Boltzmann Definition and 211 Threads

Ludwig Eduard Boltzmann (German pronunciation: [ˈluːtvɪg ˈbɔlt͡sman]; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodynamics. In 1877 he provided the current definition of entropy,



S
=

k


B



ln
⁡
Ω



{\displaystyle S=k_{\rm {B}}\ln \Omega \!}
, interpreted as a measure of statistical disorder of a system. Max Planck named the constant kB the Boltzmann constant.Statistical mechanics is one of the pillars of modern physics. It describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

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  1. 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?
  2. 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}...
  3. 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...
  4. S

    B Understanding Stefan Boltzmann Law: Derivation and Simulation Guide

    Hello, everyone, I was looking at this video () and I need to make a simulation for Stefan Boltzmann law calculating its constant. I didn't understand few things. In the video, it shows that Stefan Boltzmann law is R = e*sigma*T^4vand then says that rate of net heat transferred ΔQ/Δt =...
  5. 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...
  6. Toby_phys

    Boltzmann vs Maxwell distribution?

    So I worked through the Boltzmann distribution and got: $$ P\propto e^{\frac{-E}{k_BT}} $$ Where $E$ is the energy. So surely this means the kinetic energy (and therefore speed) of particles is distributed over a Boltzmann distribution. Or in equation: $$ P\propto e^{\frac{-mv^2}{2k_BT}} $$...
  7. T

    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...
  8. Arman777

    I Are we living in a Boltzmann box ?

    In this video,Prof.Leonard Susskind talks about Boltzmann box and the universe.He claims that we might live in a Boltzmann box,And explains why is time runs one way, etc. This video is taken 2013 and we are in 2017 , so what's the new ideas about it. Are we living in a Boltzmann box ? , Or...
  9. J

    I Boltzmann with degenerate levels

    Suppose we have some model for some system, and that model has given us a sequence \mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3,\ldots, whose values are interpreted as the energy levels of the system. Denoting the energy levels slightly redundantly for future modification soon below, we state that...
  10. J

    I How Do Symmetry Properties Affect Boltzmann Distributions for Bosonic Particles?

    Introduction and warm up Suppose H_0 is some Hamilton's operator that has eigenvectors |\psi_n\rangle for n\in\mathbb{N} with some eigenvalues E_n so that H_0|\psi_n\rangle = E_n|\psi_n\rangle,\quad \forall n\in\mathbb{N}. Suppose we define a new Hamilton's operator by setting H=H_0\otimes...
  11. G

    B Why Is the Boltzmann Factor Exponential?

    Hi community, trying to get my head around the Boltzmann factor... e^(-E/kT) It states in a book I'm reading that this is linked to the probability of a particle moving from its current energy state to an energy state E above it? So if you were looking at an energy jump of 4kT that the...
  12. G

    Entropy of ideal gas: Volume it *has* or is *allowed* to have?

    Hi. If an ideal gas of ##N## particles is allowed to expand isothermically to double its initial volume, the entropy increase is $$\Delta S=N\cdot k_B \cdot \log\left(\frac{V_f}{V_i}\right)=N\cdot k_B \cdot \log\left(\frac{2V}{V}\right)=N\cdot k_B \cdot \log\left(2\right)\enspace .$$ This can...
  13. A

    I Expanding universe and the Boltzmann brain problem

    Hello I have been reading Sean Carroll's book "From eternity to here" where he mentioned the concept of functioning brains emerging from random fluctuations on a quantum level due to the expansion of universe. They have been called Boltzmann brains https://en.wikipedia.org/wiki/Boltzmann_brain...
  14. durant35

    I Boltzmann brains and other low probability events

    I am interested in your opinion with regards to expectation values of low likelihood events in QM. For example, Boltzmann brains are suggested as a problem in cosmology despite their probability being extremely miniscule. Is it realistic to expect Boltzmann brains and other low probability...
  15. 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...
  16. durant35

    I Boltzmann Brains: Existence, Models & Opinion

    Hi guys I recently read a lot about Boltzmann brains and their possibility of existence. Are there any models in cosmology which exclude them and make them a physical impossibility? How can we safely conclude that we are not Boltzmann brains which last for a moment and then vanish? Are the...
  17. 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...
  18. 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...
  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. bananabandana

    Are the Gibbs and Boltzmann forms of Entropy equivalent?

