Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath to satisfy the large number approximations, W W sys can be small. Let us consider two derivations of the canonical ensemble, one based on the Legendre transform of the entropy, the other on explicit consideration of the microcanonical probabilities: 1) Massieu functions Let U be the average energy of an ensemble in contact with a heat bath of temperature T. Now consider the Legendre transform Just like S[U] f (T ) S 1 T U W (U) e S /k B is the number of microstates populated at energy U, Q(T ) e S[U ]/k B e(s U /T )/k B e A/k BT is the number of microstates populated at temperature T when the average energy is U. Q is known as the canonical partition function, and depends on T instead of U. Now consider a specific value of the energy E i not necessarily equal to the average. In that case Q Ei (T ) e S[E i ]/k B T e S /k B e E i /k B T W (E i )e E i /k B T The probability of the system having energy E i out of all possible energies at temperature T therefore is given by p i W (E i )e E i /k B T W (E i )e E i /k B T 1 Q W (E i )e E i /k B T. Again, Q is the canonical partition function, or the average number of microstates populated at temperature T. For states of very high energy E i, the Boltzmann factor e -E/kT is very small, and their microstates counted by W(E i ) do not contribute significantly to the sum of states. 2) Explicit treatment of the bath Let the system and bath together have constant energy U tot (overall closed system). Then the number of ways the system can be at energy E is W (E ) W bath E ). The number of ways the system can be at any energy is simply W tot ). From this follows for the probability of being at energy E that
p ρ (eq) W (E )W (U bath tot E ) ; using S k B lnω W tot ) p W (E )e { } 1 S bath E ) S tot ) k B We can simplify the exponent as follows: Let U be the (as yet unknown) average energy U E of the ensemble of canonical systems. e S tot ) S(U) + S bath U), and Taylor-expanding S bath S bath E ) S bath U) + S bath E E U S bath U) + S bath U bath (E U) +... U bath E S bath U) 1 T (E U) +... (E U) +... In the last step, we used 1 / T S U and du du bath sys to simplify the second term. Inserting both of these into the exponential at the top of the page, p ρ (eq) W (E )e W (E )e W e E /kbt Q 1 k B T E +U TS 1 1 k B T E + 1 T U S { } ; letting A U TS and defining Q e A/k B T This is the probability of being in state at temperature T. We can derive an alternate formula for the normalization factor Q by realizing that p 1 1 Q W e E /k T B 1 or Q W e E /k B T { } T r Ωe βh. where Q is the canonical partition function, A the Helmholtz free energy, and β 1 / k B T. 3) The meaning of the canonical partition function For practical use, the important consequence of both of these derivations is that
Q W e E /k B T e A/k BT This formula connects the microscopic world of individual energy levels and their degeneracies (degeneracy microcanonical partition function) with the macroscopic world of thermodynamic potentials, in this case A. Thus, once the microcanonical partition functions Ω (at energy E ) are known, Q can be calculated. Once Q is known, A can be calculated, and from A all other thermodynamics quantities. In the canonical ensemble, all energies E are allowed, but their probabilities are weighted by the Boltzmann factor, and U is no longer a variable. The transformation W (U) Q(T ) is analogous to the Legendre transform U(S,V, N) A(T,V, N) and not surprisingly Q is directly related to A. In the canonical picture, W W (E ) and Q have simple interpretations: W is the degeneracy of energy level E. Q is the effective number of states populated at temperature T. For example, let E 0 0 (ground state) and E >0 > 0 (excited states), and let W 0 1 (singly degenerate ground state). Q(T 0 ) 1: only one state (the ground state) is populated at zero temperature. As T increases, Q increases because > 0 states are populated. When k B T E, a state E makes a full contribution to Q. lim e x 1 x E /k B T 0 We can talk about levels E of a single molecule if we want: the derivations assumed that the thermal bath is large and W tot W bath, not that W of the system is large. The canonical ensemble can be applied to any size system without further approximations. If E is continuous, we replace the sum by an integration Q W (E) is the continuous density of states. 0 dew (E)e E /k BT, where Finally, Q is sometimes written as a sum over every single state, not degenerate energy levels: Q e βe i ˆρ eq (T,V ) 1 Q e βĥ i1 For example, if level 5 is 3-fold degenerate, this sum would have to contain the term e βe 5 three times, rather than explicitly writing Ω 5 e βe 5 3e βe 5
