Basic Theoretical Physics: A Concise Overview P40 pdf

53.7 Molecular Field Theories; Mean Field Approaches 407 following identity): ˆ S l · ˆ S m ≡  ˆ S l  T · ˆ S m + ˆ S l ·  ˆ S m  T +  ˆ S l −  ˆ S l  T  ·  ˆ S m −  ˆ S m  T  . Even the factor 2 in front of  m is derived in this way, i.e., from J l,m = J m,l . However, neglecting fluctuations is grossly incorrect in the critical region, which is often not small (e.g., for typical ferromagnets it amounts to the upper ∼ 20% of the region below T c ; in contrast, for conventional 3d- superconductivity the critical region is negligible). Generally the neglected terms have a drastic influence on the critical exponents. In the critical region the order parameter does not converge to zero ∝ (T c − T ) 1 2 , as the Landau theory predicts, but ∝ (T c −T ) β ,whereforad = 3-dimensional system the value of β is ≈ 1 3 instead of β Landau ≡ 1 2 . We also have, both above and below T c , with different coefficients: χ ∝|T c − T | −γ , with γ = γ 3d ≈ 4 3 instead of the value from the Landau theory, γ Landau ≡ 1 . Similarly we have ξ ∝|T c − T | −ν , with ν = ν 3d ≈ 2 3 instead of the Landau value ν Landau ≡ 1 2 . For the twodimensional Ising model, a model which only differs from the Heisenberg model by the property that the vectorial spin operators ˆ S l and ˆ S m are replaced by the z-components ( ˆ S z ) l and ( ˆ S z ) m , the difference be- tween the critical exponents and the Landau values is even more drastic than in d=3, as detailed in the following. For d = 2 the exact values of the critical exponents are actually known: β(2d) Ising ≡ 1 8 ,γ(2d) Ising ≡ 7 4 and ν(2d) Ising ≡ 1 , 408 53 Applications I: Fermions, Bosons, Condensation Phenomena and the critical temperature is reduced by the fluctuations by almost 50% compared to the molecular field approximation. (For the d = 2-dimensional Heisenberg model the phase transition is even completely suppressed by fluc- tuations). It is no coincidence 19 that the quantitative values of the critical exponents and other characteristic quantities of the critical behavior are universal in the sense that they are (almost) independent of the details of the interactions. For interactions which are sufficiently short-ranged, these quantities depend only on (i) the dimensionality d of the system, and (ii) the symmetry of the order parameter, e.g., whether one is dealing with uniaxial symmetry, as for the Ising model, or isotropic symmetry, as for the Heisenberg model. In this context one defines universality classes of systems with the same critical behavior. For different models one can also define a kind of molecular field ap- proximation, usually called mean field approximation, by replacing a sum of bilinear operators by a self-consistent temperature-dependent linear ap- proximation in which fluctuations are neglected. In this context one should mention the Hartree-Fock approximation 20 , see Part III, and the Hartree- Fock-Bogoliubov approximation for the normal and superconducting states of a system with many electrons. Without going into details we simply men- tion that the results of all these approximations are similar to the Landau theory. However, in the following respect the mean field theories are somewhat more general than the Landau theory: they lead to quantitative predictions for the order parameter, for T c , and for Landau’s phenomenological coeffi- cients A, α and b. In this context one should also mention the BCS theory of superconductivity (which we do not present, see [25]), since not only the Ginzburg-Landau theory of superconductivity can be derived from it but also one sees in particular that the carriers of superconductivity have the charge 2e,note. 53.8 Fluctuations In the preceding subsections we have seen that thermal fluctuations are im- portant in the neighborhood of the critical temperature of a second-order phase transition, i.e. near the critical temperature of a liquid-gas transition or for a ferromagnet. For example, it is plausible that density fluctuations become very large if the isothermal compressibility diverges (which is the case at T c ). 19 The reason is again the universality of the phenomena within regions of diameter ξ,whereξ is the thermal correlation length. 20 Here the above bilinear operators are “bilinear” expressions (i.e., terms con- structed from four operators) of the “linear” entities ˆc + i ˆc j (i.e., a basis con- structed from the products of two operators), where ˆc + i and ˆc j are Fermi creation and destruction operators, respectively. 