    Homework Statement Are the Gibbs and Boltzmann entropies always equivalent? Homework Equations $$ S=k_{B}ln\Omega $$ [Boltzmann entropy, where ##\Omega## is the number of available microstates $$ S=-k_{B}\sum_{i}p_{i} ln(p_{i}) $$ [Gibbs entropy, where ##p_{i}## is the probability of a...
  21. E

    How do calculate the prob of being in a state in therm eq?

    Homework Statement A pore has three configurations with the energy levels shown. The pore is in thermal equilibrium with the surroundings at temperature T . Find the probabilities ##p_1##, ##p_2##, ##p_3## of being in each level. Each level has only one microstate associated with it. Also...
  22. G

    B The shape of the body in Stefan Boltzmann Law modern use

    Hello, It looks like Stefan Boltzmann Law can be used for a lot of different purposes: to calculate the temperature of stars, sun, temperature of the Earth's sky, temperature of particular surface, wall, the radiation emitted by the body by knowing its temperature, and so on. What confuses me...
  23. M

    What is the best biography of Boltzmann?

    Hi everyone! Could you suggest a good biographic book about Boltzmann?
  24. S

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

    The analysis of the distribution of spins for a paramagnetic solid in a B field shows that the probability of a dipole being aligned/anti-aligned with the B field ##\to 0.5## as ##T \to \infty##. The intuitive justifications that I've read say that this is "expected" as thermal motion tends to...
  25. S

    I Boltzmann distribution: isothermal atmosphere error?

    There is a well-known analysis of the distribution of particles by height in an isothermal atmosphere. It states that the probability of finding a particle at height ##h## is ##p(h) \propto e^{-\beta mgh}##, and then goes on to state that the number of particles at height ##h## is ##n(h) \propto...
  26. Alltimegreat1

    B Exploring the Possibility of Boltzmann Brains in the Far Future

    A coworker of mine keeps mentioning Boltzmann Brains. I was never really interested until yesterday when reading Timeline of the Far Future on Wikipedia, which states the ultimate fate of the universe could be a Boltzmann Brain. Is this a theory scientists believe could really materialize?
  27. samjohnny

    Is My Approach to Rubber Band Boltzmann Statistics Correct?

    Homework Statement Homework Equations $$ Z(1) = \sum_{i=1}^{} e^{\frac{E_i}{K_bT}} $$ where ##E_i## is each of the possible energy states available to a single link (in this case the right and the left states). $$ P=\frac{\sum_{i=1}^{} e^{\frac{E_i}{K_bT}}}{Z} $$ The Attempt at a Solution...
  28. A

    Calculating Equations in SI Units: Boltzmann Constant & More

    Excuse me, If I want to calculate one equation in SI units. Which value of Boltzmann constant I should deal with. Also, the electron temperature it can be used in Kev or it should be in ev. Also, the nm must be converted to meter or it can be used in nanometer.
  29. P

    Probability that a magnetic dipole is oriented with theta

    1. the problem goes like this : The energy of interaction of a classical magnetic dipole with the magnetic field B is given by E = −μ·B. The sum over microstates becomes an integral over all directions of μ. The direction of μ in three dimensions is given by the angles θ and φ of a spherical...
  30. K

    Boltzmann distribution derivation.

    please check the video at 5:33. how can we find the partial derivative w.r.t n1 n2 and on? isn't each state (n1, n2 and on) one discrete state not a continuous variable? is it because we can have multiple particles in the given energy state? However its a finite discrete number. as far as I...
  31. 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...
  32. 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...
  33. Graeme M

    Simple explanation of the Boltzmann Brain paradox

    Mods: I am not sure if this is a Physics question or more appropriate for Cosmology. I read a short discussion (on another forum) about the Boltzmann Brain paradox. I did a little further reading on the web but most explanations were a bit too deep (read: over my head). I wonder if someone...
  34. O

    Relativistic Boltzmann (including QM) equation from entropy?

