What is Boltzmann equation: Definition and 40 Discussions

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.
The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element




{\displaystyle \mathrm {d} ^{3}{\bf {r}}}
) centered at the position


{\displaystyle {\bf {r}}}
, and has momentum nearly equal to a given momentum vector


{\displaystyle {\bf {p}}}
(thus occupying a very small region of momentum space




{\displaystyle \mathrm {d} ^{3}{\bf {p}}}
), at an instant of time.
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection–diffusion equation.
The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.

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  1. H

    I Boltzmann equation for annihilation

    In the Dodelson's textbook, the author introduce the Boltzmann equation for annihilation. ##a^{-3} \frac{d(n_1 a)}{dt} = \int \frac{d^3 p_1}{(2 \pi)^3 2E_1} \int \frac{d^3 p_2}{(2 \pi)^3 2E_2} \int \frac{d^3 p_3}{(2 \pi)^3 2E_3} \int \frac{d^3 p_4}{(2 \pi)^3 2E_4} \times (2 \pi)^4 \delta^3 (p_1...
  2. Dario56

    I Boltzmann Entropy Formula – Derivation

    Boltzmann entropy definition is given by: $$ S = k_B lnW $$ where ##W## is the weight of the configuration which has the maximum number of microstates. This equation is used everywhere in statistical thermodynamics and I saw it in the derivation of Gibbs entropy. However, I can't find the...
  3. RicardoMP

    A How to reduce an integral in phase space to a one-dimensional form?

    I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...
  4. B

    I Some questions about baryogenesis

    I'm trying to follow Scott Dodelson's Modern Cosmology. Specifically Chapter 3. Coverage of the subject of baryogenesis appears to be missing from Dodleson's book, so I'm trying to reconstruct things on my own. This represents the formula:$$n_p[T]=g_p\space (\frac{k_b\space m_p\space...
  5. Haynes Kwon

    I Boltzmann equation and energy level occupancy at infinitely high temp

    Let's look at the Boltzmann equation $$ \frac {p_{i}} {p_{j}} = e^{\frac{E_{j}-E_{i}} {kT}},$$ and take infinitely high temperature, the RHS becomes 1. I interpreted that this means every energy level is occupied by equal number of electrons. But if T is high enough, wouldn't the hydrogen atom...
  6. E

    A Deriving Relativistic Boltzmann Equation | Theory Physics MSc Thesis

    Hello guys, i am currently studying for my msc thesis in theoretical physics and i need to find the derivation of relativistic Boltzmann equation, any suggestions ? Because i ve searched for papers/books for it and couldn't find anything. Any ideas? Thx in advance
  7. Physics Dad

    Boltzmann equation - Why does theta = 5040/T

    Hi, I am working on a Boltzmann equation question and I know that the solution I am looking for is that: log(nij/nji)=log(gij/gji)-Eij(eV)(5040/T) The only thing I don't understand is why log(e-Eij/kt) = θ = 5040/T From what I have read in textbooks, it is just a given, but I really want to...
  8. P

    A Vlasov-Fokker-Plank(VFP) equation from Boltzmann equation

    Dear community, I am studying some equations related to the acceleration of cosmic rays(CRs) in magnetized plasma and I have seen a couple the equations I am not able to understand. First, I see that it is used as time-dependent Boltzmann equation for the CRs ∂ƒ/∂t + (vx + u)∂ƒ/∂x -...
  9. M

    I Boltzmann equation and Hamiltonian

    Hello! I read today, in the context of DM, about the Boltzmann equation: $$L[f]=C[f]$$ where ##L[f]## is the Liouville operator (basically ##\frac{df}{dt}##), with ##f(x,v,t)## being the phase-space distribution of the system and ##C[f]## being the collision operator. I am a bit confused about...
  10. A

    Scattering dynamics and viscosity

    I have been studying the statistical mechanics' viewpoint of fluid dynamics by considering the derivation of Navier-Stokes' equations from the Boltzmann equation involving the whole Chapman-Enskog expansion. It is clear that through this process, it is possible to account for the dependence of...
  11. D

    Velocity correlations and molecular chaos

    I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first...
  12. T

    For the Boltzman equation, why is df/dt=0 when collisionless?

