The average energy = the most probable energy?

[PhyMathNovice]

Hi,
Just like what the Title tells about. In the Statistical Physics, keeping the desity unchanged while taking the limit of Number(or Volume)=infinity. Does the average energy equal to the most probable energy ?

 

[RedX]

When you keep density and temperature constant, you'd expect all intensive variables to remain constant, such as energy/particle.

The most probable energy should remain the same too. If you have a bunch of independent events, then the probabilities multiply. The maximum of this product is the product of the maximums.

 

[Andrew Mason]

Hi,
Just like what the Title tells about. In the Statistical Physics, keeping the desity unchanged while taking the limit of Number(or Volume)=infinity. Does the average energy equal to the most probable energy ?

No. The speed distribution of molecules at thermal equilibrium is given by the Maxwell-Boltzmann distribution curve (lets call it the MBd). The peak of the MBd gives the most probable speed. The average speed is determined by the area under the MBd. Since the curve is not symmetrical about the peak, the vertical line that divides the area under MBd into two equal parts occurs to the right of the peak - that line marks the average or mean speed (equal numbers of molecules moving faster and slower than that speed). So the average speed is higher than the most probable speed. Since the corresponding energies are determined by the square of the speed x m/2 (m = mass of one molecule), the average energy is also greater than the most probable energy.

AM