    I'm interested in the derivation of relativistic Boltzmann equation from entropy after reading Scott Dodelson's wonderful cosmology book. Does anyone know of any good readings for this? The usual way of doing things in classical mechanics is to assume \frac{dN}{dt}= 0 and go from there; but...
  35. C

    Why does corrected Maxwell Boltzmann have decimals?

    So I have just been reading up on statistical thermodynamics and have no idea why the bose-einstein, fermi dirac and maxwell boltzman are all integers, that makes sense, but then when you make the degenerate correction to the maxwell Boltzmann by dividing by N! we get decimal answers. Why is...
  36. throneoo

    Normalization of 1D velocity boltzmann distribution

    Suppose the pdf is A*exp(-mv^2/2kT) , where A is the normalization constant. To obtain A I would integrate the pdf over the all possible values of v. The question is, should the limits be (-infinity,infinity) or [0,infinity) ? It seems that only by choosing the former can I get the correct...
  37. RyanJones

    Boltzmann Brains: Are They Taken Seriously by Scientists?

    How seriously are Boltzmann Brains taken by the scientific community? I understand there is at least some mathematical evidence for their (someday?) existence, but do many physicists think that there is anything worthwhile in examining the issue? Or, is it one of those "how many angels on the...
  38. 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...
  39. 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...
  40. 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)...
  41. U

    Differences between Boltzmann and Fermi-Dirac distributions

    Hi All, In relation to the Boltzmann distribution vs the FD/BE distributions in different applications, I have 2 basic questions: 1. The Boltzmann distribution comes most easily from the Canonical Ensemble (constant N, V,T) while the FD/BE come from the Grand Canonical ensemble (constant .mu...
  42. I

    Deriving equations for pressure & number density for centrif

    Homework Statement Consider particles in a gas centrifuge. This device is used to separate gases of different molar mass by rotating a cylinder at high rates. Derive two equations: one for the pressure (P) and one for the particle number density (nv) as functions of M, r, w and T, where r is...
  43. J

    Find Physicists' Published Papers: Max Planck, Ludwig Boltzmann

    Does anyone have any suggestions for finding lists of all papers published by individual physicists? Usually the Google machine turns up hits pretty quickly, but I've hit a brick wall looking for lists for Max Planck and Ludwig Boltzmann. netlib.org/bibnet/ is amazing, but it's pretty narrow.
  44. 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...
  45. 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)...
  46. G

    Temperature, Kinetic Energy, Boltzmann Factors, and SHO

    I am confused about the following; where am I going wrong here? 1. (1/2)kT is defined as the average kinetic energy of the molecules of a substance at temperature T, right? 2. You can derive the Boltzmann distribution/Boltzmann factors using (1/2)kT as the kinetic energy, making an argument...
  47. 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)}} -...
  48. S

    Are Boltzmann Brains More Likely Than a Fine-Tuned Universe?

    So I read this Boltzmann brain concept, and um yeah... So does this paradox imply that it is much more likely that I/you are a mind projecting an illusion universe in which we live as compared to a fine-tuned, and relatively low entropy universe in which there is many self-aware minds..?Also...
  49. 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...
  50. G

    Derivative Maxwell boltzmann distribution

    Homework Statement i need to show that the peak of the maxwell Boltzmann distribution is equal to 1/2 kt. Homework Equations maxwell Boltzmann distribution according to modern physics 3rd edition by kenneth kramer. ill try to do my best with this N(E)= \frac{2N}{√∏}...
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