A simple model of fluctuations in an ideal gas
We define fluctuation a spontaneous variation of an observable greatness around a value, which represents the most probable status.
The probability of such observable states could exists is expressed in (1):
Whereis the entropy difference and the termodinamical probability of the status.
andare referred to the most probable state, where their value is maximum. Let's suppose that fluctuations will happen at constant inner energy (i.e. Ideal gas at constant temperature).
Free energy is defined as:
and (1) becomes:
average value will be:
Fluctuation in a temperature constant gas has an energy difference , which is the one molecule medium kinetic energy for each degree of liberty at the same temperature.
With this teory we can studt density fluctuation in a volumeof ideal gas.
For an ideal gas:
whereis the condensation (e.g. relative volume variation).
Fluctuations value will be:
Note that volume relative variations are opposite in signum at density:
so we have:
So fluctuations value variates with one over the number of molecules in the gas.
Let's note that this brief theory contraddicts Carnot's Principle, because it conducts at spontaneous diminuitions of entropy in little volumes of gas. This deviation is observable in a particle in a fluid moving of brownian motion which climbs up in vertical directions; the increasing of potential energy (gravitational) is tied to a diminuition of local kinetic energy, that is to say a local diminuition of fluid's temperature: here we have total heat to work transformation.