Statistical Mechanics – Closed Systems

We now know how to express our knowledge of the system. This allows us to formulate a new set of occupation probabilities, derived in light of information available to us. All that’s left to ask is: what can we reasonably expect to know about a system?

One of the most fundamental properties of any entity is its energy. A thermodynamic system might have energy by virtue of the jostling of its constituent particles, or of their interparticular attraction, or of their interaction with some external potential field. In any case, a system’s energy is a numerical property which can in principle be measured.

For a system’s mean energy to be fixed, it must in equilibrium with its environment. This does not mean that the system’s energy is precisely fixed; since the microstate of the system is not known exactly, the system’s energy is essentially a randomly distributed variable with some known variance. But the mean energy about which the system’s energy fluctuates can  be known. It’s this mean value of the system’s energy we can measure as a macroscopic variable.

So we choose to characterise the system’s microstates \{\alpha\} by their associated energies \{E_\alpha\} and nothing more. This allows us to place a single, second constraint on the occupation probabilities. The consequences are explored below.

 

It is now asserted that something is known of  the system: its mean energy \langle E\rangle \equiv U. Then two things are known of the probabilities \{p_\alpha\}: they satisfy the equations

\displaystyle \sum_\alpha p_\alpha=1

\displaystyle \sum_\alpha p_\alpha E_\alpha = U

We appeal to our old dogma: maximise the Gibbs entropy. Using the method of Lagrange multipliers, it is necessary to extremise the function

\displaystyle -\sum_\alpha p_\alpha\ln p_\alpha-\lambda\Bigg(\sum_\alpha p_\alpha-1\Bigg)-\beta\Bigg(\sum_\alpha p_\alpha E_\alpha -U\Bigg)

The quantity \beta is our brand-new Lagrange multiplier. Taking the differential of the above yields

\displaystyle -\sum_\alpha (\ln p_\alpha +1+\lambda+\beta E_\alpha)dp_\alpha-\Bigg(\sum_\alpha p_\alpha-1\Bigg)d\lambda-\Bigg(\sum_\alpha p_\alpha E_\alpha -U\Bigg)d\beta=0

Since the \{p_\alpha\}, \lambda and \beta are independent, we must demand that

\ln p_\alpha+1+\lambda+\beta E_\alpha=0

\displaystyle \sum p_\alpha-1=0

\displaystyle \sum_\alpha p_\alpha E_\alpha-U=0

Rearranging the first equation yields

p_\alpha = e^{-(1+\lambda)} e^{-\beta E_\alpha}

Substitution into the second equation gives

\displaystyle e^{-(1+\lambda)}\sum_\alpha e^{-\beta E_\alpha}=1

We eliminate \lambda between the two equations to give

\displaystyle p_\alpha=\frac{e^{-\beta E_\alpha}}{\sum_\alpha e^{-\beta E_\alpha}}

 

How does this probability distribution differ from that of the isolated system? The equation for p_\alpha shows that some states are more probable than others, namely those with a lower energy. The probability of a system being in a given state decreases exponentially with its associated energy. Were we to play the game with the screens again, the monitors would be dominated by pictures of low-energy states. The name for this imaginary set of virtual copies of our system is the canonical ensemble. Here’s a graph of the probability distribution of microstate energies:

closed

There is nothing inherently ‘special’ about this particular ensemble, the one for which we profess to know only the mean internal energy. We could have asserted that additional properties of the system were known, and derived a different probability distribution. But it turns out that this set of probabilities is enough to reconstruct the thermodynamics of a variety of systems.

The system about which nothing was known we called the ‘isolated system’. What shall we name this one? The kind of system populating the canonical ensemble is known as a closed system. Why is this? The system is not ‘closed’ to the same extent that an isolated system is: we assumed our system had come to equilibrium with its environment, meaning energy must be allowed to flow across the system’s boundary. The system is really called closed to distinguish it from another kind of system, the open system. This kind of system is characterised by a different set of probabilities, which may be explored in the distant future!

 

The next item on our agenda is to construct the thermodynamics of the closed system. In doing so, we’ll supply the mysterious parameter \beta with physical significance. In the next post we define an important quantity known as the partition function.

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