Probability at a temperature T that a system has a particular energy

In summary, the conversation is about understanding the behavior of small systems with normal modes of vibration, specifically one with 2 normal modes of vibration and natural frequencies $$\omega_1$$ and $$\omega_2=2\omega_1$$. The question is what would be the probability at a temperature T for the system to have an energy less than $$5\omega_1/2$$, assuming that the zero of energy is taken at $$T=0$$. The speaker is seeking guidance for better understanding of these types of cases.
  • #1
Hector Triana
7
2
Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a small system that has 2 normal modes of vibration, with natural frequencies $$\omega_1$$ and $$\omega_2=2\omega_1$$. So, what would be the probability at a temperature T that the system would get an energy less than $$5\omega_1/2$$? if it's assumed the zero of energy is taken as its value at $$T=0$$
I would like any guidance for better understanding of that kind of cases because those problems are very interesting, specially how to approach them.

Thanks for your attention.
 
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  • #2
p ~ exp(-E/kT), E ~ omega^2
 
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FAQ: Probability at a temperature T that a system has a particular energy

What is the relationship between temperature and the probability of a system having a particular energy?

The higher the temperature, the higher the probability that a system has a particular energy. This is because at higher temperatures, particles within a system have more kinetic energy and therefore, are more likely to occupy higher energy states.

How does the probability of a system having a particular energy change as temperature increases?

As temperature increases, the probability of a system having a particular energy also increases. This is due to the Boltzmann distribution, which states that the probability of a particle being in a particular energy state is directly proportional to the Boltzmann factor, which increases with temperature.

Can the probability of a system having a particular energy be zero at any temperature?

No, the probability of a system having a particular energy cannot be zero at any temperature. According to the Boltzmann distribution, even at very low temperatures, there is still a non-zero probability that a particle will occupy a higher energy state.

How does the probability of a system having a particular energy change with an increase in the number of particles in the system?

As the number of particles in a system increases, the probability of the system having a particular energy also increases. This is because there are more particles available to occupy different energy states, leading to a higher probability of a particular energy being occupied.

Can the probability of a system having a particular energy be greater than 1?

No, the probability of a system having a particular energy cannot be greater than 1. The total probability of all possible energy states must add up to 1, as it is the sum of all possible outcomes. Therefore, the probability of a system having a particular energy cannot exceed 1.

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