Boltzmannn.
/ˈbɒltsmən/, U.S. /ˈbɔltsmən/, /ˈbɑltsmən/ Chiefly Physics and Chemistry.
attributive and in the genitive. Designating various concepts and formulae developed by or attributed to Boltzmann, or arising from his work in the field of statistical mechanics.
1. Boltzmann equation n. (also Boltzmann's equation) (a) = Boltzmann entropy formula n. at sense 5; (b) = Boltzmann transport equation n. at sense 7; (c) (originally) the equation pi/ pj = e(εj−εi) / kT, where pi and pj are the probabilities that a given thermodynamic system is in state i and j respectively, εi and εj are the energies of the two states, k is Boltzmann's constant, and T is the temperature of the system; (in later use also) any equation of the form A = Be−C/kT.The equation in sense (c) first appears in connection with the Boltzmann distribution (see sense 3).
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1872 tr. R. Clausius in London, Edinb. & Dublin Philos. Mag. 4th Ser. 44 366 This is the form of Hamilton's equation made use of by Szily, and which expresses the principle of least action with variable energy. It agrees with Boltzmann's equation..in the above-cited memoir.
1917 Physical Rev. 10 309 If f is the number of molecules per unit cell in the twelve dimensional region, and J is the rate of the change of this number of molecules due to encounters the Boltzmann equation may be written [etc.].
1923 Physical Rev. 21 432 Since we are dealing with equilibrium conditions, the distribution of electrons is given by Boltzmann's equation ρ = ρ1εVe / kT.
2000 R. Plonsey & R. C. Barr Bioelectricity (ed. 2) iv. 84 Boltzmann's equation expresses the ratio of open to closed channels as [open]/ [closed] = exp ( QgVm/ kT).
2015 I. H. Hutchinson Student's Guide Numerical Methods viii. 105 The rate of change of the distribution function will be precisely that given by the left-hand side of Boltzmann's equation.
2016 Physics Lett. B. 762 109/1 Thermodynamics can be linked to information theory by the Boltzmann equation, S = kB ln (W).
2. Boltzmann formula n. (also Boltzmann's formula) any of various equations developed by or attributed to Boltzmann; esp.: (a) = Boltzmann entropy formula n. at sense 5; (b) = Boltzmann equation n. (c) at sense 1.
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1876 London, Edinb., & Dublin Philos. Mag. 5th Ser. 2 477 The seventh column gives the values of the angles of rotation adopted as definitive; the eighth column, the same values calculated by Boltzmann's formula.
1916 Proc. Koninklijke Akad. van Wetenschappen te Amsterdam 18 1052 In Boltzmann's formula the dimensions no longer come right;..that the entropy of the whole is equal to the sum of the entropies of the parts, however, does hold.
1930 Monthly Notices Royal Astron. Soc. 91 163 The strength of each band depends on the relative number of atoms in the corresponding state, for which we take the Boltzmann formula.
2004 Surface Sci. 561 230/1 The adsorption energy difference between the two halves can be estimated by the Boltzmann formula.
2009 P. Atkins & J. de Paula Elements Physical Chem. (ed. 5) iv. 96/2 Boltzmann's formula is consistent with the entropy of a substance increasing with temperature.
3. Boltzmann distribution n. (also Boltzmann's distribution) a probability distribution giving the probability that a thermodynamic system will be in a particular state, given the energy of that state and the temperature of the system.
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1901 Philos. Trans. (Royal Soc.) A. 196 398 We may regard the Boltzmann distribution as unstable, in the sense that a slight deviation from perfect conservation of energy may result in a complete redistribution of the total energy.
2014 T. A. Waigh Physics Living Processes xi. 227 It is then assumed that around any ion the charge distribution is spherically symmetric..and the variation in charge follows the Boltzmann distribution.
4. Boltzmann's constant n. (also Boltzmann constant) a physical constant relating the average relative kinetic energy of particles in a gas to the temperature of that gas.In 2018 the General Conference on Weights and Measures agreed that, as part of the redefinition of the kelvin, the value of Boltzmann's constant would from May the following year be set at exactly 1.380648 × 10−23 J K−1, having previously been given as 1.38064852(79) × 10−23 J K−1 (where the figure in parenthesis represents the standard uncertainty due to errors in measurement).Boltzmann never directly posited the constant, but he developed the underlying principles from 1877.
