What is Boltzmann: Definition and 210 Discussions

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,






{\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. bhobba

    I A Bold New Take on Quantum Theory

    New Scientist recently published an article entitled 'A Bold New Take on Quantum Theory'. I found interesting Unfortunately, it is behind a paywall, but I will give my precis. How QM, which only predicts probabilities, gives rise to the solid, well-defined world around us is still a mystery...
  2. VVS2000

    A H-Theorem and Lagrange multipliers

    so I was studying H theorem from Richard Fitzpartic's site. https://farside.ph.utexas.edu/teaching/plasma/Plasma/node35.html Given H, they consider the following equation and set the constants as I want to understand how they got these particular values for a, b &c can we consider the...
  3. Gwen

    Thermal lattice Boltzmann model ignoring source term -- python code help please

    LBM model for phase change- relevant equations found here. Also here. #Thermal LBM #solves 1D 1 phase phase-change #D2Q5 Lattice nx=100 # the number of nodes in x direction lattice direction ny=5 # the number of nodes in y...
  4. Demystifier

    A Quantum analog of Boltzmann entropy?

    In classical statistical physics, entropy can be defined either as Boltzmann entropy or Gibbs entropy. In quantum statistical physics we have von Neumann entropy, which is a quantum analog of Gibbs entropy. Is there a quantum analog of Boltzmann entropy?
  5. 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 =...
  6. 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...
  7. S

    I Is the Boltzmann energy distribution an instance of energy diffusion?

    I (mechanical engineer) have researched this question but can't get to an answer. The equilibrium condition for confined particle diffusion of a solute in a solvent is reached when the solute spatial density is uniform (= zero density gradient), and entropy is max. But per Boltzmann, when...
  8. R

    A Exact Boltzmann Factor - Comparing y1 & y2 w/ 7 & 8 Particles

    Page 51 in <Moderator's note: link to copyrighted material removed, see instead http://dx.doi.org/10.1119/1.2967703> The image is showing functions: y1=e^(ln(n0)-x) y2=invDigamma(digamma(n0+1)-x)-1 where y2 is the "exact" version. How exactly is it exact? For this example (with 7 particles)...
  9. Bill Dreiss

    A Resolve Discrepancies in Boltzmann's Equilibrium Theory

    In Lectures on Gas Theory (Dover Books on Physics) (p. 74), Boltzmann states “In nature, the tendency of transformations is always to go from less probable to more probable states”, by which he means what are now called macrostates. Thus he claims that an ideal gas almost always evolves to the...
  10. 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...
  11. mohamed_a

    I Problem regarding understanding entropy

    I was reading about thermodynamics postulates when i came over the differnetial fundamental equation: I understand that the second element is just pressure and last element is chemical energy, but he problem is i don't understand what is the use of entropy and how does it contribute to a...
  12. J

    I Boltzmann brains and limitless energy....

    Hello! Amateur question alert! Please excuse any misuse of terms. Answers gratefully received. :smile: I have a question about the energy that can be drawn from the vacuum through quantum fluctuations. My understanding is there are very strict limits on how much energy can be borrowed for how...
  13. 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...
  14. sbrothy

    Popular physics: Boltzmann Brains and Sci-Fi

    I assume you people are all so preoccupied with all the hard sciencemastering going on that you're missing out on the more basic [sic] research abundantly present under the "Popular Physics" header at arxiv.org. Particularly: Futurama, Marvel's Supervillains and Boltzmann Brains. I'm not...
  15. yucheng

    Derivation of Boltzmann Factor via Reservoir Method (canonical ensemble)

    https://scholar.harvard.edu/files/schwartz/files/7-ensembles.pdf https://mcgreevy.physics.ucsd.edu/s12/lecture-notes/chapter06.pdf On page 3 of both the notes above, the author merely claims that $$P \propto \Omega_{\text{reservoir}}$$ But isn't $$P \propto...
  16. 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?
  17. Philip Koeck

    I Strange feature of BE, FD and Boltzmann distributions

    Hi. I've just come across something rather strange, I believe, about the micro-canonical derivation of the BE-distribution (as well as the Boltzmann and FD-distributions). See for example https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics#Derivation_from_the_microcanonical_ensemble...
  18. J

    I Restricted Boltzmann machine uniqueness

    I am dealing with restricted Boltzmann machines to model distributuins in my final degree project and some question has come to my mind. A restricted Boltzmann machine with v visible binary neurons and h hidden neurons models a distribution in the following manner: ## f_i= e^{ \sum_k b[k]...
  19. T

