Probability of a fluctuation/entropy decrease

• shimzz5
In summary, the conversation discusses the discrepancy between the microcanonical and canonical ensemble in determining the probability of fluctuations in a gas at room temperature or higher. In some references, the probability is given by -ΔS, while in others it is given by the Boltzmann factor plus the number of microstates -M/T + S. The question arises whether the entropy factor cancels out the -M/T factor for ordinary temperatures and why we don't observe random fluctuations into low entropy states according to the second formula. The conversation references two sources, one being an arXiv paper and the other being Chapter 4.3.2 of a publication.
shimzz5
Suppose we have a gas in the room at some temperature which is room temperature or higher.

In some references the probability is given by -ΔS, which is indeed a tiny number and makes sense.

However, in other references the probability is given by the Boltzmann factor plus the number of microstates -M/T + S, where T is the temparature of the gas and S the entropy of the fluctuated object. How to resolve this contradiction (between the microcanonical and canonical ensemble)? Shouldn't the entropy factor cancel the -M/T factor for ordinary temperatures? It seems that in the second equation (which corresponds to the free energy) the fluctuations are probable even now because the temperature is pretty high? Why don't we observe random fluctuations into low entropy states according to the second formula?

Sorry for my bad english, thanks.

shimzz5 said:
In some references the probability is given by ...

However, in other references the probability is given by
Could you provide a link to these references? The context may help.

1. What is the probability of a fluctuation?

The probability of a fluctuation refers to the likelihood of a random event occurring. In the context of entropy decrease, it is the chance that a system will spontaneously move towards a state of lower disorder.

2. How is the probability of a fluctuation calculated?

The probability of a fluctuation can be calculated using statistical mechanics and thermodynamics principles, such as the Boltzmann distribution and the second law of thermodynamics. It takes into account the number of possible microstates that can lead to a decrease in entropy and the energy barriers that must be overcome for the system to reach that state.

3. What factors affect the probability of a fluctuation?

The probability of a fluctuation is influenced by several factors, including the size of the system, the energy barriers present, and the temperature. A larger system with more particles will have a lower probability of a fluctuation, while a smaller system with fewer particles will have a higher probability. Additionally, higher energy barriers and lower temperatures will decrease the likelihood of a fluctuation occurring.

4. Can the probability of a fluctuation be predicted?

No, the probability of a fluctuation cannot be predicted with certainty. It is a probabilistic concept, meaning that it can only provide a likelihood of a certain event occurring. While statistical mechanics and thermodynamics can be used to calculate the probability, it is impossible to know the exact outcome of a random event.

5. How does the probability of a fluctuation relate to entropy decrease?

The probability of a fluctuation is directly related to entropy decrease. As the probability of a fluctuation decreases, the likelihood of a decrease in entropy also decreases. This is because a decrease in entropy requires a specific arrangement of particles, which is less likely to occur through random fluctuations compared to an increase in entropy, which can occur through a wider range of arrangements.

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