The general considerations, are they productive?


 
They say systems are quasi-independent when they either interact strong for a short time or do that weakly, e.g. through their surfaces. The Gibbs’s distribution (1901) defines the probability to find such a weakly interacting subsystem of a big main system in the state of definite energy when the two are in equilibrium. Surprisingly, the distribution does not include any dependence on the way of interaction between the subsystem and the main system. No wonder Einstein called Gibbs the brightest mind of US. The distribution is factorized in two functions of the subsystem’s energy. The first one, modulated by the temperature, exponentially decreases with energy. It happens in the the subsystem there are many quantum states of the same energy, e.g the degenerate states of different values of a spin component. The number of particles in each state should be the same because of equal probabilities of back and forth transitions between the states. Thus the second factor should be the number of quantum states of the subsystem, which increases with energy. Due to competition of the two, the distribution gets a sharp maximum whose width is inversely proportional to the square root from the number of particles in the subsystem. The position of the maximum represents the average energy of the subsystem which can be for instance gas in a thermostat.

On the two states of nature 🙂

Molecule speed distribution in gas

When considering a single molecule as a quasi-independent subsystem, the Gibbs distribution comes down to the Maxwell distribution over molecule’s speed (1860) later generalized by Boltzmann for their energy (1877).

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