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The Canonical Ensemble The Canonical Ensemble Stephen R. Addison February 12, 2001 The Canonical Ensemble We will develop the method of canonical ensembles by considering a system placed in a heat bath at temperature T: The canonical ensemble is the assembly of systems with ¯xed N and V: In other words we will consider an assembly of systems closed to others by rigid, diathermal, impermeable walls. The energy of the microstates can fluctuate, the system is kept in equilibrium by being in contact with the heat bath at temperature T: Schematically, we can view this ensemble as: State 1 E1;V;N Bath T State 2 E2;V;N Bath T State 3 E3;V;N Bath T . State º Eº ;V;N Bath T The system for which the canonical ensemble is appropriate can be thought of as a sub-system of the system for which the microcanonical ensemble is ap- propriate. Bath E0 System Eº Isolated system with E;V;N ¯xed. Basically what we do is to examine one state and consider the rest to be in the 1 heat bath. Thus the macroscopic system is speci¯ed by T;V; andN as illustrated. Isolated Temperature T V;N Let the combined energy of the system and the heat bath = E0 The system with V;N will be in one of a variety of microstates E1;E2;:::;Er: These energies could be degenerate in some cases, but we assume that they can be ordered. E1 · E2 · E3 · :::· Er · ::: We'll select ±E so that we select one energy level but several microstates. Let the system be in a state with energy Er, the energy of the reservoir is then E0 ¡ Er. What's the probability that the system will be in a microstate with energy Er? When we considered an isolated system, we found the probability of it be- ing in a macrostate speci¯ed by (E;V;N;®) was proportional to the multiplicity ­(E;V;N;®): In this situation, we could choose to analyze the system or the heat bath. It will prove to be e±cient to analyze the heat bath. The multiplicity of the heat bath is: ­(E0 ¡ Er) We have an isolated system with two sub-systems, labelling the heat bath as system 2, we have: pr = const­2(E0 ¡ Er) The ratio of probabilities for the states Ei; and Ej is pi ­2(E0 ¡ Ei) = pj ­2(E0 ¡ Ej) We could write similar expressions for all pairs of levels, if we consider three pi ­2(E0 ¡ Ei) = pi ­2(E0 ¡ Ei) 2 pi ­2(E0 ¡ Ei) = pj ­2(E0 ¡ Ej) pi ­2(E0 ¡ Ei) = pk ­2(E0 ¡ Ek) we can add them to give pi + pj + pk ­2(E0 ¡ Ei)+­2(E0 ¡ Ej +­2(E0 ¡ Ek)) = pi ­2(E0 ¡ Ei) We can generalize this result to yield ­2(E0 ¡ Er) pr = P ­2(E0 ¡ Ei) i Where the sum is taken over all levels. Next, we rewrite the expression for pr in terms of the reservoir entropy. S = k ln ­, so S=k =ln­; and ­ = eS=k. Using this we write µ ¶ S2(E0 ¡ Er) pr = (constant) £ exp k With a large reservoir, we can assume that E0 À Er. If the heat bath is large this inequality holds for all states with a reasonable chance of occurring. Now we'll expand in a Taylor series about S2(E0). Recall µ ¶ µ ¶ 2 df 1 2 d f f(x0 + a)=f(xo)+a + a 2 + ::: dx x=x0 2! dx x=x0 so 2 2 1 1 Er @S2(E0) Er @ S2(E0) S2(E0 ¡ Er)= S2(E0) ¡ + 2 ::: k k k @E0 2!k @E0 Now, we know that @S2(E0) 1 = @E T where T is the temperature of the heat bath. Thus, we have 1 1 Er S2(E0 ¡ Er)= S2(E0) ¡ : k k kT All higher order partial derivatives are zero by assumption of a large heat bath µ ¶ 2 @ S2 @ 1 = =0 @E2 @E T Now using ­2(E0 ¡ Er) pr = P ­2(E0 ¡ Ei) i 3 and ­=eS=k and substituting 1 ¯ = kT we get S2 exp( k ¡ ¯Er) pr = P S2 exp( k ¡ ¯Ei) i eS2=ke¡¯Er = P eS2=ke¡¯Ei i e¡¯Er = P e¡¯Ei i e¡¯Er pr = Z Partition Functions and the Boltzmann Distribu- tion In the last equation, I have de¯ned X Z = e¡¯Ei = Sum over States = Zustandsumme = Partition Function i We will also use an alternate de¯nition where the sum is over energy levels rather than states. In this alternate de¯nition, we let the degeneracy of the level be g(Ei): Then X ¡¯Ei Z = g(Ei)e : Ei e¡¯Er The equation pr = Z is called the Boltzmann distribution. The Boltzmann distribution gives the probability that is in a particular state when it is in a heat bath at temperature T: Z plays a central r^ole in the study of systems at ¯xed T: The term e¡¯Er is called the Boltzmann factor. It is now time to combine these ideas with our knowledge of probability, recall X X <x>= xif(xi)= pixi i i In thermal physics, in the canonical ensemble, the probability distribution (pi = f(xi) is the Boltzmann distribution, the average is called an ensemble average. 