Chapter 1Statistical Ensembles 1.1 Principle of statistical physics and ensembles Key points: All possible states appear with an equal probability. • Statistical systems are complex systems. We do not know all the information that is needed to completely characterize the systems. For example, the liter of gas may contain 1022 atoms. The completely characterize such a system we need to known the three components of the velocity for each atoms and the three components of the position for each atoms. It is impossible to obtain 6× 1022 real numbers to completely characterize the gas. However, not knowing all the information needed to characterize gas does not prevented us to develop a theory of gas. This is because we are only interested in some average properties of gas such as the pressure, volume, temperature. Those properties do not depend on every little details of each atoms. Not knowing every thing about the atoms does not prevent us from calculating those properties. This is the kind of problems facing statistical physics. In statistical physics we try to understand the properties of a complex system without know all the information of the systems. This is possible since the properties we are interested in do not depend on the details of the system. In statistical physics there is only one principle: All possible states appear with an equal probabillity Let us explain what do we mean by the above statement. Suppose we know certain properties of a complex systems. But those properties do not characterize the system completely. That means the system has a numbers of states that all have the same properties. Thus even after knowing those properties, we still do not know, among those possible states, which state the system is in. According to the principle of statistical physical, we say all the possible states are equally likely. But the system can only be in one state at a given time. What do we mean by “all the possible states are equally likely”? There are two points of view. In the first point of view, we may imagine we have many copies of the system, all have the same properties. But each copy may be in a different possible states. Then “equally likely” means that each possible state appear the same number of times among the copies of the system. The copies of the system is called ensemble. We have to have an ensemble to even define the probabilities. Under the first interpretation, statistical physical is a science that deal with ensembles, rather than individual systems. The second point of view only apply to the situation where the properties of the system is independent of time. In this case we interpret “equally likely” as that all the possible states appear 1for the same amount of time during a long period of time. The second point of view is related to the first point of view if we view the system at different times as the different copies of the system. The second point of view may be equivalent to the first point of view. The two points of view are equivalent only when the system can visit all the possible states, many times, during the long period of time. This is the ergodic hypothesis. Not all systems are ergodic. In this class, we will take the first point of view. We regard the statistical physics as a theory for ensembles. We will apply the theory for ensembles to individual systems, assuming the systems are ergodic. 1.2 Microcanonical ensemble A microcanonical ensemble is an ensemble formed by isolated systems. All the systems in the ensemble have the same energy (and possibly some other properties). Here by “same energy” we really mean all systems has an energy which lies within a small window between E and E + ΔE. 1.2.1 Number of states and entropy A simple example: N spins in magnetic field. Energy: E = �0/2 and E = −�0/2. ↑↓ How many states with energy E (ie with energy between E − �0 and E)? How many states Nwith M = E + 2 upsppin and N − M downspiins There are total of 2N states�0 (1 + 1)N =101N + N 111N −1 + ... + CM 1M 1N −M + ...N 0 N +N 1 n+ ↓)N = N −1 +... + CMN −M +...(↑ ↑↓ ↑↓ N ↑↓ (1.2.1) where N ! CM = N M !