9.1 System in contact with a heat reservoir

Transcription

1 Chapter 9 Canonical ensemble 9. System in contact with a heat reservoir We consider a small system A characterized by E, V and N in thermal interaction with a heat reservoir A 2 characterized by E 2, V 2 and N in thermal interaction such that A A 2, A has hence fewer degrees of freedom than A 2. with E 2 E N = const. N 2 N N 2 = const. E + E 2 = E = const. Both systems are in thermal equilibrium at temperature T. The wall between them allows interchange of heat but not of particles. The system A may be any relatively small macroscopic system such as, for instance, a bottle of water in a lake, while the lake acts as the heat reservoir A 2. Distribution of energy states The question we want to answer is the following: Under equilibrium conditions, what is the probability of finding the small system A in any particular microstate α of energy E α? In other words, what is the distribution function ρ = ρ(e α ) of the system A? We note that the energy E is not fixed, only the total energy E = E +E 2 of the combined system. Hamilton function. The Hamilton function of the combined system A is H(q, p) = H (q(), p()) + H 2 (q(2), p(2)), 03

2 04 CHAPTER 9. CANONICAL ENSEMBLE were we have used the notation q = (q(), q(2)), p = (p(), p(2)). Microcanonical ensemble of the combined system. Since the combined system A is isolated, the distribution function in the combined phase space is given by the microcanonical distribution function ρ(q, p), ρ(q, p) = δ (E H(q, p))) dq dp δ (E H(q, p)), where Ω(E) is the density of phase space (8.4). dq dp δ(e H) = Ω(E), (9.) Tracing out A2. It is not the distribution function ρ(q, p) = ρ(q(), p(), q(2), p(2)) of the total system A that we are interested in, but in the distribution function ρ (q(), p()) of the small system A. One hence needs to trace out A 2 : ρ (q(), p()) dq(2) dp(2) ρ(q(), p(), q(2), p(2)) = dq(2) dp(2) δ(e H H 2 ) Ω(E) Ω 2(E H ). Ω(E) (9.2) where Ω 2 (E 2 ) = Ω(E H ) is the phase space density of A 2. Small E expansion. Now, we make use of the fact that A is a much smaller system than A 2 and therefore the energy E given by H is much smaller than the energy of the combined system: E E. In this case, we can approximate (9.2) by expanding the slowly varying logarithm of Ω 2 (E 2 ) = Ω 2 (E H ) around the E 2 = E as [ ] ln Ω2 ln Ω 2 (E 2 ) = ln Ω 2 (E H ) ln Ω 2 (E) H +... (9.3) E 2 and neglect the higher-order terms since H = E E. Γ 0 Γ 0 E 2 =E Derivatives of the entropy. Using (8.4), namely that ( ) ( ) Γ(E, V, N) Ω(E)Δ S = k B ln = k B ln, (9.4) where Δ is the width of the energy shell, we find that derivatives of the entropy like T = S E = k ln Ω(E) B E (9.5) A marginal distribution function p(x) = p(x, y)dy in generically obtained by tracing out other variables from a joint distribution function p(x, y).

3 9.. SYSTEM IN CONTACT WITH A HEAT RESERVOIR 05 can be taken with respect to the logarithm of the phase space density Ω(E). Boltzmann factor. Using (9.5) for the larger system A 2 we may rewrite (9.3) as [ Ω 2 (E H ) = exp ln Ω 2 (E) ln Ω ] 2(E 2 ) E 2 H +... E2 =E [ = Ω 2 (E) exp H ]. k B T 2 The temperature T 2 of the heat reservoir A 2 by whatever small amount of energy the large system A 2 gives to the small system A. Both systems are thermally coupled, such that T = T 2 = T. We hence find with (9.2) ρ (q(), p()) = Ω H 2(E) Ω(E) e k B T e H k B T. (9.6) The factor exp[ H /(k B T )] is called the Boltzmann factor. Distribution function of the canonical ensemble. The prefactor Ω 2 (E)/Ω(E) in (9.6) is independent of H. We may hence obtain the the normalization of ρ alternatively by integrating over the phase space of A : ρ (q(), p()) = e βh (q(),p()) dq() dp() e βh (q(),p()), β = k B T. (9.7) 9.. Boltzmann factor The probability P α of finding the system A (which is in thermal equilibrium with the heat reservoir A 2 ) in a microstate α with energy E α is given by P α = e βeα α e βeα when rewriting (9.7) in terms of P α. Boltzmann distribution (9.8) The number of states Ω 2 (E 2 ) = Ω 2 (E H ) accessible to the reservoir is a rapidly increasing function of its energy. The number of states Ω 2 (E 2 ) = Ω 2 (E H ) accessible to the reservoir decreases therefore rapidly with increasing E = E E 2. The probability of finding states with large E is accordingly also rapidly decreasing. The exponential dependence of P α on E α in equation (9.8) expresses this fact in mathematical terms.