    From wikipedia, The general equation is $$\frac{df}{dt} = (\frac{∂f}{∂t})_{force}+(\frac{∂f}{∂t})_{diff}+(\frac{∂f}{∂t})_{coll}$$ where the "force" term external force, the "diff" term represents the diffusion of particles, and "coll" is the collision term. So shouldn't be df/dt=0 when it is...
  13. NFuller

    Statistical Mechanics Part I: Equilibrium Systems - Comments

    Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part I: Equilibrium Systems Continue reading the Original PF Insights Post.
  14. mertcan

    A Collision integral approximation in boltzmann equation

    Hi, as you can see at the end of the picture/attached file collision integral is approximated to a discrete sum. Could you express how this approximation is derived?
  15. T

    I Derivation of the General Boltzmann Equation and Its Validity

    I've seen the derivation where: ## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {v} \cdot \vec {\nabla} f + \vec {a} \cdot \vec \nabla_{\vec{v}} f ## Although I was told this should more generally be written as: ## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {\nabla}...
  16. tomadevil

    Validate the Stefan Boltzmann equation

    Homework Statement You are performing an experiment to validate the Stefan Boltzmann equation. What irradiance would you measure at a temperature of 109C? The emissivity of your thermal heat source is 0.81 and your thermopile measures 0 W/m2 at 27 C when directed towards a blackbody. Submit...
  17. M

    Chemical potential using Boltzmann equation

    Homework Statement I must calculate chemical potential using the Boltzmann equation in relaxation time approximation $$f=f^0-\tau v_z^2 \partial f^0/\partial z,$$ where ##f^0## is given as $$f^0 = 2(\frac{m}{2\pi\hbar})^3 \frac{1}{\exp{\beta(z)(\frac{mv^2}{2}-\mu(z))}+1}$$ I have to consider...
  18. S

    A Lattice Boltzmann simulation with arbitrary equilibrium func

    Hi everyone, I plan to do a simulation of a Boltzmann equation with experimentally known scattering between two particles. Initially I intend to incorporate the scattering into the collision integral and use Lattice Boltzmann Equation (LBE) afterwards. But I only see LGBK (DnQb) which requires...
  19. P

    Classical Book recommendations for Boltzmann equation

    Im trying to learn transport processes and the Boltzmann transport equation. What books do you guys recommend for beginners? Thanks!
  20. F

    Elements of plasma kinetic theory, Bittencourt

    Homework Statement Consider the motion of charged particles, in one dimension only, in the presence of an electric potential V ( x). Show, by direct substitution, that a function of the form f=f(1/mv^2 + qV) is a solution of the Boltzmann equation under steady-state conditions. Homework...
  21. E

    Boltzmann Equation: Entropy & Thermodynamics

    There was an equation I saw before and I think it pertains to Boltzmann and thermodynamics. I think it describes the entropy of a system. From what I can remember, it involves the symbol omega to denote micro states, k for a constant, and a logarithm somewhere. Anyways, hopefully some one knows...
  22. C

    Boltzmann equation and Kinetic equilibrium

    In section 3.1 of Dodelson's "Modern Cosmology", after introducing the Boltzmann equation, in the second paragraph of page 60 the author states that: "The first, most important realization is that scattering processes typically enforce kinetic equilibrium. That is, scattering takes place so...
  23. C

    Boltzmann equation Kinetic equilibrium (Dodelson)

    In section 3.1 of Dodelson's "Modern Cosmology", after introducing the Boltzmann equation, in the second paragraph of page 60 the author states that: "The first, most important realization is that scattering processes typically enforce kinetic equilibrium. That is, scattering takes place so...
  24. H

    Scott Dodelson Modern Cosmology 4.2

    Hi guys, I recently started reading/working through Scott Dodelson's Modern Cosmology in preparation for a Masters course I'm taking next year and one of the exercises has stumped me and (arghhh!) its not one of the solved ones in the back! It is in Chapter 4 (The Boltzmann equations) and is...
  25. T

    Fick's Second law from Boltzmann Equation

    Homework Statement Starting from the kinetic equation for the distribution function F*(t, r, v) of some labelled particle admixture in a gas, derive the self-diffusion equation ∂n*/∂t = D∇2n* for the number density n*(t,r) = ∫d3vF*(t,r,v) of the labelled particles. Derive also the expression...
  26. W

    Introductory Material for Relativistic Boltzmann Equation

    I’m searching for an introduction to relativistic Boltzmann equation. (Sorry, I know this is a question about learning material) I’ve read an excellent script from David Tong about non-relativistic kinetic theory (from Liouville to Navier-Stokes using Boltzmann equation (see link below)...
  27. D

