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:








{\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:















{\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










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






{\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. O

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

    Hi. I'm not sure where to put this question, thermodynamics or the quantum physics forum (or somewhere else). For a system in equillibrium with a heat bath at temperature T, the Boltzman distribution can be used. We have the probability of finding the system in state n is given by ##p_n =...
  2. rogdal

    Deriving the kinetic energy flux in an effusion process

    I could not find any derivations in the litterature, except for the expected value of the energy flux expression itself: $$\overline{\Phi_{effusion,\epsilon}} = \overline{\dot{N_{ef}}}\overline{\epsilon_{ef}}=\frac{3Nl}{2A}\sqrt{\frac{(k_BT)^3}{2\pi m}}$$ I've started off by calculating the...
  3. rogdal

    Gas in a box with Maxwell-Boltzmann distribution

    I have considered two scenarios: 1) A particle that has just collided with the wall at ##z=L## is moving with a velocity ##v_z<0## moving away from the wall. Hence, the probability that this particle has of colliding again is ##0##, so its distribution is also ##0##. 2) A particle moving with...
  4. sol47739

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

    1.Does the Maxwell Boltzmann distribution change depending on the shape of the container? Pressure and the volume is constant. How is the Distribution affected whether the gas is in: a,sphere b,cube c,cuboid? Why does/doesn’t the distribution change depending on the shape of the container...
  5. S

    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. R

    I Boltzmann Distribution: Formula & Fig 2a in Document

    Hello Can anyone explain what formula (or parameters) was used to create the exponential Boltzmann distribution in fig 2a of this document? http://image.sciencenet.cn/olddata/kexue.com.cn/upload/blog/file/2009/5/20095251352697121.pdf I figure it must be something like y=e^(ln(600)-b*x) for some b?
  7. J

    Finding the g-value in the Boltzmann distribution

    Hello, I was wondering if someone could show me how to determine the number of orbitals available for a state and the number of electrons in that state. For calcium in the ground state, the electron config is 1s2 2s2 2p6 3s2 3p6 4s2. For the first excited state I assumed 1s2... 4s1 3d1. From...
  8. R

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

    Hi With the exact Boltzmann distribution, ni = InverseDigamma(-α-β*εi)-1: https://studyres.com/doc/269738/revision-of-boltzmann-statistics-for-a-finite-number-of-p... Shouldn't I be able to get (n0, n1, n2, n3, n4, n5, n6, n7) = (6, 3, 2, 0, 0, 0, 0, 0) for some α and β, if N=11, E=7 and Δε=1...
  9. M

    Energy landscape in the two state model (Boltzmann distribution)

    From an excel file I can get the probability of each energy state Εi and I saw at Wikipedia that the probability of each energy is proportional with e^−Εi/KT, from this I find the energy of every micro state. Also from the formula which I found on a paper I can get a curve like the curve...
  10. Rzbs

    I Maxwell Boltzmann distribution

    In the Aschcroft & Mermin solid state book there is a curve to compare F.D and M.B distribution. I can't understand the concept of M.B curve; what does mean exactlly when x =0? It means the probability of zero energy for particles is most or ...?
  11. dRic2

    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. PainterGuy

    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. R

    A What is the derivation of the exact Maxwell-Boltzmann distribution?

    I would like to see a derivation of the exact Maxwell-Boltzmann distribution shown as (16) in this document: https://www.researchgate.net/publication/222670999_Exact_Maxwell-Boltzmann_Bose-Einstein_and_Fermi-Dirac_Statistics This is my starting point (f being the function to maximize, g and h...
  14. I

    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. Viona

    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. R

    I Deriving the Boltzmann distribution

    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...
  17. S

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

    In Vol III, 14-4 and 14-5 of the Feynman Lectures (http://www.feynmanlectures.caltech.edu/III_14.html), Feynman gives a discussion of the p-n junction, in which he derives the diode characteristic equation via a nice, simple and convincing application of the Boltzmann distribution to the...
  18. Philip Koeck

    A Can indistinguishable particles obey Boltzmann statistics

    Many textbooks claim that particles that obey Boltzmann statistics have to be indistinguishable in order to ensure an extensive expression for entropy. However, a first principle derivation using combinatorics gives the Boltzmann only for distinguishable and the Bose Einstein distribution for...
  19. T

    Boltzmann Partition Function of H_2

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  20. danyull

    Average energy for n-state system

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  21. K

    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. Konte

    I Stationary states -- Boltzmann distribution

    Hello everybody, - In quantum mechanics, the state ## | \psi \rangle ## of a system that is in thermodynamic equilibrium can be expressed as a linear combination of its stationary states ## | \phi _n \rangle ## : $$ | \psi \rangle = \sum_n c_n | \phi _n \rangle $$ It permit us to express the...
  23. S

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

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  24. S

    I Boltzmann distribution: isothermal atmosphere error?

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  25. K

    Average height of molecules in a box as a function of temperature

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  26. K

    Boltzmann distribution derivation.

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  27. RaulTheUCSCSlug

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  28. D

    Possible to derive Boltzmann distribution using W, not lnW?

    Hi all, in following the many available derivations of the Boltzmann distribution I was trying to do it by maximizing W, which is N choose n1,n2,...nt., instead of lnW, because it should give the same answer (since W is monotonically increasing with lnW, am I wrong?). So given the two...
  29. throneoo

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  30. S

    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. E

    Quantum harmonic oscillator: average number of energy levels

    Homework Statement I must find the average number of energy levels of quantum harmonic oscillator at temperature T, and the answer is given as I must use Boltzmann distribution and the sum of geometric progression. For finding the average value I must use the equation <F>=trace(F*rho)...
  32. C

    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. G

    Derivative Maxwell boltzmann distribution

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  34. T

    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. T

    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. T

    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. S

    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. S

    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. M

    Maxwell boltzmann distribution

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  40. lonewolf219

    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. D

    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. A

    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. K

    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. T

    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. A

    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. A

    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. A

    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. P

    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. edpell

    Boltzmann Distribution with two gasses

    What is the Boltzmann distribution look like if we have a mix of two gasses with very different molecular weights? Say hydrogen at 2 amu and xenon at 132 amu. Do they have equal average momentum and hence the hydrogen has an average velocity 66 times the velocity of the xenon and an energy...
  50. O

    Boltzmann distribution of two different gases

    hi everyone, consider two different masses of ideal gases with different molar masses, we're putting them in a uniform gravitational field and wait until they come to their equilibrium states. how would the density distribution change with height in this case? ( i came out with this question...