4 06 CHAPTER 9. CANONICAL ENSEMBLE Example. Suppose a certain number of states accessible to A and A 2 for various values of their respective energies, as given in the figure, and that the total energy of the combined system is 007. Let A be in a state α with energy 6. E 2 is then in one of the states with energy 00. If A is in a state γ with energy 7, the reservoir must be in one of the 0 5 states with energy 000. The number of realizations of states with E = 6 the ensemble contains is hence much higher than the number of realization of state with E = 7. Canonical ensemble. An ensemble in contact with a heat reservoir at temperature T is called a canonical ensemble, with the Boltzmann factor exp( βe α ) describing the canonical distribution (9.8). Energy distribution function. The Boltzmann distribution (9.8) provides the probability P α to find an individual microstates α. There are in general many microstates in a given energy, for which P (E) = E<E α<e+δ P α Ω(E) e βe, (9.9) is the corresponding energy distribution function. Ω(E) = Ω (E) is, as usual, the density of phase space. P (E) is rapidly decreasing for increasing energies due to the Boltzmann factor exp( βe α ). P (E) is rapidly decreasing for decreasing energies due to the decreasing phase space density Ω(E). The energy density is therefore sharply peaked. We will discuss the the width of the peak, viz the energy fluctuations, more in detail in Sect. 9.6.

5 9.2. CANONICAL PARTITION FUNCTION Canonical partition function We rewrite the distribution function (9.7) of the canonical ensemble as ρ(q, p) = e βh(q,p) d 3N q d 3N p e βh(q,p), where we dropped all the indices for simplicity, though in fact we are still describing the properties of a small system (which is nevertheless macroscopically big) in thermal equilibrium with a heat reservoir. Partition function. The canonical partition function ( kanonische Zustandssumme ) Z N is defined as d 3N q d 3N p Z N = e βh(q,p). (9.0) h 3N N! It is proportional to the canonical distribution function ρ(q, p), but with a different normalization, and analogous to the microcanonical space volume Γ(E) in units of Γ 0 : Γ(E) Γ 0 = = where Θ is the step function. d 3N q d 3N p h 3N N! E<H(q,p)<E+Δ d 3N q d 3N p( ) Θ(E + Δ H) Θ(E H), h 3N N! Free energy. We will show that it is possible to obtain all thermodynamic observables by differentiating the partition function Z N. We will prove in particular that F (T, V, N) = k B T ln Z N (T ), Z N = e βf (T,V,N), (9.) where F (T, V, N) is the Helmholtz free energy. Proof. In order to proof (9.) we perform the differentiation β ln Z N = Z N Z N β [ ] dqdp / [ ] dqdp = β h 3N N! e βh h 3N N! e βh dqdp ( H) e βh = dqdp e βh = H = U. where we have used the shortcut dqdp = d 3N qd 3N p and that H = E = U is the internal energy.