53.8 Fluctuations 409 We also expect that the fluctuations in magnetization for a ferromagnet are especially large when the isothermal susceptibility diverges, and that especially strong energy fluctuations arise when the isothermal specific heat diverges. These qualitative insights can be formulated quantitatively as follows (here, for simplicity, only the second and third cases are considered): |η k | 2  T −|η k  T | 2 ≡ k B T · χ k (T ) and (53.10) H 2  T − (H T ) 2 ≡ k B T 2 · C (V,N,H, ) , (53.11) where C (V,N,H, ) = ∂U(T,V,N,H, ) ∂T , with U(T,V,N,H, )=H T , is the isothermal heat capacity and χ k (T )thek-dependent magnetic suscep- tibility (see below). For simplicity our proof is only performed for k = 0 and only for the Ising model. The quantity η is (apart from a proportionality factor) the value of the saturation magnetization for H → 0 + , i.e., η(T ):=M s (T ) . We thus assume that H = −  l,m J l,m s l s m −h  l s l , with s l = ±1 , and define M :=  l s l , i.e., H = H 0 − hM. In the following, “tr” means “trace”. We then have M T = tr  M ·e −β·(H 0 −hM)   tr  e −β·(H 0 −hM)  , and hence χ = ∂M T ∂h = β · ⎛ ⎝ tr  M 2 e −β(H 0 −hM)  tr  e −β(H 0 −hM)  −  tr  Me −β(H 0 −hM)  tr  e −β(H 0 −hM)   2 ⎞ ⎠ , which is (53.10). Equation (53.11) can be shown similarly, by differentiation w.r.t. β = 1 k B T of the relation U = H T = tr  He −βH  tr {e −βH } . 410 53 Applications I: Fermions, Bosons, Condensation Phenomena A similar relation between fluctuations and response is also valid in dy- namics – which we shall not prove here since even the formulation requires considerable effort (→ fluctuation-dissipation theorem, [49], as given below): Let Φ ˆ A, ˆ B (t):= 1 2 ·  ˆ A H (t) ˆ B H (0) + ˆ B H (0) ˆ A H (t)  T be the so-called fluctuation function of two observables ˆ A and ˆ B, represented by two Hermitian operators in the Heisenberg representation, e.g., A H (t):=e i Ht  ˆ Ae −i Ht  . The Fourier transform ϕ ˆ A , ˆ B (ω) of this fluctuation function is defined through the relation Φ ˆ A , ˆ B (t)=:  e iωt dω 2π ϕ ˆ A , ˆ B (ω) . Similarly for t>0thegeneralized dynamic susceptibility χ ˆ A , ˆ B (ω) is defined as the Fourier transform of the dynamic response function X ˆ A, ˆ B (t −t  ):= δ ˆ A T (t) δh ˆ B (t  ) , 21 where H = H 0 − h B (t  ) ˆ B, i.e., the Hamilton operator H 0 of the system is perturbed, e.g., by an alter- nating magnetic field h B (t  ) with the associated operator ˆ B = ˆ S z ,andthe response of the quantity ˆ A on this perturbation is observed. Now, the dynamic susceptibility χ ˆ A , ˆ B (ω) has two components: a reactive part χ  ˆ A, ˆ B and a dissipative part χ  ˆ A, ˆ B , i.e., χ ˆ A, ˆ B (ω)=χ  ˆ A, ˆ B (ω)+iχ  ˆ A, ˆ B (ω) , where the reactive part is an odd function and the dissipative part an even function of ω. The dissipative part represents the losses of the response pro- cess. Furthermore, it is generally observed that the larger the dissipative part, the larger the fluctuations. Again this can be formulated quantitatively us- ing the fluctuation-dissipation theorem [49]. All expectation values and the 21 This quantity is only different from zero for t ≥ t  . 53.9 Monte Carlo Simulations 411 quantities ˆ A H (t) etc. are taken with the unperturbed Hamiltonian H 0 : ϕ ˆ A , ˆ B (ω) ≡  ·coth ω 2k B T ·χ  ˆ A, ˆ B (ω) . (53.12) In the “classical limit”,  → 0, the product of the first two factors on the r.h.s. of this theorem converges to 2k B T ω , i.e., ∝ T as for the static behavior, in agreement with (53.10). In this limit the theorem is also known as the Nyquist theorem; but for the true value of  it also covers quantum fluctuations. Generally, the fluctuation-dissipation theorem only applies to ergodic systems, i.e., if, after the onset of the pertur- bation, the system under consideration comes to thermal equilibrium within thetimeofmeasurement. Again this means that without generalization the theorem does not apply to “glassy” systems. 53.9 Monte Carlo Simulations Many of the important relationships described in the previous sections can be visualized directly and evaluated numerically by means of computer simu- lations. This has been possible for several decades. In fact, so-called Monte Carlo simulations are very well known, and as there exists a vast amount of literature, for example [51], we shall not go into any details, but only describe the principles of the Metropolis algorithm, [50]. One starts at time t ν from a configuration X(x 1 ,x 2 , ) of the system, e.g., from a spin configuration (or a fluid configuration) of all spins (or all positions plus momenta) of all N particles of the system. These configurations have the energy