Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby

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1 Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby Lecture 1: Probabilities Lecture 2: Microstates for system of N harmonic oscillators Lecture 3: More Thermodynamics, Boltzmann and Entropy Lecture 4: Entropy Lecture 5: Entropy and applications of statistical mechanics Lecture 6: 2-Level System Lecture 7: Lecture 8: Lecture 9: Lecture 10: Lecture 11: Lecture 12: Lecture 13: Lecture 14: Lecture 15: Lecture 16: Lecture 17: Lecture 18:

2 Lecture 1: Probabilities {Note lots of graphs in this course} Probability of obtaining score S on: 1 die [Figure 1. Plotting P vs. S, dots on the P=1/6 line] 2 dice: Total Score Average Score Number of Configurations Total: [Figure 2: Prob vs. avg score, 1 to 6 on x-axis, 0 to 1/36 to 1/6 on y-axis, triangle dot formation] 7 Dice: Total Average Score Number of ways Probability / /7 21* *5x(1 dot) + 2x(2 dots): 6x(1 dot) + 1x(3dot): [Figure 3: Prob vs. score plotted as a Gaussian between scores 1 to 6, FWHM=245660] Scores obtained: dice: [Figure 4: same as figure 3 but very thin (width ~10-12 )] With a large number of components we can describe properties with precision. We can make predictions about the collective behaviour of large systems without needing to predict the detailed motion of its components. Equally likely elementary outcomes 1.1 Microstates

3 Microstates are QM eigenstate solutions to S.E. but for a whole system rather than an elementary particle. For an isolated system the energy is fixed but this energy has degenerate states: W (large number) microstates with energy, E. [Figure 4: E on y axis then lots of dashed lines which I suppose is energy levels, circle enclosing some of the dashes labelled W microstates.] From Fermi s Golden Rule: Let p i be the probability of the system being in state I (among the W states): (1.1) (1.2) When a steady state has been reached, for all states. This can happen only if, i.e. all probabilities are equal and all equal. This is the Principle of Equal Equilibrium Probability (PEEP). Refresh: Possible E of single SHM, think about E of N SHM s. Lecture 2: Microstates for System of N harmonic oscillators Classically, Where is classical angular frequency (2.2) Energies where [Figure 2.1: of parabola with lines across it showing energy levels, at going up] A microstate of system of N oscillators is given by the states each oscillator is in. The energy of N oscillators is:

4 ( ) ( ) ( ) ( ) ( ) (2.3) What is the number of microstates W corresponding to a given value of Q (i.e. to a given total energy)? Consider: : Q dots, (N-1) partitions The no ways W is equal to no ways of arranging the Q dots and (N-1) partitions: 2.1 Thermal Equilibrium (2.4) Two systems of N harmonic oscillators The no microstates corresponding to distribution of energy is: From PEEP each of these W microstates is equally likely. (2.5) In the general case rather than W 1 we shall focus on the density of states, i.e. the number of microstates per unit energy: Note that where the band of energies within which E is fluctuating. (2.6) (2.7) Lecture 3: More thermodynamics System: E 1 W 1 energy E 1 W 1 G 1 G 1

The probability of this discussion of energy: For most likely partition, maximise, that is, maximise: (3.1) (3.2) ( ) Thus for most likely partition of energy (i.e. equilibrium): (3.3) (3.

5 The probability of this discussion of energy: For most likely partition, maximise, that is, maximise: (3.1) (3.2) ( ) Thus for most likely partition of energy (i.e. equilibrium): (3.3) (3.4) This quantity: is therefore equalised between two systems when they can exchange energy. [Figure 3.2: Ok I ll try and draw it!] lng E lng E Ok that went well. Define Statistical Mechanical temperature as: *For a large system This T is identical to the ideal gas temperature. (3.5)* 3.2 Boltzmann Distribution Visa vies systems of constant T. E R g R energy E 1 [Figure 3.3] T Microstate i Heat Reservoir Because the reservoir is large, T is too good an approximation slowly varying with E R.

6 By PEEP the probability of the situation shown is: (3.6) But: (3.7) Since (3.8) (3.9) (3.10) Z is the sum over all microstates j of the system of the Boltzmann Factor. Mean energy of a system at constant T The mean energy (3.11) Recall: (3.12) 3.3 Entropy (3.13) For an isolated system we define:

7 (3.14) For a non-isolated system exploring different microstates with probabilities p i (assumed to be known): {ends} Lecture 4: Entropy (4.1) For the general case where our system explores its microstates with given (generally equal) probabilities p i, consider N replicas of the system, where N is arbitrarily large. Then ~Np 1 systems will be in microstate 1, ~Np 2 in microstate 2, etc. Total Energy Np 1 E 1 + Np 2 E 2 + is effectively fixedm thus the system is thermally isolated in its behaviour, exploring states only with a total fixed energy. Thus the effective W is the number of distinct permutations of Np 1 identical objects, Np 2 identical objects, etc. (4.2) Using Stirling Approximation: (4.3) So the entropy of N replicas is: (4.4) For N=1, then: Entropy of an isolated system (4.5) (4.6) { (4.7)

8 (4.8) Entropy of a system at constant temperature (4.9) ( ) Recall Helmholtz free energy: (4.10) Thermodynamic entropy (4.11) Consider a system thermally isolated on which work can be done. The total internal energy is: When the volume of the system is changed: The RHS (and last term) must be associated with heat. (4.12) (4.13) Lecture 5: Entropy and applications of statistical mechanics If a system is insulated (thermally isolated): (5.1)

9 Now, (5.2) For a slow change in constraints, FMG shows that no additional transitions between microstates are induced, i.e. dp i = 0 for all For an insulated system: Since, the other term in (4.2) must be the heat: At constant temperature, the probabilities, p i, are given by the Boltzmann distribution: (5.3) (5.4) (5.5) Also (4.6): { } { } Thus statistical mechanical and thermodynamic entropy are identical. (5.6) 5.1 Applications of Statistical Mechanics Vacancies in crystals For N atoms, how many vacancies will there be at temperature T? Each vacancy costs an amount of energy ( ) due to not being bonded fully. Free energy F is to be minimised:

10 So when volume and temperature are held constant, hence we minimise not (5.7) What are and for a given? (5.8) Provided a system is sufficiently large, it s energy is effectively fixed as if it were thermally isolated so we may use: W is the number of arrangements of N atoms, and vacancies on a total of sites: { } (5.9) { } { } ( ) ( ) Since, (5.10) Note: At 300K, (5.11) At 3000K, Practical crystals at room temperature are not yet at thermal equilibrium but reflect vacancy concentrations from temperatures ( ) where the crystal first formed.

11 Lecture 6: 2-Level System E.g. each magnetic atom/ion in a dilute magnetic semi-conductor If then Because (6.1) B 0 μb μb Take (6.2) Low T: High T: P 1 p 1 (6.3) 0 KT μb KT μb T So, ( ) (6.4)

12 Diagram: E T μb Heat capacity C Schottky Anomaly (6.5) kt μb T Mean magnetic dipole moment μ μ T 0 So the magnetic polarisation with n magnetic ions per unit volume of crystal is ( ) Harmonic Oscillator ω ( ) ω ω [ ]

13 So This is a geometric series (6.6) (6.7) Lecture 7: More Harmonic Oscillator and Heat Capacity

Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby

Statistical Mechanics Victor Naden Robinson vlnr500 3 rd Year MPhys 17/2/12 Lectured by Rex Godby Lecture 1: Probabilities Lecture 2: Microstates for system of N harmonic oscillators Lecture 3: More Thermodynamics,