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Part II: Statistical Physics Chapter 6: Boltzmann Statistics

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Part II: Statistical Physics Chapter 6: Boltzmann Statistics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Part II: Statistical Physics Chapter 6: Boltzmann Statistics X Bai SDSMT, Physics Fall Semester: Oct. - Dec., 2014 X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem 1 Introduction: Very brief 2 Boltzmann Factor Isolated System and System of Interest Boltzmann Factor The Partition Function Z 3 Average Values in a Canonical Ensemble Applications 4 The Equipartition Theorem X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Introduction In the first part of Thermal Physics, the Thermodynamics, we have learned: (1) Bulk properties of a large system, equations of state; (2) Microscopic picture of a thermal system: multiplicity, entropy, and the 2nd Law, which includes simple statistical treatment of an isolated system. (3) Thermodynamic treatment of systems interacting with each other or in contact with the heat reservoir =) the maximum entropy, the minimum free energy principles, and their applications in engine and refrigerators. (4) How enthalpy (H=U+PV), Helmholtz free energy (F=U-TS), and Gibbs free energy (G=U-TS+PV) govern the processes toward equilibrium and phase transformations. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Introduction We tried to connect (2) and the rest contents by showing simple examples such as Ideal Gas, Einstein solid, and van der Waals gas/fluid =) impressive connections between macroscopic and microscopic properties. In doing so, we based all arguments on a fundamental assumption: a closed (isolated) system visits every one of its microstates with equal frequency. In other words, all allowed microstates of the system are equally probable. In this course, we will develop more complicated models based on the same fundamental assumption for the study of a greater variety of physical systems. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem From an isolated system to a non-isolated system We will introduce the most powerful tool in statistical mechanics to find the probability of finding a system in any particular microstate. To start, let's revisit the Isolated System and System of Interest. Reservoir R Combined system U0 - ε U0 = const System S ε A combined (isolated) system: (1) a heat reservoir (2) a system of interest in thermal contact with the heat reservoir X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Isolated System- cnt. Some fundamental assumptions for an Isolated System: An isolated system in thermal equilibrium will pass through all the accessible microstates states at the same recurrence rate as it evolves over time, i.e. all accessible microstates are equally probable. Probability of a particular microstate of a microcanonical ensemble 1 = (Total number of all accessible microstates) . (Ω of a particular macrostate) Probability of a particular macrostate = (Total number of all accessible microstates) The energy inside the system is conserved. A set of hypothetical systems with this probability distribution is called a mirocanonical ensemble These provides us with the basis for the study of a system we are interested in. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem A system in thermal contact with a heat reservoir The system of interest can be any small macroscopic or microscopic object - a box of gas, a piece of solid, an atom or molecule, etc. Assuming: Interactions between the system and the reservoir are weak: with heat exchange but no affect on the microscopic structure inside the system of interest; Total energy conservation: U0 = UR + US = const: Energies in the system (and therefore in the reservoir) may fluctuate by a "small" amount δ: U0 = (UR − δ) + (US + δ) = const: This system of interest can also be a "small" system: with small number of particles, small volume, ... At the equilibrium between the system and reservoir, our question would be "What is the probability P(Ei ) of finding the system S in a particular (microscopic) quantum state i of energy Ei ?" X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Boltzmann