# What is Boltzmann distribution: Definition and 70 Discussions

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure 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. The distribution is expressed in the form:

p

i

e

ε

i

/

k
T

{\displaystyle p_{i}\propto e^{-{\varepsilon _{i}}/{kT}}}
where pi is the probability of the system being in state i, εi is the energy of that state, and a constant kT of the distribution is the product of Boltzmann's constant k and thermodynamic temperature T. The symbol

{\textstyle \propto }
denotes proportionality (see § The distribution for the proportionality constant).
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 .
The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference:

p

i

p

j

=

e

(

ε

j

ε

i

)

/

k
T

{\displaystyle {\frac {p_{i}}{p_{j}}}=e^{{(\varepsilon _{j}-\varepsilon _{i})}/{kT}}}
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"
The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The Gibbs entropy formula

S
=

k

B

i

p

i

log

p

i

{\textstyle S=-k_{\mathrm {B} }\sum _{i}p_{i}\log p_{i}}
) and the thermodynamic definition of entropy (

d
S
=

δ

Q

rev

T

{\textstyle dS={\frac {\delta Q_{\text{rev}}}{T}}}
, and the fundamental thermodynamic relation).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; in contrast, the latter is used to describe particle speeds in idealized gases.

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1. ### I Is there a Boltzmann distribution for a system with continuous energy?

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2. ### Deriving the kinetic energy flux in an effusion process

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3. ### Gas in a box with Maxwell-Boltzmann distribution

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4. ### B Shape & Dimensions of Containers: Impact on the Maxwell Boltzmann Distribution

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5. ### I Validity of the Boltzmann Distribution

I (mechanical engineer) have researched this question but can't get to an answer. My question concerns the validity of the Boltzmann distribution. We start with "particles in a box". These particles (at t-zero) may exhibit a range of energies. We place this box of particles in a heat bath for...
6. ### I Boltzmann Distribution: Formula & Fig 2a in Document

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7. ### Finding the g-value in the Boltzmann distribution

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8. ### I Can the Exact Boltzmann Distribution Yield Specific Quantum State Populations?

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9. ### Energy landscape in the two state model (Boltzmann distribution)

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10. ### I Maxwell Boltzmann distribution

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11. ### B Question about the Boltzmann distribution

I was reading about the Debye-Huckle theory for electrolytes solutions (https://en.wikipedia.org/wiki/Debye–Hückel_theory). In all the books, notes, and in the wikipedia age too, there is this statement that troubles me: Shouldn't I have the "normalization factor" (i.e ##1/Z##) in the above...
12. ### Boltzmann distribution and the number of molecules with a certain velocity

Hi, You could skip these details and find the main question at the bottom. I added the details for the sake completeness and context. Thanks. Boltzmann distribution of molecular speeds provides an insight into the different speeds the molecules of a gas are moving around with. It provides you...
13. ### A What is the derivation of the exact Maxwell-Boltzmann distribution?

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14. ### I Boltzmann Distribution and microstate probabilities

For a canonical ensemble the probability of occupying a certain microstate varies depending on the energy, however I thought that every microstate has an equal chance of being occupied. So what part of the canonical ensemble have I misunderstood?
15. ### Average Velocity of gas molecules calculated with a Boltzmann distribution

Hello What is the meaning of the average velocity of gas molecules calculated by Boltzmann distribution (in kinetic theory of gases)? Does all molecules have the same average velocity?
16. ### I Deriving the Boltzmann distribution

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17. ### I Boltzmann Distribution: Feynman's treatment of p-n junction

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18. ### A Can indistinguishable particles obey Boltzmann statistics

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19. ### Boltzmann Partition Function of H_2

Homework Statement In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibration levels of an ##H_2## molecule, for example, are more accurately described by the approximate...
20. ### Average energy for n-state system

Homework Statement Find the average energy ##\langle E \rangle## for (a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε. (b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit). Homework Equations Definition of...
21. ### I Boltzmann Distribution Derivation Question

Hello, I have a question about Boltzmann Distribution. I wonder why partial N of Nj is 1 and partial U of Nj=Ej. because N is constant, partial N of Nj has to be 0 and Partial Nj of U has to be 0 as well. They are constants so, to make sense of the equation, alpha and beta have to be 0 but...
22. ### I Stationary states -- Boltzmann distribution

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23. ### I Boltzmann distribution for spin-1/2 dipole: high T limit

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24. ### I Boltzmann distribution: isothermal atmosphere error?

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25. ### Average height of molecules in a box as a function of temperature

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26. ### Boltzmann distribution derivation.

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27. ### Average Speed for Maxwell's Distribution of Molecular Speed

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28. ### Possible to derive Boltzmann distribution using W, not lnW?

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29. ### Normalization of 1D velocity boltzmann distribution

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30. ### Derivation of the Boltzmann distribution (Dr. David Tong)

Hello! Dr. David Tong, in his statistical physics notes, derives the Boltzmann distribution in the following manner. He considers a system (say A) in contact with a heat reservoir (say R) that is at a temperature T. He then writes that the number of microstates of the combined system (A and R)...
31. ### Quantum harmonic oscillator: average number of energy levels

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32. ### Why does photons of a given frequency satisfy the Boltzmann distribution?

