332 42 Phase Changes, van der Waals Theory and Related Topics and different coefficients for T<T c and T>T c . The values given above for β = 1 2 ,γ(= 1) and δ(= 3) are so-called molecular field exponents. Near the critical point itself they must be modified (see below). 42.3 Critical Behavior; Ising Model; Magnetism and Lattice Gas Molecular field theory (and thus the van der Waals expressions) are no longer applicable in the neighborhood of the critical point, because thermal fluc- tuations become increasingly important, such that they may no longer be neglected. The spontaneous magnetization is then given by m H ∝ (T 0 − T) β with β ≈ 1 3 not : 1 2 for a three-dimensional system, and β ≡ 1 8 in a two-dimensional space. As a consequence of thermal fluctuations, instead of the Arrott equation, for T ≥ T 0 the following critical equation of state holds 1 : H = a ·|T − T 0 | γ m H + b ·|m H | δ + , (42.4) where γ ≈ 4 3 and δ ≈ 5ford =3, and γ ≡ 7 4 and δ ≡ 15 for d =2. However, independent of the dimensionality the following scaling law applies: δ = γ + β β , i.e., the critical exponents are indeed non-trivial and assume values dependent on d, but – independently of d – all of them can be traced back to two values, e.g., β and γ. 2 1 Here the interested reader might consider the exercises (file 8, problem 2) of winter 1997, [2], 2 The “ scaling behavior ” rests on the fact that in the neighborhood of T c there is only a single dominating length scale in the system, the so-called thermal correlation length ξ(T ) (see e.g. Chap. 6 in [38]). 42.3 Critical Behavior; Ising Model; Magnetism and Lattice Gas 333 The (not only qualitative) similarity between magnetism and phase changes on liquefaction is particularly apparent when it comes to the Ising model. In this model one considers a lattice with sites l and m,onwhich a discrete degree of freedom s l , which can only assume one of the values s l = ±1, is located. The Hamilton function (energy function) of the system is written: H = − l,m J l,m s l s m − l h l s l (Ising model) . (42.5) The standard interpretation of this system is the magnetic interpretation, where the s l are atomic magnetic moments: s l = ±1 correspond to the spin angular momentum s z = ± 2 . The energy parameters J l,m are “ exchange integrals ” of the order of 0.1eV, corresponding to temperatures around 1000 Kelvin. The parameters h l cor- respond to external or internal magnetic fields at the position r l . However, a lattice-gas interpretation is also possible, in which case l and m are lattice sites with the following property: s l = ±1 means that the site l is either occupied or unoccupied, and J l,m is the energy with which atoms on the sites r l and r m attract (J l,m > 0) or repel (J l,m < 0) each other. The similarity between these diverse phenomena is therefore not only qualitative. One obtains the above-mentioned molecular field approximation by re- placing the Hamilton function in the Ising model, (42.5), by the following expression for an (optimized 3 ) single-spin approximation: H ≈ −→ H MF := − l m 2J l,m s m T + h · s l + l,m J l,m ·s l T s l T . (42.6) Here, s m T is a thermodynamic expectation value which must be de- termined in a self-consistent fashion. In the transition from (42.5) to (42.6), which looks more complicated than it is, one has dismantled the cumbersome expression s l s m , as follows: s l s m ≡s l T s m + s l s m T −s l T s m T +(s l −s l T ) ·(s m −s m T ) , where we neglect the last term in this identity, i.e., the “ fluctuations ”, so that from the complicated non-linear expression a comparatively simple self- 3 Here the optimization property, which involves a certain Bogoliubov inequality, is not treated. 334 42 Phase Changes, van der Waals Theory and Related Topics consistent linear approximation is obtained: h l → h l + m 2J l,m s m T . Here, s l is formally an operator, whereas s l T is a real number. Analogous molecular field approximations can also be introduced into other problems. A good review of applications in the theory of phase transi- tions is found in [41]. 