Basic Theoretical Physics: A Concise Overview P36 pptx

48.2 The Entropy of an Ideal Gas from the Microcanonical Ensemble 365 The Stirling approximation was used again, since Ω(3N) ≈  2eπ 3N  3N 2 . Using N! ∼ =  N e  N one thus obtains: s  0 ≡ ln   4πm h 2  3 2 ·e 5 2 ·V 0 U 3 2 0  . The entropy constant therefore only depends on the type of gas via the log- arithm of the particle mass m. The above result is reasonable because, with the factor N, it explicitly expresses the additivity for the entropy of an ideal gas, which (as already mentioned) depends crucially on the permutation factor 1 N! for identical par- ticles. Furthermore the law 1 T = ∂S ∂U yields the relation between U and T : U = 3 2 Nk B T. Similarly p T = ∂S ∂V yields pV = Nk B T . Only the expression for the chemical potential μ is somewhat less apparent, but we shall require it later: From μ T = − ∂S ∂N it follows that μ = 5 2 k B T − T · S N ≡ U + pV − T · S N . Thus μ ≡ G N , i.e. the chemical potential μ is identical to the free enthalpy G = U + pV − T · S per particle. This identity is generally valid for a fluid system. 366 48 Canonical Ensembles in Phenomenological Thermodynamics 48.3 Systems in a Heat Bath: Canonical and Grand Canonical Distributions In the previous section we treated closed systems. Now we shall concentrate on closed systems that consist of a large system, a so-called heat bath, to which a small partial system (which is actually the system of interest) is weakly coupled. For a canonical ensemble the coupling refers only to energy, and only serves to fix the temperature. For a so-called grand canonical en- semble on the other hand not only exchange of energy takes place with the large system but also particles are exchanged. In this case not only the tem- perature T of the small system is regulated by the heat bath but also the chemical potential μ. In the next section we shall consider how the transition from a micro- canonical ensemble to a canonical and grand canonical ensemble is achieved mathematically. For a canonical ensemble the appropriate variable of state is the Helmholtz free energy F (T,V,N). For a magnetic system it is the Gibbs free energy F g (T,V,H,N). They can be obtained from the corresponding partition func- tions Z(T,V,N):=  j e −βE j (V,N) and Z  (T,V,H,N):=  j e −βE j (V,H,N) , where F (T,V,N)=−k B T · ln Z(T,V,N)and F g (T,V,H,N)=−k B T · ln Z  (T,V,H,N) . In a grand canonical ensemble not only the energy fluctuates but also the number of particles of the “small system” ˆ Hψ (k) j = E (k) j ψ (k) j and ˆ Nψ (k) j = N k ψ (k) j . One therefore has, in addition to the energy index j, a particle-number index k. Thus, in addition to the reciprocal temperature β = 1 k B T the chemical potential μ appears as a further distribution parameter. Both parameters control the expectation values, i.e.,  ˆ N β,μ =   j,k N k p (k) j  β,μ and 48.4 From Microcanonical to Canonical and Grand Canonical Ensembles 367 U =  ˆ H β,μ :=   j,k E (k) j p (k) j  β,μ , where p (k) j (β,μ)= e −β “ E (k) j −μN k ” Z(β,μ) , with the grand canonical partition function Z(β,μ):=  j,k e −β “ E (k) j −μN k ” . The grand canonical Boltzmann-Gibbs distribution p (k) j is therefore very simi- lar to the canonical Boltzmann-Gibbs distribution. In particular the grand canonical partition function Z is related to the grand canonical thermody- namic potential Φ in a similar way as the free energy F (T,V,N) is related to the usual partition function Z(T,V,N): Φ(T,V,μ)=−k B T · ln Z(T,V,μ) . The quantity Φ is the Gibbs grand canonical potential; phenomenologically it is formed from the free energy by a Legendre transformation with respect to N: Φ(T,V,μ)=F(T,V,N(T,V,μ)) − μN , and dΦ = −pdV − N dμ − SdT. 48.4 From Microcanonical to Canonical and Grand Canonical Ensembles For an ergodic 2 system one can calculate the results for observables ˆ A 1 ,which only involve the degrees of freedom of the small system “1”, according to the microcanonical distribution for E I ≈ U − ε i . 3 2 A classical system in a given energy range U −dU<E≤ U is called ergodic if for almost all conformations in this energy region and almost all observables A(p, q) the time average 1 t 0 t 0 R 0 dtA(p(t), q(t)) for t 0 →∞is almost identical with the so-called ensemble average: A(p, q) = R U−dU<H(p,q)≤U d f pd f qA(p,q) R U−dU<H(p,q)≤U d f pd f q . Most fluid systems are ergodic, but important non-ergodic systems also exist, for example, glasses and polymers, which at sufficiently low temperature often show unusual behavior, e.g. ageing phenomena after weeks, months, years, decades or even centuries, because the investigated conformations only pass through untypical parts of phase space, so that for these systems application of the principles of statistical physics becomes questionable. 