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64-479: Maxwell–Boltzmann may refer to: Maxwell–Boltzmann statistics , statistical distribution of material particles over various energy states in thermal equilibrium Maxwell–Boltzmann distribution , particle speeds in gases See also [ edit ] Maxwell (disambiguation) Boltzmann (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

128-440: A {\displaystyle N_{a}} balls from a total of N {\displaystyle N} balls to place into box a {\displaystyle a} , and continue to select for each box from the remaining balls, ensuring that every ball is placed in one of the boxes. The total number of ways that the balls can be arranged is As every ball has been placed into a box, ( N − N

192-436: A − N b − ⋯ − N k ) ! = 0 ! = 1 {\displaystyle (N-N_{a}-N_{b}-\cdots -N_{k})!=0!=1} , and we simplify the expression as This is just the multinomial coefficient , the number of ways of arranging N items into k boxes, the l -th box holding N l items, ignoring the permutation of items in each box. Now, consider

256-884: A body or system , such as one or more particles , with total energy E , invariant mass m 0 , and momentum of magnitude p ; the constant c is the speed of light . It assumes the special relativity case of flat spacetime and that the particles are free. Total energy is the sum of rest energy E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}{\textrm {c}}^{2}} and relativistic kinetic energy : E K = E − E 0 = ( p c ) 2 + ( m 0 c 2 ) 2 − m 0 c 2 {\displaystyle E_{K}=E-E_{0}={\sqrt {(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}}}-m_{0}c^{2}} Invariant mass

320-512: A Minkowski four-vector , namely the four-momentum; (these are the contravariant components). The Minkowski inner product ⟨ , ⟩ of this vector with itself gives the square of the norm of this vector, it is proportional to the square of the rest mass m of the body: a Lorentz invariant quantity, and therefore independent of the frame of reference . Using the Minkowski metric η with metric signature (− + + +) ,

384-608: A corresponding number of microstates available to the reservoir. Call this number Ω R ( s 1 ) {\displaystyle \;\Omega _{R}(s_{1})} . By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable. Therefore, for instance, if Ω R ( s 1 ) = 2 Ω R ( s 2 ) {\displaystyle \;\Omega _{R}(s_{1})=2\;\Omega _{R}(s_{2})} , we can conclude that our system

448-451: A fixed energy ( E = ∑ N i ε i ) {\textstyle \left(E=\sum N_{i}\varepsilon _{i}\right)} in the container. The maxima of W {\displaystyle W} and ln ⁡ ( W ) {\displaystyle \ln(W)} are achieved by the same values of N i {\displaystyle N_{i}} and, since it

512-412: A marking on each one, e.g., drawing a different number on each one as is done with lottery balls. The particles are moving inside that container in all directions with great speed. Because the particles are speeding around, they possess some energy. The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy. More precisely,

576-553: A massive object moving at three-velocity u = ( u x , u y , u z ) with magnitude | u | = u in the lab frame : is the total energy of the moving object in the lab frame, is the three dimensional relativistic momentum of the object in the lab frame with magnitude | p | = p . The relativistic energy E and momentum p include the Lorentz factor defined by: Some authors use relativistic mass defined by: although rest mass m 0 has

640-490: A more fundamental significance, and will be used primarily over relativistic mass m in this article. Squaring the 3-momentum gives: then solving for u and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity: Inserting this form of the Lorentz factor into the energy equation gives: followed by more rearrangement it yields ( 1 ). The elimination of

704-399: A system. Alternatively, one can make use of the canonical ensemble . In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat capacity as to maintain constant temperature, T , for the combined system. In the present context, our system is assumed to have

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768-488: Is where: Equivalently, the number of particles is sometimes expressed as where the index i now specifies a particular state rather than the set of all states with energy ε i {\displaystyle \varepsilon _{i}} , and Z = ∑ i e − ε i / k T {\textstyle Z=\sum _{i}e^{-\varepsilon _{i}/kT}} . Maxwell–Boltzmann statistics grew out of