4) Computation of thermodynamic quantities from Q We can use the formalism to compute any average we want. Example 1: Heat capacity of and RNA hairpin Consider the model for the reaction F U corresponding to the interconversion of a folded RNA hairpin to an unfolded RNA hairpin. The figure below shows a simple 2-D lattice model for an RNA with four bases, two of which can base pair in the stem. In this model, we can count all the states that are non-superimposable. F U Fig. 12.1 Folding/unfolding reaction of an RNA hairpin in a simple 2-D lattice model that allows only 90 hinge motions of the bases, and counts only one interaction energy (base pairing energy), between the two black bases. Let the base pairing energy between the two black bases be ε kj/mole. Renormalizing the minimum energy to 0, we have E f 0 W f 1 and E U +ε and W U 5. The partition function for this system is Q W F + W U e ε / RT. The average energy of the system is given by U E F p F + E U p U W F Q 0 + W U Q ε W U εe ε / RT. ε / RT W F + W U e We now have the energy as a function of temperature and the parameters W F, W U and ε. Taking the derivative to compute the heat capacity one obtains C V U T 1 2 ε R T p U p F. This heat capacity has a maximum when p U p F 1/2, in the middle of the folding/unfolding transition. Thus the melting temperature of an RNA piece could be determined by calorimetry by looking for the heat capacity maximum. Note that a simple model like the above will not be perfectly accurate. For example, by making it 2-D, we are grossly underestimating the number of unfolded configurations compared to the folded ones. We could easily remedy this by writing p U as p U 1 p F W / W / RT U F e ε ε / RT 1+ W U / W F e
and making the ration W U /W F an adustable parameter in our fitting model. Or we could do a better calculation by, say, running molecular dynamics to count configurations and evaluate energies. The important thing is that statistical mechanics lets you derive realistic model functions for your experiment, instead of fitting the data by a polynomial or some other such function that has no physical meaning and provides no physical insight. In the above equation for C V, the physical insight is that the heat capacity is directly related to the base pairing energy ε, depends on the ratio (ε/t) 2, has a peak when the folded an unfolded states are equally populated, and depends on the ratio W U /W F (which is probably much larger than 5 in reality). Example 2: a compact formula for the energy Q e βa e β (U TS) e βu +ln Ω(U ) Q lnq UQ or U β β Example 3: a formula for pressure P A V k B T lnq T,N,... V T,N,... Example 4: entropy Once we have U and A, it is easy to obtain S: S 1 (U A) T Thermodynamics provides us with the manipulations to get any variable we want once we have computed the partition function. Q (or W) contain all knowledge about the equilibrium properties of a system. Example 5: harmonic oscillator Consider a linear spring with FkL, where L is the displacement from equilibrium. E ωn if we shift E n0 0 to the zero point. Therefore we have Q 1 e ωn/k BT e ω /k BT n0 n0 ( ) A k B T lnq +k B T ln 1 e ω /k BT U TS FL for a spring A U TS FL; n ( ) 1 1 e ω /k BT V,N,... F kl L 2 A k B T lnq L rms k T B k lnq. This remarkable result tells us how the root-mean-square thermal fluctuations of a spring depend on temperature T, spring constant k and number of states populated Q. For a macroscopic spring, L rms would be rather small, but for a molecule-sized spring, it could be quite large relative to the length of the spring. This is the reason we cannot build a Maxwell daemon by making a little door between two chambers held tight by a spring on
one side so particles can only go through one way. In reality, the thermal motion of the particles would make the spring vibrate so much that particles could get through either way (or if k is too large, neither way). Example 6: electronically excited oscillator For a spring (vibrating molecule) in an excited state with lowest vibrational energy level starting at E D instead of E0, the partition function would be E D + ωn Q e ωn/kbt e D /k BT e D /kbt 1 e β ω. This is the same result as before, but multiplied by a Boltzman factor. As expected, there are fewer states populated because they are at higher energy D where thermal excitation cannot reach them easily. A typical chemical example would be if the excited state is a transition state with energy D E. Such a state would be less populated by a factor e E /k B T, correspondingly slowing down the chemical reaction, which depends on the molecule getting to the transition state. The rate would thus be given by k reaction Ae E /k B T, where A is the rate if the molecule were already at the transition state, and the second factor gives the decrease in rate because the probability of the molecule being at such a high energy is very small. This is the Arrhenius equation for the rate. In the chapter on chemical equilibrium, we will derive what A is from first principles, thus providing a complete derivation of the activated rate law from statistical mechanics.