(N − M )! (1.2.2) CM is the number of ways to pick M objects from N objects. We find N number of states with M ↑ spins = CM (1.2.3)N The entropy is a function of energy E which is defined as kB ln CE/�0S(E)= kB ln(number of states) = (1.2.4)N When n is large 11ln(n!) =(n +2)ln(n + 1) − (n + 1) +2 ln(2π)+ ... (1.2.5) Thus (see Fig. 1.1) k−1S(E)= ln CM BN ≈N ln N − M ln M − (N − M )ln(N − M ) M = − M ln()− (N − M )ln(N − M )NN =N (−f ln f ↑ − f ln f ) (1.2.6)↑ ↓↓2 e/e0E/E =0−0.50.510S(E)/NFigure 1.1: The entropy per spin, S(E)/N, as a function of E or � the average energy per spin. The maximum entropy of a spin112 spin is kB ln(2) =0.69314718056kB . M = 1 + Ewhere f (or f ) is the probability for a spin to be up (or down). Using f = N 2 E0 and ↑↓Ef = M ↑ = 1 N 2 −E0 where E0 = N�0, we find ↓ E 1 EE 1 E k−1S(E)=N� −(1+ )ln(+ )−(2 −E0 )ln((1.2.7)B 2 E0 2 E0 12 −E0 )� Clearly, from the definition, the physical meaning of the entropy is number of states with energy E = e S(E)/kB (1.2.8) 1.2.2 Concept of temperature To introduce the concept of temperature, let us put two systems of spins together. System 1 has N1 spins and System 2 has N2 spins. Let ˜E1,2 be the energies of the two systems at the beginning. The total energy is E = E1 +˜˜E2. If we allow the two systems to exchange their energy, then the spins in the two systems may wondering around and sample all the possible states with total energy E. The question is what is the probability for system 1 to have an energy E1 The number states with system 1 having an energy E1 is k−1S1(E1)e k−1S2(E−E1) k−1[S1(E1)+S2(E−E1)]N(E1)= e BB = e B (1.2.9) Every possible states are equally possible. Probability for system 1 to have an energy E1 k−1[S1(E1)+S2(E−E1)]P (E1)∝e B (1.2.10) From Fig. 1.2, we see that when N →∞, P (E1)is almost like a δfuncction We can sayfor sure that the energy of system 1 has such a value ¯ E1 that it maximizes the total entropy S1(E1)+S2(E −E1), or ¯¯ S1�(E1)= S2�(E −E1) (1.2.11) ¯If ˜E1 at the beginning is not equal to E1, then after we bring the two spin systems together, ˜¯ E1 will shift from E1 to E1. We see that Eq. (1.2.11) is a condition for equilibrium. It is also maximum entropy condition.We have derived the second law of thermodynamics: as an isolated system approach to the equilibrium state, its entropy always increase (if we define the entropy as in Eq. (1.2.4)). If we define the temperature as 1 ∂S(E)= βkB = (1.2.12)T ∂E 3 /e e 0 T T b b −0.5 0.5 0 P(E )/P1max101000100100001E /E||00.20.40.60.810−0.5−0.4−0.3−0.2−0.1Figure 1.2: For a system of N1 spins and a system of N2 spins with total energy E, we plot the probability P (E1) for the N1sppi system to have an energy E1. Here N2 =2N1 and N1 = 10, 100, 1000, 10000. E is chosen to be −N1�0. P (E1)reach its maximum when E1 = E/3. Figure 1.3: The relation between temperate T , the inverse temperature β with the average energy per spin �. then the equilibrium condition Eq. (1.2.11) becomes T1 = T2 (1.2.13) For our spin system 111 2�0 + � = βkB = kB ln � 1 � (1.2.14)T�02�0 − � where � = E/N is the average energy per spin. 1.2.3 Curie’s law For a spin112 system in magnetic field B, �0 = gµB B. The total magnetic energy is MB where M is the magnetic moment. The energy per spin is � = MB/N. From Eq. (1.2.14), we find a relation between the Bfiiel induced magnetic moment M and the temperature T 1= kB ln � gµB N − 2M � T gµB B gµB N + 2M 4 cCa Y Cu O10225experimentCurie law00100200300T (K) (emu/mole Cu)0.008Figure 1.4: Curie’s law for a magnetic material. kB TFor B � gµB , we have M�gµBN and 22gµNBBM=4kBT 22 BWe find magnetic susceptibility χ = g4kµB TN ∝1/T. This is the Curie’s law (see Fig. 1.4). 1.2.4 Properties of entropy Entropy is an extensive quantity From � 1 �� 1 � k−1S(E)=N� −(1+ )ln(+ )−(2 −�0 )ln((1.2.15)B 2 �0 2 �0 12 −�0 )� we see that entropy is proportional to N, the size of system. Thus S is extensive quantity. In contrast, �, as the average energy per spin, is intensive quantity. The total energy E is a extensive quantity and the temperature T is an intensive quantity. Entropy and energy window From the definition of entropy S(E, ΔE)= kB ln(number of states with energy between E and E + ΔE) (1.2.16) we see that entropy also depend on the energy window ΔE. However, in the thermodynamical limit N →∞, sucha dependence can be dropped and we can regard S as a function of E only. To see this, we consider S(E, αΔE)=kB ln(number of states with energy between E and E + αΔE) =kB ln[α ×(number of states with energy between E and E + αΔE)] =S(E, ΔE)+ kB ln α (1.2.17) Since S(E, ΔE)∼N, as long as α = O(Nn), kB ln α term can be dropped. 