43 The Kinetic Theory of Gases 43.1 Aim The aim of this section is to achieve a deeper understanding on a microscopic level of both the thermal equation of state p(T,V,N) and the calorific equa- tion of state U(T,V,N) for a fluid system, not only for classical fluids, but also for relativistic fluids by additionally taking Bose and Fermi statistics (i.e., intrinsic quantum mechanical effects) into account. Firstly, we shall make a few remarks about the history of the ideal gas equation. The Boyle-Mariotte law in the form p · V = f(T ) had already been proposed in 1660, but essential modifications were made much later. For example, the quantity “mole” was only introduced in 1811 by Avogadro. Dalton showed that the total pressure of a gas mixture is made up additatively from the partial pressures of the individual gases, and the law V (Θ C ) V 0 ◦ C = 273.15 + Θ C 273.15 , which forms the basis for the concept of absolute (or Kelvin) temperature, was founded much later after careful measurements by Gay-Lussac († 1850). 43.2 The General Bernoulli Pressure Formula The Bernoulli formula applies to non-interacting (i.e., “ideal”) relativistic and non-relativistic gases, and is not only valid for ideal Maxwell-Boltzmann gases, but also for ideal Fermi and Bose gases. It is written (as will be shown later): p = N V · 1 3 m(v)v 2 T . (43.1) m(v)= m 0 1 − v 2 c 2 is the relativistic mass, with m 0 as rest mass, c the velocity of light and v the particle velocity. The thermal average T is defined from the distribution 336 43 The Kinetic Theory of Gases function F (r, v) which still has to be determined: A(r, v) T := d 3 rd 3 vA(r, v) ·F (r, v) d 3 rd 3 vF(r, v) . (43.2) F (r, v)Δ 3 rΔ 3 v is the number of gas molecules averaged over time with r ∈ Δ 3 r and v ∈ Δ 3 v, provided that the volumes Δ 3 r and Δ 3 v are sufficiently small, but not too small, so that a continuum approximation is still possible. In order to verify (43.1) consider an element of area Δ 2 S of a plane wall with normal n = e x .LetΔN be the number of gas molecules with directions ∈ [ϑ, ϑ+Δϑ) and velocities ∈ [v, v+Δv) which collide with Δ 2 S in a time Δt, i.e., we consider an inclined cylinder of base Δ 2 Sn and height v x Δt containing molecules which impact against Δ 2 S at an angle of incidence ∈ [ϑ, ϑ + Δϑ) in a time Δt,where cos 2 ϑ = v 2 x v 2 x + v 2 y + v 2 z , for v x > 0 (note that v is parallel to the side faces of the cylinder). These molecules are elastically reflected from the wall and transfer momentum ΔP to it. The pressure p depends on ΔP , as follows: p = ΔP/Δt Δ 2 S , and ΔP = i∈Δt 2m i (v x ) i , i.e, ΔP ∝ Δ 2 S · Δt , with a well-defined proportionality factor. In this way one obtains explicitly: p = ∞ v x =0 dv x ∞ v y =−∞ dv y ∞ v z =−∞ dv z 2mv 2 x F (r, v) , or p = R 3 (v) d 3 vF(r, v)mv 2 x . Now we replace v 2 x by v 2 /3, and for a spatially constant potential energy we consider the spatial dependence of the distribution function, F (r, v)= N V ·g(v) , with R 3 (v) d 3 vg(v)=1; this gives (43.1). a) We shall now discuss the consequences of the Bernoulli pressure formula. For a non-relativistic ideal gas mv 2 T = m 0 v 2 T =: 2 ·ε kin., Transl. T , 43.2 The General Bernoulli Pressure Formula 337 with the translational part of the kinetic energy. Thus, we have p = 2 3 n V ·ε kin., Transl. T , where n V := N V ; i.e., p ≡ 2U kin.,Transl. 3V . On the other hand, for a classical ideal gas: p = n V k B T. Therefore, we have ε kin., Transl. T = 3 2 k B T. This identity is even valid for interacting particles, as can be shown with some effort. Furthermore it is remarkable that the distribution function F (r, v) is not required at this point. We shall see later in the framework of classical physics that F (r, v) ∝ n V · exp −β mv 2 2 is valid, with the important abbreviation β := 1 k B T . This is the Maxwell-Boltzmann velocity distribution, which is in agree- ment with a more general canonical Boltzmann-Gibbs expression for the thermodynamic probabilities p j for the states of a comparatively small quantum mechanical system with discrete energy levels E j , embedded in a very large so-called microcanonical ensemble of molecules (see below) which interact weakly with the small system, such that through this inter- action only the Kelvin temperature T is prescribed (i.e., these molecules only function as a “thermostat”). This Boltzmann-Gibbs formula states: p j ∝ exp(−βE j ) . The expression p ≡ 2U kin.,Transl. 