Equal a priori probablities

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In summary, the conversation discusses the postulate about equal a priori probabilities in thermal equilibrium in statistical mechanics, specifically in the case of N non-interacting simple harmonic oscillators. It is questioned whether phase space states with particles near p=0 would be more probable due to spending more time in that neighborhood, and whether states with minimized total kinetic energy would also be more likely. However, it is concluded that the number of available states outweighs the probabilities of finding a specific state, up to the average energy of the particles.
  • #1
bobloblaw
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Hey guys a simple question about the postulate about equal a priori probabilites in thermal equilibrium in statistical mechanics.

I was thinking about the case of N non-interacting simple harmonic oscillators in thermal equilibrium. Shouldn't those phase space states that have single particle states with p near zero be more probable than those states with p near pmax? SInce particles near p = 0 spend more time in that neighborhood. I guess more generally wouldn't those phase states for which the total kinetic energy is minimized being more likely? I assume I'm not thinking about this properly but I can't
 
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  • #2
bobloblaw said:
SInce particles near p = 0 spend more time in that neighborhood.
Why?
bobloblaw said:
I guess more generally wouldn't those phase states for which the total kinetic energy is minimized being more likely?
The number of available states wins against probabilities to find some specific state - up to the average energy of the particles.
 

What is the concept of equal a priori probabilities?

Equal a priori probabilities refer to the idea that all possible outcomes of an event are equally likely to occur. This means that each outcome has an equal chance of happening, with no bias towards any particular outcome.

Why is the concept of equal a priori probabilities important in science?

In science, equal a priori probabilities are important because they allow for unbiased and objective analysis of data. By assuming that all outcomes are equally likely, scientists can eliminate potential biases and make more accurate conclusions based on the evidence.

What are some examples of situations where equal a priori probabilities are assumed?

One example is in the flipping of a fair coin, where there are only two possible outcomes - heads or tails - and each outcome has an equal chance of occurring. Another example is in a controlled experiment, where all variables are held constant and the outcomes are assumed to have equal probabilities.

How do scientists determine if equal a priori probabilities are appropriate in a given situation?

Sometimes, it may not be clear if all outcomes are truly equally likely. In these cases, scientists use statistical analysis and experimental design to assess the likelihood of each outcome and determine if equal a priori probabilities are appropriate. They may also gather data from previous studies or conduct pilot experiments to inform their assumptions.

What are the limitations of assuming equal a priori probabilities?

While the concept of equal a priori probabilities is useful in many scientific contexts, it may not always accurately reflect the real world. In certain situations, some outcomes may be more likely to occur than others due to external factors or biases. Additionally, the assumption of equal probabilities can sometimes oversimplify complex systems or phenomena, leading to inaccurate conclusions.

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