Question

How do I find the change in temperature in an ensemble?

Answer 1

by converting into energy u can find the changing of temp in an ensemble

Answer 2

The temperature is the measure of the average kinetic energy.

In the canonical ensemble, different from the microcanonical ensemble, the total energy E=Ekin+V=p2/2m+V(x) fluctuates. The average kinetic energy is given by

⟨Ekin⟩=∫d3xd3pEkine−E/kBT∫d3xd3pe−E/kBT=∫d3pEkine−Ekin/kBT∫d3pe−Ekin/kBT=32kBT.

For N particles, we have the result

⟨Ekin⟩=32NkBT.

The fluctuations in the kinetic energy are described by the standard deviation δEkin. They describe the effect that the kinetic energy is not fixed but fluctuates due to the presence of the reservoir. Statistical mechanics determines the fluctuations to be given by

(δEkin)2=⟨(Ekin−32kBT)2⟩=∫d3p(Ekin−32kBT)2e−Ekin/kBT∫d3pe−Ekin/kBT=32(kBT)2.

For N particles, we have the result

⟨Ekin⟩=32N−−√kBT.

Thus we have that the kinetic energy is a stochastic variable. It is fluctuating and given by

Ekin=32NkBT(1±23N−−−√).

For large systems (N→∞) the fluctuations are small and thus the kinetic energy of the system is well described by the average.