Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th Finish Ch. 3 - Statistical distributions Statistical mechanics - ideas and definitions Quantum states, classical probability, ensembles, macrostates... Entropy Definition of a quantum state Reading: All of chapter 4 (pages ) ***Homework 3 due Fri. Feb. 1st**** Assigned problems, Ch. 3: 8, 10, 16, 18, 20 Homework 4 due next Thu. Feb. 7th Assigned problems, Ch. 4: 2, 8, 10, 12, 14 Exam 1: Fri. Feb. 8th (in class), chapters 1-4

Statistical distributions nini xixi 16 Mean:

Statistical distributions nini xixi 16 Mean:

Statistical distributions nini xixi 16 Mean:

Statistical distributions nini xixi 16 Standard deviation

Statistical distributions Gaussian distribution (Bell curve) 64

Statistical Mechanics (Chapter 4) What is the physical basis for the 2nd law?What is the physical basis for the 2nd law? What is the microscopic basis for entropy?What is the microscopic basis for entropy? Boltzmann hypothesis: the entropy of a system is related to the probability of its state; the basis of entropy is statistical. Statistics + Mechanics Statistical Mechanics Thermal Properties

Statistical Mechanics Use classical probability to make predictions.Use classical probability to make predictions. Use statistical probability to test predictions.Use statistical probability to test predictions. Note: statistical probability has no basis if a system is out of equilibrium (repeat tests, get different results). How on earth is this possible? How do we define simple events?How do we define simple events? How do we count them?How do we count them? How can we be sure they have equal probabilities?How can we be sure they have equal probabilities? REQUIRES AN IMMENSE LEAP OF FAITH

Statistical Mechanics – ideas and definitions A quantum state, or microstate A unique configuration.A unique configuration. To know that it is unique, we must specify it as completely as possible...To know that it is unique, we must specify it as completely as possible... e.g. Determine:Position MomentumEnergySpin of every particle, all at once!!!!! THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM

Statistical Mechanics – ideas and definitions A quantum state, or microstate A unique configuration.A unique configuration. To know that it is unique, we must specify it as completely as possible...To know that it is unique, we must specify it as completely as possible... Classical probability Cannot use statistical probability.Cannot use statistical probability. Thus, we are forced to use classical probability.Thus, we are forced to use classical probability. An ensemble A collection of separate systems prepared in precisely the same way.A collection of separate systems prepared in precisely the same way.

Statistical Mechanics – ideas and definitions The microcanonical ensemble: Each system has same:# of particles Total energy VolumeShape Magnetic field Electric field and so on These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many different microstates might correspond to the same macrostate.

64 Statistical Mechanics – ideas and definitions An example: Coin toss again!! width

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Volume V 10 particles, 36 cells

Ensembles and quantum states (microstates) Cell volume,  V Many more states look like this, but no more probable than the last one Volume V There’s a major flaw in this calculation. Can anyone see it? It turns out that we get away with it.

Entropy Boltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.