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the world > matter > physics > mechanics > dynamics > thermodynamics > [noun] > entropy > specific constant or equation
Boltzmann's constant1915
1915 London, Edinb., & Dublin Philos. Mag. 6th Ser. 30 296 This theory leads to the following expression for the electrical conductivity c at temperature T°K..where..k = Boltzmann's constant.
2019 R. A. Serway & J. W. Jewett Physics for Scientists & Engineers (ed. 10) xxi. 573 The units of entropy are those of Boltzmann's constant, J/K.
5. Boltzmann entropy formula n. (also Boltzmann's entropy formula) the equation S = k logW relating the entropy S of an ideal gas to the number W of microstates (microstate n. 1) of that gas.
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1915 London, Edinb., & Dublin Philos. Mag. 6th Ser. 29 299 He [sc. Einstein] finds, using Boltzmann's entropy-formula..that this produces a gain of entropy.
2018 D. E. Barrick Biomolecular Thermodynamics iv. 158 We will also apply the Boltzmann entropy formula to the heat flow model developed at the beginning of this chapter.
6. Boltzmann factor n. (also Boltzmann's factor) a mathematical expression of the form e−ε/kT, where k is Boltzmann's constant, T is the temperature of a given thermodynamic system in a particular state, and ε is the energy of the state, used to express the probability of the system occupying that state; cf. Boltzmann equation n. (c) at sense 1.
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1927 Physical Rev. 30 182 The superior intensity of the R1 branch just noted may very likely be attributed to the differential effect of the Boltzmann factor.
2002 B. Widom Statist. Mech. ii. 29 At low temperatures, because of the Boltzmann factor.., almost all the oscillators are in their ground states.
7. Boltzmann transport equation n. (also Boltzmann's transport equation) (originally) an integro-differential equation used to study statistically the distribution of velocities of the atoms or molecules of an idealized gas that is not in a state of equilibrium; (in later use also) any of various modified or analogous versions of this equation used to study other quantities in different contexts.
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1946 Physical Rev. 70 367/1 The more standard approach afforded by the Boltzmann transport equation was first applied to the d.c. problem by Morse, Allis, and Lamar.
2007 Physics Lett. B. 655 126/2 We propose an alternative realization of the recombination picture by formulating hadron production in terms of a Boltzmann transport equation.
8. Boltzmann machine n. Computing a recurrent neural network which employs probabilistic methods and a Boltzmann distribution as its sampling function.Pattern and speech recognition are common applications of Boltzmann machines.
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1983 S. E. Fahlman et al. in Proc. National Conf. Artific. Intelligence 111/2 This is the kind of pattern completion task which the Boltzmann machine can solve in a single setting.
2016 J. Hearty Adv. Machine Learning with Python (e-book, accessed 21 Jan. 2019) ii. 33 The Boltzmann machine has seen particular success in processing image data.
9. Boltzmann brain n. (also Boltzmann's brain) a hypothetical self-aware entity produced by random thermodynamic fluctuations.The notion of a Boltzmann brain developed as a refinement of an earlier hypothesis that the observable universe is the product of random fluctuations in a space that is otherwise at thermal equilibrium. This hypothesis first appeared in a paper published by Boltzmann in 1896, in which he attributes it to a former assistant.The possibility for the existence of Boltzmann brains in a given theory is generally taken as an indication that it is flawed.
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2006 Stud. Hist. & Philos. Mod. Physics B 37 410 A fluctuation that encompasses our neighbourhood and produces the orderly patterns we perceive here is much less likely than a fluctuation that simply creates Boltzmann's brain.
2017 L. A. Barnes in K. Chamcham et al. Philos. of Cosmol. xxii. 462 Boltzmann Brains can cause problems for a multiverse theory.
This entry has been updated (OED Third Edition, June 2019; most recently modified version published online December 2021).
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