We want to calculate the grand canonical potential Ω = −kBT ln ZG. We use the expan- sion 1 X (−1)n−1xn ln(1 + x) = ; (23) n n=1 which leads to (Z eβµ + Z e2βµ + ::: )2 ln Z = (Z eβµ + Z e2βµ + ::: ) − 1 2 : (24) G 1 2 2 Collecting the powers of the fugacity z ≡ eβµ we get Z2 ln Z = Z eβµ + Z − 1 e2βµ + :::: (25) G 1 2 2 For the grand potential we obtain Z2 Ω = −k TZ eβµ − k T Z − 1 e2βµ + :::: (26) B 1 B 2 2 This expression allows us to calculate the particle number: @Ω βµ 2 2βµ N = − = Z1e + (2Z2 − Z1 ) e + :::: (27) @µ V;T From (26) and (27) we obtain Z2 PV = −Ω = k T N − Z − 1 e2βµ + ::: : (28) B 2 2 With eβµ ≈ N we get Z1 2 2 Z1 N PV = −Ω = kBT N − Z2 − 2 + ::: : (29) 2 Z1 Finally PV = kBTN [1 + Bn + ::: ] ; (30) where Z2 1 B ≡ −V 2 − : (31) Z1 2 7 A. Van der Waals gas N X p2 1 X H = i + V (r − r ) : (32) 2m 2 i j i=1 i6=j 1 Z d3N p Z Z = d3N r e−βH(fpig;frig) N N! (2πh¯)3 1 1 Z 1 P 3N −β 2 i6=j V (ri−rj ) = 3N d r e : (33) N! λT For N = 1 this gives V Z1 = 3 : (34) λT Limiting the virial expansion at N = 1 we would obtain βµ Ω = −kBTZ1e : (35) This would then give @Ω βµ Ω PV N = − = Z1e = − = : (36) @µ T;V kBT kBT Thus we recover the ideal gas relation PV = NkBT . For N = 2 we obtain 1 1 Z 1 V Z 3 3 −βV (r1−r2) 3 −βV (r) Z2 = 6 d r1d r2 e = 6 d r e : (37) 2 λT 2 λT 1 Z B = − d3r e−βV (r) − 1 : (38) 2 We choose the interaction potential as shown in Fig. 1. For r > 2r0 the potential is week, so that βV (r) 1. Then −βV (r) e − 1 ≈ −1 for r < 2r0 ; −βV (r) e − 1 ≈ −βV (r) for r > 2r0 : (39) This gives 1 4π β Z B = (2r )3 + d3r V (r) = b − aβ ; (40) 2 3 0 2 jrj>2r0 8 FIG. 1: Interaction potential in VW gas. where 1 4π 1 Z b = (2r )3 > 0 and a = − d3r V (r) > 0 : (41) 2 3 0 2 jrj>2r0 Finally, we obtain an PV = kBTN 1 + bn − : (42) kBT This is equivalent to 2 (P + an )V = kBTN[1 + bn] : (43) Since bn 1 we can replace (1+bn) ≈ (1−bn)−1 and we obtain the van der Waals equation of state 2 2 (P + an )(1 − bn)V = (P + an )(V − bN) = kBTN: (44) The physical meaning of the condition bn 1 can be understood if we notice that b is roughly the "excluded" volume per particle.