6 08 CHAPTER 9. CANONICAL ENSEMBLE With (5.3), namely that U = (βf )/ β, we find that β ln Z N = U = β (βf ), ln Z N = βf, Z N = e βf, which is what we wanted to prove. Integration constant. Above derivation allows to identify ln Z N = βf only up to an integration constant (or, equivalently, Z N only up to a multiplicative factor). Setting this constant to zero results in the correct result for the ideal gas, as we will show lateron in Sect Thermodynamic properties. Once the partition function Z N and the free energy F (T, V, N) = k B T ln Z N (T, V, N) are calculated, one obtains the pressure P, the entropy S and the chemical potential µ as usual via P = ( F V ) T,N, S = ( F T ) V,N, µ = ( ) F N T,V Specific heat. The specific heat C V is given in particular by ( ) C V S T = = 2 F ( ) T T = 2 k 2 T 2 B T ln Z N, (9.2) where we have used F = k B T ln Z N. V. 9.3 Canonical vs. microcanonical ensemble We have seen that the calculations in the microcanonical and canonical ensembles reduce to a phase space integration and a calculation of a thermodynamic potential: Microcanonical ensemble Canonical ensemble Phase space Density of states: Partition function: integration Ω N (E) = d 3N d 3N q d 3N q d 3N p p δ(e H) Z N (T ) = e βh(q,p) h 3N N! Thermodynamic potential S(E, V, N) = k B ln ( ) ΩN (E)Δ h 3N N! F (T, V, N) = k B T ln Z N (T ) Laplace transforms. The relation between the density of states Ω N (E) and the partition function Z N (T ) can be defined as a Laplace transformation in the following way. We use the definition (9.) of the density of states Ω(E), dq dp δ(e H) = Ω(E), H = H(q, p),

7 9.4. ADDITIVITY OF F (T, V, N) 09 in order to obtain 0 de e βe h 3N N! Ω N(E) = d 3N q d 3N p 0 de e βe δ(e H) h 3N N! = d 3N q d 3N p e βh(q,p) = Z h 3N N (T ). (9.3) N! We have thus shown that Z N (T ) is the Laplace transform of Ω N (E). Additive Hamilton functions. In both the microcanonical and in the canonical ensemble we have to perform an integration which is usually difficult. When the Hamilton function is additive, H = i H i, the integration in the canonical ensemble can be factorized, which is not the case for the microcanonical ensemble. Therefore, it is usually easier to calculate in the canonical ensemble than in the microcanonical ensemble. 9.4 Additivity of F (T, V, N) An important property of the free energy is that it has to be additive. Non-interacting systems. Let us consider two systems in thermal equilibrium. Neglecting the interaction among the systems, the total Hamilton function can be written as a sum of the Hamiltonians of the individual systems, H = H + H 2, N = N + N 2. Multiplication of partition functions. The partition function of the total system is Z N (T, V ) = d 3N q d 3N p e β(h +H 2 ), h 3N N!N 2! where have made use of the fact that there is not exchange of particles between the two systems. The factor in the denominator is therefore proportional to N!N 2! and NOT to N!. It then follows that the partition function factorizes, Z N (T, V ) = d 3N q d 3N p e βh (q,p ) h 3N N! d 3N 2 q d 3N 2 p e βh 2(q 2,p 2 ) h 3N 2 N2! = Z N (T, V ) Z N2 (T, V 2 ), and that the free energy F = k B T ln Z N is additive: F (T, V, N) = F (T, V, N ) + F 2 (T, V 2, N 2 ). The Laplace transform F (s) of a function f(t) is defined as F (s) = 0 f(t) exp( st)dt.

