Macroscopic and Microscopic Quantities
Macroscopic quantities do not change over time — Equilibrium state
Equal Probability Hypothesis
For an isolated system in equilibrium, each microscopic state has the same probability of being reached
Thermodynamics: Bottom-up phenomenological approach
Statistical Physics: Top-down theoretical approach
Measurable macroscopic quantities are actually the statistical averages of unmeasurable microscopic quantities
However, since t is a time scale that is extremely small relative to a macroscopic system but still large compared to the microscopic world, the concept of an ensemble is introduced
Replicate the system N times while ensuring their macroscopic states remain the same
Time averaging is equivalent to ensemble averaging (Ergodic Hypothesis ensures this)
(For an isolated system, starting from any microscopic state, after a sufficiently long time, the system will traverse all possible microscopic states
On the other hand, if the number of ensembles N is large enough, it will also traverse all possible microscopic states. Hence, they are equivalent)
Statistical averaging is assumed to be ensemble averaging by default
Core of Statistical Physics: Solving the probability Pi that a system falls into each microscopic state
The partial derivative of Pi with respect to time equals zero → Equilibrium statistical mechanics; otherwise, it describes non-equilibrium states
Microcanonical Ensemble
Isolated system, fixed N, V, E
Each microscopic state has a definite energy
Canonical Ensemble
Fixed N, V, T
The probability of a system in a canonical ensemble taking a particular microscopic state with a specific energy can be determined
The system's energy (microscopic quantity) is uncertain, but its average energy (internal energy, macroscopic quantity) is definite
Assume a system is coupled to a large heat reservoir, ensuring a fixed temperature
After reaching equilibrium, both the system and the reservoir have the same definite temperature
The system and the heat reservoir together form an isolated system, which is an element of the microcanonical ensemble, with a definite energy E0
Let the system's energy be ES and the heat reservoir's energy be E0 - ES
The total number of microscopic states: Ωtot(E0) = Σ Ωs(ES)ΩR(E0-ES)
Only depends on total energy E0 and is independent of ES
The probability of the system taking a particular microscopic state is proportional to ΩR(E0-ES)
Since the heat reservoir is large, taking the logarithm and expanding leads to... the partition function
Using the partition function, all macroscopic quantities of the system can be expressed
U, S, F
All other macroscopic quantities can be derived from U and F
Grand Canonical Ensemble
Fixed μ, V, T
Similarly, the grand partition function can be obtained
Using it, any macroscopic quantity can be calculated
Quantum Statistics
For quantum systems, not only statistical averaging is needed, but also quantum averaging
...Conclusion: ⟨A⟩_quantum = Σ ⟨k| ρA |k⟩
Key point: Solve for the density matrix
From this, the expressions for the average values of physical quantities in the canonical and grand canonical ensembles under the quantum statistical framework can be obtained
Von Neumann Equation can be compared to the Liouville Equation