5 EEE2E1(a)(b)(c)2 EEE1EG()EG()E1G()G()E2Figure 1.5: Total entropy of combined systems. Figure 1.6: Total numbers of states in the combined system with the total energy 2¯ E and the system 1 energy E1. Additive property of entropy Consider two systems both with N spins. The first system has total energy E1 and the second E2. E2/�0The first system has Γ1 = E1/�0 ≡ Γ(E1)possible states and the second Γ2 = =Γ(E2)CN CN possible states. If we put the two systems together, but forbid any exchange of energy between them (see Fig. 1.5a), then the combined system will has Γ =Γ1Γ2 possible states. The entropy of the combined system S= kB ln Γ is the sum of the sub systems S= S1 + S2 (1.2.18) If we allow the two system to exchange energy, the two systems will reach an equilibrium state. ¯The subsystem will have the same average energy E=(E1 + E2)/2 in the equilibrium state. The equilibrium state of the combined system will have a total energy 2¯ E. The number of possible states 2E/�0¯¯¯E− E1), it is clear that the ¯become Γ = C2N . Since Γ = �E1 Γ(E1)Γ(2¯ Γ >Γ =Γ(E1)Γ(E2) and the equilibrium state has a higher entropy (see Fig. 1.6). Thus reaching equilibrium always increase entropy (the second law of thermodynamics). After the two systems reach the equilibrium, we now forbid the energy exchange. The total number states is then reduced to ¯¯ E). We like to show that ln Γ( E)Γ(¯ Γ in the Γ� =Γ(E)Γ(¯ ¯ E)=ln¯thermodynamical limit, ie the system Fig. 1.5b and the system Fig. 1.5c have the same entropy. ¯Γ� ¯As the maximum of the Γ(E1)Γ2(E− E1), we find ¯>Γ/2N, where 2N is the number of possible distinct values of E1. We also have ¯<Γ. Thus Γ� ¯ln¯Γ� >ln(¯Γ >ln¯Γ/2N) (1.2.19) or ¯¯ ¯ S>S� >S− kB ln(2N) (1.2.20) 6 E4 states7 states4 states4 states(a)(b)Figure 1.7: The lines represent possible states. The thick lines represent states that actually appear in the ensembles. ¯ ¯¯Since SS ¯ and S� is of order N. In large N limit, we can regard S = S�. From Fig. 1.6 we also see that as system go from Fig. 1.5a to the equilibrium state Fig. 1.5b or Fig. 1.5c, the entropy of the system is maximized. Or equilibrium state has maximum entropy. Reversible and irreversible processes The system in Fig. 1.5b is evolved from that in Fig. 1.5a. Thus there are only Γ(E1)Γ(E2)possible initial states, and there will be only Γ(E1)Γ(E2) possible final states. Those the system Fig. 1.5b ¯has ¯Γ states with energy 2E, it will only be in one of Γ(E1)Γ(E2)possible final states. But we have no clue about which are the Γ(E1)Γ(E2) possible final states. We lost the information. We only ¯know the total energy of the system, and we only know the state can be in one of the Γ states. This is how the entropy get increased. The evolution from Fig. 1.5a to Fig. 1.5b is also presented in Fig. 1.7a. The Fig. 1.7b representa reversible (or adiabatic) evolution, say, caused bya change in �0. We see that reversible (or adiabatic) processes do not change the entropy, since the number of possible states is not changed. 1.3 Application to classical ideal gas Each degree of freedom is described by a point in phase space (q,p). A particle has three degrees of freedom and its state is described by (x,px,y,py ,z,pz ). Consider a Npartticl system. How many states with total energy below E. The answer is infinity. We need quantum physics to get a sensible result. Each state in a degree freedom occupies a finite area ΔqΔp = h. For the N particle system, the phase space is 6N dimensional. Each h3N volume in the 6N dimensional phase space correspond to one state. Thus the number of states with total energy below E is given by VN S3N (√2mE)3N /3N (1.3.1) �N<(E)= h31 N � � d3N qd3N p =2 i /2m