A mode of frequency ##\nu## has energy ##E_n = h \nu##. In terms of photons, the interpretation that I have read several places, is that this correspond to ##n## photons of energy ##h \nu##. Furthermore, it is stated that the probabilty of finding ##n## photons at frequency ##\nu## is given by...
33. ### Derivative Maxwell boltzmann distribution

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34. ### Boltzmann Distribution: Solving 1D Ideal Gas Homework

Homework Statement I have to find the Boltzmann ditribution of a 1 dimensional ideal gas. The answer is given as: \frac{dn}{n}=\sqrt{\frac{m}{2piKT}}e^{(\frac{-mc^2}{2KT})} For the second part I have to find the mean kinetic energy. 2. Homework Equations / Attempt For part 1...
35. ### Maxwell Boltzmann Distribution

I don't know how to integrate the Maxwell-Boltzmann distribution without approximation or help from Maple. Given the Maxwell-Boltzmann distribution: f(v) = 4\pi\left[\frac{m}{2\pi kT}\right]^{3/2}v^2\textrm{exp}\left[\frac{-mv^2}{2kT}\right] Observe the appearance of the Boltzmann factor...
36. ### Maxwell Boltzmann Distribution

I don't know how to integrate the Maxwell-Boltzmann distribution without approximation or help from Maple. Given the Maxwell-Boltzmann distribution: f(v) = 4*pi*[m/(2*pi*k*T)]^(3/2)*v^2*exp[(-m*v^2)/(2*k*T)] Observe the appearance of the Boltzmann factor exp[(-m*v^2)/(2*k*T)] with E =...
37. ### Boltzmann distribution problem

A system has two non-degenerate energy levels E1 and E2, where E2>E1>0. The system is at tempreture T. The Average energy of the system is = E1+E2e^(-B*deltaE) / 1+e^(-B*deltaE) where deltaE= E2 -E1 and B=1/kT (k=Boltzmann constant). show that for very low temperatures kT<<deltaE, average...
38. ### Boltzmann Distribution: Calculate Probability of Particle in 4 States

A certain particle is interacting with a reservoir at 500 k and can be in any four possible states. The ground state has energy 3.1 eV and three excited states all have the same energy. what is the probability that the particle is in ground state? what is the probability that the particle is in...
39. ### Maxwell boltzmann distribution

The question is in the image file. I am stuck, I don't know how to start. Can someone guide me please?
40. ### Planck and Boltzmann Distribution

Does anyone know if Max Planck knew about the Boltzmann distribution before he published his results in 1900? Also, when Planck introduced h, did he also give the value?
41. ### Using Maxwell Boltzmann distribution to find number of atoms

Homework Statement You will recall from our discussion of the Franck-Hertz experiment that the energy difference between the first excited state of mercury and the ground state is 4.86 eV. If a sample of mercury vaporized in a flame contains 1.06×1020 atoms in thermal equilibrium at 1563 K...
42. ### Boltzmann distribution for particle energy

I'm puzzled by the appearance in the literature of 2 conflicting forms: P(E)=√(E)*exp(-E), which I understand as derived from the Maxwell distribution for speed. It is a chi -square distribution with 3 degrees of freedom. P(E)=exp(-E), which seems wrong to me. But it is not simply a...
43. ### Boltzmann Distribution (Chemistry/ Physics)

Homework Statement Okay i want to ask 2 questions. Question 1 Hyperstoichiometric compound Al3O has a vibrational energy spacing of 3.08x 10^-21J how many molecules are present in the ground state at 300k and 3000K Question 2 Calirimetric data for protein unfolding yielded following...
44. ### How to derivate Maxwell Boltzmann Distribution

Hi PFers. I am interested in Maxwell Boltzmann Distribution. I have searched in Internet for the derivation but I am not satified with them. Can somebody show me the way to derivate it? Thanks. ψ(v)=(m/2*pi*kT)^(3/2) e^(-mv^2/2kT)
45. ### I remember I read somewhere about the boltzmann distribution that it

I remember I read somewhere about the Boltzmann distribution that it is only valid for high temperatures. Why is that?
46. ### Boltzmann Distribution: Exploring Energy in High Density Reservoirs

Consider a resevoir of N atoms in contact with a single atom. Obviously, if the atom is in a high energy state then the multiplicity left for the resevoir is significantly lower. So this is in agreement with the fact that looking at the single atom, the probability for the ground state is very...
47. ### Why Does Boltzmann Distribution Depend Only on Temperature?

Why is the Boltzmann distribution for a collection of atoms independent of their total energy? (it only depends on their temperature) One would assume that if the energy is high there'd be a greater tendency to be in excited states or am I wrong?
48. ### Using Boltzmann distribution law to find Temperature (1% of photons> 1eV)

Homework Statement Use the Boltzmann distribution function to calculate the temperature at which 1.00% of a population of photons will have energy greater than 1.00 eV. The energy required to excite an atom is on the order of 1 eV. The Attempt at a Solution I attached my attempt but...
49. ### Boltzmann Distribution with two gasses

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50. ### Boltzmann distribution of two different gases

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