E(X). Then a new state X  is proposed (but not yet accepted) by some systematic procedure involving random numbers. If the proposed new state has a lower energy, E(X  ) <E(X) , then it is always accepted, i.e., X(t ν+1 )=X  . In contrast, if the energy of the proposed state is enhanced or at least as high as before, i.e., if E(X  )=E(X)+ΔE , with ΔE ≥ 0 , then the suggested state is only accepted if it is not too unfavorable. This means precisely: a random number r ∈ [0, 1], independently and identically 412 53 Applications I: Fermions, Bosons, Condensation Phenomena distributed in this interval, is drawn, and the proposed state is accepted iff e −ΔE/(k B T ) ≥ r. In the case of acceptance (or rejection, respectively), the next state is the proposed one (or the old one), X(t ν+1 ) ≡ X  (or X(t ν+1 ) ≡ X) . Through this algorithm one obtains a sequence X(t ν ) → X(t ν+1 ) → of random configurations of the system, a so-called Markoff chain, which is equivalent to classical thermodynamics i.e., after n equilibration steps, thermal averages at the considered temperature T are identical to chain- averages, f(X) T = M −1 n+M  ν≡n+1 f(X(t ν )) , and ν is actually proportional to the time. In fact, it can be proved that the Metropolis algorithm leads to thermal equilibrium, again provided that the system is ergodic, i.e., that the dynamics do not show glassy behavior. Monte Carlo calculations are now a well-established method, and flexible enough for dealing with a classical problem, whereas the inclusion of quantum mechanics, i.e., at low temperatures, still poses difficulties. 54 Applications II: Phase Equilibria in Chemical Physics Finally, a number of sections on chemical thermodynamics will now follow. 54.1 Additivity of the Entropy; Partial Pressure; Entropy of Mixing For simplicity we start with a closed fluid system containing two phases. In thermal equilibrium the entropy is maximized: S(U 1 ,U 2 ,V 1 ,V 2 ,N 1 ,N 2 ) ! =max. (54.1) Since (i) V 1 + V 2 = constant, (ii) N 1 + N 2 = constant and (iii) U 1 + U 2 = constant we may write: dS =  ∂S ∂U 1 − ∂S ∂U 2  · dU 1 +  ∂S ∂V 1 − ∂S ∂V 2  ·dV 1 +  ∂S ∂N 1 − ∂S ∂N 2  · dN 1 . Then with ∂S ∂U = 1 T , ∂S ∂V = p T and ∂S ∂N = − μ T it follows that in thermodynamic equilibrium because dS ! =0: T 1 = T 2 ,p 1 = p 2 and μ 1 = μ 2 . Let the partial systems “1” and “2” be independent, i.e., the probabilities  for the states of the system factorize: (x “1 + 2” )= 1 (x 1 ) ·  2 (x 2 ) . Since ln(a · b)=lna +lnb. we obtain S k B = −  ∀statesof“1 + 2” (“1 + 2”) · ln (“1 + 2”) 414 54 Applications II: Phase Equilibria in Chemical Physics = −  x 1  1 (x 1 ) ·  x 2  2 (x 2 ) · ln  1 (x 1 ) −  x 2  2 (x 2 ) ·  x 1  1 (x 1 ) · ln  2 (x 2 ) , and therefore with  x i  i (x i ) ≡ 1weobtain: S “1 + 2” k B = 2  i=1 S i k B , (54.2) implying the additivity of the entropies of independent partial systems.This fundamental result was already known to Boltzmann. To define the notions of partial pressure and entropy of mixing,letus perform a thought experiment with two complementary semipermeable mem- branes, as follows. Assume that the systems 1 and 2 consist of two different well-mixed fluids (particles with attached fluid element, e.g., ideal gases with vacuum) which are initially contained in a common rectangular 3d volume V . For the two semipermeable membranes, which form rectangular 3d cages, and which initially both coincide with the common boundary of V ,letthefirst membrane, SeM 1 ,bepermeable for particles of kind 1, but nonpermeable for particles of kind 2; the permeability properties of the second semipermeable membrane, SeM 2 , are just the opposite: the second membrane is nonperme- able (permeable) for particles of type 1 (type 2). 1 On adiabatic (loss-less) separation of the two complementary semiper- mable cages, the two kinds of particles become separated (“de-mixed”) and then occupy equally-sized volumes, V , such that the respective pressures p i are well-defined. In fact, these (measurable!) pressures in the respective cages (after separation) define the partial pressures p i . The above statements are supported by Fig. 54.1 (see also Fig. 54.2): Compared with the original total pressure p,thepartial pressures are reduced. Whereas for ideal gases we have p 1 + p 2 ≡ p (since p i = N i p N 1 +N 2 ), this is generally not true for interacting systems, where typically p 1 +p 2 >p,since through the separation an important part of the (negative) internal pressure, i.e., the part corresponding to the attraction by particles of a different kind, ceases to exist. 