    Boltzmann equation for the early Universe

    I'm currently reading about the Boltzmann equation, used for the early Universe. The equation I end up with, after some simplifications is the following: \begin{equation} a^{-3}\frac{d}{dt}\left(n_1a^3\right) = n_1^{(0)}n_2^{(0)}\langle\sigma v\rangle\left[\frac{n_3 n_4}{n_3^{(0)}n_4^{(0)}} -...
  28. WannabeNewton

    Boltzmann equation for photons

    Hi there. I have a couple of questions regarding the derivation of the Boltzmann equation in Dodelson for photons when given scalar overdensity perturbations to the FRW metric. To start with, let ##\Theta(\vec{x})## denote the temperature perturbations to the Bose-Einstein distribution of the...
  29. vead

    Boltzmann equation for carrier transport

    I am little bit confused about derivation for Boltzmann equation for electron look at this link http://relativity.livingreviews.org/Articles/lrr-2008-10/articlesu25.html which is final boltazmann equation ?
  30. A

    Understanding the Boltzmann Equation Derivation in Electric and Magnetic Fields

    Below is part of derivation of the Boltzmann equation in an electric and magnetic field. I don't understand how to arrive at the bottom equation though. It is known that the dependence of the original distribution function is the given. My idea is to use chain rule but I don't see how to get a...
  31. U

    Boltzmann equation with collision term

    Homework Statement Solve the Boltzmann equation for a homogeneous plasma with not external forces present when the collision term is \frac{\partial f(v,t)}{\partial t} = -\nu (f(v,t) - f_0(v)), where \nu and f_0 are constants. Homework Equations Boltzmann equation \frac{\partial...
  32. D

    Getting to the Boltzmann Equation from a Nonequilibrium Distribution Function

    Homework Statement Hi. I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function. At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)Homework Equations The nonequilibrium...
  33. A

    Why Can't I Get the Same Answer for the Boltzmann Equation Problem?

    Why I can't calculate the same answer as the solution below? I use the value as what the below solution showed. Q: Where Nj is the number of atoms in excited state, No is the number of atoms in the ground state, Pj and Po are constants determined by the number of states having equal energy...
  34. N

    How does the coefficient in the Boltzmann equation comes from

    In the Boltzmann equation, {\bf{L}}\left[ f \right] = {\bf{C}}\left[ f \right], the right which is the collision term and in general it is{\bf{C}}[f]=\sum\limits_{p,p_1} {|Amplitiude|^2}{f(p)}. and explicitly, the collision term for decaying process is \begin{eqnarray} {\bf{C}}\left[ f...
  35. Q

    Derivation of Boltzmann Equation

    Hi: How do you derive the Boltzmann equation? Thanks.
  36. L

    Solving Boltzmann Equation: Guidance for Partial Differentiation

    Hi I am trying to make the following equation to get Boltzmann equation which I write below. f(\mathbf{x}+\frac{\mathbf{p}}{m}dt,\mathbf{p} + \mathbf{F}dt,t+dt) \,d\mathbf{x}\,d\mathbf{p} - f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} = \left. \frac{\partial...
  37. F

    140-year-old Boltzmann equation solved

    I've never heard of this before. Maybe I should learn more physics: http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors http://www.pnas.org/content/107/13/5744.abstract?sid=08ced618-f372-4c0d-a80c-64786d92c3ff
  38. X

    Study Kinetic Theory: Deriving Boltzmann Equation from Liouville

    I am studying kinetic theory. The Boltzmann equation can be derived from the liouville equation, I want to know how to deduce it , which books explore the question. I am reading a book( the title is kinetic theory, the author is R.L.Liboff), in this book the method proposed by Prigogine is used...
  39. S

    Boltzmann equation/ Statistical Mechanics

    Homework Statement If we assume entropy is a function of the multiplicity, \Omega, (S=k*f(\Omega)) show that that function f(\Omega) is ln(\Omega).Homework Equations The Attempt at a Solution \Omega can be written as N!/ni!. By using stirling's approximation, this becomes \Omega=...
  40. B

    Temperature at which ground state and first excited state have equal populations

    Homework Statement Consider a pure hydrogen hydrogen gas ata tempeature of 10080 K. What is the ratio of the populations of the ground state(n=1) to the first excited state(n=2). Note that the energy difference is 1.634e-18 joules between these two states. At what temperature would both...