3 Here and in the following we shall systematically use small letters for the small system and large letters for the large system (or heat bath). For example, U ≈ E I + ε i . 368 48 Canonical Ensembles in Phenomenological Thermodynamics A 1  =  U− dU−ε i <E I ≤U−ε i ψ i | ˆ A 1 ψ i N (2) (U −ε i )  U− dU−ε i <E I ≤U−ε i N (2) (U − ε i ) . (48.1) In the following we shall omit the indices 1 and (2) . We now introduce a Taylor expansion for the exponent of N(U − ε i ) , viz: N(U − ε i ) ≡ e S(U−ε i ) k B =e S(U) k B ·e − ε i k B · dS dU ·e − ε 2 i d 2 S 2k B dU 2 · . The first term is a non-trivial factor, the second term on the right-hand-side of this equation, gives e − ε i k B T ; one can already neglect the next and following terms, i.e., replace the factors by 1, as one sees, for example, for N →∞in the term ε 2 i k B d 2 S dU 2 , with U ≈ 3 2 Nk B T, thus obtaining e − ε 2 i d 3Nk B dT ( 1 k B T ) → 1 , i.e., if one uses a monatomic ideal gas as heat bath and replaces factors e − const. N by unity. Inserting this Taylor expansion into the above formula one obtains the Boltzmann-Gibbs distribution for a canonical system. One may proceed similarly for the case of a grand canonical ensemble. 49 The Clausius-Clapeyron Equation We shall now calculate the saturation vapor pressure p s (T ), which has already been discussed in the context of van der Waals’ theory; p s (T ) is the equilib- rium vapor pressure at the interface between the liquid and vapor phase of a fluid. We require here the quantity dp s dT . There are two ways of achieving this. a) The first method is very simple but rather formal. Inversely to what is usually done it consists of replacing differential quotients where necessary by quotients involving differences, i.e. the non-differentiable transition function from gas to liquid state from the Maxwell straight line is ap- proximated by a gently rounded, almost constant transition function in such a way that for the flat parts of a curve one may equate quotients of differences with corresponding differential quotients. Having dealt with mathematical aspects, we shall now consider the physics of the situation: We have V = N 1 v 1 + N 2 v 2 ,wherev 1 and v 2 are the atomic specific volumes in liquid and vapor phase respectively, and N 1 and N 2 are the corresponding numbers of molecules. We therefore have ΔV = ΔN ·(v 2 −v 1 ), since N = N 1 + N 2 is constant. For N 2 → N 2 + ΔN we also have N 1 → N 1 − ΔN, and thus from the first law: ΔU = δQ + δA = l · ΔN −pΔV , with the (molecular) specific latent heat of vaporization l(T )(≈ 530 cal/g H 2 O ). Using the Maxwell relation, which is essentially a consequence of the second law, we obtain  ∂U ∂V  T = ΔU ΔV = T ∂p ∂T − p = T dp S dT − p s . Thus ∂p ∂T = ,or dp s (T ) dT = l(T ) T · (v 2 − v 1 ) . (49.1) 370 49 The Clausius-Clapeyron Equation This is known as the Clausius-Clapeyron equation. We shall now derive some consequences from this equation, and in doing so we must take the sign into account. In the case of water boiling, everything is “normal” provided one is not in the close vicinity of the critical point: v 2 (vapor)  v 1 (liquid), and thus from (49.1) we obtain dp s dT ≈ l T · v 2 , and with v 2 ≈ k B T p s : dp s dT ≈ l ·p s k B T 2 , i.e., p s (T ) ≈ p 0 · e − l k B T , with constant p 0 . As a result there is a very fast drop in saturation vapor pressure with increasing temperature. There is no peculiarity here with regard to the sign; however, the Clausius- Clapeyron equation (49.1) is valid not only for a boiling transition but also for melting. For water-ice transitions, v 2 , the atomic specific volume of the liquid, is 10% smaller than v 1 , the atomic specific volume of the ice phase 1 . As a result of this, dp s dT = l T · (v 2 − v 1 ) is now negative, in agreement with the anomalous behavior of the phase diagram of H 2 O mentioned earlier. We now come to a second derivation of the Clausius-Clapeyron equation: b) The method is based on an ideal infinitesimal Carnot process,whichone obtains by choosing for a given liquid or gas segment in the