1 In Fig. 54.1 the permeabilty properties on the l.h.s. (component 1) are somewhat different from those of the text. 54.1 Additivity of the Entropy; Partial Pressure; Entropy of Mixing 415 Fig. 54.1. Partial pressures and adiabatic de-mixing of the components of a fluid. The volume in the middle of the diagram initially contains a fluid mixture with two different components “1” and “2”. The wall on the l.h.s. is non-permeable for both components, whereas that on the r.h.s. is semipermeable, e.g. non-permeable only to the first component (see e.g. Fig. 54.2). The compressional work is therefore reduced from δA = −p · dV to δA = −p 1 · dV ,wherep 1 is the partial pressure. Moreover, by moving the l.h.s. wall to the right, the two fluid-components can be demixed, and if afterwards the size of the volumes for the separate components is the same as before, then S(T,p) |fluid ≡ P i=1,2 S i (T,p i ) Since the separation is done reversibly, in general we have S “1 + 2” (p) ≡ S 1 (p 1 )+S 2 (p 2 ) (note the different pressures!), which is not in contradiction with (54.2), but should be seen as an additional specification of the additivity of partial en- tropies, which can apply even if the above partial pressures do not add-up to p. Now assume that in the original container not two, but k components (i.e., k different ideal gases) exist. The entropy of an ideal gas as a function of temperature, pressure and particle number has already been treated earlier. Using these results, with S(T,p,N 1 , N k )= k  i=1 S i (T,p i ,N i )and p i = N i N ·p, i.e., p i = c i ·p, with the concentrations c i = N i N ,wehave S(T,p,N 1 ,N 2 , ,N k ) k B = k  i=1 N i ·  ln 5k B T 2p − ln c i + s (0) i k B  . (54.3) Here s (0) i is a non-essential entropy constant, which (apart from constants of nature) only depends on the logarithm of the mass of the molecule consid- ered (see above). 416 54 Applications II: Phase Equilibria in Chemical Physics In (54.3) the term in −ln c i is more important. This is the entropy of mixing. The presence of this important quantity does not contradict the ad- ditivity of entropies; rather it is a consequence of this fundamental property, since the partial entropies must be originally calculated with the partial pres- sures p i = c i ·p and not the total pressure p, although this is finally introduced via the above relation. It was also noted earlier that in fluids the free enthalpy per particle, g i (T,p i )= 1 N · G i (T,p i ,N i ) , is identical with the chemical potential μ i (T,p i ), and that the entropy per particle, s i (T,p i ), can be obtained by derivation w.r.t. the temperature T from g i (T,p i ), i.e., s i (T,p i )=− ∂g(T,p i ) ∂T . From (54.3) we thus obtain for the free enthalpy per particle of type i in a mixture of fluids: g i (T,p)=g (0) i (T,p)+k B T · lnc i , (54.4) where the function g (0) i (T,p) depends on temperature and pressure, which corresponds to c i ≡ 1, i.e., to the pure compound. This result is used in the following. It is the basis of the law of mass action, which is treated below. 54.2 Chemical Reactions; the Law of Mass Action In the following we consider so-called uninhibited 2 chemical reaction equilib- ria of the form ν 1 A 1 + ν 2 A 2 ↔−ν 3 A 3 . Here the ν i are suitable positive or negative integers (the sign depends on i, and the negative integers are always on the r.h.s., such that −ν 3 ≡|ν 3 |); e.g., for the so-called detonating gas reaction, the following formula applies: 2H +O ↔ H 2 O, whereas the corresponding inhibited reaction is 2H 2 + O 2 ↔ 2H 2 O (slight differences, but remarkable effects! In the “uninhibited” case one is dealing with O atoms, in the “inhibited” case, however, with the usual O 2 molecule.) The standard form of these reactions that occur after mixing in fluids is: k  i=1 ν i A i =0, 2 The term uninhibited reaction equilibrium means that the reactions considered are in thermodynamic equilibrium, possibly after the addition of suitable cat- alyzing agencies (e.g., Pt particles) which decrease the inhibiting barriers. . same critical behavior. For different models one can also define a kind of molecular field ap- proximation, usually called mean field approximation, by replacing a sum of bilinear operators by a. generally observed that the larger the dissipative part, the larger the fluctuations. Again this can be formulated quantitatively us- ing the fluctuation-dissipation theorem [49]. All expectation values. we have seen that thermal fluctuations are im- portant in the neighborhood of the critical temperature of a second-order phase transition, i.e. near the critical temperature of a liquid-gas transition or