The average energy = the most probable energy?

In summary: Statistical Physics. In summary, when keeping density and temperature constant, the most probable energy and average energy are not equal. The most probable energy is determined by the peak of the Maxwell-Boltzmann distribution curve, while the average energy is determined by the area under the curve, which is not symmetrical around the peak. Therefore, the average energy is higher than the most probable energy.
  • #1
PhyMathNovice
2
0
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 ?
 
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  • #2
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.
 
  • #3
PhyMathNovice said:
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
 

1. What is the average energy?

The average energy refers to the average amount of energy possessed by a system or particle. It is calculated by taking into account all the possible energy states and their respective probabilities.

2. What is the most probable energy?

The most probable energy is the energy state that has the highest probability of occurring in a system. It is the most likely energy that a particle will possess at a given time.

3. How does the average energy relate to the most probable energy?

The average energy is equal to the most probable energy in a system when the system is in thermal equilibrium, meaning that all energy states have equal probabilities of occurring. In other cases, the two values may be different.

4. Why is the most probable energy important in thermodynamics?

The most probable energy is important in thermodynamics because it represents the state of maximum entropy, which is the most stable state for a system. It also helps in predicting the behavior of particles in a system.

5. Can the average energy and most probable energy be the same in all systems?

No, the average energy and most probable energy can differ in different systems depending on the probabilities of different energy states. In some systems, the two values may be close or even equal, but in others, they can be significantly different.

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