832-417: Is a total of k {\displaystyle k} boxes labelled a , b , … , k {\displaystyle a,b,\ldots ,k} . With the concept of combination , we could calculate how many ways there are to arrange N {\displaystyle N} into the set of boxes, where the order of balls within each box isn’t tracked. First, we select N

896-438: Is closely related to the energy–momentum relation. The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc relates total energy E to the (total) relativistic mass m (alternatively denoted m rel or m tot ), while E 0 = m 0 c relates rest energy E 0 to (invariant) rest mass m 0 . Unlike either of those equations,

960-571: Is considered to be a continuous variable, the Thomas–Fermi approximation yields a continuous degeneracy g proportional to ε {\displaystyle {\sqrt {\varepsilon }}} so that: which is just the Maxwell–Boltzmann distribution for the energy. In the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of

1024-683: Is easier to accomplish mathematically, we will maximize the latter function instead. We constrain our solution using Lagrange multipliers forming the function: Finally In order to maximize the expression above we apply Fermat's theorem (stationary points) , according to which local extrema, if exist, must be at critical points (partial derivatives vanish): By solving the equations above ( i = 1 … n {\displaystyle i=1\ldots n} ) we arrive to an expression for N i {\displaystyle N_{i}} : Substituting this expression for N i {\displaystyle N_{i}} into

1088-517: Is essentially a division by N ! of Boltzmann's original expression for W , and this correction is referred to as correct Boltzmann counting . We wish to find the N i {\displaystyle N_{i}} for which the function W {\displaystyle W} is maximized, while considering the constraint that there is a fixed number of particles ( N = ∑ N i ) {\textstyle \left(N=\sum N_{i}\right)} and

1152-423: Is invariant.) where the index s runs through all microstates of the system. Z is sometimes called the Boltzmann sum over states (or "Zustandssumme" in the original German). If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy ε i {\displaystyle \varepsilon _{i}}

1216-409: Is mass measured in a center-of-momentum frame . For bodies or systems with zero momentum, it simplifies to the mass–energy equation E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}{\textrm {c}}^{2}} , where total energy in this case is equal to rest energy. The Dirac sea model, which was used to predict the existence of antimatter ,

1280-432: Is massless, as is the case for a photon , then the equation reduces to This is a useful simplification. It can be rewritten in other ways using the de Broglie relations : if the wavelength λ or wavenumber k are given. Rewriting the relation for massive particles as: and expanding into power series by the binomial theorem (or a Taylor series ): in the limit that u ≪ c , we have γ ( u ) ≈ 1 so

1344-515: Is necessary to assume that the particles are non-interacting, and that multiple particles can occupy the same state and do so independently. Suppose we have a container with a huge number of very small particles all with identical physical characteristics (such as mass, charge, etc.). Let's refer to this as the system . Assume that though the particles have identical properties, they are distinguishable. For example, we might identify each particle by continually observing their trajectories, or by placing

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1408-450: Is not valid for massless particles, since the expansion required the division of momentum by mass. Incidentally, there are no massless particles in classical mechanics. In the case of many particles with relativistic momenta p n and energy E n , where n = 1, 2, ... (up to the total number of particles) simply labels the particles, as measured in a particular frame, the four-momenta in this frame can be added; and then take

1472-411: Is simply the sum of the probabilities of all corresponding microstates: where, with obvious modification, Energy%E2%80%93momentum relation In physics , the energy–momentum relation , or relativistic dispersion relation , is the relativistic equation relating total energy (which is also called relativistic energy ) to invariant mass (which is also called rest mass) and momentum . It

1536-447: Is the occupation number of the energy level i . {\displaystyle i.} If we know all the occupation numbers { N i ∣ i = 1 , 2 , 3 , … } , {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \},} then we know the total energy of the system. However, because we can distinguish between which particles are occupying each energy level,

