Lecture I

Isolated measures of a closed system can be done, but it sometimes can have no utility. Imagine have the velocity of a particle in a room, an ambiente with tons of particles. In these cases, it is better to have statistical measures of a system. In this way, statistical mechanics can be applied to solids, gases and other.

Basic Statistics

We are going to study things in certain states \(i\). This way, the probability of a state can be defined as \[ \lim_{N \rightarrow \infty} \dfrac{N_i}{N_{total}} \]

We can also specify the quantity associated with the state \(i\) as \(F(i)\). Thus, the average value of this measure \(\langle F(i) \rangle\) is the summation over all possible state times the corresponding probability: \(\sum_i F(i)P(i)\).

Mechanics

Symmetry

Is an important characteristic of these systems. If present, we may use statistical mechanics to uncover final states. Elseway, we should do by experimenting.

Conservation of Energy

System law’s can be represented as Markov Chains: you have a bunch of states and their transitions. More specifically, if the chain is irredutible, we can assure there is a conservation law. Else, there is loss of information.

Entropy

Entropy is measured over the number of states that has no null probability. \[ S = \log M \] Where \(M\) is the number of stationary states.

A more general entropy formula is given by \[ S = - \sum_i P(i) \log(P(i)) \] Example: \(n\) coins with equal probability (each coin can take two states) achieve \(2^n\) possible states, and the total probability is \(S=n \log 2\)

Boltzmann Distribution

The Boltzmann distribution is found by minimizing entropy with respect to some constraints. It describes a system in equilibrium.

\[ -S(P_i) = \sum_i \log P_i = F(P_i) \]

with respect to

\[ \sum_i P_i = 1 \text{ and } \sum_i P_iE_i = \langle E \rangle \]

This is done using Lagrange Multipliers. Recover the \(F\) above to write: \[ F + \alpha(\sum_i P_i - 1) + \beta(\sum_i P_i E_i - \langle E \rangle) = F'(P_i) \]

\[ \sum_i \log P_i + \alpha(\sum_i P_i - 1) + \beta(\sum_i P_i E_i - \langle E \rangle) = F'(P_i) \]

For a specific \(i\) we have \(\frac{\partial F}{\partial P_i} = \log P_i + 1 + \alpha + \beta E_i\). Now we equal this to 0, obtaining

\[\log P_i + 1 + \alpha + \beta E_i = 0\] \[\log P_i = -(1 + \alpha) - \beta E_i\] \[P_i = e^{-(1+\alpha)}e^{-(\beta E_i)}\] The value \(z=e^{-(1+\alpha)}\) is called the partition function, and finally we obtain \[P_i = \dfrac{e^{-\beta E_i}}{z}\]

We obtain a probability function where \(z\) acts as a normalizing factor and \(\beta\) has something you tune to change the average energy (below).

As the temperature increases (yellow to red), all states start to become equally probable.

Energy

We still have to discover unknown parameters. Lets use our constraints

\[ \sum_i P_i = \sum_i \dfrac{e^{-\beta E_i}}{z} \implies z(\beta) = e^{-\beta E_i} \]

by differentiating \(z\) with relation to \(\beta\)

\[\frac{\partial z}{\partial \beta} = - \sum_i E_i \dfrac{e^{-\beta E_i}}{z}=\langle E \rangle\]

So, it is possible to obtain the energy by deriving the partition function with respect to \(\beta\):

\[ E(\beta) = - \frac{\partial \log z}{\partial \beta} \]

Beta

\[S = - \sum_i \dfrac{e^{-\beta E_i}}{z} [-\beta E_i - \log z] \] thus \[ S = \beta \langle E \rangle + \dfrac{\log z}{z}\sum_i e^{-\beta E_i} \implies S = \beta \langle E \rangle + \log z(\beta) \] Finally, deriving it gives \[ dS = \beta dE + Ed\beta + \dfrac{\partial \log z}{\partial \beta} d\beta \] Last terms cancel out, giving \[ dS = \beta dE \implies T = \dfrac{1}{\beta} \]