equation of state the line p s (T ) corresponding to the Maxwell construction as the lower Carnot path (i.e. T 2 ≡ T ), whereas one chooses the saturation pressure line p s (T + ΔT )astheupper Carnot path (i.e. T 1 ≡ T + ΔT ). We then find ΔA = ⎛ ⎝  1 −  2 ⎞ ⎠ p s dV = p s ·(v 2 − v 1 )ΔN ! = ΔT T · Q 1 , since η = ΔA Q 1 = ΔT T . Using Q 1 = l · ΔN and ΔA = Δp s · (v 2 − v 1 )ΔN = ΔT T · Q 1 , the Clausius-Clapeyron equation is obtained: (49.1). These derivations imply – as we already know – that the Maxwell relations, the second law, and the statement on the efficiency of a Carnot process are all equivalent, and that in the coexistence region the straight-line Maxwell section, e.g., p s (T ), is essential. 1 Chemists would again prefer to use the specific molar volume. 50 Production of Low and Ultralow Temperatures; Third Law of Thermodynamics Low temperatures are usually obtained by a process called adiabatic demag- netization. Ultralow temperatures are achieved (in Spring 2004 the record was T min =0.45 ×10 −9 Kelvin) in multistage processes, e.g., firstly by adiabatic demagnetization of electron-spin systems, then by adiabatic demagnetization of nuclear spins, thirdly by laser cooling, and finally by evaporation methods. (Many small steps prove to be effective.) Evaporation cooling is carried out on atomic and molecular gas systems, mainly gases of alkali atoms, that are held in an electromagnetic “ trap ”. The phenomenon of Bose-Einstein condensa- tion is currently being investigated on such systems at extreme temperatures ( < ∼ 10 −7 K and lower powers of ten). This will be discussed later. In 2001 the Nobel Prize was awarded for investigations of the Bose-Einstein conden- sation of ultracold gases of alkali atoms (see below). These investigations could only be performed after it had been discovered how to obtain ultralow temperatures in a reproducible and controllable manner. 1 Next we shall consider the production of low temperatures in general. The techniques usually depend on “ x-caloric effects ”, e.g., the magnetocaloric effect. We shall consider the following examples: a) Gay-Lussac’s experiment on the free expansion of a gas from a container (see above). This occurs at a constant internal energy, such that  dT dV  U = − ∂U ∂V ∂U ∂T . With the Maxwell relation ∂U ∂V = T ∂p ∂T − p and van der Waals’ equation of state p = − a v 2 + k B T v −b we obtain  dT dV  U = − a c (0) v v 2 , 1 Nobel Prize winners: Cornell, Ketterle, Wiemann. 372 50 Production of Low and Ultralow Temperatures; Third Law i.e. the desired negative value (see above). In this connection we should recall that the exact differential dU = ∂U ∂T dT + ∂U ∂V dV ! =0. b) The Joule-Thomson effect has also been discussed above. This involves a pressure drop at constant internal enthalpy per particle.Oneobtains the expression  dT dp  I/N = − ∂I/N ∂p ∂I/N ∂T = = 2a k B T − b c (0) p ·(1 − ) , i.e., giving a negative value above and a positive value below the so-called inversion temperature T Inv. .Thus,forT<T Inv. (this is the normal case) a drop in temperature occurs for a reduction in pressure. c) Thirdly we shall consider the magnetocaloric effect or the phenomenon of temperature reduction by adiabatic demagnetization, i.e., dH<0for constant entropy S(T,H). Here one obtains as above:  dT dH  S = − ∂S ∂H ∂S ∂T , so that one might imagine reducing the temperature indefinitely, if the entropy S(T,H) behaved in such a way that for T = 0 at finite H also S(0,H)werefinite,with ∂S ∂H < 0 . One would then only need to magnetize the magnetic sample in a first stage (step 1) isothermally (e.g., H → 2H) and subsequently (step 2) to demagnetize it adiabatically, i.e. at constant S, in order to reach absolute zero T = 0 immediately in this second step. This supposed behavior of S(T,H) is suggested by the high temperature behavior: S(T,H) ∝ a(T ) − b H 2 . However, it would be wrong to extrapolate this behavior to low temper- atures. In fact, about 100 years ago the third law of thermodynamics was proposed by the physico-chemist Walter Nernst. This is known as Nernst’s heat theorem, which can be formulated, as follows: Let S(T,X) be the entropy of a thermodynamic system, where X repre- sents one or more of the variables of state, e.g., X = V , p , m j or H. Then, for X>0, the limit as S(T → 0,X) is zero; and the convergence to zero is such that the absolute zero of temperature in Kelvin, T =0, is unattainable 50 