Intro to Statistical Mechanics Flashcards
What are the two approaches to thermal physics?
Thermodynamics and statistical mechanics.
What is thermodynamics?
Assumes a unique normal state for any system called thermal equilibrium.
What are 2 thing in thermodynamics?
- Can state 4 laws relating quantities p,T,u and s
- Laws are empirical and are justified experimentally
What is statistical mechanics?
Defines the concepts of thermal equilibrium and entropy in terms of microscopic probabilities.
What are 4 things in statistical mechanics?
- Gives probability of a particular state
- Determines measurable properties using averages of these states
- Derives the results of thermodynamics
- Makes predictions about quantum systems
What is an isolated system?
System which is not connected to a reservoir so E,N,V fixed.
What is a model system?
The simple harmonic oscillator.
What is the equation for the potential energy of the harmonic oscillator?
V = 1/2 mω^2 *x^2
What is the equation for En of the harmonic oscillator?
En = (n+1/2)ћω, where n is a quantum number
If we have multiple SHOs, what is the total energy equal to?
E = E1+E2+E3+E4
What is the equation for the zero point energy?
Nћ*ω/2, where N is the number of SHOs
What is the zero point energy?
- Energy of the ground state
- Evidence that things move even at zero temperature
What is a microstate?
Any microscopic configuration of the system e.g. for 1l of N2 gas this would specify the position and velocity of every molecule (very difficult to measure)
What is a macrostate?
Any configuration with particular measurable properties e.g. for 1l of N2 gas this would specify u & N & V (can be made up of very many microstates)
What ways can 3SHOs with E=2ћω be arranged?
2-0-0 with E(1) = 2ћω, 1-1-0, 1-0-1 with E(1) = 1ћω, 0-1-1 , 0-0-2, 0-2-0 with E(1) = 0ћω -> 1 macrostate (E = 2ћω) made up of 6 microstates
What is the symbol for microstates?
Ω
How does the measure ment effect the number of macrostates?
Normal measurement = 1 macrostate, sensitive to E(1) = 3 macrostates (3 values of E(1), sensitive to E(1), E(2) and E(3) = 6 macrostates
What is the first assumption of SM?
Each one of the possible microstates of a system in thermal equilibrium is equally likely to occur.
What is the second assumption of SM?
The system’s dynamics are such that the microstate is constantly changing.
What is the third assumption of SM?
Given enough time, the system will explore all possible microstates and spend equal time in each of them -> known as the Ergodic Hypothesis.
What are the 2 consequences of our assumptions in SM?
- System will appear to choose macroscopic configuration that maximises number of microstates
- measure value of an observable is equal to the average over the microstates.
For the 3 SHOs with E = 2ћω, how do we find the expected value of E(1)?
= sum over i of Ei(1)P(i) = 1/Ω * sum over i of Ei(1) = 1/6(2+1+1+0+0)ћω = 2/3 *ћω
What is the micro-canonical ensemble?
An ensemble of isolated systems with fixed E & N.
What is the canonical ensemble?
Ensemble of systems each of which can exchange energy with a large reservoir.
What is the grand-canonical ensemble?
Ensemble of system each of which can exchange both energy and particles with a large reservoir.