3V is (as mentioned) not only valid for a classical ideal gas (ideal gases are so dilute that they can be regarded as interaction-free), but also for a non- relativistic ideal Bose and Fermi gas. These are gases of indistinguishable quantum-mechanical particles obeying Bose-Einstein and Fermi statistics, respectively (see Part III). In Bose-Einstein statistics any number of par- ticles can be in the same single-particle state, whereas in Fermi statistics a maximum of one particle can be in the same single-particle state. 338 43 The Kinetic Theory of Gases In the following we shall give some examples. Most importantly we should mention that photons are bosons, whereas electrons are fermions. Other examples are pions and nucleons (i.e., protons or neutrons) in nuclear physics; gluons and quarks in high-energy physics and as an example from condensed-matter physics, He 4 atoms (two protons and two neutrons in the atomic nucleus plus two electrons in the electron shell) and He 3 atoms (two protons but only one neutron in the nucleus plus two electrons). In each case the first of these particlesisaboson,whereas the second is afermion. In the nonrelativistic case, the distribution function is now given by: F (r, v)= 1 exp β mv 2 2 − μ(T ) ∓ 1 , (43.3) with −1 for bosons and +1 for fermions. The chemical potential μ(T ), which is positive at low enough temperatures both for fermions and in the Maxwell-Boltzmann case (where ∓1 can be replaced by 0, since the exponential term dominates). In contrast, μ is always ≤ 0forbosons. Usually, μ can be found from the following condition for the particle density: F (r, v)d 3 v ≡ n V (r) . Here for bosons we consider for the time being only the normal case μ(T ) < 0, such that the integration is unproblematical. b) We now come to ultrarelativistic behavior and photon gases. The Bernoulli equation, (43.1), is valid even for a relativistic dependence of mass, i.e., p = n V 3 m 0 v 2 1 − v 2 c 2 T . Ultrarelativistic behavior occurs if one can replace v 2 in the numerator by c 2 , i.e., if the particles almost possess the speed of light. One then obtains p ≈ n V 3 m 0 c 2 1 − v 2 c 2 T and p ≈ U(T,V ) 3V , with U (T,V )= m 0 c 2 N j=1 1 1 − v 2 j c 2 T , where the summation is carried out over all N particles. Now one performs the simultaneous limit of m 0 → 0andv j → c, and thus obtains for 43.2 The General Bernoulli Pressure Formula 339 a photon gas in a cavity of volume V the result p ≡ U 3V . Photons travel with the velocity of light v ≡ c, have zero rest mass and behave as relativistic bosons with (due to m 0 → 0) vanishing chemical potential μ(T ). A single photon of frequency ν has an energy h·ν.AtatemperatureT the contribution to the internal energy from photons with frequencies in the interval [ν, ν +dν) for a gas volume V (photon gas → black-body (cavity) radiation)is: dU(T,V )= V · 8πν 2 dν · hν c 3 · exp hν k B T − 1 , which is Planck’s radiation formula (see Part III), which we state here without proof (see also below). By integrating all frequencies from 0 to ∞we obtain the Stefan-Boltzmann law U = VσT 4 , where σ is a universal constant. The radiation pressure of a photon gas is thus given by p = U 3V = σT 4 3 . c) We shall now discuss the internal energy U(T,V,N)foraclassicalideal gas. Previously we have indeed only treated the translational part of the kinetic energy, but for diatomic or multi-atomic ideal gases there are addi- tional contributions to the energy. The potential energy