Lagrange multipliers - isothermal-isobaric ensemble

Question

I know that the entropy of isothermal-isobaric ensemble is given by:

$$S = -k \sum_{i=1}^M p_i \ln p_i \quad \textrm{where $p_i$ must be normalized} \quad \sum_{i=1}^M p_i = 1 \, .$$

The average energy is

$$\sum_{i=1}^M p_i \varepsilon_i = \langle E \rangle$$

and the average volume is

$$\sum_{i=1}^M p_i V_i = \langle V \rangle \, .$$

Some authors say that the probability of finding and state $i$ is given by

$$p_i = \frac{1}{Q} \, \exp (-\beta \varepsilon_i - \gamma V_i)$$

where $\beta$ and $\varepsilon$ are Lagrange multipliers.

I need to physically interpret these two terms. I compared

$$S = k \, \ln \, Y + k \beta \langle E \rangle + k \gamma \langle V \rangle$$

with

$$S = - \frac{G}{T} + \frac{\langle E \rangle}{T} + \frac{P \langle V \rangle}{T}$$

Where I can obtain that

$$G = -kT \, \ln Y, \quad \gamma = \frac{P}{kT} \quad \textrm{and} \quad \beta = \frac{1}{kT} \, .$$

How can I obtain this equation using Lagrange?

$$p_i = \frac{1}{Q} \, \exp (-\beta \varepsilon_i - \gamma V_i)$$

I need some idea to open this equation, given that the physical interpretation of this parameters were done.

Answer 1

I am not sure I clearly get the question. You can derive the expression of $p_i$ by maximizing the entropy $S$ under the constraints of your system (here being fixed average energy and volume) by Lagrange multipliers. Solving the saddle points equations leads to your result. Is that what you need ?

Answer 2

The answer is given in the well-known article by Jaynes: Information theory and statistical mechanics. I will not go into the details, but summarize the line of thought, showing how the Lagrange multipliers enter the picture.

The entropy in information theoretical sense is defined as $$ S = -\sum_ip_i\log p_i $$ Taking $p_i$ to be the probabilities of the microstates of a physical system, we can calculate these probabilities by maximizing the entropy with the appropriate constraints: normalization of probability, $$ \sum_i p_i=1 $$ and the constraints imposed by ensemble of interest. Thus, if we demand that the system has constant energy and volume, we demand that $$ \sum_i p_i \epsilon_i = E,\\ \sum_i p_i v_i = V $$ We then maximize te entropy with these constraints. The method of Lagrange multipliers is one of the possible ways to do this, which consists in maximizing the function $$ f(\{p_i\}) = S(\{p_i\}) + \alpha \left(\sum_ip_i-1\right) +\beta \left(\sum_i p_i \epsilon_i - E\right) + \gamma \left(\sum_i p_i v_i - V\right) $$ in respect to $p_i$ and the Lagrange multipliers $\alpha,\beta,\gamma$. The rest is math.