1600-758: Is the temperature , P is pressure, V is volume , and μ is the chemical potential . Boltzmann's equation S = k ln ⁡ W {\displaystyle S=k\ln W} is the realization that the entropy is proportional to ln ⁡ W {\displaystyle \ln W} with the constant of proportionality being the Boltzmann constant . Using the ideal gas equation of state ( PV  =  NkT ), It follows immediately that β = 1 / k T {\displaystyle \beta =1/kT} and α = − μ / k T {\displaystyle \alpha =-\mu /kT} so that

1664-410: Is the basis for constructing relativistic wave equations , since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant . In relativistic quantum field theory , it is applicable to all particles and fields. The energy–momentum relation goes back to Max Planck 's article published in 1906. It

1728-400: Is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be formulated as: E 2 = ( p c ) 2 + ( m 0 c 2 ) 2 {\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,} This equation holds for

1792-413: Is twice as likely to be in state s 1 {\displaystyle \;s_{1}} than s 2 {\displaystyle \;s_{2}} . In general, if P ( s i ) {\displaystyle \;P(s_{i})} is the probability that our system is in state s i {\displaystyle \;s_{i}} , Since the entropy of

1856-491: Is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation , such as relativistic particles (resulting in Maxwell–Jüttner distribution ), and to other than three-dimensional spaces. Maxwell–Boltzmann statistics is often described as the statistics of "distinguishable" classical particles. In other words,

1920-442: Is zero. Similarly, d V R = 0. {\displaystyle dV_{R}=0.} This gives where U R ( s i ) {\displaystyle U_{R}(s_{i})} and E ( s i ) {\displaystyle E(s_{i})} denote the energies of the reservoir and the system at s i {\displaystyle s_{i}} , respectively. For

1984-568: The g i {\displaystyle g_{i}} boxes, the second object can also go into any of the g i {\displaystyle g_{i}} boxes, and so on). Thus the number of ways W {\displaystyle W} that a total of N {\displaystyle N} particles can be classified into energy levels according to their energies, while each level i {\displaystyle i} having g i {\displaystyle g_{i}} distinct states such that

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2048-403: The i -th level accommodates N i {\displaystyle N_{i}} particles is: This is the form for W first derived by Boltzmann . Boltzmann's fundamental equation S = k ln ⁡ W {\displaystyle S=k\,\ln W} relates the thermodynamic entropy S to the number of microstates W , where k is the Boltzmann constant . It

2112-423: The magnetic field B in the same unit ( Gauss ), using the cgs (Gaussian) system of units , where energy is given in units of erg , mass in grams (g), and momentum in g·cm·s . Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first atomic bomb liberated about 1 gram of heat , and

2176-466: The Lorentz factor also eliminates implicit velocity dependence of the particle in ( 1 ), as well as any inferences to the "relativistic mass" of a massive particle. This approach is not general as massless particles are not considered. Naively setting m 0 = 0 would mean that E = 0 and p = 0 and no energy–momentum relation could be derived, which is not correct. In Minkowski space , energy (divided by c ) and momentum are two components of

2240-588: The Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is occupied. In general, there may be many particles with the same amount of energy ε {\displaystyle \varepsilon } . Let the number of particles with the same energy ε 1 {\displaystyle \varepsilon _{1}} be N 1 {\displaystyle N_{1}} ,

2304-411: The Maxwell–Boltzmann distribution, most likely as a distillation of the underlying technique. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system. Maxwell–Boltzmann statistics

2368-420: The beginning, see metric tensor (general relativity) for more information. In natural units where c = 1 , the energy–momentum equation reduces to In particle physics , energy is typically given in units of electron volts (eV), momentum in units of eV· c , and mass in units of eV· c . In electromagnetism , and because of relativistic invariance, it is useful to have the electric field E and

2432-547: The case where there is more than one way to put N i {\displaystyle N_{i}} particles in the box i {\displaystyle i} (i.e. taking the degeneracy problem into consideration). If the i {\displaystyle i} -th box has a "degeneracy" of g i {\displaystyle g_{i}} , that is, it has g i {\displaystyle g_{i}} "sub-boxes" ( g i {\displaystyle g_{i}} boxes with