of the interaction between molecules is, however, still zero, except during direct collisions, whose probability we shall neglect. So far we have only the Maxwell rela- tion ∂U ∂V = T ∂p ∂T − p =0. This means that although we know p = N V k B T and thus ∂U/∂V =0, it is not yet fully clear how the internal energy depends on T .For a monatomic ideal gas, however, U ≡ U kin., transl. . From the Bernoulli formula it follows that U ≡ 3 2 Nk B T. 340 43 The Kinetic Theory of Gases However, for an ideal gas consisting of diatomic molecules the following is valid for an individual molecule ε m 0 2 v 2 s T + ε rot. + ε vibr. , where v s is the velocity of the center of mass, ε rot. is the rotational part of the kinetic energy and ε vibr. the vibrational part of the energy of the molecule. (The atoms of a diatomic molecule can vibrate relative to the center of mass.) The rotational energy is given by ε rot. = L 2 ⊥ T /(2Θ ⊥ ) . 1 A classical statistical mechanics calculation gives ε rot. T = k B T, which we quote here without proof (→ exercises). There are thus two ex- tra degrees of freedom due to the two independent transverse rotational modes about axes perpendicular to the line joining the two atoms. Fur- thermore, vibrations of the atom at a frequency ω = k m red. , where k is the “spring constant” and m red. = m 1 m 2 m 1 + m 2 the reduced mass (i.e. in this case m red = m atom 2 ), give two further de- grees of freedom corresponding to vibrational kinetic and potential energy contributions, respectively. In total we therefore expect the relation U(T )=Nk B T · 3+f 2 to hold, with f = 4. At room temperature, however, it turns out that this result holds with f = 2. This is due to effects which can only be under- stood in terms of quantum mechanics: For both rotation and vibration, there is a discrete energy gap betweenthegroundstateandtheexcited state, (ΔE) rot. and (ΔE) vibr. . Quantitatively it is usually such that at room temperature the vibrational degrees of freedom are still “ frozen-in ”, because at room temperature 1 The analogous longitudinal contribution ˙ L 2 || ¸ T /(2Θ || )wouldbe∞ (i.e., it is frozen-in, see below), because Θ || =0. 43.3 Formula for Pressure in an Interacting System 341 k B T (ΔE) vibr. , whereas the rotational degrees of freedom are already fully “ activated ”, since k B T (ΔE) rot. . In the region of room temperature, for a monatomic ideal gas the internal energy is given by U(T )= 3 2 Nk B T, whereas for a diatomic molecular gas U(T )= 5 2 Nk B T. 2 This will be discussed further in the chapter on Statistical Physics. 43.3 Formula for Pressure in an Interacting System At this point we shall simply mention a formula for the pressure in a classical fluid system where there are interactions between the monatomic particles. This is frequently used in computer simulations and originates from Robert Clausius. Proof is based on the so-called virial theorem. The formula is men- tioned here without proof: p = 2 3 U kin. transl. V + 1 6 N i,j=1 F i,j · (r i − r J ) T V . In this expression F i,j is the internal interaction force exerted on particle i due to particle j. This force is exerted in the direction of the line joining the two particles, in a similar way to Coulomb and gravitational forces. 2 Only for T 1000K for a diatomic molecular gas would the vibrational degrees of freedom also come into play, such that U(T )= 7 2 Nk B T . . [38]). 42.3 Critical Behavior; Ising Model; Magnetism and Lattice Gas 333 The (not only qualitative) similarity between magnetism and phase changes on liquefaction is particularly apparent when it. formula applies to non-interacting (i.e., “ideal”) relativistic and non-relativistic gases, and is not only valid for ideal Maxwell-Boltzmann gases, but also for ideal Fermi and Bose gases. It. formula states: p j ∝ exp(−βE j ) . The expression p ≡ 2U kin.,Transl. 3V is (as mentioned) not only valid for a classical ideal gas (ideal gases are so dilute that they can be regarded as interaction-free),