Production of Low and Ultralow Temperatures; Third Law 373 Fig. 50.1. Third Law of Thermodynam- ics (schematically). The low temperature behavior of the entropy S between 0 and 4 units is presented vs. the absolute tem- perature T (here between 0 and 1.2 units); only the 2nd and 4th curve from above (i.e., with S(T =0)≡ 0) are realis- tic, whereas the 1st and 3rd lines repre- sent false extrapolations suggested by the high-T asymptotes in a finite number of steps. In the case of, for example, adiabatic demagneti- zation this results in a countably-infinite number of increasingly small steps (see Fig. 50.1): Figure 50.1 shows the qualitative behavior of the entropy S(T,H)of a paramagnetic system as a function of T for two magnetic field strengths. The first and third curves from the top correspond to extrapolations sug- gested by high-temperature behavior; but they do not give the true behavior for T → 0. This is instead represented by the second and fourth curves, from which, for the same high-temperature behavior, we have for all H =0,as postulated by Nernst: S(T =0,H) ≡ 0 . The third law, unattainability of absolute zero in a finite number of steps –whichisnot a consequence of the second law – can be relatively easily proved using statistical physics and basic quantum mechanics, as follows. Consider the general case with degeneracy, where, without loss of gener- ality, E 0 = 0. Let the ground state of the system be g 0 -fold, and the first excited state g 1 -fold; let the energy difference (= E 1 − E 0 )beΔ(X). Then we obtain for the x-caloric effect:  dT dX  S = − ∂S ∂X ∂S ∂T , where S = − ∂F ∂T and F = −k B T · ln Z, with the following result for the partition function: Z = g 0 + g 1 · e −βΔ + . Elementary calculation gives S(T,X) k B =lng 0 + g 1 g 0 · Δ k B T · e − Δ k B T + , where the dots describe terms which for k B T  Δ can be neglected. If one assumes that only Δ, but not the degeneracy factors g 0 and g 1 , depend on 374 50 Production of Low and Ultralow Temperatures; Third Law X, it follows strictly that  dT dX  S ≡ T · ∂Δ ∂X Δ + , since the exponentially small factors ∝ e −βΔ in the numerator and denominator of this expression cancel each other out. In any case, for T → 0 we arrive at the assertion of unattainability of absolute zero. Furthermore, we find that the assumption, S → 0, in Nernst’s heat theorem is unnecessary. In fact, with g 0 ≡ 2 for spin degeneracy of the ground state, one obtains: S(T =0,H ≡ 0) = k B ln 2(=0). In spite of this exception for H ≡ 0, the principle of unattainability of abso- lute zero still holds, since one always starts from H =0,whereS(0,H)=0. In this respect one needs to be clear how the ultralow temperatures men- tioned in connection with Bose-Einstein condensation of an alkali atom gas are achieved in a reasonable number of steps. The deciding factor here is that ultimately only the translational kinetic energy of the atoms is involved, and not energetically much higher degrees of freedom. Since we have M  v 2  T 2 = 3k B T 2 , the relevant temperature is defined by the mean square velocity of the atoms, k B T = M ·  v 2  T 3 , where we must additionally take into account that the relevant mass M is not that of an electron, but that of a Na atom, which is of the order of 0.5 × 10 5 larger. One can compare this behavior with that of He 4 ,whereat normal pressure superfluidity (which can be considered as some type of Bose- Einstein condensation for strong interaction) sets in at 2.17 K, i.e., O(1) K. The mass of a Na atom is an order of magnitude larger than that of a He 4 atom, and the interparticle distance δr in the Na gas considered is three to four orders of magnitude larger than in the He 4 liquid, so that from the formula k B T c ≈  2 2M(δr) 2 one expects a factor of ∼ 10 −7 to ∼ 10 −9 , i.e. temperatures of O  10 −7  to O  10 −9  K are accessible. . to a canonical and grand canonical ensemble is achieved mathematically. For a canonical ensemble the appropriate variable of state is the Helmholtz free energy F (T,V,N). For a magnetic system. In particular the grand canonical partition function Z is related to the grand canonical thermody- namic potential Φ in a similar way as the free energy F (T,V,N) is related to the usual partition. of a grand canonical ensemble. 49 The Clausius-Clapeyron Equation We shall now calculate the saturation vapor pressure p s (T ), which has already been discussed in the context of van der Waals’