8 0 CHAPTER 9. CANONICAL ENSEMBLE Convolution of densities of states. That the overall partition function factorizes follows also from the fact that the density of states Ω(E) of the combined system, Ω(E) = d 3N q d 3N p δ(e H H 2 ) = d 3N q d 3N p d 3N 2 q d 3N 2 p de 2 δ(e H E 2 ) δ(e 2 H 2 ) = de 2 Ω (E E 2 ) Ω 2 (E 2 ), is given by the ( convolution) of the density of states Ω i (E i ) of the individual systems. Using the representation (9.3) for the partition function we obtain de e βe Z N = Ω(E) h 3N N!N 2 de e β(e +E 2 ) = de h 3N 2 Ω (E E 2 ) Ω N!N 2! }{{} 2 (E 2 ). Ω (E ) A change of the integration variable from de to de then leads again to Z N (T, V ) = Z N (T, V ) Z N2 (T, V 2 ). (9.4) Note that this relation is only valid if H = H + H 2 and H 2 = Ideal gas in the canonical ensemble We consider now the ideal gas in the canonical ensemble, for which the Hamilton function, H = N i= contains just the kinetic energy. p i 2 2m, Z N(T, V ) = Factorization. The integral leading to Z N factorizes in (9.5): Z N (T, V ) = V N N! = V N N! ( + d 3N q d 3N p e β N h 3N i= p2 i /(2m), (9.5) N! dp h e β ( + where we have used the variable substitution x 2 = ) 3N p 2 2m p 2 2k B T m, dx = dp + 2kB T m, 3N 2kB T m e dx) x2, (9.6) h dx e x2 = π. (9.7) Note that a variable transfomation (E, E 2 ) (E, E 2 ) with a Jacobian determinant, de de 2 = de de 2 J, where J is the respective Jacobian.

9 9.5. IDEAL GAS IN THE CANONICAL ENSEMBLE Thermal wavelength. Evaluating (9.6) explicitly with the help of (9.7) we get Z N (T, V ) = V N N! ( 2πmkB T h ) 3N ( ) N V, (9.8) N! λ 3 T where we have defined the thermal wavelength λ T as λ T = h 2πmkB T. For air (actually nitrogen, N 2, with m = kg) at T = 298 K, the thermal wavelength is 0.9 A, which is actually smaller than the Bohr radius. Quantum mechanical effects start to play a role only once λ T becomes larger than the typical interparticle separation. Thermal momentum. Heisenberg s uncertainty principle Δx Δp h allows to define a thermal momentum p T as p T = h λ T = 2πmk B T, p 2 T 2m = πk BT = 2π 3 E kin, E kin = 3 2 k BT, where we have used (3.5) for the average energy E kin per particle. The thermal momentum p T is hence of the same order of magnitude as the average momentum p of the gas, as defined by E kin = p 2 /(2m), but not identical. Free energy. From (9.8) we obtain (with log N! N log N N) ( ( ) ) N V F (T, V, N) = k B T ln N! λ 3 T ( ) ) N = k B T ln ( V N! + ln λ 3 T ( = k B T N ln N + N + N ln V ) λ 3 T (9.9) and hence F (T, V, N) = Nk B T { ( ) } V ln + Nλ 3 T for the free energy of the ideal gas. Entropy. Using we then have λ T T = λ T 2T, ( ) F S = T V,N ( ) V T ln Nλ 3 T = 3 ln λ T T = 3 2T ( ) ] [ ] V 3 = Nk B [ln + + Nk Nλ 3 B T T 2T,

10 2 CHAPTER 9. CANONICAL ENSEMBLE which results in the Sackur-Tetrode equation S = Nk B { ln V Nλ 3 T }. (9.20) Comparing (9.20) with (8.25), namely with the microcanonical Sackur-Tetrode equation { [ (4πmE ) ] } 3/2 V S = k B N ln + 5, 3h 2 N N 2 one finds that they coincide when E/N = 3k B T/2. Chemical potential. The chemical potential µ is ( ) [ F µ = = k B T ln V ] [ V + + Nk B T N T,V Nλ 3 T ( ) V = k B T ln Nλ 3 T. Nλ 3 T λ3 T V ] The previous expressions were much simpler obtained than when calculated in the microcanonical ensemble. Equivalence of ensembles. In the thermodynamic limit the average value of an observable is in general independent of the ensemble (microcanonical or canonical). N, V, N V = const. is taken. One therefore usually chooses the ensemble that is easier to work with. Fluctuations of observables. Fluctuations of observables, A 2 A 2, may however be ensemble dependent! An example for an observable for which this is the case is the energy, which is constant, by definition, in the microcanonical ensemble, but distributed according to (9.9) in the canonical ensemble. 9.6 Energy fluctuations We evaluated the representation (9.2) for the specific heat in a first step: C V T = 2 T 2 [ = T = T ] k B T ln Z n [ k B ln Z n + k BT Z N Z n β [ k B ln Z n Z n T Z N β ] β T ]. β T = k B T 2