2496-453: The characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type (e.g., electrons, protons,photon etc.) as principally indistinguishable. Once this assumption is made, the particle statistics change. The change in entropy in the entropy of mixing example may be viewed as an example of a non-extensive entropy resulting from

2560-451: The configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs paradox . At the same time, there are no real particles that have

2624-551: The distinguishability of the two types of particles being mixed. Quantum particles are either bosons (following Bose–Einstein statistics ) or fermions (subject to the Pauli exclusion principle , following instead Fermi–Dirac statistics ). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density. Maxwell–Boltzmann statistics can be derived in various statistical mechanical thermodynamic ensembles: In each case it

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2688-461: The energy levels ε i {\displaystyle \varepsilon _{i}} with degeneracies g i {\displaystyle g_{i}} . As before, we would like to calculate the probability that our system has energy ε i {\displaystyle \varepsilon _{i}} . If our system is in state s 1 {\displaystyle \;s_{1}} , then there would be

2752-453: The energy–momentum equation ( 1 ) relates the total energy to the rest mass m 0 . All three equations hold true simultaneously. A more general form of relation ( 1 ) holds for general relativity . The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However

2816-434: The energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, E ′ and p ′ as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. E and p as determined by particle physicists in a lab, and not moving with the particles). In relativistic quantum mechanics , it

2880-547: The equation for ln ⁡ W {\displaystyle \ln W} and assuming that N ≫ 1 {\displaystyle N\gg 1} yields: or, rearranging: Boltzmann realized that this is just an expression of the Euler-integrated fundamental equation of thermodynamics . Identifying E as the internal energy, the Euler-integrated fundamental equation states that : where T

2944-429: The factorial: to write: Using the fact that ( 1 + N i / g i ) g i ≈ e N i {\displaystyle (1+N_{i}/g_{i})^{g_{i}}\approx e^{N_{i}}} for g i ≫ N i {\displaystyle g_{i}\gg N_{i}} we can again use Stirling's approximation to write: This

3008-506: The inner product is and so or, in natural units where c = 1, In general relativity , the 4-momentum is a four-vector defined in a local coordinate frame, although by definition the inner product is similar to that of special relativity, in which the Minkowski metric η is replaced by the metric tensor field g : solved from the Einstein field equations . Then: Performing

3072-399: The largest thermonuclear bombs have generated a kilogram or more of heat. Energies of thermonuclear bombs are usually given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT). For a body in its rest frame, the momentum is zero, so the equation simplifies to where m 0 is the rest mass of the body. If the object

3136-409: The momentum has the classical form p ≈ m 0 u , then to first order in ( ⁠ p / m 0 c ⁠ ) (i.e. retain the term ( ⁠ p / m 0 c ⁠ ) for n = 1 and neglect all terms for n ≥ 2 ) we have or where the second term is the classical kinetic energy , and the first is the rest energy of the particle. This approximation

3200-524: The number of particles possessing another energy ε 2 {\displaystyle \varepsilon _{2}} be N 2 {\displaystyle N_{2}} , and so forth for all the possible energies { ε i ∣ i = 1 , 2 , 3 , … } . {\displaystyle \{\varepsilon _{i}\mid i=1,2,3,\ldots \}.} To describe this situation, we say that N i {\displaystyle N_{i}}

3264-444: The number of possible states of the system, we must count each and every microstate, and not just the possible sets of occupation numbers. To begin with, assume that there is only one state at each energy level i {\displaystyle i} (there is no degeneracy). What follows next is a bit of combinatorial thinking which has little to do in accurately describing the reservoir of particles. For instance, let's say there

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3328-520: The number of ways of distributing the N i {\displaystyle N_{i}} objects in the g i {\displaystyle g_{i}} "sub-boxes". The number of ways of placing N i {\displaystyle N_{i}} distinguishable objects in g i {\displaystyle g_{i}} "sub-boxes" is g i N i {\displaystyle g_{i}^{N_{i}}} (the first object can go into any of

