Boltzmann Distribution (also called Gibbs Distribution)
- is a probability distribution that gives the probability that a system will be in a certain state as a function of that state’s energy and the temperature of the system
- should not be confused with the Maxwell-Boltzmann Distribution. The former gives the probability that a system will be in a certain state as a function of that state’s energy; in contrast, the latter is used to describe particle speeds in idealized gases
Formulation
The Boltzmann Distribution is expressed in the form:
𝑝𝑖 ∝ 𝑒-ε𝑖/𝑘𝑇
where:
- 𝑝𝑖 is the probability of the system being in state 𝑖
- ε𝑖 is the energy of that state
- constant 𝑘𝑇 of the distribution is the product of Boltzmann’s constant 𝑘 and thermodynamic temperature 𝑇
- ∝ denotes proportionality
Boltzmann Factor
The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the state’s energy difference
- 𝑝𝑖/𝑝𝑗∝ 𝑒-ε𝑖/𝑘𝑇/ 𝑒-ε𝑗/𝑘𝑇
- 𝑝𝑖/𝑝𝑗 ∝ 𝑒ε𝑗-ε𝑖/𝑘𝑇
Boltzmann Distribution
The Boltzmann Distribution is a probability distribution that gives the probability of a certain state as a function of that state’s energy and temperature of the system to which the distribution is applied. It is given as
where:
- 𝑝𝑖 is the probability of the system being in state 𝑖
- ε𝑖 is the energy of that state
- constant 𝑘𝑇 of the distribution is the product of Boltzmann’s constant 𝑘 and thermodynamic temperature 𝑇
- 𝑀 is the number of all states accessible to the system of interest
The normalization denominator 𝑄 (denoted by some authors by 𝑍) is the canonical partition function
Subpages
- PGM - Gibbs Distribution
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