11 9.6. ENERGY FLUCTUATIONS 3 Second derivatives. The remaining derivative with respect to the temperature T are T k B ln Z n = Z n T T Z N β = Z n T 2 Z N β T 2 Z N Z n β + [ T Z 2 N ( ) ] 2 Zn 2 Z n β T Z N β 2 k B T. 2 With the first two terms canceling each other We find [ C V = ( ) ] 2 2 Z n k b T 2 Z N β Z n 2 Z N β for the specific heat C v as a functions of derivatives of the partition function Z N. (9.2) Derivatives of the partition function. The definition (9.0) for the partition function corresponds to Z d 3N q d 3N p n Z N β = H e βh(q,p) h 3N N! d 3N q d 3N p d, Z 3N q d 3N p N = e βh(q,p), e h 3N N! βh(q,p) h 3N N! viz to Z N Z n β = E, Z N 2 Z n β 2 = E 2. (9.22) Specific heat. Our results (9.2) and (9.22) lead to the fundamental relation between the specific heat C V C V = k b T 2 [ E 2 E 2 ] (9.23) and the fluctuations E 2 E 2 of the energy. Both the specific heat C V N and the right-hand side of (9.23) are extensive. The later as a result of the central limit theorem discussed in Sect. 8.6, which states that the variance of independent processes are additive. The specific heat describes the energy exchange between the system and an heat reservoir. It hence makes that sense that C V is proportional to the size of the energy fluctuations. Relative energy fluctuations. The relative energy fluctuations, E 2 E 2 E N (9.24) vanish in the thermodynamic limit N. The scaling relation (9.24) if a direct consequence of (9.23) and of the fact that both C V and the internal energy U = E are extensive. Eq. (9.24) is consistent with the demand that the canonical the microcanonical ensembles are equivalent in the thermodynamic limit N. Energy fluctuations are absent in the microcanonical ensemble.

12 4 CHAPTER 9. CANONICAL ENSEMBLE 9.7 Paramagnetism We consider a system with N magnetic atoms per unit volume placed in an external magnetic field H. Each atom has an intrinsic magnetic moment µ = 2µ 0 s with spin s = /2. Energy states. In a quantum-mechanical description, the magnetic moments of the atoms can point either parallel or anti-parallel to the magnetic field. state alignment moment energy probability (+) parallel to H +µ µh P + = c e βε + = c e +βµh ( ) anti-parallel to H µ +µh P = c e βε = c e βµh We assume here that the atoms interact weakly. One can therefore a single atom as a small system and the rest of the atoms as a reservoir in the terms of a canonical ensemble. Mean magnetic moment. We want to analyze the mean magnetic moment µ H per atom as a function of the temperature T : where we used that µ H = µ eβµh µ e βµh e βµh + e βµh, µ H = µ tanh µh k B T, tanh y = ey e y µh, y = βµh = e y + e y k B T. Magnetization. We define the magnetization, i.e. the mean magnetic moment per unit volume, as M = N µ H and analyze its behavior in the limit of high- and of low temperatures. High-temperature expansion. Large temperatures correspond to y and hence to e y = + y +..., e y = y Then, so that tanh y = ( + y +...) ( y +...) 2 µ H = µ2 H k B T. y,

13 9.7. PARAMAGNETISM 5 Curie Law. For the magnetic susceptibility χ, defined as M = χh, we then have χ = Nµ2 k B T. At temperatures high compared to the magnetic energies, χ T which is known as the Curie law. Low-temperature expansion. Low temperatures correspond to y, and hence e y e y, tanh y, µ H = µ, M = Nµ. The magnetization saturates at the maximal value at low temperatures independent of H.

14 6 CHAPTER 9. CANONICAL ENSEMBLE