Boltzmann Distribution

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A Boltzmann Distribution is a probability distribution for particles in a system over various possible microstates.



  • (Wikipedia, 2017) ⇒ Retrieved:2017-9-16.
    • In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution[1] ) is a probability distribution, probability measure, or frequency distribution of particles in a system over various possible states. The distribution is expressed in the form

      [math] F({\rm state}) \propto e^{-\frac{E}{kT}} [/math]

      where [math] E [/math] is state energy (which varies from state to state), and [math] kT [/math] (a constant of the distribution) is the product of Boltzmann's constant and thermodynamic temperature.

      In statistical mechanics, the Boltzmann 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. It is given as

      [2] [math] p_i={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}} [/math] where pi is the probability of state i, εi the energy of state i, k the Boltzmann constant, T the temperature of the system and M is the number of states accessible to the system.[3] The sum is over all states accessible to the system of interest. The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a natural gas storage tank. Because of this the Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy.

      The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference.

      [math] \frac{F({\rm state2})}{F({\rm state1})} = e^{\frac{E_1 - E_2}{kT}} [/math]

      The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium." [4]

      The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.[5] The Boltzmann distribution 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. Whereas, the latter is used to describe particle speeds in idealized gases.

  1. Translated by J.B. Sykes and M.J. Kearsley. See section 28
  2. McQuarrie, A. (2000) Statistical Mechanics, University Science Books, California
  3. Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York
  5. Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.