    Definition of a system in Boltzmann entropy

    Context Boltzmann first defined his entropy as S = k log(W). This seems to be pretty consistently taught. However, the exact definitions of S & W seem to vary slightly. Some say S is the entropy of a macrostate, while others describe it as the entropy for the system. Where the definition of...
  20. 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...
  21. 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...
  22. J

    Restricted Boltzmann machine understanding

    Suppose you have an experiment of 2 possible outcomes 0 and 1 with probabilities p and 1-p respectively. I've been told in University that Restricted Boltzmann machines (RBM) can be used to infer probability distributions so I guess that one could built a RBM in order to infer p in the example...
  23. J

    A Restricted Boltzmann machine for Quantum state tomography

    I'm struggling with my Final Degree Project. I would like to perform a quantum simulation and perform quantum tomography for a single-qubit using a resrticted Boltzmann machine. In order to do so I'm trying to follow the recipe in the paper "Neural Network quantum state tomography, Giacomo...
  24. J

    I Born rule in classical Ising model: Feynman -> Boltzmann ensemble?

    Quantum mechanics is often said to be equivalent with Feynman path ensemble, which "after Wick rotation" becomes Boltzmann path ensemble, also called euclidean path integrals (popular for numerical calculations), or random walk/diffusion MERW (maximal entropy random walk). But Boltzmann path...
  25. 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 ...?
  26. 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...
  27. C

    A Use of the Stefan Boltzmann Law in a Non Equilibrium system?

    Hi everyone, I am currently trying to work something out in regards to non-equilibrium thermodynamics. If I have a block of metal in vacuum that is being heated by a laser with a constant power P, is it even possible to be able to describe the emission of radiation by the block via the Stefan...
  28. 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...
  29. TheQuestionGuy14

    B Why are Boltzmann Brains considered a problem for physics?

    When Boltzmann brains are mentioned in physics, they're often seen as a problem. An idea that is forbidden. Physicists like Sean Carroll actively tried to come up with new theories that avoid Boltzmann brains. Others like Leonard Susskind have also labled them as problems. Why is this, what's so...
  30. C

    B How do we resolve the Boltzmann Brain problem

    I think this refutes the standard picture since if ylour universe would just keep expanding we should be a Boltzmann brain. https://en.wikipedia.org/wiki/Boltzmann_brain https://arxiv.org/pdf/0802.0233 I would bet on the "big rip" (dark energy getting stronger) since that would also resolve...
  31. 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...
  32. 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...
  33. T

    Speed usage in Lattice Boltzmann Method

    One thing that confuses me is the physical speed and sound speed. The lattice sound speed cs=1/sqrt{3} corresponds to the physical sound speed for isothermal flow (sqt{RT}). Why isn't the physical speed (e.g. inlet speed up of lid cavity) converted and use accoringly? $$c_p=\sqrt{RT}≈330m/s...
  34. J

    I Value of beta in Boltzmann Statistics taking degeneracy into account

    Hello, The relationship between entropy ##S##, the total number of particles ##N##, the total energy ##U(β)##, the partition function ##Z(β## and a yet to be defined constant ##β## is: $$S(\beta)=k_BN \cdot \ln(Z(\beta)) - \beta k_B \cdot U(\beta)$$ Which leads to: $$\frac{dS}{d\beta} =...
  35. 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?
  36. 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
  37. 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...
  38. R

    I Boltzmann particle distinguishability

    What justifies the use of multinomial coefficient in the combinatorics used by Boltzmann? The particles are distinct but counts as identical when they are in the same energy state?
  39. 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 -...
  40. W

    I What Are the Differences Between Boltzmann and Gibbs Entropies?

    Hi everyone, I have a few questions I'd like to ask regarding what I have read/heard about these two definitions of entropy. I also believe that I have some misconceptions about entropy and as such I'll write out what I know while asking the questions in the hope someone can correct me. Thanks...
  41. 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?
  42. 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...
  43. QuarkDecay

    When is the Boltzmann equation applicable in a Fermi plasma?

    When do we use the Boltzmann equation for density in a Fermi plasma? n in [cm-3] and when do we use the ρ=m/V, ρ in [Kg/m3 ] (this is not an example, I just added the equations to make my question more understandable) Is the ideal gas only when we have electron and ions? Is the Boltzmann...
  44. D

    I Exploring the Possibility of Boltzmann Brain Formation

    ‘Everything can happen, will happen’ so the saying goes. So given infinite time a universe is meant to generate infinite Boltzmann brains. But I was thinking about this: - Things that are possible can become impossible (example: heat death making Boltzmann brains formation impossible) -...
  45. 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...
  46. 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...
  47. NFuller

    Statistical Mechanics Part II: The Ideal Gas - Comments

    Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part II: The Ideal Gas Continue reading the Original PF Insights Post.
  48. 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...
  49. 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?