Rules for Large Number and Microstate
More is difference. What is the mathematical shape?
Collections
Rule of The Large Number
To describe equilibrium properties of macroscopic bodies, stastical mechanics has to deal with very large number \(N\) of microscopic freem degrees. Actually, taking the thermodynamic limit of \(N\to\infty\) leads to a number of simplifications, some of which are describe in three types:
- Intensive quantities, such as temperature T, generalized forces.
- Extensive quantities, such as energy E, entropy S, and generalized displacements.
- Exponential dependence, that is \(O(exp(N \phi))\), is encountered in enumerating discrete micro-states, or computing volumes in phase space.
In statistical mechanics, two simplify are used: summation of exponential quantities and saddle point integration.
When \(N\to\infty\),
\[ S=\sum_{i=1}^\aleph\varepsilon_i = \sum_{i=1}^\aleph \exp(N\phi_i) \approx \exp(N\phi_{max}) \] \[ I=\int{\exp(N\phi(x))}{dx} =\sqrt{\frac{2\pi}{N|\phi^{''}(x_{max})|}}e^{N\phi(x_{max})} \]
- 正则系统中一个假设:一个系统随时间演化称为是对时间的平均，其与多个 copy 的平均（也就是对系统的平均）是等价的。
- 极大似然
- Relationship between the genetics map distance and frequency. The genetics map distance always less than 50%. When genes number is large, the cross over is happen less.
The MPD
Ensemble Theory: Assuming a distribution of many copies of the same system(ensemble) in accessible phase space(deterimined by the connection between the system with the environments). The macroscopic properities of the system is just the average values of such ensemble. Different macroscopic environmental constrains different types of ensemble:
| Constraints | Fixed Variables | Ensemble |
|---|---|---|
| Isolated | E, N | Microcanonial Ensemble |
| Thermal Contact | N | Canonical Ensemble |
| Particle Diffusion | Grandcanonical Ensemble |
吉布斯现象
Reading: More is difference. Anderson