Factor Now let's figure out what this P(Ei ) is. Assume two microstates s1 and s2 in the system, corresponding to two different energy levels E(s1) and E(s2). The probability to find the system at these two states are P(s1) and P(s2). Ωcomb:(s1; U − E(s1)) = ΩS (s1) × ΩR (U − E(s1)) (1) * ΩS (s1) = 1 (system is now on a fix known state - no degeneracy case.) Ωcomb:(s1; U − E(s1)) = ΩR (U − E(s1)) = ΩR (s1) (2) Ωcomb:(s1; U − E(s1)) P(s1) = (3) Ωcomb:(U; N; T ) And same for state s2. So, P(s ) Ω (s ) 2 = R 2 (4) P(s1) ΩR (s1) This is the ratio. So, what is P(s1)? X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Boltzmann Factor- cont. Since S = klnΩ, Ω = eS=k . SR (s1)=k ΩR (U − E(s1)) = e (5) SR (s2)=k P(s2) e [SR (s2)−SR (s1)]=k = S (s )=k = e (6) P(s1) e R 1 The difference SR (s2) − SR (s1) is the entropy change in the reservoir. It must be tiny. So, we can use the thermodynamic identity to find the answer: TdSR = dUR + PdVR − µdNR (7) 3 dVR ≈ A˚ ≈ 0: Volume change due to re-distribution of particles on microstates dNR = 0, for system consisting of single atom, for example 1 1 S (s ) − S (s ) = [U (s ) − U (s )] = − [E(s ) − E(s )] (8) R 2 R 1 T R 2 R 1 T 2 1 −E(s2)=(kT ) P(s2) −[E(s2)−E(s1)]=(kT ) e = e = −E(s )=(kT ) (9) P(s1) e 1 Note: Let's see a case in which the PdV is not negligible. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Partition Function for a hydrogen atom An example with PdVR being big enough and requires a new Boltzmann Factor: When high-n states are occupied because the approximate radius of the 2 −11 electron wave function is a0n , a0 = 5 × 10 m is the Bohr radius. When keep PdV in Atomic model and Energy-level diagram for a hydrogen atom 1 dSR = T (dUR + PdVR − µdNR ), the new Boltzmann Factor becomes (at constant pressure): BoltzmannFactor = e−(E+PV )=kT . Hydrogen at ground state: 5 −30 3 −6 PV0 ≈ 10 Pa × 10 m ≈ 10 eV . When n = 10, 2 3 PV10 ≈ (10 ) PV0 ≈ 1 eV . comparing with kT at room temperature: kT ∼ 8:617 × 10−5 eV =K × 300 K ∼ 2:6 × 10−2 eV : Low temperature: kT ∼ 8:617 × 10−5 eV =K × 1 K ∼ 8:617 × 10−5 eV : X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Boltzmann Factor- cont. The Boltzmann factor is Boltzmann factor = e−E(s)=(kT ) (10) Rewrite Eq. (9), −E(s2)=(kT ) P(s2) −[E(s2)−E(s1)]=(kT ) e = e = −E(s )=(kT ) (11) P(s1) e 1 P(s ) P(s ) 1 2 = 1 = = const: (12) e−E(s2)=(kT ) e−E(s1)=(kT ) Z 1 P(s) = e−E(s)=(kT ) (13) Z We arrived at the most useful formula in all of statistical mechanics. Please memorize it. It is also called the Boltzmann distribution, or the canonical distribution. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Partition Function Z 1 −E(s)=(kT ) We arrived at P(s) = Z e To calculate the probability, we still need to know Z. The formula for Z can be easily obtained by the fact that X X 1 −E(s)=(kT ) 1 X −E(s)=(kT ) 1 ≡ P(s) = e = e (14) Z Z s s s X −E(s)=(kT ) Z = e (15) s Z is just the sum of all Boltzmann Factors. Several remarks: 1 Z is a constant - independent of particular state s. But it depends on temperature. 2 Assuming the energy of ground state is zero, the Boltzmann Factor of ground state is 1. Boltzmann Factors for excited states are less than 1. 3 At very low temperature, Z ≈ 1. 4 At high temperatures, Z can be very big. X Bai Part II: Statistical Physics Outline Introduction: Very brief Boltzmann Factor Average Values in a Canonical Ensemble The Equipartition Theorem Boltzmann Factor- Remarks. Several remarks about the Boltzmann factor: P(s ) 1 For the ratio 2 = e−[E(s2)−E(s1)]=(kT ), only the energy difference P(s1) E(s2) − E(s1) makes contributions. 2 We do not have to know anything about the reservoir except that it maintains a constant temperature T. 3 We made the transition from "the fundamental assumption for an isolated system" to "The system of interest which is in thermal equilibrium with the thermal reservoir": The system visits each microstate with a frequency proportional to the Boltzmann factor. 4 An ensemble of identical systems all of which are in contact with the same heat reservoir and distributed over states in accordance with the Boltzmann distribution is called a canonical ensemble.
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