3392-716: The population [B,A] while for indistinguishable particles, they are not. If we carry out the argument for indistinguishable particles, we are led to the Bose–Einstein expression for W : The Maxwell–Boltzmann distribution follows from this Bose–Einstein distribution for temperatures well above absolute zero, implying that g i ≫ 1 {\displaystyle g_{i}\gg 1} . The Maxwell–Boltzmann distribution also requires low density, implying that g i ≫ N i {\displaystyle g_{i}\gg N_{i}} . Under these conditions, we may use Stirling's approximation for

3456-417: The populations may now be written: Note that the above formula is sometimes written: where z = exp ⁡ ( μ / k T ) {\displaystyle z=\exp(\mu /kT)} is the absolute activity . Alternatively, we may use the fact that to obtain the population numbers as where Z is the partition function defined by: In an approximation where ε i

3520-400: The reservoir S R = k ln ⁡ Ω R {\displaystyle \;S_{R}=k\ln \Omega _{R}} , the above becomes Next we recall the thermodynamic identity (from the first law of thermodynamics ): In a canonical ensemble, there is no exchange of particles, so the d N R {\displaystyle dN_{R}} term

3584-410: The same energy ε i {\displaystyle \varepsilon _{i}} . These states/boxes with the same energy are called degenerate states.), such that any way of filling the i {\displaystyle i} -th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i -th box must be increased by

3648-445: The same energy and momenta. Although we still have, in flat spacetime: The quantities E , p , E ′ , p ′ are all related by a Lorentz transformation . The relation allows one to sidestep Lorentz transformations when determining only the magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to; Since m 0 does not change from frame to frame,

3712-417: The second equality we have used the conservation of energy. Substituting into the first equation relating P ( s 1 ) , P ( s 2 ) {\displaystyle P(s_{1}),\;P(s_{2})} : which implies, for any state s of the system where Z is an appropriately chosen "constant" to make total probability 1. ( Z is constant provided that the temperature T

3776-400: The set of occupation numbers { N i ∣ i = 1 , 2 , 3 , … } {\displaystyle \{N_{i}\mid i=1,2,3,\ldots \}} does not completely describe the state of the system. To completely describe the state of the system, or the microstate , we must specify exactly which particles are in each energy level. Thus when we count

3840-417: The summations over indices followed by collecting "time-like", "spacetime-like", and "space-like" terms gives: where the factor of 2 arises because the metric is a symmetric tensor , and the convention of Latin indices i , j taking space-like values 1, 2, 3 is used. As each component of the metric has space and time dependence in general; this is significantly more complicated than the formula quoted at

3904-650: The title Maxwell–Boltzmann . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Maxwell–Boltzmann&oldid=515199088 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Maxwell%E2%80%93Boltzmann statistics The expected number of particles with energy ε i {\displaystyle \varepsilon _{i}} for Maxwell–Boltzmann statistics

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3968-456: The total energy of the particle E and its relativistic momentum p are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures E and p , while the other frame measures E ′ and p ′ , where E ′ ≠ E and p ′ ≠ p , unless there is no relative motion between observers, in which case each observer measures

4032-411: Was pointed out by Gibbs however, that the above expression for W does not yield an extensive entropy, and is therefore faulty. This problem is known as the Gibbs paradox . The problem is that the particles considered by the above equation are not indistinguishable . In other words, for two particles ( A and B ) in two energy sublevels the population represented by [A,B] is considered distinct from

4096-423: Was used by Walter Gordon in 1926 and then by Paul Dirac in 1928 under the form E = c 2 p 2 + ( m 0 c 2 ) 2 + V {\textstyle E={\sqrt {c^{2}p^{2}+(m_{0}c^{2})^{2}}}+V} , where V is the amount of potential energy. The equation can be derived in a number of ways, two of the simplest include: For

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