Deriving the Boltzmann distribution

In summary, the derivation of the Boltzmann distribution using the reservoir model states that the probability for a system to be at a specific energy level is proportional to the number of states in the reservoir. The temperature of the system is constant, so the number of states for the entire system is also constant. The systems A and R are independent, so the number of states for the entire system is equal to the product of the number of states for A and R. If the number of states in the reservoir increases, the number of states for A decreases, meaning they are inversely proportional. This results in the probability for A to be at a certain energy level being proportional to the number of states in the reservoir. However, this explanation may not
  • #1
raeed
8
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I was reading the derivation of Boltzmann distribution using the reservoir model.
lets call the reservoir by index R and the tiny system by index A.
In the derivation they proposed that the probability for being at energy e (for A) is proportional to the number of states in reservoir. I didn't understand this completely and i would be happy to get some help!
here is my take on it, and please correct me if I'm wrong.
- The temperature of the whole system is T and it's constant therefor the number of states for the whole systems g is also constant
- both A and R are independent of each other therefor g = gA ⋅ gR
- if gR goes up then gA has to go down meaning gA ∝ gR
- P(e) ∝ 1/gA → P(e) ∝ gR
I'm not really convinced by my explanation so if someone could explain it and perhaps give me an intuitive physical explanation, I'd be happy. Thank you
 
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  • #2
The starting point is that all microstates are equally probable. Then if the system A is made up a single particle, its state has no influence on the total probability of the state of A + R (the multiplicity of A is always 1), so you can focus on the states of the reservoir only.
 

What is the Boltzmann distribution in simple terms?

The Boltzmann distribution is a probability distribution that describes the distribution of particles in a system at thermal equilibrium. It states that the probability of a particle occupying a certain energy state is proportional to the negative exponential of the energy state divided by the temperature.

What is the significance of the Boltzmann distribution?

The Boltzmann distribution is significant because it allows us to understand and predict the behavior of particles in a system at thermal equilibrium. It is also the basis for many important laws and principles in statistical mechanics, such as the law of equipartition of energy and the ideal gas law.

How is the Boltzmann distribution derived?

The Boltzmann distribution is derived by considering the statistical mechanics of a system at thermal equilibrium. It involves calculating the probability of a particle being in a certain energy state using the principles of statistical mechanics and then finding the distribution that maximizes this probability.

What are the assumptions made in deriving the Boltzmann distribution?

The Boltzmann distribution is derived under the assumptions of a non-interacting system, where the particles do not interact with each other, and a constant temperature throughout the system. It also assumes that the particles are in thermal equilibrium, meaning that there is no net energy flow between them.

How can the Boltzmann distribution be applied in real-world situations?

The Boltzmann distribution can be applied in various fields, such as physics, chemistry, and engineering. It is commonly used to analyze and predict the behavior of gases, liquids, and solids in thermal equilibrium. It is also used in the study of chemical reactions, phase transitions, and thermodynamics.

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