4 Average Energy in the Canonical Ensemble X X 1 ¡¯Er <E>= piEr = Ere i Z r P Let's simplify this result, consider Z = e¡¯Er r @Z @ X = e¡¯Er @¯ @¯ r X ¡¯Er @ = e (¡¯Er) @¯ rX ¡¯Er = ¡ Ere r so X 1 ¡¯Er <E> = Ere Z r 1 @ X = ¡ e¡¯Er Z @¯ r 1 @Z = ¡ Z @¯ @ ln Z <E> = ¡ @¯ This is an average over the states of the system that exchange energy with the reservoir. The fluctuations around this energy are small. Gibbs Entropy Formula Consider a general macroscopic system with state labelled 1; 2; 3;:::;r;:::P . The probability that a system is in a state r is pr: Without constraints, pr =1 is all about the way we can say about the system. The general de¯nition of entropy is then X S = ¡k pr ln pr: r This is the Gibbs entropy formula. We can deduce this formula for a generalized ensemble. Consider an ensemble of º replicas of our system. We'll assume that each replica has the same probability p2;p2;p3;:::;pr;::: of being in the state i: Provided º is large enough, the number or systems in the ensemble in state r is ºr = ºpr 5 The multiplicity ­º for the ensemble with º1 subsystems in state 1, º2 subsys- tems in state 2, etc., is the number of ways the distribution can be realized: º! ­º = º1!º2! :::ºr! Now º! Sº = k ln ­º = k ln º1!º2! :::ºr! Sº = k ln º! ¡ k(ln º1!+º2!+:::+ ºr!) Recall Stirling's approximation ln N!=N ln N ¡ N,so X Sº = k(º ln º ¡ ºr ln ºr) r But ºr = ºpr X Sº = kº ln º ¡ k ºpr ln(ºpr) rX = kº ln º ¡ kº pr ln(ºpr) Xr X = kº ln º ¡ kº pr ln º ¡ kº pr ln pr r X Xr = kº ln º ¡ kº ln º pr ¡ kº pr ln pr X r r Sº = ¡k ln º pr ln pr r But entropy is extensive so the entropy of one system (replica) is X S = Sº = ¡k pr ln pr r The Gibbs entropy formula is consistent with the Boltzmann entropy formula S = k ln ­: In an isolated system with energy in the range E to E + ±E; the number of microstates in the interval is ­(E;V;N): The probability of ¯nding the system in one of the microstates is 1=­(E;V;N) in the range and 0 outside the range. So 1 pr = (there are ­ terms in an isolated system) ­(E;V;N) X X 1 X S = ¡k pr ln pr = ¡k pr ln = k ln ­ pr = k ln ­ r r ­ r 6 Entropy of a System in a Heat Bath To ¯nd the entropy of a system in a heat bath, we can use the Gibbs entropy formula and the Boltzmann distribution for pr: X S = ¡k pr ln pr r and e¡¯Er pr = : Z Combining these expressions, and simplifying S X ¡ = pr ln pr k r µ ¶ X e¡¯Er e¡¯Er = ln r Z Z X e¡¯Er ¡ ¢ X e¡¯Er = ln e¡¯Er ¡ ln Z r Z r Z X e¡¯Er Z = (¡¯Er) ¡ ln Z r Z Z X ¡¯E ¡¯Ere r = ¡ ln Z r Z but, by de¯nition, we have X ¡¯E Ere r <E>= ; r Z so we can simplify the above result to yield S ¡ = ¡¯<E>¡ ln Z; k and ¯nally we ¯nd <E> S = + k ln Z: T We will usually assume that the system energy is well de¯ned and use <E> and E interchangeably, that is the system's mean energy (which is an estimate) and the system's energy are interchangeable. Our development of the partition function through its ensemble tells us that Z = Z(T;V;N), thus S and <E> are also functions of T; V; and N: 7 Summary X S = ¡k pr ln pr r e¡¯Er pr = Z X Z = e¡¯Er r X 1 ¡¯Er <E>= Ere Z r @ ln Z <E>= ¡ @¯ <E> S = + k ln Z = S(T;V;N) T In these expressions, Z and <E>, and as a consequence S are de¯ned as functions of T; V; and N: Contrast this with an isolated system where S = S(T;V;N): For a macroscopic system at temperature T , energy fluctuations are negligible | i.e.
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